Finite element modeling of the human head under impactconditionsClaessens, M.H.A.

DOI:10.6100/IR501340

Published: 01/01/1997

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Citation for published version (APA):Claessens, M. H. A. (1997). Finite element modeling of the human head under impact conditions Eindhoven:Technische Universiteit Eindhoven DOI: 10.6100/IR501340

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Finite Element Modeling of the

Human Head under

Impact Conditions

Maurice Claessens

Finite Element Modeling of the Human Head

under Impact Conditions

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Finite element modeling of the human head under impact conditions I Maurice Henri Andre Claessens. - Eindhoven : Technische Universiteit Eindhoven, 1997.- XIV, 137 p. Proefschrift. - ISBN 90-386-0369-X NUGI 834 Trefw.: hoofd I letselmechanica I eindige-elementenmethode I botsingsmechanica I biomechanica Subject headings: injury biomechanics I finite element method I head models

Printed by Febodruk BV, Enschede, The Netherlands

This research wa..'l performed within a co-operative research project between the Eindhoven University of Technology and the TNO Cra..'lh-Safety Research Centre.

This work was sponsored by the Stichting Nationale Computerfaciliteiten (National Com­puting Facilities Foundation, NCF) for the use of supercomputer facilities, with financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for Scientific Research, NWO).

Finite Element Modeling of the Human Head

under Impact Conditions

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven,

op gezag van de Rector Magnificus, prof.dr. M. Rem, voor een commissie

aangewezen door het College van Dekanen in het openbaar te verdedigen op

donderdag 18 september 1997 om 16.00 uur

door

Maurice Henri Andre Claessens

geboren te Eindhoven

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr.ir. J.D. Janssen prof.dr.ir. J.S.H.M. Wismans

en de copromotor:

dr.ir. A.A.H.J. Sauren

voor mijn ouders

Summary

Notation

1 Introduction 1.1 Basic anatomy of the human head .

1.1.1 Skull ....... . 1.1.2 Brain ........ . 1.1.3 Meninges and CSF .

1.2 Epidemiology of head injury 1.2.1 Head injury incidence.

1.3 Literature survey . . . . . . . 1.3.1 Numerical models ... 1.3.2 Generation of finite element meshes 1.3.3 Boundary and interface conditions . 1.3.4 Constitutive properties 1.3.5 Validation methods 1.3.6 Discussion

1.4 Scope ..... . 1.4.1 Objective 1.4.2 Application 1.4.3 Research strategy

1.5 Outline of this thesis . .

2 Contact mechanics under transient dynamic conditions 2.1 Introduction ........ . 2.2 Governing equations .... .

2.2.1 Conservation of mass . 2.2.2 Balance of momentum 2.2.3 Balance of moment of momentum

2.3 Numerical solution ...... . 2.3.1 Spatial discretization .. 2.3.2 Temporal discretization.

Contents

v

vii

1 2 2 3 5 6 7 8 9

11 12 15 16 17 18 18 19 19 21

23 23 24 25 25 25 25 25 27

Contents ii

2.3.3 Contact conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.4 Combination of contact algorithm with Newmark-,6 time integration 33

2.4 Testing and parametric study of 2-D contact algorithm 34 2.4.1 Model description . 34 2.4.2 Analytical solution 35 2.4.3 Numerical solution 38 2.4.4 Parametric study 41

2.5 3-D contact 44 2.6 Conclusions . . . . . . . 45

3 Development of a 3-D human head model 3.1 Introduction ...... . 3.2 3-D model development ..

3.2.1 Mesh generation 3.2.2 Material properties 3.2.3 Characteristics of the 3-D model . 3.2.4 Impact load .......... .

3.3 Modal response of the model ...... . 3.4 Simulation of impact and comparison with experiments

3.4.1 Direct comparison with experimental data 3.4.2 Shear response . 3.4.3 Dilatational response

3.5 Time step refinement . . . 3.6 Mesh refinement . . . . . . 3.7 Discussion and conclusions

4 Parametric study 4.1 Introduction .......................... . 4.2 Material properties . . . . . . . . . . . . . . . . . . . . . . .

4.2.1 Variation of Young's modulus in elastic brain model . 4.2.2 Variation of linear viscoelastic brain properties .

4.3 Boundary and interface conditions . . . . . . . . 4.3.1 Relative motion between skull and brain .. 4.3.2 Influence of foramen magnum ....... . 4.3.3 Modeling of a kinematic boundary condition

4.4 Anatomical detail of the model . 4.4.1 Modeling of substructures 4.4.2 Modeling of frontal sinus

4.5 Discussion and conclusions . 4.5.1 Discussion . 4.5.2 Conclusions . . . . .

47 47 48 48 51 51 52 53 55 56 58 59 61 62 64

67 67 68 70 73 76 76 77 83 88 88 93 96 96 96

iii

5 Discussion, Conclusions and Recommendations 5.1 Discussion ........ .

5.1.1 Contact algorithm 5.1.2 3-D head model . 5.1.3 Parametric study

5.2 Conclusions .... 5.3 Recommendations .

Bibliography

A Derivation of contact conditions for velocity and acceleration A.1 Introduction ............. . A.2 Derivation of velocity constraint .. . A.3 Derivation of acceleration constraint .

B Application of contact conditions B.1 Introduction ........... . B.2 Coupling of only kinematic conditions . B.3 Correct modeling of impact ... B.4 Correct modeling of 3-D contact .

C Mesh generation C.1 Introduction . C.2 Generation of elements

D Von Mises stress D.1 Introduction . D.2 Von Mises stress related to distortion

Samenvatting

Acknowledgments

Curriculum Vitae

Contents

99 99

100 100 101 103 104

107

117 117 117 118

119 119 120 123 124

127 127 127

131 131 131

133

135

137

Contents iv

Summary

Impact loads on the head, occurring for example when a person is involved in a car or sports accident, may cause tissues to deform beyond recoverable limits resulting in in­juries. Knowledge of the mechanisms by which an impact load results in an injury is still incomplete. Mathematical models provide a powerful tool in the analysis of the mechanics of head impact. In particular the finite element method lends itself for the construction of a mathematical head model because of its capability to describe complex geometries and physical and geometrical nonlinearities.

The objective of this thesis is to develop a three-dimensional finite element model of the human head and to assess the importance of boundary and interface conditions, the level of geometrical detail to which the various structures are modeled, and the influence of variation of the material properties of brain tissue on the results.

First a reference model has been developed using digital CT and MRI data, con­sisting of a skull filled with a homogeneous continuum representing the brain. The modal characteristics of the model compare well with experimentally determined modes. The transient response of the model has been validated by comparing the calculated pressure responses, as a result of an impact to the frontal bone, with the experimental results re­ported by Nahum et al. (1977) in impact experiments on post-mortem human subjects. The numerical results for the frontal and occipital regions of the brain are comparable to the experimental results.

In the reference model - as opposed to reality - no relative motion at all is possible between brain and skull at their interface. Therefore the skull-brain interaction has been investigated in a parametric study using a contact algorithm. Test simulations with the algorithm of the impact of two beams demonstrated the importance of properly applying the three contact constraints at the point in time the two structures come into contact. Simulating relative motion between brain and skull under the same impact conditions gave pressure levels in the frontal region that were approximately a factor three lower compared to the reference model. These results indicate that the simulation of the interaction between the various structures is important when trying to understand the mechanisms responsible

v

vi

for brain injuries. Inclusion of the foramen magnum in the skull in combination with the possibility of relative motion at the skull-brain interface affects only the results in the foramen region when compared to the first model with relative motion included. Variation of the Young's modulus of brain tissue over the range of data found in literature, shows to have a large effect on the pressure and von Mises stress responses for various regions in the brain. This result indicates that comparing the findings of various authors, who each use different geometries, loading conditions and material properties in their models, is already questionable when considering only the material properties. The modeling of the various structures of the brain, such as cerebrum, cerebellum and brainstem, and the inclusion of the three meningeal partitions in the reference model results in lower pressures in the occipital region of the brain in comparison to the reference model. The difference in pressure just above and below the tentorium indicates the possible supportive function of this structure.

Conclusively, the simulation of head impact is actually possible and numerical results have been shown to agree with experimental findings. The inclusion of relative motion between the various structures of the head seems to be critical and should have a high priority in future investigations. In addition to the relative motion the material properties also have a large influence on the calculated results. Implementation of constitutive properties of brain tissue determined in well-controlled laboratory experiments under impact conditions will make the calculated results with numerical models more reliable. Only then can calculated field parameter distributions deliver new leads in the identification of injury mechanisms.

Convention

a, A, a a g

A a A A4 a·b A:B Ac AT v det(A)

Indices

Symbols

r 6

p A A

scalar vector column matrix time derivative second order tensor fourth order tensor inner product double dot product conjugate of A transpose of A gradient operator determinant of A

initial value of a value of a at timet normal component of vector a

Newmark parameter penalty factor Newmark Parameter strain mass density Lagrange multiplier wave length

Notation

vii

viii Notation

v Poisson's ratio (}" Cauchy stress

~ material coordinates A area a=ii acceleration vector b specific body force vector c4 elasticity tensor c wave propagation velocity E Young's modulus F deformation tensor

f external load vector

f frequency G shear modulus h element length I identity tensor J determinant of deformation tensor K stiffness matrix K bulk modulus L length

f column of Lagrange multipliers lv/ mass matrix N shape function n normal vector T_ transformation matrix T period time t time !lt time increment u displacement vector v volume V=U velocity vector w internal energy w weighting function X position vector

Chapter 1

Introduction

The human head is one of the most vulnerable parts of the human body, when subjected to an impact loading. The study of the human head under impact conditions may be split in two tasks. Firstly, the calculation of the deformation patterns of the skull and its contents as a result of an impact loading and secondly the identification of possible relationships between a particular deformation in a tissue and an injury in this tissue. For the first task, numerical models provide a powerful tool for the simulation of the deformations. They reduce the necessity of performing large numbers of experiments while model predictions may suggest experiments which otherwise might not have come to mind. Moreover they enable the calculation of tissue loads and deformations that cannot possibly be determined in experiments. For the second task, experiments need to be performed to determine the level of response at which the biological tissues fail to recover. Once both tasks are sufficiently apprehended the link between loads and sustained injuries may be made in the identification of injury mechanisms and improvement of injury criteria. In the research into injury mechanisms, relationships are to be identified between physiological, pathophysiological and/or biochemical tissue damage and mechanical parameters produced in an impact event by impact loads. Impact loads are classified as either contact loads, that are directly applied to the head, or non-contact (inertial) loads, that are transferred to the head via the neck. An injury criterion (Wismans et al., 1994) is a physical parameter or function of a set of physical parameters which correlates well with the injury severity. An example of a well-known injury criterion is the HIC criterion by Versace (1971). The focus

1

2 Introduction

in this study will be on the first task, the simulation of the deformations of the human head using numerical models.

This chapter provides a background on the research of the numerical modeling of the human head under impact conditions. In section 1.1 a brief description of the anatomy of the human head is given. The injuries that are distinguished and their incidence are described in section 1.2 on the epidemiology of head injury. The literature survey in sec­tion 1.3 presents a description of the state of the art concerning the numerical simulation of head injury. The scope and the objective of this study are presented in section 1.4. The chapter is concluded with an outline of this thesis.

1.1 Basic anatomy of the human head

The human head is a hard shell (skull) which is covered by a soft layer on the outside. The central nervous system (CNS) consists of the brain and spinal cord. Inside the shell the brain is separated from the skull by a layer built up of membranes (meninges) and cerebrospinal fluid (CSF). The major parts of the human head are briefly described in the next sections. More detailed descriptions of the anatomy of the human head can be found in DeMyer (1988) and Martin (1989).

1.1.1 Skull

The skull consists of two main parts, the cranial vault and the facial bones. The facial anatomy is defined as; the area from the forehead to the lower jaw and includes fourteen bones. The cranial vault ((neuro )cranium) can be subdivided in the braincase (calvarium) and the base of the braincase. The braincase forms a shell and is composed of eight bones (Figure 1.1) that are separated by sutures. The eight bones are called, ethmoid, sphenoid, temporal (2), parietal (2) and occipital bone. During infancy and early childhood, the articulations (joints, sutures) are composed of cartilage, which gradually disappears as the bones grow together and ossify. In the full-grown state no movement is possible between the bones.

The thickness of the skull varies between 4 and 7 mm. The sandwich structure of the skull consists of a compact internal and external layer (lamina intern a and lamina extern a) with in between a porous layer (diploe), which is the thickest of the three layers. The brain case is covered on the outside by a 5 to 7 mm thick tissue layer, that can be subdivided into 5 separate layers. From outside to inside; hair-skin layer, a subcutaneous connective tissue layer, a muscle and fascial layer, a loose connective tissue layer and the fibrous tissue lining the bone (periosteum). The thickness, firmness and mobility of the three outer layers serve as a protection for the head. The base of the brain case is an irregular plate of bone. The base contains various small holes for arteries, veins and nerves and one large opening (foramen magnum) that forms the transition area between the spinal cord and the brain.

Basic anatomy of the human head 3

frontal bone

temporal bone

sphenoid bone

occipital bone

(a) (b)

Figure 1.1: The neuro-cranium in frontal (a) and medial (b) view. Adapted from: Tortora (1980).

1.1.2 Brain

The head of an adult human being has a mass of about 4.5 kg of which the brain con­tributes about 1.5 kg. The Central Nervous System (CNS) is built up of various structures (Figure 1.2). The brain constitutes 98% of the weight of the CNS and it represents about 2 % of the weight of the human body.

The cerebrum consists of two hemispheres, the cerebr-al hemispheres, which represent the largest structure of the brain. The hemispheres can be subdivided into various regions; the cerebral cortex, that consists of gray matter, and the white matter or the nerve fibers. The gray matter forms a 1.5 to 4.5 mm thick layer located on the surface of the hemispheres, and is built up of nerve cells and support cells (glia). The glia or neuroglial cells provide structural and metabolical support for neurons during development as well as in the mature brain. The neuron, or nerve cell, is the functional cellular unit of the nervous system. The cerebral cortex is arranged in a number of folds (gyri), which are separated by fissures. Due to these fissures each hemisphere can be subdivided into four lobes (Figure 1.3), named by their association to the nearest cranial bone. The fibers in the white matter are arranged in tracts and serve to connect one part of a cerebral hemisphere with another, to connect the cerebral hemispheres to each other, and to connect the cerebral hemispheres to the other parts of the CNS. In addition, within these areas of white matter are a number of areas of gray matter. The fibers are wrapped by a grease-like substance, that forms the so-called myelin sheath. Beneath the longitudinal fissure the two hemispheres are interconnected by the corpus callosum. In each of the cerebral hemispheres there exists a fluid-filled space, called the lateral ventricle.

The midbrain forms a fibrous connection with the cerebral hemispheres above and the pons below. Within the midbrain there is a canal (cerebral aquaduct of Sylvius) that forms the connection between the third and fourth ventricle. The third ventricle is located

4

Central nervous system Brain (encephalon)

Cerebrum -Hemispheres

ECortex • - - -

Diencephalon -Brainstem

White matter ... - - -Basal nuclei • - - - - -

'----Midbrain (mesencephalon)

----Pons ... - - - - -

=Medulla Oblongata -

Foramen magnum

Cerebellum

Cervical ... --- -------- ---

Spinal cord (myelon) Thoracic ... - - - - - - - - - -

Sacral - - - - - -

Introduction

Figure 1.2: Lateral view of the CNS. Adapted from: DeMyer (1988).

along the midline between the two halves of the diencephalon and the fourth ventricle is located in the brain stem. The diencephalon consists of two major components the thalamus and the hypothalamus.

The medulla oblongata appears to be continuous with the pons above and the spinal cord below and forms the connection between the brain mass and the spinal cord.

The pons lies below the midbrain, in front of the cerebellum, and above the medulla oblongata. It is composed of white matter fibers that form the connection between the two cerebral hemispheres. Lying deep within the white matter are areas of gray matter that are nuclei for some of the cranial nerves.

The cerebellum lies behind the pons and the medulla oblongata. Its two hemispheres are joined by a narrow strip-like structure called the vermis. The outer cortex of the two cerebellar hemispheres consists of gray matter. The outer surface of the cerebellum forms narrow folds separated by deep fissures.

The spinal cord comprises two percent by weight of the CNS and averages 45 em in length. Thirty-one pairs of nerves originate at the spinal cord. The spinal cord is protected

Basic anatomy of the human head 5

frontal lobe

corpus callosum

diencephalon

Figure 1.3: Medial surface of cerebral hemisphere, brain stem and spinal cord. Adapted from: Martin (1989}.

by the bony spinal column, the meninges and pressurized CSF. The three structures; midbrain, pons and medulla oblongata are together called the brain stem.

1.1.3 Meninges and CSF

The brain and spinal cord are covered by three membranes (meninges), which have im­portant protective and circulatory functions (Figure 1.4). The dura mater is the thickest and outer most of the three membranes. It covers the cerebral hemispheres and brain stem and actually contains two layers; the outer periosteal layer and the inner meningeal layer. The periosteal layer is attached to the inner surface of the skull. Within the dura there are large low-pressure blood vessels, which are part of the return path for cerebral venous blood. This path is termed the dural sinus. The meningeal layer forms three parti­tions; the falx cerebri between the two cerebral hemispheres, the falx cerebelli between the cerebellar hemispheres and the tentorium cerebelli between the cerebral hemispheres and the cerebellum. The dura mater that covers the spinal cord is continuous with both the meningeal layer of the cranial dura and the epineurium of the cranial peripheral nerves. The epineurium constitutes the outside layer of the peripheral nerve fibers. The head is innervated by cranial nerves, whereas the limbs and trunk are innervated by the spinal nerves.

The arachnoid mater adjoins but is not tightly bound to the dura mater, thus al­lowing a potential space to exist between them. This space, called the subdural space is important clinically, because the dura mater contains blood vessels. Breakage of one of

6 Introduction

cerebral cortex

Figure 1.4: Coronal view of the three. meninges with the dural sinus and falx cerebri. Adapted from Martin (1989).

these vessels can lead to subdural bleeding and the formation of a blood clot (subdural hematoma). Within the subdural space, a thin film of watery fluid, known as cerebrospinal fluid, encloses and bathes the brain. In the sagittal sinus and transverse sinuses, the arach­noid mater forms structures called arachnoid granulations, which reabsorb CSF into the blood. Since the arachnoid mater bridges small and large fissures in the brain there are a number of cavities (cisterns) below the membrane filled with CSF.

The subarachnoid space separates the arachnoid mater from the innermost layer, the pia mater. Filaments of arachnoid mater pass through the subarachnoid space and connect to the pia mater. The subarachnoid space is filled with CSF and also contains the veins and arteries that overlie the surface of the CNS. The pia mater is a very delicate thin membrane of fine connective tissue invested with numerous small blood vessels that adhere to the surface of the brain, dipping well into its fissures.

The CSF is found in the four ventricles of the brain, the cisterns around the brain and in the subarachnoid space around both the brain and the spinal cord. The fluid spaces are all interconnected with each other and the pressure is regulated at a constant level. The CSF provides some nutrients for the brain and is assumed to protect the brain from mechanical shock. The brain and CSF have approximately the same mass density, so that the brain simply floats in the fluid. The 150 ml of CSF constantly circulate through and around the brain.

1.2 Epidemiology of head injury

In the event of an accident, the head is exposed to inertial and contact loads, that may cause the various tissues to deform beyond recoverable limits, resulting in injury. Mechanical

Epidemiology of head injury 7

impact resulting in head injury is one of the leading causes of injury, fatality and disability in our society (Viano et al., 1989; Waxweiler et al., 1995)

In the field of injury biomechanics extensive research is being conducted in the iden­tification of injury mechanisms. An injury mechanism describes the process of how a load applied to body tissues and systems leads to deformations of tissues beyond a recoverable limit, resulting in injuries of these tissues (Viano et al., 1989). Following the classification presented by Bandak (1995), three types of neural injury (traumatic brain injury) may be distinguished. The first category represents the diffuse type where a substantial volume of the brain is affected. Injuries included are axonal, neural and microvascular injuries. Dif­fuse injuries result from loadings that produce deformation patterns in various parts of the brain. One type of neural damage, that accounts for a large portion of all head injury cases, affects the axon and is referred to as diffuse axonal injury (DAI) (Gennarelli et al., 1982; Thibault and Gennarelli, 1985). The second type of injuries are focal in nature and occur in localized regions of the brain, subjected to tensile or compressive stresses. Focal injuries are the result of a localized mechanical response at either the loading site or other remote areas. The loadings may also cause vascular injury associated with brain motion. Partic­ularly vulnerable are the bridging veins that cross the subdural space. The last category is concerned with those injuries that are primarily dependent on movements of the brain relative to the cranial cavity. Typical injuries encountered in this category are subdural hematoma and brain stem injuries. Results of the relative contributions of inertial loading, by Ommaya et al. (1994), have shown, that both translational and rotational components cause focal injuries, while diffuse injuries were only seen, when a rotational component was present.

1.2.1 Head injury incidence To assess the effectiveness of countermeasures and to determine causative factors, the occurrence and severity of injuries recorded in accident data bases provides a valuable source. The reported incidences of traffic injuries vary widely from one data base to another, due to, among other factors, the used definitions, for example the population involved. Further, some countries define a traffic accident fatality only if it occurs at the scene of the accident, while others include a certain period after the accident. Data bases commonly used to determine injury incidences are those of the police, hospitals and motor vehicle insurance companies, and none of those is fully representative. In police data bases, severe and fatal accidents are over-represented (van Kampen and Harris, 1995; Stearn, 1988) since police intervention is not always necessary or called for in minor and moderate accidents.

A rough estimate of traumatic brain injury in the United States in 1990 is about 2 million cases, including about 51,600 fatalities and 200,000 hospital admissions (Waxwei­ler et al., 1995). Faverjon et al. (1988) determined the frequency and severity of head and neck injuries for front seat occupants restrained by retractor seatbelts in frontal impacts. Results showed 30% of the front seated car occupants sustained some kind of head injury, while 14% of the car drivers and 10% of the right-front seated passenger sustained injuries

8 Introduction

of AIS1 ~ 2. Dutch figures (van Kampen and Harris, 1995) showed that 36% of the people injured in traffic suffered some kind of head injury and 9% also sustained a brain injury. Deutscher (1993) analyzed the data of 10,736 car-to-car accidents provided by a German in­surance company and found 254 (2.4%) accidents with fatalities and 9,102 (85%) accidents where one of the occupants had sustained an injury of MAIS 12•

Although life is its own measure of value, society as a whole has a certain "quan­tity" of functional capacity (Luchter and Walz, 1995). Injured people have a diminished functional capacity, decreasing also the overall "amount" of societal capacity. The human capital costs provide an estimate for the costs of accidents on society. In the United States the costs for traffic injuries in 1990 were estimated at US$70.6 billion and traffic fatalities at US$31.3 billion (Blincoe and Faigin, 1992). The total costs of traffic accidents in the Netherlands are estimated to be Dfl.6.0 billion (Flury, 1995).

1.3 Literature survey

The primary task of this thesis is ~the simulation of the deformations in the human head as a result of an impact loading, using numerical models. For the construction of a numerical model various input parameters need to be available, like the geometry, constitutive prop­erties and the boundary and interface conditions the model is subjected to. The realism of the constructed model is dependent on two factors; the available data and the com­putational resources, necessary for simulating the r-esponse. Aim of the literature survey presented here is to provide an overview of the state of the art, and by no means a com­plete and detailed survey of all the models reported in literature, concerning the numerical modeling of the human head under impact conditions. A more historic overview of the research field may be found in the review papers by Khalil and Viano (1982b) and Sauren and Claessens (1993).

The section on numerical models will present some of the models that have been presented lately and their major findings and illustrates the capabilities of numerical models nowadays. The section on the generation of meshes will discuss how the models have been constructed and what possible techniques are available for building a finite element grid of the human head. Next the boundary and interface conditions are discussed and the methods used to represent them in the finite element models. Section 1.3.4 on constitutive properties discusses the material properties of the structures of the human head as used by various authors. In section 1.3.5 the necessary validation of numerical models is discussed and the methods and data used to validate numerical models of the human head are presented. The last section summarizes the presented topics and points out where future research with numerical models may aid in understanding and identifying injury mechanisms inside the human head.

1. AIS stands for Abbreviated Injury Scale (Assn. for the Advancement of Automotive Medicine, (1995). 2. MAIS is the maximum of all AIS scores in case more than one body region is injured.

9

1.3.1 Numerical models The mathematical models used to analyze the response of the head to impact or inertial loading can be subdivided into three types; lumped parameter models, continuum models and finite element models. Lumped parameter models (Stalnaker and Fogle, 1971; Stal­naker et al., 1985; Willinger and Cesari, 1990; Willinger et al., 1991) are in particular useful in providing a first prediction of the frequency response of the human head with very low computational costs. Their main limitation is constituted by the fact that they do not lend themselves to describing the distribution of field parameters in the model, such as stress, strain and pressure. With continuum models (Engin, 1969; Advani and Owings, 1975; Akka, 1975) closed-form solutions can be obtained describing field-parameter distri­butions. However, these models are only tractable after introducing idealizations concerning geometry, constitutive properties and boundary conditions. Finite element models are in principle capable of handling complex geometries and different kinds of nonlinearities, the computational costs however, are much higher than for lumped parameter models.

Both two-dimensional and three-dimensional finite element models have been devel­oped. Two-dimensional models representing various cross-sections of the human head have long been the only means to simulate the deformations that occur when the head is sub­jected to an impact loading, due to the lack in computational resources. The drawback of plane-strain models is that -especially under large deformations- they may provide unreal­istic results, because under all circumstances they will maintain the plane character of the strain distribution whereas these in reality are likely to be of a three-dimensional nature. With the increasing computational power of hardware and software more detailed three­dimensional models are being constructed and used in impact simulations, that incorporate the various structures of the human head to a greater detail.

The 3-D model, containing a skull, brain and falx, developed by DiMasi et al. (199la) was modified by Bandak and Eppinger (1994) to allow for a variable traction boundary condition on the surface of the brain, that releases beyond specified tensile and shear load levels. To account for the occurence of DAI a so-called cumulative damage strain measure was developed, based on maximum principal strain. The measure monitors the accumu­lation of strain damage by calculating the volume of the brain experiencing strain levels greater than a specific level. Computational results showed mostly rotational acceleration to affect the measure, which is consistent with the experimental findings by Ommaya et al. (1994), who reported on the presence of diffuse injuries only when a rotational component was present in the loading.

Ruan et al. (1993) constructed a 3-D model partially based on the geometric data from the work by Shugar (1977). The model contained next to the brain and skull, repre­sentations for the dura mater, falx cerebri and scalp. Impact simulations with the model showed the Head Injury Criterion (HIC) to be generally proportional to impact force, coup pressure and brain maximum shear stress. Coup site and contrecoup site are defined in head injury biomechanics as, respectively, the regions where the impact occurs and the re­gion located diametrically opposed to the impact site. An extension of the model was made by coupling the finite element model, see Figure with a multilink rigid body model

10 Introduction

of the Hybrid III dummy (Ruan and Prasad, 1995). This enabled the analysis of cadaver

finite element model

Hybrid Ill dummy neck

torso (rigid body) - ---.

Figure 1.5: Finite element head model coupled with rigid body dummy model. Adapted from Ruan and Prasad (1995).

tests and crash barrier tests. Results based on the skull tensile stress corresponded with the tolerance curve for skull fracture as presented by Ono et al. (1980). A parametric study by Ruan and Prasad (1996), using modal analysis of the complete head model showed the base of the cranium, brain stem and occipital lobe to be structurally weak in vibration. Further the element type and skull thickness proved to significantly influence the lower skull modes, while the properties of the CSF layer did not affect the modal frequencies of the full head model to a large extent. The modeling of a neck restraint influenced in particular the frequencies of the first two eigenmodes.

Using Magnetic Resonance Imaging (:v!RI) data of a human head Willinger et a!. (1995) constructed a 3-D model that consisted of the skull, the brain, the falx and an elastic solid layer representing the CSF. The material properties for the CSF were tuned, such that an eigenmode of the model was comparable to the head's natural frequency of approximately 100.0 Hz (Willinger et al., 1992). The results obtained with the model were used to construct an alternative dummy head model.

An example of a detailed 3-D model was presented by Zhou et al. (1995a,b). The model was constructed from an anatomical atlas and incorporated representations of the scalp, skull, gray and white matter, cerebellum, ventricles and the meninges. The finite element mesh contained 22,995 elements and it is probably the most detailed model to date with which impact simulations have been performed. Direct frontal impact was simulated for the purpose of validation and a rotational impulse was prescribed to delineate the differences between homogeneous and inhomogeneous brain models. The inhomogeneous brain model contained different material properties for the white and gray matter, in that the Young's modulus of the white matter was chosen 60% higher than for the gray matter. Similar studies (Zhou et al., 1994) with the prescribed rotational impulse of a porcine model, indicated that the numerical results were in good agreement with experimental

11

findings when a 60% higher Young's modulus for the white matter was taken. In modeling white and gray matter in the inhomogeneous model, the resulting pressure response was almost the same as for the homogeneous brain. However, the shear-stress responses were different for the two models. In a parametric study (Zhou et al., 1996) using this model both sagittal and lateral rotational accelerations were prescribed, by scaling the experimental data of Abel et al. (1978). Using the same loading, a lateral rotation was shown to lead to overall higher shear stresses in the brain than a sagittal rotation, although the maximum shear stresses in the genu of the corpus callosum were higher in the latter. The modeling of the brain as being viscolelastic showed similar trends as for the elastic case.

Krabbe! and Appel (1995) developed a 3-D model of a human skull using CT scans. Using the global kinematic data, obtained from multi-body simulations with dummies, impact simulations were announced, to assess the stresses and strains in areas where skull fracture is likely to occur.

Similarly, Bandak et al. (1995) also used CT-data to develop various head models with different resolutions utilizing a mapping method. This mapping method was also used by Shugar (1977) in one of the first ever developed 3-D finite element head models. With the models impact simulations were performed, by driving the head at an assigned initial velocity of 16.8 ms-1 into a rigid wall. The results showed maximum principal stresses to occur on the inner surface of the skull at the point of impact. On the outer surface the maximum stresses were found on a ring with a particular radial distance from the point of impact, consistent with the findings by Shugar (1977).

Next to the complete head models, also brain models have been developed, by as­suming the skull to be rigid and prescribing kinematic trajectories for the nodes located at the outer perimeter of the brain. Ward et al. (1978) made various finite element models of the cerebrum, cerebellum and partitioning folds, that represented the cranial contents of a baboon, a human and a small primate. The results of prescribing accelerations showed in particular the locations and magnitude of maximum stresses to differ in the human and animal models.

To investigate the tolerance of the human head to angular acceleration Mendis (1992) created both a brain finite element model of a primate and a human. Using the results of the experimental brain models obtained by Margulies (1987) the primate model was validated. Characteristic strain responses in the finite element model were found to correlate with the intensity of induced DAI in the experimental models. Using the human finite element model, tolerance levels for DAI found for the primate were extrapolated to estimate corresponding human tolerance levels.

1.3.2 Generation of finite element meshes

The calculation of the deformations of the various tissues inside the head requires among other factors an element mesh that describes the geometry of the tissues with sufficient detail. The process of mesh generation of the models presented in literature is often done using an anatomical atlas. This method, as mentioned by Zhou et al. (1995a), is cumber­some and time consuming and, because of the amount of work involved, the inclusion of details will be hampered.

12 Introduction

(a) (b)

Figure 1.6: (a) MRI parasagittal image of the human head (Adapted from SoftLab software systems laboratory (1993)) and (b) view of a rendered image of the bone structure using CT-data of a woman used in the Visible Human Project (Adapted from Lorensen (1996)).

The contrary is true for Magnetic Resonance Imaging (:VIRI) and Computed Tomo­graphy (CT). These imaging techniques are extensively used to aid in medical diagnostics. Examples of CT and !\1RI images are given in Figure 1.6. These techniques provide the user the geometrical description of the various structures at such a detail , that for them to be useful in finite element analysis some kind of data reduction has to be performed, to keep these models manageable in size. Examples of the usage of these imaging methods for the purpose of mesh generation have been presented by Keyak et al. (1990) and Kang et al. (1993) for the generation of grids of bones, like tibia and femur. Bandak (1995) and Krabbe! and Appel (1995) have shown that CT-data can also be used to obtain a mesh of the human head. Bandak (1995) used a transformation method where an initial regular grid of elements was transformed to fill a particular closed volume, for instance the interior of the skull. The method used by Krabbe! and Appel (1995) was similar to the one demonstrated by Keyak et al. (1990) and Kang et al. (1993) in the generation of meshes of bones. The contours of adjacent images are stacked and elements are generated inbetween the contours. The first method generates a very smooth mesh due to the type of equations used to describe the transformation, but this method is only applicable for closed convex volumes. The second method has the disadvantage that it can easily lead to distorted elements when adjacent contours differ greatly in size or shape.

1.3.3 Boundary and interface conditions

Boundary and interface conditions constitute a very important feature of head impact models, since they have a significant influence on the model's response. Presented mod­eling efforts in this respect relate to kinematic and dynamic boundary conditions at the skull-brain interface, dynamic boundary conditions at the foramen magnum and kinematic boundary conditions at the head-neck junction. Further the applied loads may also be

13

regarded as boundary conditions. They may have either a kinematic nature, when a tra­jectory is prescribed, or a dynamic nature when the head is impacted with an impactor or a force is prescribed on a part of the head.

Looking at the anatomy, the skull-brain interface is only one of the interfaces where relative motion may occur. All the major parts of the central nervous system are separated from one another by either a membrane-like structure or a fluid layer or a combination of the two. In most numerical models the skull-brain interface is coupled either directly using one set of nodes for both the skull and brain or by a layer with a low shear modulus representing the CSF layer. The first type of model allows no relative motion at all, whereas in the second relative motion between brain and skull is possible, though from a computational point of view one must be careful that the layer with low shear modulus does not cause badly conditioned elements, making the solution at least questionable. An alternative method to include the relative motion between different structures is the application of contact algorithms. A contact algorithm monitors the kinematic state between the surfaces of two bodies and if overlapping occurs, proper conditions are applied that prohibit the further penetration of the two bodies.

Concerning the modeling of the skull-brain interface using contact algorithms a dis­tinction can be made between three methods. The first method only considers tangential relative motion with or without friction. In the second, tangential relative motion is mod­eled with or without friction in combination with possible separation of the two structures and the third method simulates only separation between structures. The contact algorithm for this last method is also known as gap element. The first modeling method was utilized by Galbraith and Tong (1988) and Tong et al. (1989) in a 3-D model >vith different interface conditions ranging from pure slip via various levels of friction to a no-slip condition. These models simulated the half-cylinder physical models of Margulies (1987). Results showed that for the higher friction levels higher shear strain levels were induced and that for the no-slip condition maximum principal strains were located near the boundary whereas for the slip conditions more spatial variations were found. Including a falx made the maximum shear strains to occur at the tip of the falx, and these appeared to be relatively insensitive to the level of friction. This modeling method was also used by DiMasi et al. (1991a) and Bandak and Eppinger (1994) for simulating a sliding motion of the brain along the skull.

In numerical models a ferret head, Ueno et al. (1991, 1995) simulated the skull­brain interface using a gap element. The studies compared the numerical results with the experimental findings by Lighthall et al. (1989) and showed the calculated pressure history to correspond favorably with the experimentally measured pressure and the pattern of contusion seen in the impact region corresponded with the high shear stress pattern in the model. In all the studies mentioned above, the contact algorithm was applied between a deformable brain and a rigid skull. To the author's knowledge, no studies have been reported, where both skull and brain were modeled as deformable and a contact algorithm was active inbetween, except for the work by Kuijpers et al. (1995). They used the second method to simulate a possible forming of a gap between brain and skull in the contre-coup region.

Various numerical methods exist for incorporation of a contact algorithm in the

14 Introduction

finite element method under transient dynamic conditions. The utilization of a contact algorithm under transient dynamic conditions together with the fact that within the head contact occurs between structures with a large difference in Young's modulus, for instance between brain and skull, make the numerical simulations highly non-linear.

The foramen magnum is often simulated as a force-free opening (Lighthall et al., 1989; Ueno et al., 1989; Chu and Lee, 1991; Trosseille et al., 1992; Chu et al., 1994) in combination with a no-slip condition along the skull-brain interface. This combination of a no-slip condition at the skull-brain interface with a force-free opening is likely to pose a too strong constraint on movement of the brain through the spinal canal opening. In particular when one realises that the brain stem is surrounded by various fluid-filled spaces (cisterns) which are probably incapable of restricting the brain to slide through the foramen magnum. To understand the effect of movement of the brain through the foramen magnum, Kuijpers et al. (1995) studied the variation of modeling the foramen as a force-free opening an:d omitting the hole in the skull, in combination with either a slip or no-slip condition at the skull-brain interface. Results showed that the model used at the foramen magnum affected especially the pressure response in the foramen region.

Still, further research is needed on the precise modeling of the foramen magnum. When considering a model for the foramen an issue of consideration is that the pressure in the brain may change not only due to motion of brain tissue through the foramen opening, but also through change in volume of the CSF-filled ventricles along with inflow or outflow of CSF and blood during impact.

In modeling the kinematic condition at the head-neck junction, most studies model a free boundary, assuming the neck does not restrain the head immediately after impact. Chu and Lee (1991), Chu et al. (1994) and Kuijpers et al. (1995) modeled the influence of the neck by constructing a simple model of the neck, using beams, and assuming the head rotated around a point located a specific distance below the centre of gravity of the head. Various kinematic boundary conditions at the head-neck junction were investigated by Ruan et al. (1991) and used in their 3-D model (Ruan and Prasad, 1996). In their study on side impact (Ruan et al., 1991) the infuence of a free boundary condition, a single hinge and a double support, made up of a single hinge and a simple support, were studied. Results showed that the free-boundary and single-hinge conditions gave similar responses, i.e. compressive stresses at the coup site and tensile stresses at the contre-coup site. Whereas the double support imposed a too strong constraint on the motion of the head.

Some authors (Lin, 1986; Kumaresan and Radhakrishnan, 1996; Ruan and Prasad, 1996) included a model that represented the complete neck to accurately simulate the motion of the head-neck system. An example of such a model is shown in Figure 1.7. The neck models, presented to allow for a more realistic motion of the head under impact conditions, only incorporate passive material behaviour, i.e. the influence of the active behaviour of muscles has probably not been taken into account. De Jager (1996) has shown, using multibody models of the head-neck system, that the inclusion of the active behaviour of muscles allows for a more realistic motion of the head-neck system that is in

Literature 15

Figure 1.7: Sagittal view of a cross-section of a 3-D model of the human head-neck system. Adapted from Liu (1986).

good agreement with sled acceleration tests with human volunteers.

The applied loads on the head models can be grouped in explicitly modeled impacts be­tween the head model and some kind of external structure, and prescribed motions of the skull. In the modeling of impact, DHviasi et al. (199la,b) simulated the impact of a head model with the A-pillar of an automobile, to allow for a loading representative of automo­bile collisions. The trajectory of the head was prescribed according to experimental data obtained from head forms launched into free-flight and impacting an A-pillar. Comparing the results of padded and unpadded A-pillars, the strain and strain rates of the former were substantially larger. Other authors (Ruan et al., 1993, 1994; Zhou et al., 1995a,b) have fo­cussed on prescribing impacts that were comparable to the experimental data presented by Nahum et al. (1977).

The assessment if rotational motion of the head leads to particular injuries has made some authors (Bandak and Eppinger, 1994; Zhou et al., 1994) prescribe a rotational motion of the head model. In particular the work by Bandak and Eppinger (1994) showed the rotational accelerations to be consistent with the experimental findings by Ommaya et al. (1994).

1.3.4 Constitutive properties

In all models presented to date, the material properties were assumed to be homogeneous and isotropic. The used constitutive models were predominantly linear elastic or viscoelas­tic. A possible cause for this might be that up to a few years ago, because of the limited ca­pacity of computational hardware and software, only 2-D models could be used with linear elastic descriptions for the various tissues. Therefore from the experimental side, for quite a long period, not very many initiatives were undertaken to characterize the constitutive properties of the various tissues with more sophisticated material models. Mendis (1992)

16 Introduction

made a significant step towards a more realistic representation of the material properties by utilizing a non-linear stress-strain relationship for viscoelastic incompressible materials under large deformations. Stalnaker and Mendis (1991) and Mendis (1992) have proven this model to be suitable for modeling the large strains and high strain-rate behaviour of the full-cylinder silicone-filled physical models by Margulies (1987). In a first approach the gel was said (Margulies, 1987) to mimic the mechanical behaviour of brain tissue un­der impact loadings. Mendis (1992) obtained the material properties for the baboon and human head models by fitting the constitutive model to experimental data of Estes and McElhaney (1970) and Galford and McElhaney (1970).

Apart from the experimental data presented by the University of Pennsylvania (Gal­braith et al., 1993; Arbogast and Meaney, 1995; Thibault and Margulies, 1996) no new experimental data has been presented since the early seventies. Although so very little experimental data is available, the Young's moduli for brain tissue for example used in the head models vary by as much as one order of magnitude. Galbraith et al. (1993) and Ar­bogast and Meaney (1995) showed that the constitutive properties of CNS tissue can also be described by the quasi-linear viscoelastic model (Fung, 1993). Arbogast and Meaney (1995) further showed the material properties of brain stem to depend on axonal fiber direction. Incorporation of the anisotropy in the constitutive model was suggested by em­ploying a similar approach as presented by Pamidi and Advani (1978). To quantify the age-dependence of the material properties of porcine brain material, Thibault and Mar­gulies (1996) measured the complex shear modulus of samples obtained from the frontal cortex. The results showed the elastic and viscous components of the complex shear mod­ulus to change significantly with the development of the cortical region of the brain over time.

1.3.5 Validation methods

To assess the quality of the calculated deformations in the human head using numerical models, the numerical results have to be compared with experimental results, for exam­ple strain distributions. The amount of available experimental data is limited, since no techniques exist to date that allow for the in vivo measurement of deformation patterns inside a biological structure, like the human head, under impact conditions. Alternatives have been sought in the usage of experimental data obtained in experiments using animals and cadavers. ·with these experimental models it was only possible to retrieve pressure responses of certain locations inside the head, and kinematic data and contact forces of in particular the skulL Apart from the experimental drawback, that not every mechanical quantity can be measured, and the ethical aspects, there are also other drawbacks in the use of animals and cadavers.

To correlate the results from animal experiments with the results of human head models, a conversion method has to be used. This method primarily compensates for the differences in size and geometry between the heads of animals and humans. The scaling method proposed by Holbourn (1943) for example, is based on the mass ratio of the nu­merical and experimental models. Lighthall et al. (1989) and Ueno et al. (1989) used a scaling method based on the area ratio of their 2-D models.

17

The use of cadavers for the purpose of validation has the drawback of not knowing to what extent a cadaver (Kahum and Smith, 1976; Nahum et al., 1977) shows the same behavior as a human being under impact conditions, see also Wismans et al. (1994).

To allow for a first assessment of the quality of numerical models in the calculation of deformation patterns, physical models (Margulies, 1987; Tong et al., 1989; Cheng et al., 1990) may provide a first step in the validation of numerical models.

Various authors have presented numerical models that were validated using one or more of the experimental methods described above. A short overview of the results will be presented here.

Nahum et al. (1977) measured the pressures in the CSF-layer at various locations in the heads of cadavers, which were subjected to an impact. For the purpose of a first validation Zhou et al. (1995a,b) compared the contact forces between head and impactor, and the coup and contre-coup pressures between model and experiment. Ruan et al. (1993) also used the experimental data by Nahum et al. (1977) for his 3-D finite element model, but compared the numerical with the experimental results at more locations in the CSF-layer.

In an experimental modal analysis of a human volunteer, Willinger and Cesari (1990) observed a decoupling effect at a frequency of 100.0 Hz between brain and skull for motion in the sagittal plane. To validate their 3-D model, Willinger et al. (1995) tuned the CSF properties so as to obtain this same decoupling effect in the numerical model.

Lighthall (1988) and Lighthall et al. (1989) performed controlled brain deformations in ferrets by displacing the surface of the brain with a rigid impactor with a controlled velocity and stroke. Ueno et al. (1991, 1995) used these experimental results for comparison with the numerical results obtained with both 2-D and 3-D models.

1.3.6 Discussion

The state of the art concerning the numerical modeling of head injury under extreme loading conditions has shown considerable progress on the detail with which 3-D models are constructed nowadays. The question arises to what extent there is any benefit on the inclusion of all the details if the material properties that are prescribed for the various structures vary over a range of an order of magnitude. This issue will remain unresolved as long as no further experimental data on constitutive properties becomes available. Hmv­ever, in the mean time parametric studies with detailed models will help to understand how the various structures and interactions between structures affect the field parameter distributions.

The way of modeling the skull-brain interface remains a controversial issue. Ueno and Melvin (1995) used the experimental data by Nusholtz et al. (1984), who using high-speed cineradiographic film of the impact of repressurized cadavers observed internal movement of the brain in the sagittal plane, but noticed only a small amount of relative motion of the skull with respect to the brain. Ueno and Melvin (1995) therefore concluded that a rigidly attached brain in the numerical models was a proper approach of the skull-brain interface. Remarks concerning this conclusion are; firstly that although the displacements might be small along the skull-brain interface, due to the incompressible nature of brain

18 Introduction

tissue, a small displacement may lead to a large change in pressure. Secondly, Holbourn {1945) reports on the experimental evidence of the sliding of the brain along the base of the skull, letting the brain execute a S\virling movement (Shelden et al., 1944). Referring to the same experimental data by Nusholtz et al. (1984), another experiment with a cadaver showed substantial motion of the brain, indicating that a stick-slip effect might be present. Further, although Ueno et al. (1995) preferred a rigidly attached brain to the skull, in other work (Ueno and Melvin, 1995) they modelled the skull-brain interface using the so-called gap elements. To resolve the issue a combined effort of both experiments and numerical research is needed.

On the subject of material properties, Thibault and Margulies (1996) mentioned that the range in reported tissue properties may also be related to the anisotropic and inhomogeneous nature of brain tissue, for which experimental evidence was reported by Shuck and Advani (1972). Further the reported material properties have often been mea­sured at lower shear rates, than the shear rates associated with traumatic: head injury loads.

The field parameter distributions inside the head cannot be mea.'lured experimen­tally, making the direct validation of numerical models on these quantities actually im­possible. Validation is at the moment only possible using pressure responses at various locations in the head, with most locations lying in the CSF-layer (Nahum et al., 1977).

In numerical analyses finite element simulations of structures, that contain nonlinearities of physical and geometrical nature, which are definitely encountered in the analysis of head impact, care should be taken. For example correct strain measures should be used for the possible large deformations and rotations that occur and time steps should be selected such that they allow for a proper representation of the frequency contents of the response. It is often difficult to deduce from published reports to what extent attention has been paid to these aspects. Often the basic dynamical characteristics such as the location of the center of gravity, the relevant moments of inertia and the eigenfrequencies are not reported.

1.4 Scope

1.4.1 Objective

The project presented in this thesis is aimed at the development of a three-dimensional finite element model of the human head to assess the importance of both the degree of geometrical detail to which the components are modeled and the conditions at their in­terfaces. The applied impact loading is comparable to the load a human is subjected to when involved in a car accident. To obtain a qualitative indication of the validity of the numerical results a first comparison will be made with data available in literature. A more thorough validation of the numerical models will be the subject of a future project.

Scope 19

1.4.2 Application

Once the model is validated various applications are possible. The identification of injury mechanisms is possible through the combination of the results of three research fields; iso­lated tissue studies, clinical diagnostics, and accident reconstruction. Isolated tissue studies focus on the macroscopic deformations occurring in the brain under impact situations and histologically interpret these at the microscopic level. In clinical diagnostics, real-life tissue injuries are encountered, when patients are treated for traumatic injuries. The process of accident reconstruction registers parameters that describe the cause of an accident, such as speed of the vehicle and braking distance, and identifies the loads a person was subjected to in the accident. For the identification of the loads, multibody models of car occupants inside a vehicle may provide the kinematic trajectories, that could be used as input for a finite element simulation of the human head, to assess the associated deformations of the various tissues.

Using the finite element model more appropriate head injury criteria may be de­veloped. The current empirical head injury criteria are all based on one single kinematic parameter, such as translational acceleration of the head's center of gravity. It has been dis­cussed in literature (Hardy et al., 1994; Ommaya et al., 1994) that injury criteria, based on macroscopic or microscopic tissue damage might be appropriate to replace or complement current criteria based on gross head motion.

Neck-injury research may benefit from the combination of head and neck (de Jager, 1996) models, where possible interaction between head and neck structures may provide new insights.

Finally in the research of systems aimed at injury prevention, more detailed studies are possible to permit the development and 'virtual testing' of airbags, helmets and various kinds of padding, such as for car-interiors.

1.4.3 Research strategy

The numerical modeling in this research was done using the finite element method. Other theoretical and numerical techniques are also available, as discussed in section 1.3, but they show limitations when modeling the human head. The finite element method is appropriate in modeling complex geometries, interaction between structures and non-linear material behavior of the various tissues.

The visualization tools for creating an accurate geometrical description of the human head are provided by Magnetic resonance imaging (MRl) and Computed tomography (CT) systems. For the creation of a finite element mesh based on MRI and/or CT data only just recently techniques have been presented (Miiller and Ruegsegger, 1995), and further devel­opment is still necessary. Moreover, incorporation of geometrical details in a finite element model has the disadvantage that the model quickly becomes large and computationally ex­pensive. Therefore it is not desirable to aim for the construction of a finite element model with the highest possible degree of detail incorporated.

As to the interaction between structures, the interaction between structures within the head, governed by interface conditions is distinguished from the interaction of the head

20 Introduction

with its environment, described by the boundary conditions. In addition to fluid-solid interaction inside the human head there is also a solid-solid interaction, where the various structures inside the head are not rigidly connected, but have the ability to move relative to one another. The modeling of fluid-solid interaction is only possible using special finite element codes (Bathe et al .. 1995), that are able to explicitly model the fluid behavior and the deformations of the solid part. Little attention has been paid on the importance of the relative motion between structures (Khalil and Viano, 1982b; Sauren and Claessens, 199:3). The use of different interface conditions in numerical simulations may provide an indication of their importance in relation to injury mechanisms. The description of the interface conditions between the various structures in the human head remains a problem to be solved. In a finite element model, contact algorithms, that monitor the state of contact and apply the proper conditions may provide a tool to model the relative motion between structures. The boundary conditions at the head-neck junction describe the connection of the head to the rest of the human body by the neck. The neck interacts with the head in a complex manner, with muscles, ligaments and the spinal cord playing an important role. In addition to the boundary conditions at the head-neck junction the loads applied to the head are also determined by the interaction of the head with its environment. Loads are either transferred via the neck (non-contact loads) or directly applied to the head (contact loads). The duration and magnitude of the latter are often unknown.

Like other biological materials the properties of the soft tissues of the human head are also non-linear and time-dependent. The previously mentioned literature reviews have shown that the majority of the constitutive models used are still based on the assump­tion that brain tissue is a homogeneous and isotropic linearly ( visco )elastic material. Only just recently new experimental data on intracranial tissue properties is becoming available (Arbogast and Meaney, 1995; Galbraith et al., 1993). The data on hand though lack con­sistency and completeness, despite the fact that over the past twenty years considerable progress has been made in the description of non-linear nonhomogeneous materials using mixed numerical-experimental methods (van Ratingen, 1994), and by the introduction of multiphasic formulations (Bowen, 1976; Oomens et al., 1987). As Hardy et al. (1994) men­tioned, more accurate and representative constitutive models for head tissues are required, particularly for the brain, at deformation levels and strain rates typical of real-life impact conditions.

Wave propagation ha.s been presented in literature (Fung, 1990; Liu and Chandran, 1975) as a possible cause for injuries inside the head. The simulation of wave propagation in a finite element simulation puts demands on time size and element size.

Since so little is known about the material properties, and the proper interfaces and boundary conditions, the construction of a detailed model is impossible without a number of assumptions. Strategy: the strategy in this research has been to start off with simple models, that represent only a few structures of the human head. Focus in this study was on the geo­metrical parameters and not on an accurate material description. The material properties may be assessed experimentally, whereas the significance of the geometrical parameters, the amount of detail needed for modeling the structures and the importance of the condi-

Outline of this thesis 21

tions at the interfaces and boundaries can only be determined using numerical or physical models. Numerical models have the advantage over physical models in that boundary and interface conditions can be quickly altered. Direct and detailed validation of a numerical model on these aspects is hardly feasible, because the deformations of tissues inside the head can not be measured. Physical models allow for the measurement of deformations of the surrogate brain under impact conditions. Once it is sufficiently understood which parameters are important, these physical models may be set up in which the numerically predicted deformations of the head are validated, and by accident reconstruction, accidents may be numerically simulated to link injuries encountered in patients with numerically cal­culated deformation patterns. In the end effect, this process may result in the identification of injury mechanisms.

Another strategy for the identification of injury mechanisms concentrates on ani­mals, both in experiments and in numerical models. The experiments with animals offer more possibilities as to in vivo experimental methods and physical models for obtaining data concerning the deformations in the brain due to a specific load. Studies of this kind have been undertaken by .Vlargulies (1987); Margulies et al. (1990); Meaney et al. (1993, 199-5) and by Ueno et al. (1989, 1995). Once it is understood from the animal models how deformations are related to impact loads and possibly what injury mechanisms play a role, the results have to be scaled to the human head, for which various methods exist (Holbourn, 1943; Lighthall et al., 1989; Ueno et al., 1989). In this thesis the choice has been made to use the first proposed strategy.

To date relative motion between in particular the skull and brain was. in most cases, approximated by assuming a very compliant layer, representing the CSF . to exist between brain and skull. Contact algorithms provide a more suitable tool in the modeling of interface conditions between structures. Since the working of these algorithms determine for a large extent how the input energy is transferred from one body to the other, a lot of attention has been paid to the modeling of contact under transient dynamic conditions using the finite element method.

As was discussed in the literature survey (section 1.:3.6); the deformations inside the human head can not be measured and direct validation of numerical models in head injury research is actually impossible. The results obtained with the developed three-dimensional model, based on CT and MRI data, will be compared with the experimental data by Nahum et al. (1977) to obtain an indication of the va.lidity of the numerical results. Proper validation will be the topic of a future study. The parametric study with simple models delineates to what extent various parameters influence results, like for example strain distributions, which are suspected to be a cause for head injury.

1.5 Outline of this thesis

The work presented in this thesis focuses on four topics: • Assessment of the modeling of relative motion between structures using contact

algorithms under transient dynamic conditions.

22 Introduction

• Generation of a three-dimensional finite element model of the human head. • Validation of the finite element model using calculated and measured pressure re­

sponses. • Performing a parametric study using three-dimensional models where various pa-

rameters are varied, such as interface conditions and material properties. Chapter 2 describes the application of contact algorithms under transient dynamic condi­tions. The combination of specific contact algorithms with particular time-discretization schemes is presented. Chapter 3 describes the generation of a three-dimensional finite el­ement model using CT and MRI data. To assess the quality of the model, a comparison is made with experimental data to obtain a first indication. The results of a parametric study using the generated three-dimensional model are presented in Chapter 4. Finally in Chapter 5 the results obtained in this study are discussed and recommendations for further research are given.

Chapter 2

Contact mechanics under transient dynamic conditions

2.1 Introduction

In the field of injury biomechanics two types of dynamic loading of the human head are distinguished; contact and non-contact loading. Contact loading causes injuries to the head because of contact phenomena and inertial effects. With non-contact loading the head as a mass system is accelerated or decelerated resulting in inertial loads.

In case of both contact and non-contact impact, contact between the intracranial contents and the skull occurs. In case of contact impact there is also contact between the head and the environment. Typical time duration of these loading conditions is between 5 and 200 ms.

To simulate the response of the human head to an extreme loading the finite ele­ment method is an appropriate tool in the analysis of a geometrical complex structure, like the human head. Geometrical non-linearities as a result of large displacements, physical non-linearities due to non-linear material behavior, and constraint non-linearities caused by changing contact interactions require the use of iterative procedures. The contact inter­actions are dealt with by a contact algorithm. This algorithm monitors the state of contact and ensures application of the proper conditions.

The equations of motion are solved in time either an explicit or an implicit direct integration scheme. Explicit methods are characterized by their small time step

23

24 Contact mechanics under transient dynamic conditions

as a result from stability requirements, but have low computational costs per time step. Implicit methods are unconditionally stable, their time step being limited only by accu­racy demands. The implicit method is computationally more expensive, than the explicit method, due to the satisfaction of the equation of motion at each new point in time.

Several strategies to handle the contact conditions in the contact algorithm have been reported in literature; the penalty method and the Lagrange multiplier method are the most common used methods. A mathematically attractive method is the direct con­straint method, because the number of equations in the system is reduced as the contact conditions are applied. The chosen strategy implemented in the algorithm will affect the resulting numerical solution. An overview of strategies to handle contact conditions, with their advantages and disadvantages, has been given by Man (1994).

In a preliminary study by Kuijpers (1994) the finite element codes MARC and DYNA3D were compared, concerning their capabilities of modeling contact between two structures. The simulations modeled the impact of two deformable beams. The main conclusions were firstly, that the overall results generated by the two solvers were comparable, but that the stresses and strains calculated by DYN A 3D at the contact interface deviated from the analytical solution. Second finding, was that the modeling of contact using the finite element method is not a foregone simulation method. Since, inside the human head contact may occur between structures having a large difference in Young's modulus, for example between brain and skull, and the purpose of this thesis is the accurate reconstruction of deformations occurring inside the human head, in this chapter the working of contact algorithms is analyzed.

The focus will be on the simulation of frictionless contact between deformable con­tinua for transient dynamic loading conditions. First in this chapter the continuum me­chanics for a continuum in space are presented. Next the equation of motion is discretized in space and time using the finite element method. The most common constraint meth­ods for prescribing the contact conditions between two continua are briefly described and one method is chosen. Further a brief description is given of how the contact conditions are taken into account in the finite element method. In the remainder of this chapter a parametric study is presented to assess the sensitivity of the contact algorithm to vari­ous parameters and the analysis of the contact algorithm is extended to three-dimensional problems.

2.2 Governing equations

The deformation of a continuum is governed by a set of three equations, describing con­servation of mass, balance of momentum and balance of moment of momentum (Malvern, 1969). In this section these equations will be discussed.

Numerical solution 25

2.2.1 Conservation of mass

The conservation of mass states that the mass of a certain volume dV of matter remains the same:

dV p dVo Po (2.1)

where p is the density of the continuum and V the volume occupied by the material. The

subscript 0 refers to a reference configuration. Often J = * is used, where J = det(F), with F the deformation tensor.

2.2.2 Balance of momentum

The equation of motion (balance of momentum) states that the time rate of change of the total momentum of a continuum equals the vector sum of all the external forces acting on the continuum:

V · uc + pb = pu (2.2)

where u denotes the stress tensor, b the body force vector per unit mass and u the dis­placement vector.

2.2.3 Balance of moment of momentum

The balance of moment of momentum states that the time rate of change of the total moment of momentum of a continuum is equal to the vector sum of the moments of the external forces acting on the continuum. Using the equation of motion (2.2) this relation can be reduced to:

u" = u (2.3)

2.3 Numerical solution

The finite element method (FEM) is used to solve the governing equations. In this section a brief description of the finite element method is given, followed by a short overview of the solution in time of the spatially discretized equations, using direct integration methods. This section is concluded with an overview of the required contact conditions to describe the contact between two deformable continua and how these contact conditions are imple­mented in the FEM.

2.3.1 Spatial discretization

The finite element method (Bathe, 1996; Johnson, 1987; Zienkiewicz and Taylor, 1989) starts with a reformulation of the partial differential equation (2.2) as an equivalent vari­ational equation. The variational formulation for the domain f! is solved in the function space Vh, which is spanned by simple functions depending only on a finite number of pa­rameters. Since the function space ~'h is finite, an approximation of the exact solution will be obtained.

26 Contact mechanics under transient dynamic conditions

In this section the FK\1 is applied to the equation of motion (2.2) assuming linear elastic material behavior (u = C 4

: t:(u), with t:{u) H(Vu)c + (Vu)] the infinitesimal strain tensor) and no material damping.

The first step in the FEM consists of multiplying the equation of motion with a weighting function w

(2.4)

and integration by parts of the first term in the resulting equation over the domain n, yields

J.w; (pu)df! + t.(Vw)": C 4: E(u)df! t. W · (pb)df! + lr W • (u · n)df (2.5)

where r represents the boundary of the domain n. This equation is written in the weak form. The equation only contains first order derivatives, as opposed to the second order derivative in equation (2.4), because the continuity requirements for the stress u have been reduced by one order.

Next the domain n is divided into a finite number N.z of subregions (elements). The displacement field u(re, t) within each element is approximated by interpolation between the nodal displacements Un(t) with the use of polynomials Nn(re), the so-called shape or interpolation functions

n!d u(re,t) = I>iVn(re) ·Un(t) {2.6)

n=l

where n~d represents the number of nodes of element e. The shape functions are chosen such that N;( re) 1 in node i ( re = re;) and N;( re) = 0 for all other nodes. For a solu­tion to equation (2.5) to exist the shape functions are required to have a non-zero first derivative and to satisfy inter-element continuity. The shape functions therefore have to be C0-continuous.

When the Galerkin method is applied the weighting functions are interpolated in the same way as the displacement field

n~d

w(re) = L Nn(x) · Wn (2. 7) n=l

Substituting equation (2.6) and (2. 7) in equation (2.5), the integral is evaluated for each element (with domain ne) using a standard finite element procedure Bathe, 1996, Chapt. 4). This procedure yields

(2.8)

where M_, is the element mass matrix, !£, the element stiffness matrix and f the column -e

with nodal forces for element e. This equation must hold for every weighting function w and We may therefore be omitted. Repeating this procedure for all elements of the domain n

Numerical solution 27

and assembling the element matrices and vectors in the global system matrices accordingly, the following system matrix equation is obtained,

(2.9)

where M is the system mass matrix, K the system stiffness matrix and f the nodal load vector. The dynamic boundary conditions (Neumann boundary conditions) like body forces and external forces, are taken into account in the force vector whereas the known displace­ments (Dirichlet conditions) are taken into account in the displacement vector y.

2.3.2 Temporal discretization

The matrix equation (2.9) represents a system of linear differential equations. To solve this system in time two methods are available; direct integration methods and mode su­perposition. The discussion here will be limited to direct integration methods. The term di1·ect implies that prior to the numerical integration no transformation of the equations is carried out. In the simulation time interval [0, T] a solution is sought for the displacement,

velocity and acceleration at the equidistant points in time !:::..t, 2!:::..t, •.. , n!:::..t with !:::..t = ?;;-, for given initial conditions at time 0. An overview of the use of direct integration methods in finite element analysis has been given by Bathe (1996).

In the analysis of transient dynamic problems both so-called explicit and implicit methods are used. methods determine the displacement at the new point in time (t + !:::..t) using the equation of motion at timet, whereas implicit methods consider satis­faction of the equation of motion at timet + !:::..t. An example of an explicit method is the central difference method, in which it is assumed that

.. 1 ( 1.!t = !:::..t2 Yt-At 2yt + 1Jt+At) (2.10)

and 0 1 ( ) li --- -Il + U .t- 2!:::..t -1-!!.t -t+!!.t (2.11)

The displacement solution is sought for timet+ !:::..t by considering equation (2.9) at timet. Substituting the relation for ~~ in (2.9) yields

'l!:t-!!.t (2.12)

from which 'l!t+At is solved. ~~ is not used in this equation due to the absence of damping. The advantage of this method is that if the mass matrix is diagonal (lumped.) a set of uncoupled equations results and the system can be solved without factorization of the left­hand side of the equation, only matrix multiplications are required. A major drawback of explicit methods is that in order for an analysis to be stable the time step !:::..t is required to be smaller than a critical value 6-tcrit· The critical value is determined from the highest eigenfrequency (f max) of the spatially discretized system

(2.13)

28 Contact mechanics under transient conditions

Explicit methods are therefore said to be conditionally stable. An example of an implicit method is the Newmark-,8 method (Newmark, 1959). As­

suming no material damping is present, the equations of motion together with the Newmark equations are,

Mtit+~t + K Yt+~t

!Jt+~t 1!-t + ?Jt6.t + [(~ a)~t + ai.it+~t] /:;,.t2

Yt+~t ~~ + [ ( 1 &)tit + 6iJtMt]6.t

(2.14)

(2.15)

(2.16)

where a and 6 are the Newmark parameters. With the values a = ! and o ~ the integration method is unconditionally stable and no numerical damping is introduced. In the further analysis these values will be used. Integration of the matrix equation (2.14) is performed by solving equation (2.1.5) for and substituting the result into equa­tion (2.14)

(2.17)

For each time this equation is solved for Yt+~t' and tit+At and "itt+tlt are calculated from equation (2.15) and equation (2.16), respectively. The mass matrix Af is either non-diagonal (consistent) or diagonal (lumped), but the stiffness matrix K is always non-diagonal. Con­sequently the matrix on the left-hand side of equation (2.17) is non-diagonal, so (2.17) con­stitutes a set of coupled equations. This is a major drawback of the Newmark-,8 method; solving equation (2.17) requires factorization of the matrix on the left-hand side and ren­ders the Newmark-,8 method computationally more expensive than an explicit integration scheme.

With the incremental displacement and nodal force defined as 6.1Jt+~t Yt+~t 1ft

and 6.[t+At ft+~t- [1

, respectively, equation (2.17) can be rewritten in incremental displacement form

( 4 . 2"") 6.t 1ft+. ~~ (2.18)

The matrix on the left-hand side is denoted as the effective stiffness matrix whereas the right-hand side of the equation is denoted as the effective force r:{~1 •

The time step 6.t in an implicit method can be selected without stability consider­ations. The admissible time step can often be orders of magnitude higher than the critical time step according to equation (2.13) and the time step is only limited by accuracy re­quirements.

The choice, which integration method to use depends strongly on the type of problem un­der consideration. The problems can be divided into two categories; wave propagation and structural dynamic problems, characterized by the relation between a characteristic load time and the characteristic time periods of the system ( eigenfrequencies) under consider­ation. Wave propagation problems (Graff, 1975; van Hoof, 1994) are those in which the

Numerical solution 29

behavior at the wavefront and an accurate replication of the wavefront is of importance. Usually a large number of frequencies in the structure is excited and identifying the rele­vant frequencies to obtain the desired accuracy is a major problem. To capture the high frequency components in the solution a small time step and often a fine mesh are required. In structural dynamic problems low frequencies dominate the response.

A rough criterion for categorizing the problem states that the problem is of the structural dynamic type if the rise time and duration of the load exceed several traversal times of the waves across the structure. The necessity in wave propagation problems to use small time steps makes the use of the Newmark-t3 method less favorable; often the time step required by accmacy demands is of the same order of magnitude as the time step according to stability requirements in the explicit method. In structural dynamic problems the Newmark-,B method is preferred.

Khalil and Viano (1982a) have shown that the eigenfrequencies of skulls lie in the range of 1.0 kHz to 5.0 kHz. These frequencies have characteristic time periods of a few milliseconds ( m.s). The duration of the applied load on the head has the same order of magnitude. The relation between the characteristic load times and characteristic system times make that the analysis of the human head under extreme loading conditions can not be characterized as specifically belonging to one particular category. To be on the safe side considering the numerical stability of an analysis and taking the greater computational costs for granted, the Newmark-p method will be used in this study.

2.3.3 Contact conditions

The interactions between two or more continua that are in a state of contact are mechani­cally accounted for by applying a set of kinematic and dynamic conditions for each of the continua. These conditions are prescribed for those parts of the boundary that actually come into contact. This part of the boundary of a continuum A is denoted by r:. For frictionless contact the contact conditions are

1. no penetration may occur during the period of time the two continua are in a state of contact. For two points P and Q (with material coordinates (P and (Q ), belonging to body A. and B, respectively, the requirement of no penetration can he formulated as

(2.19)

where x denotes the initial position of a point and nA((P,t) = -n8 ((Q,t) are the outward normals of the continua at the points of conta.ct~ The fact that- the outward normal vectors are equal but opposite in sign at the point of contact also states a kinematic condition. From kinematic condition (2.19). similar kinematic conditions for the velocity and the acceleration are derived in Appendix A.

2. the contact forces in the point of contact are equal but opposite in sign for each of the bodies,

(2.20)

30 Contact mechanics under transient conditions

3. only compressive forces are transmitted between the two continua in the point of contact,

(2.21)

Solving the equation of motion subjected to the contact conditions presented above can mathematically be viewed as a constraint optimization process. The implementation of the constraint (contact) conditions in the finite element method can be achieved using various methods. An overview of constraint methods has been given by Man (1994), while an overview of contact algorithms used in the finite element analysis of impact problems has recently been presented by Zhong and Mackerle (1994). To show the consequences of the implementation of the contact conditions for the discretized system of equations of motion a briefdescription is given of three possible methods, the penalty method (Hallquist eta!., 1985) and the Lagrange multiplier method (Chaudhary and Bathe, 1986), both often encountered in literature (Kikuchi and Oden, 1988; Zhong, 1993) and the direct constraint method (Karami, 1989; .Paris and Garrido, 1985), because of its mathematical advantage of reducing the number of equations as more points come into contact. The method is often applied in boundary element methods.

The major difference between the Lagrange multiplier method and the penalty method is that in the former the kinematic contact condition (2.19) is enforced exactly and the contact force is treated as an unknown variable, whereas in the latter contact condition {2.19) is violated and the contact force is obtained by multiplying the resulting penetration by a penalty factor. This calculated force is used to satisfy the kinematic con­tact constraints, always allowing some penetration when two bodies are in contact. In the direct constraint method the contact conditions are also exactly satisfied.

Both the Lagrange multiplier method and the penalty method start off by restating the contact conditions (2.19) and (2.21 ), and they will be discussed first. First the unknown contact force f~ acting on a point P is decomposed into normal and tangential components, according to

(2.22)

where n, s 1 and s 2 are chosen such that they form an orthonormal basis, and A = f~ · n, t 1 f~ · s 1 and t 2 = f~ · s2 • Second the distance between the two points P and Q that come into contact is calculated using

(2.23)

where g is called the gap function. The contact conditions for contact without friction are now restated as

9 0 (2.24)

The third relation in (2.24) expresses the fact that if A 2: 0 then g must be 0, and vice versa. To implement conditions (2.24) in a variational formulation each method uses its own strategy.

Numerical solution 31

Lagrange multiplier method

Let q be a function of two variables such that the solutions of the equation q( >.,g) = 0 satisfy conditions (2.24). Assuming no frictional force to be present (t1 = t 2 = 0 in equation (2.22), the contact force fc is treated as an external force and included in the equation of motion (2.4), as follows

fnw·(V·u+pb-pii)drl+ frw·fcdf=O (2.25)

and the equation q( >.,g) = 0 is added to the system as an additional constraint equation

£ w(q().,g))df = 0 (2.26)

Using standard finite element procedures (see also Eterovic and Bathe (1991)) a new set of matrix equations is obtained

[ Q

(2.27)

(2.28)

where t: = >.r/ and r~ = q(>.k,gk), for each node k that comes into contact. The specific formulation of the additional constraint equation !:c depends on the chosen function q. Next to the displacement degrees of freedom y, a Lagrange multiplier >.k is added to the column of unknown degrees of freedom for each node that comes into contact, where this extra unknown represents the normal contact force.

For every node that comes into contact an equation is added to the total system of equations. The advantage of this method is that the contact conditions are exactly met. However, the size of the system increases as more points of the continua come into contact.

Penalty method

In the penalty method, the second and third conditions in (2.24) are explicitly violated by allowing some penetration to occur. Assuming no friction is present the contact force is calculated as

fc = "(g (2.29)

once penetration has occurred (g < 0). The penalty factor 'Y is a user-defined constant that can be seen as a spring-stiffness between the contacting points. This contact force is added to the equation of motion (2.4)

fnw·(V·u+pb-pii)dfl+ frw·fcdf=O (2.30)

Substituting the equation for the gap function g defined in equation (2.23) in the additional term for fc in equation (2.30) and using standard finite element procedures a new matrix equation is obtained

M · y + [ K + £] · 1! = f + P - -c

(2.31)

32 Contact mechanics under transient dynamic conditions

where Kc · y, is obtained from discretization of

(2.32)

and p is obtained from applying the finite element procedure to -c

(2.33)

The number of equations remains the same in the new matrix equation (2.31) since no additional unknowns are introduced. A disadvantage of this method is that elastic energy is accumulated in the spring-stiffness between the contacting points. This energy has no physical meaning and the total energy of a system decreases as a point belonging to continuum A penetrates continuum B. The elastic energy is recovered when the point releases (Zhong, 1993). Further the penalty factor cannot be calculated, but is chosen based on the experience of the user.

Direct constraint method

In this method the constraint equation (2.19), now formulated in terms of nodal displace­ments, is explicitly substituted in the matrix equation of motion (2.9). This leads to the elimination of one row and one column for each degree of freedom that is constrained. Contact condition (2.21) is used to check if two points are in a state of contact or if the points should be released. The direct constraint method has the advantage that the actual number of degrees of freedom is reduced, compared with the original system where no constraint equations were present, and that the contact conditions are exactly met. It has the disadvantage that all system matrices have to be updated each time a node comes into contact or is released.

The efficiency with which the equations of motion together with the applied contact con­ditions are solved is influenced by the combination of the chosen time integration scheme and contact algorithm. With the time integration schemes described in section 2.3.2 and the contact conditions discussed in this section, various combinations are possible. A few combinations often encountered in literature are discussed here.

Explicit time integration methods can only be used to full advantage in combination with the penalty method. This combination yields a system of equations that fully uses the advantage of an explicit time integration scheme, namely the ability to solve a system using only matrix multiplications. The combination is often used in crash analyses, where a large number of elements and time steps are required for an accurate analysis. The disadvantage that a small amount of penetration is necessary, remains. Combination of an explicit method with the Lagrange multiplier method results in a set of coupled equations, due to the extra introduced degrees of freedom, which makes the solution using only matrix multiplications impossible and the advantage the explicit method had over the implicit method is set aside. To enable a solution a computationally expensive matrix decomposition is necessary.

Numerical solution 33

An implicit method in combination with either the direct constraint method or the Lagrange multiplier method, satisfies the condition of no-penetration. The reduction in number of equations and of freedom in the direct constraint as opposed to the increase in number of equations and degrees of freedom in the multiplier method, makes the direct constraint method computationally more efficient. Since the implicit Newmark-!' method was chosen for time integration, the direct constraint method will be used in the contact algorithm.

2.3.4 Combination of contact algorithm with Newmark-p time integration

Contact condition (2.19) states that no penetration may occur of two continua that are in contact. Time derivatives of this condition state that velocities and (under certain condi­tions) accelerations normal to the surface of the continuum at the point of contact should be the same (see Appendix A). This last condition also follows from the second contact condition, that the contact forces are equal but opposite in sign.

Hughes et al. (1976), Jiang and Rogers (1988) and Taylor and Papadopoulos (1993) have shown that standard time integration schemes as described in section 2.3.2 are un­successful in modeling the imposed kinematic constraints. The above described methods (section 2.3.3) for implementing the contact conditions in combination with the Newmark-!' method for time integration cause errors in the solution for the velocity and the acceler­ation for the points that come into contact. The velocity and acceleration for each of the points of a pair of contacting points have a different value. The difference in velocity is independent of the time step size and for the acceleration the difference is inversely propor­tional to the time step thus consistency is not obtained as the time step gets smaller. These differences are a result of the fact that the solution is sought at discrete points in time and that often only a condition for the displacements (contact condition (2.19)) is included at the time of contact. Appendix B shows an example of the calculation of the velocity and acceleration where only displacement conditions have been used.

To model impact between continua correctly extra contact conditions are required for the velocity and the acceleration. The extra condition for the velocity can be derived from contact condition (2.20). This condition states that the contact forces between two contact points are equal but opposite in sign. As a result the sum of the contact forces is zero and the contact forces do not contribute to momentum. The total amount of momentum in the system as a result of the coming into contact remains unchanged at the point in time tc the two continua come into contact (from t;; to tt). From this balance of momentum before and after the coming into contact a velocity for the contacting points can be derived. This new velocity at tt will serve as a new initial condition after the application of the contact conditions.

A correction for the accelerations is obtained by weighting the accelerations by the masses of the contacting points so that after the implementation of the contact conditions the equation of motion is satisfied and the contact forces are equal but opposite in sign. In Appendix B, the example of the impact of a node on an element and the algorithm for correcting the velocity and acceleration, is described using the above presented method.

34

The finite element calculations in this study have been done using the commercial finite ele­ment program MARC K6.1 (MARC Analysis Research Corporation, 1994). In this program amongst other methods the direct constraint method is available and the extra conditions for the velocity and accelerations have been implemented using a lumped mass matrix. Using the direct constraint method in MARC, contact is established between a node and an element side (segment), by considering a tying relation between the node and the nodes of the segment (Appendix B). The lumped mass matrix is used to make the calculation of the new velocities and accelerations at the point of contact, computationally less expensive.

2.4 Testing and parametric study of 2-D contact algorithm

In this section the contact algorithm is tested by comparing the results obtained with the finite element method (FEM) with an analytical solution. To analyze the sensitivity of the contact algorithm to (1) the difference in stiffness between the two bodies, (2) the time step and (3) the mesh size1 a parametric study is carried out. The problem considered is the impact of two rods, where for the analytical solution one-dimensional wave theory is assumed.

2.4.1 Model description

The rods (Figure 2.1) have the same geometry and material properties. They are 10.0 m

I I

I I I

Jo X

Figure 2.1: Impact oftwo rods (above). Incident, reflected and transmitted waves at the junction between two rods (below).

long and have a height of 1.0 m. Their material properties are linear elastic with elastic modulus 1.0 ·103 Njm2

, Poisson's ratio 0.0 and mass density 1.0 · 10-3 kgjm3 • For rod 1 an initial velocity of v~ = 1.0 m/ s is prescribed.

Testing and parametric study of 2-D contact algorithm 35

2.4.2 Analytical solution

The analytical solution for the problem given above is based on one-dimensional longitu­dinal wave propagation theory (Goldsmith, 1960; Graff, 1975). In case of the impact of the two rods, waves will start traveling from the impact interface in positive and negative x-direction, in rod 2 and 1, respectively (Figure 2.1 ). At the impact interface continuity of force and velocity is required. The incident, reflected and transmitted waves have to satisfy the following conditions

V[- VR

where subscripts /, Rand T refer to the initial, reflected and transmitted waves.

(2.34)

(2.35)

A one-dimensional equation of motion for longitudinal wave propagation is obtained by considering the motion of an element dx of the rod. It is assumed that plane, parallel cross-sections, perpendicular to the axis of the rod, remain plane and parallel, further that a uniform distribution of stress throughout a cross-section exists and that radial inertia may be neglected. When the material behavior is assumed to be linear elastic and no body forces are present, the governing equation is

(2.36)

with c the one-dimensional longitudinal wave speed, defined as c !J. The coordinate x

refers to the cross-section of the rod with the origin on the left-hand side of rod 1, while the longitudinal displacement with respect to the undeformed configuration is given by u(x, t). The solution for this hyperbolic equation is of the type

u(:r, t) = f(x ct) + g(x + ct) (2.37) .

where f and g are arbitrary functions depending on the initial conditions. The function f corresponds to a wave traveling in positive x-direction and g corresponds to a wave traveling in negative x-direction. For each of the rods an equation is defined for the displacement and the velocity,

Ua(x,t) f(x- Cat)+ g(x +Cat)

t) = -caf'(x Cat)+ Ccxg'(x +Cat)

(2.38)

(2.39)

where f' is defined as ~f(x cat) and a G(x Cat) 1, 2 corresponds with the first and the second

rod, respectively. It is assumed that at t = 0 s the rods come into contact. La denotes the length of rod a. The initial conditions are

v1(x,t) Ct [-f' + g'] v? for 0 $ x $ Lt and t = 0

v2(x,t) c2 +g']=OforLt$x$L1+L2anclt=O

o-1(x, t) = Et Gut Ptci (f' + g') = 0 for 0 :S ;r. $ Lt and t = 0

P2C~ [f' + g'] = 0 for

(2.40)

(2.41)

(2.42)

(2.43)

36 Contact mechanics under transient dynamic conditions

and the boundary conditions are

a 1(.r,t)

a 2 (.r,t) PIc~ [f' + g'] = 0 for x = 0

pzc~ [f' + g'] 0 for x L1 + L2

(2.44)

(2.45)

with E = pc2• From these boundary and initial conditions f' and g' are calculated. Solving

f' and g' from equations (2.40) and (2.42) for t 0 and 0 :::; ;r :::; Lh yields

J'(x,t) g'(x, t) = ') -CJ

(2.46)

and solving f' and g' from equations (2.41) and (2.43) for t = 0 and L1 :::; x :::; L1 + L2 ,

yields f'(x, t) = 0 g'(x, t) 0 (2.47)

The boundary conditions for x = 0 and x L 1 + L 2 and 0 :::; t :::; oo come down to

f'(x,t) = -g'(x,t) (2.48)

>vhich states that a reflected wave from the free-end of the rod will have the same shape as the incident wave but is opposite in sign.

The solution of the wave propagation problem can now be visualized for any posi­tion x and point in timet. The solutions for the displacement and the velocity are plotted in Figures (2.2) and (2.3) for both rods at various points in time. The results show that

through each of the beams a wave propagates and that after time 2 ~/1 = 0.02 s the wave is back at the point of impact between the two rods. At this point in time rod 2 has a uniform speed of vr = 1.0 mfs and rod 1 has a uniform velocity 0.0 rn/ s.

When the material properties of the two rods are the same, an incident wave and a transmit­ted wave will travel from the contact interface in negative direction in rod 1 and in positive direction in rod 2 respectively. In case the material properties differ next to the incident wave a reflected wave has to travel in rod l, to satisfy the conditions (2.34) and (2.;3.5). The general solutions of the three wave fields are

UJ f1(x c1t)

'UR 92(x + Ctt) UT = fz(x- Czt)

(2.49)

(2.50)

(2.51)

where ur represents the incident wave, UR the reflected wave and tty the transmitted wave. Substituting the three wavefield solutions in the conditions (2.34) and (2.35), the following solutions for g; and f~ are found,

I 9z

EzAzc1 E1A1c2 . J' El.4Icz + EzAzcl 1 (2.52)

f~ 2E1A1c1 . f~

E1A1cz + EzAzcz (2.53)

Testing and parametric study of 2-D contact algorithm 37 .~~-----------------------

0.01-

..., 0.008-

=

t=O.O [s] t=0.0025 [s]

~ ...., =:

t=0.005 [s] O.Ql

0.008

9 o.oo6= ~ 0.004

a. 0.002

E O.Dl­

..., 0.008-

g 0.006-

8 0.004-.a o.ooz-J---,

"' 0.006 a "' 0.004 u

"' -a 0.002

"' "' ~ Of=~~~~~~ I I I

'"0 0 ~ o- ........ 1---..--.---,r-----l I I I

E O.ot

..., 0.008 c ~ 0.006

~ 0.004

'"3. 0.002

~ 0

E O.ot

...., 0.008

~ 0.006

1;; 0.004

'" '"3. 0.002

"' ~ 0

0 5 10 15 20 0 s 10 15 20 0 5 10 15 20

position x [m) position x (m] position x [m]

t=0.0075 [s] t=O.Ol [s] t=0.0125 [s]

E 0.01 E O.Dl

...., 0.008 ..., 0.008 =: =:

"' 0.006 a "' 0.006 a "' 0.004 u "' 0.004 u .a 0.002 "' 0. 0.002 00

0 ~ ifJ

0 ~

0 5 10 15 20 () 5 10 15 20 0 5 10 15 20

position x [m] position x [m] position x [m]

t=0.015 [s] t=0.0175 [s] t=0.02 [s]

s 0.01 ::: 0.01

...., 0.008-~ 0.006-s

-;::: 0.008-<l.l 0.006-a

"' 0.004-~ "' 0.004-u

"' c. 0.002- a. 0.002-.~ 0-· ,, ..... , ....... , "0

00 o-~ .......

I I I I l l 0 10 15 20 0 5 10 15 20 0 10 15 20

position x [m] position x (m] position x [m]

Figure 2.2: Analytical solution of the displacement for the impact of two rods for 9 points in time. The x-axis represents both rods, each 10.0 m in length.

As an example, suppose the elastic modulus of rod 1 is four times the elastic modulus of rod 2 (E1 4E2 ) and all other properties are identical. Then for the wave propagation speeds holds c1 = 2c2 , and for g~ and f~ the following values are found; g~ -V~ and f~ = tr;. The velocity immediately after the coming into contact should be the same for both rods at the contact interface ( v1 (a; £ 1 ) v2 ( x = £ 1 )}. The velocity is calculated as

V1 = c1(-f~ + g~) , I 1 I 2 0 2c2(-jl- :/1) = JV1 (2.54)

V2 = cz(-f~) c2 · ~!' 3 1

•) :.vo 3 1 (2.55)

The numerical solution of this example is shown in the parametric study.

38 Contact mechanics under transient conditions

t=O.O (s] t=0.002.5 [s] t=0.005 [s] 1 1

"' 0.8- 0.8 -... 0.8

0.6- 0.6 ..§.

0.6 >. ~ ~ '"" 0.4- 0.4 0.4 ·a ·r; ·a .£ 0.2- .£ 0.2 0 0.2 ..,

"' " > o- ····-- > 0 > 0 I I I

0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

position x [m] position x [m] position x [mj

t=0.0075 [s] t=O.Ol [s] t=0.0125 [s] 1 1-

0.8 0.8- 0.8

0.6 0.6- 0.6 ~ ~ >.

0.4 0.4-...,

0.4 ·;::; ·o ·;::; 0 0.2 .£ 0.2- .£ 0.2

" g: "' > 0 o- .... > 0 I I I

0 5 10 1.5 20 0 5 10 15 20 0 5 10 15 20

position x (m] position x (m] position x [m]

t=0.015 [s] t=0.0175 [s] t=0.02 [s] 1 1-

0.8 0.8 0.8-

0.6 0.6 0.6->. >. ~ ..,

0.4 ..,

0.4 0.4: ·~ ·o ·;::; .£ 0.2 0 0.2 0 0.2

0.1 ...., -.;

> 0 > 0 > 0 ..............

I I I 0 10 15 20 0 5 10 15 20 0 5 10 15 20

position x [m] position x [m] position x [m]

Figure 2.3: Analytical solution of the velocity for the impact of two rods for 9 points in time. The x-axis represents both rods, each 10.0 m in length.

2.4.3 Numerical solution

The assumptions made for the analytical solution in section 2.4.2 are taken into account in the numerical solution by using a value of 0.0 for Poisson's ratio. To model the corning into contact of the two rods an initial gap between the two rods of 0.02 m and a contact tolerance of 0.01 m is used. This tolerance sets the contact conditions once a node is within a distance of another body smaller than the contact tolerance. Release between the rods occurs once contact condition (2.21) is not satisfied anymore.

From the point in time where contact occurs the finite element solution can be compared with the analytical solution. The impact of the two rods is simulated using two­dimensional linear quadrilateral plane-strain elements. The rods are uniformly discretized in space using 20 elements each over the length of the rods. There are no other boundary conditions prescribed for the rods. Rod 1 has an initial velocity of v~ = 1.0 m/ s and due to

Testing and parametric study of 2-D contact algorithm 39

the contact tolerance will impact rod 2 at t = 0.01 s. For the Newmark-;3 time integration a time step of 5.0 · 10-4 s is used and the simulation time is 7.5 · 10-2 s. With the chosen material properties the longitudinal wave speed c0 is 1,000.0 m/ s. The traversal time of the wave through a rod is 0.01 s. The analytical results showed the wave to be back at the impact interface 0.02 s after the coming into contact. If contact occurs at 0.01 s, the rods should release at 0.03 s.

The numerical and analytical results for the displacement in x-direction are shown in Figure 2.4 and the results for the velocity in x-direction are plotted in Figure 2.5. The results are shown for nodes located at the lowerside of the rods, with node 1(3) 1 at the left face of rod 1(2) and node 2(4) at the right face of rod 1(2). The results for the velocity

0.06-y----------------,

" 0.05-4

~~ ,;j E ¥

., 0.04- /~~ .::

~ 0.03- //~/ "' ]. 0.02-L // -:.a

0.01- //

0 ~'-~~----------------I I I I I I I

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

time [s]

:: ., = " E " (_)

"' "Z. 00

:.a

0.06

0.05

0.04

0.03

0.02

0.01

0

~/~ " ,

./ .. "' / ~·

/ 2 /

-----------

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

time [s]

Figure 2.4: Displacement in x-direction. (-):numerical solution rod 1; (--):numerical solution rod 2; (---):analytical solution. The numbers refer to the node numbers.

" ---.. :: ~ ·u 0

Q) >

1.4

1.2

0.8

0.6

0.4

0.2

0

-0.2

-0.4

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

time [s]

" -::;. .£ :>,

·u 0

Q) >

1.4

1.2

0.8

0.6

0.4

0.2

0

-0.2

-0.4

0 O.Dl 0.02 0.03 0.04 0.05 0.06 0.07

time [s]

Figure 2.5: Velocity in x-direction. (-) : numerical solution rod 1; ( --) : numerical solution rod 2; (---) :analytical solution. The numbers refer to the node numbers.

are, apart from the oscillations after the instant of release, in good agreement with the

1. numbers between parentheses denote the numbers of the other rod

40 Contact mechanics under transient conditions

analytical solution (Figure 2.3). The oscillations appear periodically. The displacement solution obtained with the FEM is also similar to the analytical solution (Figure 2.2). The quality of the agreement between the analytical and numerical results makes that the results for both the displacement and velocity partially overlap. The numerical solution shows slight deviations from the analytical solution for all quantities, particularly towards the end of the analysis, but is best seen in the results for the velocity where the numerical solution oscillates around the analytical solution.

Concerning the oscillation in the numerical results of the velocity Holmes and Be­lytschko (1976) observed the presence of spurious oscillations to be inherent in a transient analysis. Chin (1974, 1975) has shown that the analytical solutions of only spatially dis­cretized .equations of motion also have oscillations, which are known as Gibbs phenomenon (Kreyszig, 1993, pg. 602). The frequency of the oscillations is governed by the spatial dis­cretization of the finite element mesh, compare for instance Figure 2.5 and Figure 2.12. As the oscillations are caused by the numerical solution of the FEM they may be removed from the solution either by using some sort of damping (Jiang and Rogers, 1988) or by post-processing the numerical solution using digital filters (Holmes and Belytschko, 1976) or local error estimates (Wiberg and Li, 1993).

Wang et al. (1992) mentioned that next to the existence of spurious oscillations, the wave propagation velocity attained in the numerical solution will be different from the physical velocity. They presented numerical analyses where it was shown that both the spatial and temporal discretization influence the arrival time of the wavefront. Wang et al. (1992) have deduced the relative velocity errors for linear elements, illustrated in Figure 2.6.

10 15 20 25 30 35 40 45 so It Figure 2.6: Contours of the relative error of velocity for linear elements (Adapted from Wang et al. {1992)).

The relative velocity error c, is defined as

CN- C

c (2.56)

Testing and paramE!~ric study of 2-D C()n_t_a_c_t_a_l..::g=-o_r_it_h_m __________ 4_1

where eN denotes the numerical wave propagation velocity and c the physical velocity. In the figure the wavelength to element size ratio is plotted against the ratio of the period time to time step size for constant levels of the relative velocity error. The wavelength A and period time T are coupled via the wave propagation speed c, A c · T. Each of the harmonic wave components, that composes the transient wave, has a different velocity error and hence travels at a different speed. This phenomenon is called velocity dispersion (Wang et al., 1992; Goldsmith, 1960). As can be seen from Figure 2.6 the contour with velocity error zero coincides with a straight line. Thus velocity errors can be suppressed by choosing a combination of mesh size and time step that lies on this line. The line can be approximated by the relation

A 2 T h= 3'

With the same mesh a critical time step can now be calculated

!lt 2 h

3 c

(2.57)

(2.58)

Using the element length of 0.5 m and the wave propagation speed of 1,000.0 m/ s results in a critical time step of 3.3333 . w-4 s.

Wang et al. (1992) found that spurious results associated with improper coarse meshes or too small time-step magnitudes generally occur before the arrival of the wave­front, whereas the spurious results due to too fine meshes or large time-step magnitudes are visible after the arrival of the wavefront. The used time step in the analyses presented in Figures 2.4 and 2.5 is than the critical time step and the point in time where oscillations occur is attributed to velocity dispersion as described by Wang et al. (1992).

2.4.4 Parametric study

For a set of parameters, that are thought to be of importance in the analysis of the human head under extreme loading conditions, a parametric study has been conducted. Three parameters have been varied. First, in the human head contact occurs between the stiff skull and the compliant central nervous system. To assess the influence of a difference in Young's moduli, an analysis using the two rods has been performed where the Young's modulus of rod 1 is 4 times the Young's modulus of rod 2. Using this value of 4 a comparison can be made with the analytical solution presented in section 2.4.2. Second an analysis has been conducted using a time step of 1.0. w-4 s instead of 5.0. w-4 s, in particular to see if the numerical solution converges to the analytical solution. The solution using the smaller time steps can also be related to the remarks Wang et al. (1992) made concerning the presence of oscillations before or after the arrival of the wavefront. Further, to assess the influence of mesh refinement both rods have been refined by splitting each element in four new elements. The original rods were discretized using 20 elements over the length of each rod, the refined mesh has 40 elements over the length and 2 elements over the height of each rod. The analysis was conducted using a time step of 1.0 · 10-4 s, the same as was used in the analysis with the smaller time step. The simulation time in all studies was

42 Contact mechanics under transient conditions

Table 2.1: Parametric study of the impact of two rods. h denotes the element length, ( dsp.) the displacement and (vel.) the velocity.

Description of time step rod E v p h result Figure parameter [s] # (N/m2

] H [kg/m3 ] (m] reference 5.0. 10 4 1 1.0. 103 0.0 1.0. 10 3 0.5 dsp. 2.4

2 1.0. 103 0.0 1.0. 10-3 0.5 vel. 2.5 elastic modulus 5.0 ·10-4 1 4.0. 103 0.0 1.0. 10-3 0.5 dsp. 2.7

2 1.0. 103 0.0 1.0. w-3 0.5 vel. 2.8 time step 1.0 ·10-4 1 1.0. 103 0.0 1.0. 10-3 0.5 dsp. 2.9

2 1.0. 103 0.0 1.0 ·10-3 0.5 vel. 2.10 mesh refinement 1.0. w-4 1 1.0. 103 0.0 1.0. w-3 0.25 dsp. 2.11

2 1.0. 103 0.0 1.0. w- 3 0.25 vel. 2.12

7.5 · w-2 s. Table 2.1 an overview of the used parameters in the various simulations including the reference simulations in paragraph 2.4.3. The results of the parametric study are shown for the displacement and the velocities of the rods.

Results: difference in stiffness

The results, for the displacement and the velocity of the analysis with a difference in elastic moduli between the two rods, are presented in Figures 2. 7 and 2.8. The results are plotted for the same nodes as in Figure 2.5.

0.06 0.06

0.05 /- 0.05

~ 4, / '"2 0.04

/ ...:::.- 0.04 / .., / ~ :::: --<!) - (j)

s 0.03 ;- s 0.03 / Q) / Q)

~ / u / <ll

-a 0.02 -a 0.02

"' "' :e :e 0.01 0.01

0 0

0 O.Ql 0.02 0.03 0.04 0.05 0.06 0,07 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

time [s] time [s)

Figure 2. 7: Displacement in x-direction for the simulation with difference in elastic moduli. (-) ; numerical solution rod 1; (--) :numerical solution rod 2. The numbers refer to the node numbers.

The result for the velocity shows that just after the coming into contact (t 0.01 s) the nodes at the contact interface (node 2 and 3) have the same velocity of approximately 0.667 m/ s. In contrast to the constant velocity of 0.5 m/ s in Figure 2.5 during the time period the two rods are in contact, the velocity now oscillates around the value of 0.667 m/ s.

Testing and parametric study of 2-D contact algorithm 43

1.8..,---------------., 1.6

1.4 1.2

1-+----"'-1-..

0.8

0.6

0.4

0.2

0

-0.2+--r--r--r--r--r--r--r....l

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

time [s]

l.B...,.------------:------, 1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

I It '• ! Itt\ II I II II~~

I ~ v ~\ I I I I l I I \ I f

' I

}, ,, '• I• I I I I f

-0.2 +-.,--,.---,r--.--r-"'T'"-T""""' 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

time [s]

Figure 2.8: Velocity in x-direction for the simulation with difference in elastic moduli. (-) numerical solution rod 1; ( --) : numerical solution rod 2. The numbers refer to the node numbers.

The value of 0.667 m/ s corresponds with the result found in the analytical solution. The results show that in rod 1 the wavefront propagates with a velocity about twice as high as in rod 2.

Results: time step variation

Figures 2.9 and 2.10 show the results for the displacement and velocity, respectively, for an analysis with an integration time step of 1.0 · 10-4 s. The results are shown for the

0.06-.----------------. 0.06,----------------,

0.05

0.04

m s 0.03

"' u ..$ .~

0.02

"0 O.Dl

0

/ /

// /

3/ /

/" / 2

/ /

0.01 0.02 0.03 0.04 0.05 0.06 0.07

time [s]

Figure 2.9: Displacement in x-direction for the simulation with a time step of 1.0 · 104 s. (-) numerical solution rod 1; ( --) : numerical solution rod 2. The numbers refer to the node numbers.

same locations as in Figure 2.5. Comparing these results with the analytical solution in Figures 2.2 and 2.3 for respectively the displacement and the velocity, show that both the results for the displacement and the velocity are in good agreement with the analytical solution. There are less oscillations present compared to Figure 2.5 and consequently the deviation of the numerical displacement solution from the analytical solution is also smaller. The points in time where the oscillations at the wavefront occur in the result for the

44

1.4

1.2

0.8

0.6 ;,.., ...,

0.4 ·o ~ >

0.2

0

-0.2

0 0.01 0.02 0.03 0.04 0.05 0.06 O.D7

time [s]

conditions

1.4...,----------------, 1.2

0.8

0.6

0.4

0.2

0

-0.2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

time [s]

Figure 2.10: Velocity in x-direction for the simulation with a time step of 1.0 · 104 s. (-) numerical solution rod 1; ( --) : numerical solution rod 2. The numbers refer to the node numbers.

velocity are in agreement with the remark made by Wang et al. (1992), that when a time step ( 1.0 · 10-4 s) smaller than the critical time step ( 3.333 · w-4 s) is used the oscillations are expected to occur before the arrival of the wavefront.

Results: mesh refinement

The results obtained with the refined mesh are shown in Figures 2.11 and 2.12. The devia­tion of the displacement solution from the analytical solution is smaller than the deviation seen in Figure 2.4, the results for the displacement almost fully overlap. The oscillations in the velocity have a higher frequency and still appear periodically, with the same period (approximately 0.02 s) as in the reference simulation in Figure 2.5. The higher frequen­cies are caused by the mesh refinement. Dll.e to the smaller elements with the same mass density higher eigenfrequencies exist in the mesh. Closer analysis of the period with which the oscillations appear shows that the period time equals twice the traversal time (0.02 s) of a single rod, this period of time is also the period time of the smallest longitudinal eigenfrequency.

2.5 3-D contact

This section describes how contact between three-dimensional continua is established. The contact conditions presented in section 2.3.3 were given for the general case with no ref­erence to the kind of problem analyzed. The major difference between two and three­dimensional contact lies in the detection of contact, implemented in the contact algorithm. In two dimensional contact problems, contact is detected between a node and a segment, whereas in a 3-D analysis, contact is detected between a node and a so-called surface patch. A patch is defined by the four nodes of the face of an element. The tying relation between the single node and the nodes of the patch now depends on the quantities of four nodes instead of two nodes in the 2-D case.

Conclusions 45

0.06 0.06.....-----------------.

0.05 0.05

0.04 1:

0.04 1:

"' "' s 0.03 "'

~ 0.03 '-' "' ~ 0. 0.02

"' ]. 0.02

:a :a 0.01 0.01

0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

time [s] time [s]

Figure 2.11: Displacement in x-direction for the simulation with a refined mesh. (-) :numerical solution rod 1; (--) :numerical solution rod 2. The numbers refer to the node numbers.

1.4

1.2

~ 0.8 -.... .£ 0.6 >, ..... 0.4 ·a 0

1! 0.2

0

-0.2

0 O.oJ 0.02 0.03 0.04 0.05 0.06 O.o7

time [s]

~ -.... E >,

·-= '-'

l

1.4

1.2

0.8

0.6

0.4

0.2

0

-0.2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

time [s]

Figure 2.12: Velocity in x-direction for the simulation with a refined mesh. (-) : numerical solution rod 1; : numerical solution rod 2. The numbers refer to the node numbers.

As there are no other differences between 2-D and 3-D contact analyses no example will be given here.

2.6 Conclusions

The balance equations presented in section 2.2 are discretized in both space and time. The contact conditions were given for two continua that come into contact. The direct constraint method provides a numerically attractive method for prescribing the contact conditions in the system equation of motion. The combination of contact conditions with the Newmark-;3 time integration method leads to erroneous results in the velocities and the accelerations of the nodes that come into contact. Taylor and Papadopoulos (1993) have shown similar errors to occur when Lagrange-multiplier and penalty methods are used in combination with Newmark-p time integration. Explicit time integration methods

46 Contact mechanics under transient conditions

have only been considered in a preliminary study (Kuijpers, 1994). Main conclusion con­cerning the choice of method for modeling contact, looking at the analysis presented in this chapter, and the amount of literature published on this topic, is that the simulation of the interaction between structures has not converged to the usage of one single method for prescribing contact. Other commercial codes may therefore show deficiencies, similar to the ones encountered here.

Correction of the errors in the velocity and acceleration is possible by satisfying the condition that the contact forces are equal but opposite in sign for the contacting nodes (balance of momentum). The results of FEM simulations of the coming into contact of two continua agreed well with their analytical counterparts. Oscillations in the velocity are the result of the so-called Gibbs-phenomenon and the point in time where these oscillations occur is influenced by the time step size. After the rods have released oscillations keep ap­pearing periodically, with period time 0.02 s, which indicates that the rods are not stress free.

Guidelines on two topics may be deduced from the study presented in this chapter on the working of contact algorithms under impact conditions. Firstly satisfying the condition that the contact forces are equal but opposite in sign (balance of momentum) at the point in time of impact for the; nodes that contact, provides an accurate method to simulate contact between two deformable structures. Since the balance of momentum is satisfied at the node level this method can be used regardless of the geometries of the structures involved. The only requirement to be met is that the time step of the time integration scheme should be smaller than the smallest traversal time of the wave through one of the structures. If this is not the case the coming into contact and release of the two structures all take place within one time step.

Secondly, concerning wave propagation it has been shown that both the spatial and temporal discretization influence the arrival time of the wavefront in the finite element simulation. For linear elements there exists a relation between mesh size and time step with which the velocity errors can be suppressed. For simple problems for which analytical solutions exist, one can take this relation into account, however for more complex 3-D structures with various element sizes and orientations this is virtually impossible. For these complex structures it might therefore be worthwhile to use experiments to assess the velocity error between the numerical solution and the physical measurements. Further for the removing of spurious oscillations from the calculated response, damping 'or filtering techniques could be utilized. However care must be taken that only oscillations related to the numerical solution are damped or removed, oscillations that characterize the physical behavior must not be influenced.

Chapter 3

Development of a 3-D huma11 head model

3.1 Introduction

In this chapter an anatomically realistic 3-D model of the human head is presented. The model is anatomically realistic in the sense that in particular attention has been paid to the geometric description of the cranial vault. Since the emphasis in this thesis is on parameters that influence deformations of the contents of the skull, for these are responsible for traumatic brain injury, the tissues covering the skull will be left out of the model.

For the creation of a model various kinds of input data are necessary; the geometric description of the head, the material properties of the various tissues, and the boundary and interface conditions of the head in relation to its environment and within the head itself.

The objective of the development of the numerical human model is to accurately reconstruct the deformations of the human head when impacted. In order to assess which geometrical details, and boundary and interface conditions influence these deformations the first step is the creation of a simple model consisting of a skull, that contains a repre­sentation for the cranium and a general description of the facial bones. The contents of the cranium is a homogeneous continuum and will be denoted as brain. Using this model and step by step increasing its complexity the influence of each of the previously mentioned parameters can be studied. With this knowledge complete human head models can be

47

48 Development of a 3-D human head model

constructed that fully represent the mechanical behavior of the human head under impact conditions.

Another option, as discussed in Section 1.4.3 for the identification of injury mecha­nisms concentrates on animals, both in experiments and in numerical models. In this thesis the choice has been made to develop a numerical human head model and to assess the im­portance of both the degree of geometrical detail to which the components are modeled and the conditions at their interfaces.

The quality of the calculated results is first studied by comparing the results of modal analyses of separate numerical models representing only the skull, and the complete head, with the results of experiments on the same parts. Due to the absence of experimen­tal studies of modal analyses performed on the brain and to obtain an indication of the correct order of magnitude of the calculated eigenfrequencies, the results of the modal anal­ysis of the brain will be compared with the results of another numerical model presented in literature. Next to using modal analyses, the quality of the model is further investi­gated by comparing the transient response of the model with the measured pressures and accelerations in the experiments performed by Nahum et al. (1977).

In th,!s ch,apter .first the development of the model starting from MRI and CT data will be described. Next the material properties and inertial properties are presented and the impact loads derived. from the experiments by Nahum et al. (1977) are discussed. The quality of the model is delineated. in section 3.3 by comparing the modal characteristics and in section 3.4 on the basis of pressures measured by Nahum et al. (1977) at various locations in the heads of cadavers. The sensitivity of the results to time-step sizes and element sizes is presented in section 3.5 and 3.6.

3.2 3-D model development

The 3-D ·reference model will be composed of a skull filled with a homogeneous continuum denoted as brain. A general description of the facialbones will be added to simulate the inertia of the facial structure.

For the reconstruction of the geometry of the structures of the human head use was made of the Visible Human data set (U.S. National Library of Medicine, 1996). This data set contains MRI (Magnetic Resonance Imaging) and CT (Computed Tomography) transversal images of a male person. MRI and CT imaging techniques provide the geometry of the various structures at such a level of detail that for the generation of a finite element mesh, some kind of data reduction has to be performed.

3.2.1 Mesh generation

In the field of mechanical design and analysis, finite element meshes are in general con­structed using commercial CAD-packages like Unigraphics (1996) and I-DEAS (1991) or finite element pre-processors like Mentat (1996) and Patran (1996). To generate a mesh these packages need some sort of geometrical description; for instance a surface descrip­tion or a solid model of the construction. To the author's knowledge no general available

3-D model 49

software packages exist to date that allow for the extraction of a surface description from a set of CT or MRI data and conversion to a format that is readable by one of these CAD-packages or FEM pre-processors. Therefore for the meshing of the head an algorithm was developed.

CT-data have been extensively used in the development of finite element models of bones, like the femur (e.g. Kang et al., 1993; Keyak et al., 1990). In the field of head injury biomechanics CT-data seem to have been used only in two models (Bandak et al., 1995; Krabbel and Appel, 1995), for the generation of a mesh.

Various approaches for the generation of a mesh can be distinguished in literature. Muller and Ruegsegger (1995) and Frey et al. (1994) generated a tetrahedral mesh with characteristic element sizes on the voxel-scale level 1 mm) using a method similar to the so-called marching cubes algorithm (Lorensen, 1987), which was originally developed for the visualization of voxel-based data. In the work of Keyak et al. (1990) and Keyak and Skinner (1992) only straight-sided hexahedral elements were used and therefore the surface of the model did not represent the actual surface. A method often encountered in the modeling of long bones, like tibia and femur, is the stacking of contours of adjacent images (Kang et al., 1993). Krabbel and Appel (1995) used this method in the development of a model of the human skull. A disadvantage of this method is that due to the necessary equal number of elements for each contour, distorted elements can easily occur when adjacent contours differ in size or shape.

In this study it is attempted to construct a mesh of the human head using only hexahedral (brick) elements, because to achieve a particular numerical accuracy, less brick elements than tetrahedral elements are needed, making the usage of hexahedral elements cheaper from a computational point of view. The most promising method to date for obtain­ing a hexahedral mesh of the volume inside the skull, appears to be the projection method as used for example by Bandak et al. (1995). This projection method or transformation is extensively described by Knupp and Steinberg (1993).

This section briefly describes the projection method presented by Bandak et al. (1995), because this method is in particular useful for meshing convex domains using hexahedral elements. Starting point for the development of the finite element model are the inner and outer surfaces of the cranium of the skull.

Surface generation

The generation of the skull surfaces was done using the software Persona (Institute of Cranio-Facial Studies Inc., 1995), originally developed for the visualization of CT and MRI data. The description of the skull's inner and outer surfaces was generated from CT slices by growing a spheroid from a point in the center of the CT data. The spheroid surface consists of a user-defined number of triangles. Growing of the spheroid is achieved by increasing the radius of each of the corner points of a triangle and checking whether the color value of the particular voxel at that location lies within a specific color range, characteristic for the skull. Once a point has reached the bone surface it is restrained from growing any further. The process of growing is continued until all the corner points have reached the surface or the user is satisfied with a particular surface.

50 Development of a 3-D human head model --------------------------------------~~---

The outer surface is generated using the same method, except that the initial spheroid now lies outside the skull and is shrunk instead of grown. Particular care had to be taken in generating the outside surface when~ the facial bones connect to the cra­nium. The corner points of the triangles reached the facial bones, instead of the desired outside surface of the cranium. A workaround to this problem, was to manually shift the concerned points of the spheroid over a particular distance and skip over the facial bones.

Generation of elements

The generation of the skull mesh and brain mesh is performed using two different methods. T he brain mesh is determined with the so-called projection method and the skull mesh is generated by expanding the faces of the elements of the brain, that lie on the inside surface, to the outside surface of the skull. The projection method is actually a transformation from logical space, defined by a rectangular initial grid , to physical space. The method is further elaborated in Appendix C.

An initia l grid was used with 10 elements in each direction, giving a mesh with 1,000 elements for the brain and 600 for the skull, 6 faces with 100 clements per face that are expanded .. '\ representation of the facial bones consisting of 156 elements, was constructed and added to the previously generated skull mesh. The mesh was divided in two along the mid-sagittal plane, and half of the model was used , to obtain a completely symmetrical model , by mirroring it relative to the mid-sagittal plane. The resulting model is shown in Figure 3.1. Since the skull model is based on the interior and exterior surfaces

Figure 3.1: Oblique view of the right half of the 3-D model that has been cut along the mid­sagittal plane. The part in light gray denotes the brain.

of t he skulL the frontal sinus is not accounted for and this makes the frontal bone look

3-D model development 51

thicker than normal. The effect of including a description of the frontal sinus in the skull is investigated in Chapter 4. The complete model consists of 1,756 linear brick elements and 2,257 nodes.

3.2.2 Material properties

The skull is represented as a homogeneous isotropic structure with linear elastic material behavior, as is standard procedure in literature. The cranial contents were modeled as one single entity and is referred to as brain. As discussed in section 1.3.4, the presented con­stitutive models for the brain in literature, are predominantly linear elastic or viscoelastic. Since the actual data on the properties vary over quite a range, the data was averaged. The properties used in the simulations with the 3-D model are listed in Table 3.1.

Table 3.1: Linear elastic material properties used in the simulation with the 3-D model.

E p v [N/m2

] [kgjm3] H skull 6.5. 109 2.07. 103 0.2 brain 1.0 . 106 1.04 . 103 0.48 face 6.5. 109 5.0. 103 0.2

The influence of the magnitude of the brain's Young's modulus is studied in a parametric study presented in Chapter 4. Further to assess the influence on the response of time-dependent material behavior, additionally a linear viscoelastic constitutive model to describe the brain will also be investigated.

The mass, moments of inertia and center of gravity of the head of the male in the Visible Human data set are unknown. To provide the finite element model with realistic dynamic characteristics the density of the face was altered to make the finite element model comply with the averaged anthropometric data presented by Pheasant (1986). This resulted in a value for the density of 5.0 · 103 kgjm3

• The mass of the model, the location of the center of gravity and the inertial properties are presented in the next section.

3.2.3 Characteristics of the 3-D model

The characteristic dimensions of the head model used in the simulation are shown in Figure 3.2.

The center of gravity and moments of inertia were calculated using the I-DEAS VI (1991) CAD-package. As mentioned in the section on material properties the density of the facial bone was tuned such that the center of gravity and moment of inertia around they-axis were comparable to the data reported by Pheasant (1986). Pheasant (1986) only determined the moment of inertia around the y-axis for the head and neck together. This makes comparison of the model with the experimental data difficult. The moment of inertia around the y-axis is 1.265 · 104 kg· m2

• The mass of the 3-D model, is 3.1 kg, which is

52 Development of a 3-D human head model

193.3 ... ..... 97.9 .. : I I

N-++-1-t--t-H-. - - - - : -r 3

216.6

Figure 3.2: Sagittal and top view of the head model; with the characteristica.l dimensions (in mm). The$ sign shows the location of the center of gravity.

about 1.5 kg lower than the mass given by Pheasant (1986). This lack of mass is attributed to the fact that not all the structures of the human head are accounted for in the model. The scalp is not represented and the modeled facial bone only represents the outline of the face.

On the topic of boundary conditions, for the modal analyses performed using the various parts, neither kinematic or dynamic boundary conditions were prescribed. In the impact simulation also no kinematic boundary conditions were set, the impact load, described in the next section, represents the dynamic boundary condition.

3.2.4 Impact load

Impact simulations were performed similar to the experiments described by Nahum et al. (1977). Nahum et al. (1977) carried out impact experiments on human cadavers to examine the feasibility of using unembalmed cadavers as experimental model for closed head impact injury. The experiments involved the impact loading of the frontal bone of a stationary cadaver. The skull was impacted along the mid-sagittal plane by a rigid mass traveling at a constant velocity. To vary the magnitude and the duration of load application various padding materials were attached to the impactor surface. The age of the male and female specimen varied between 42 and 87. During the impact event the contact force, the biaxial acceleration-time history of the skull, and a series of intracranial pressure-time histories

Modal of the model 53

were measured. The fluid pressure was measured in the subdural space of the CSF layer surrounding the brain.

The contact force is the result of the interaction between the surface of the impactor and the surface of the head of the cadaver. Nahum et al. (1977) only reported on the mass and the initial velocity of the impactor, the type of padding used remains unknown. Since in the model no scalp was included and the material properties of the padding used in the experiments are unknown, it was decided to use the measured contact force as input for the numerical model. The impact force-time history measured by Nahum et al. (1977) in their experiment 37 (Nahum et al., 1977) was, for ease of the simulation, approximated by a sine-function. Since the element formulation required a pressure input the force was divided by the impact area (1774.3 mm2), constituted by the faces offour elements of the frontal bone, to obtain a pressure.

3.3 Modal response of the model

The eigenfrequencies of the human head in vivo and of the dry human skull in vitro have been determined experimentally. To have the numerical model behave similar to a live human head under transient dynamic conditions, it is necessary that the eigenfrequencies of the numerical model lie in the same range as those of the real system. Modal analysis has proven to be an effective method for characterizing the frequency response of a system, and has been used extensively -both in numerical models and experiments- to determine the vibrational characteristics of the human head.

Gurdjian et al. (1970) measured the natural frequency of a living head and a skull filled with silicon gel. They found the skull to essentially move as a rigid body system below a frequency of 150.0 Hz, while at 313.0 and 880.0 Hz the first antiresonance and resonance mode were seen, respectively. Stalnaker and Fogle ( 1971) determined the me­chanical impedance of a human cadaver head and in vivo and in vitro in a monkey's head. In the human head an antiresonance frequency was registered at 166.0 Hz and a resonance frequency at 820.0 Hz. Khalil et al. (1979) and Khalil and Viano (1979) reported frequen­cies, measured using dry skulls, in the range 1,385.0 to 4,245.0 Hz. Tzeng et al. (1993) measured the first six resonance frequencies of a dry human skull and found them to lie in the frequency range 1,286.0 to 2,539.0 Hz. From seven human male volunteers the average frequency of the first mode was determined at 161.0 Hz.

In various numerical studies the frequency responses of 2-D and 3-D finite element models have been presented. Ward and Thompson (1975) calculated the first six mid­sagittal symmetric modes of the human brain with and without supporting structures like falx and tentorium, and found the frequencies for the brain with falx and tentorium to lie in the range from 23.1 to 37.8 Hz and in the range 19.3 to 34.5 Hz without falx and tentorium. Ruan et al. (1991) performed a parametric study of the vibrational characteristics of a 2-D model of a coronal section of the human brain with and without membranes and in which the Young's modulus of the brain was varied. Results showed in particular the lower modes to be affected by the presence of the membranes. Ruan and Prasad (1996)

54 of a 3-D human head model

presented the results of modal analyses performed with a full human head model, analyses with only the brain modeled and only the skull. In the complete head model in particular the base of the cranium, brain stem and occipital lobe were shown to be structurally weak. The eigenfrequencies for the full human head model (baseline) were in the range 154.0 to 198.0 Hz. Parametric variations, in which the membranes were excluded, resulted in a decrease of the eigenfrequencies by 10.0 to 13.0%. The data reported in literature is summarized in Table 3.2.

Table 3.2: Eigenfrequencies determined in experiments with cadavers, dry human skulls, and in numerical models. EXP points to experimentally found modes and NUM denotes calculated modes using the finite element method. FR denotes the first resonance frequency and AR the first anti-resonance. Eigenfrequencies of models with membranes included and without membranes are denoted with WM and WO, respectively.

Analysis Author Description with frequencies in Hz EXP:cadaver head Gurdjian et al. (1970) 313.0 (FR), 880.0 (AR)

Stalnaker and Fogle (1971) 166.0 (FR), 820.0 (AR)

EXP:dry skull Khalil and Viano (1979) 1,385.0- 4,245.0 (10 modes) Tzeng et al. (1993) 1,286.0- 2,539.0 (6 modes)

EXP:human volunteer Tzeng et al. (1993} 161.0 (FR) Willinger and Cesari (1990) 100.0 200.0 (FR)

NUM:brain model Ward and Thompson (1975) 23.1- 37.8 (WM) (6 modes) 19.3- 34.5 (WO) (6 modes)

NUM:skull model Ruan and Prasad (1996) 325.0 2,167.0 (10 modes)

NUM:head model Ruan et al. (1991) 70.0 - 118.0 (WM) (5 modes) 48.0- 90.0 (WO) (5 modes}

Ruan and Prasad (1996) 154.0- 198.0 (10 modes)

Since modal analysis is based on linear theory care should be taken in trying to predict responses, using modal characteristics, of a model that is subjected to large dis­placements and possibly large deformations, since these are not accounted for in the linear theory. A modal analysis was performed of only the skull model, the brain model and the complete head model. The material of the facial structure has the same Young's modulus as the skull, but due to the construction of the facial bone the geometrical stiffness is much lower than the rest of the skull. To prevent eigenmodes of the facial structure from domi­nating the lower modes, the facial bone was omitted in the modal analyses. The analyses were carried out using the Lanczos method (Bathe, 1996). This method is in particular useful when extracting a large number of eigenfrequencies from a large model. For the modal analyses the material properties as presented in Table 3.1 were used and the results of the three analyses are given in Table 3.3.

To obtain an indication of the validity of the modal characteristics of the presented

Simulation of impact and comparison with experiments 55

Table 3.3: Eigenfrequencies of the lower modes of the skull model, brain model and complete head model, respectively.

mode Skull model Brain model Head model no: [Hz] [Hz] [Hz] 1 1,994.0 79.3 219.4 2 2,121.0 80.9 231.4 3 2,472.0 89.5 237.6 4 2,731.0 103.2 248.2 5 2,749.0 104.7 257.7 6 2,967.0 114.7 262.6 7 3,073.0 118.3 264.1 8 3,265.0 120.1 275.7 9 3,347.0 124.7 283.8 10 125.6 294.0

3-D model, the frequencies of the skull modes are compared with the experimental frequen­cies measured by Khalil and Viano (1979), using dry human skulls. The first ten modes presented by Khalil and Viano (1979) were in the interval between 1,385 and 4,245 Hz. The calculated modes are almost in the same range, and although the presented 3-D model has a relatively coarse mesh, the agreement is quite good. This comparison only included the frequencies. Experimental data on mode shapes do not exist.

The calculated frequencies for the head are of the same order of magnitude as the frequencies measured by Willinger and Cesari (1990) between 100.0 and 200.0 Hz. In particular the seventh mode showed a behavior where the core of the brain moved relative to the skull in a translational manner. No real relative motion between skull and brain was possible at the skull-brain interface, because the brain and skull are rigidly coupled and no layer is present representing the system of membranes and fluid.

For lack of experimental data about modal characteristics of the brain only a com­parison can be made with the results presented by Ward and Thompson (1975). The first mode reported by Ward and Thompson (1975) was 19.3 Hz for the model without mem­branes. This frequency is substantially lower than the frequency calculated here, which is explained by the fact that Ward and Thompson (1975) used a Young's modulus of 68.0 · 103 N/m2 for the cerebrum. This value is significantly lower than the value that is on average encountered in literature.

3.4 Simulation of impact and comparison with experiments

For the 3-D head model the boundary conditions, as described in Section 3.2.3 are applied. Due to the expected large displacements and rotations a total Lagrangian approach with large displacement formulation was used. The fundamental stress and strain measures used, are second Piola-Kirchhoff stress (P) and Green-Lagrange strain {E), respectively. Due

56 of a 3-D human head model

to the expected small strains and linear elastic isotropic material behavior, as described in section 3.2.2, a relation between stress and strain is assumed according to Hooke's law, P = C 4

: E. Nonlinear terms in the strain-displacement relation are incorporated, because of the use of the large displacement formulation.

For the simulation a period of 15.0 ms was taken. The time step was calculated by considering relevant frequencies of the system that should be incorporated in the so­lution. Khalil et al. (1979) and Khalil and Viano (1979) have measured the vibrational characteristics of dry skulls and found the eigenfrequencies to lie in the range from 1,000 to 5,000 Hz. The results of the modal analyses on the skull model showed frequencies in the same range. To describe an oscillation with sufficient accuracy a time step of roughly 10% of the period time of the highest frequency is admissible. Using this rule an initial time step for the simulations of 2.0 · w-5 s was used. The simulations were performed with the commercial finite element code MARC K6.1 (1994). This finite element code has been used because in an extension of the model, where the interface between skull and brain (Chapter 4) is modeled, the contact algorithm as presented in Chapter 2 and implemented in the MARC code, will be used.

The next sections describe the results of the simulations with the 3-D head model using the prescribed pressure-time history on the frontal bone. In the first section (3.4.1) the results will be described of a comparison between the measurements and simulated response. Since the magnitudes of stress and strain depend on the coordinate system of reference, it is more practical to use invariants rather than individual components as param­eters for analysis. In Section 3.4.2 the von Mises stress, a representative of the maximum shear stress is shown, and in Section 3.4.3 minimum and maximum principal strains are presented, that describe the dilatational behavior or change of volume of an infinitesimal cube of material. The von Mises stress and principal strains are used to delineate the distortion or deformation seen in the brain during impact.

3.4.1 Direct comparison with experimental data

The simulation of the 750 time steps of the 3-D model took 12.6 CPU hours on a CRAY C98/4256 supercomputer.

For a comparison with experimental data the pressure for elements in the frontal and occipital regions of the brain was calculated asp -~(a.,.,+ ayy + azz) The pressure­time histories presented in Figure 3.3 were calculated as follows. In each of the regions four elements were chosen. For each element the calculated pressures in all eight integration points were averaged. The coup and contre-coup pressures were obtained by averaging the element pressures in each region.

Pressure is a quantity, that only describes the dilatational behavior, whereas the distortion that is often responsible for damage in tissues, is not accounted for. Since distor­tion of brain tissue under impact conditions has never been measured in human subjects, pressure remains as yet the only measured quantity with which comparison is possible with the numerical modeL The agreement of the measured and calculated pressures at the coup site is quite reasonable. The simulation results for the contrecoup pressure are much higher (-141.6 kPa) than the measured pressure (-59.8 kPa).

Simulation of impact and comparison with experiments 57

150

100 ~ 0.. 50 .::s Q) 0 .... ;:l

"' 00 -50 Q) .... 0..

-100

-150

0 0.005 0.01

time [s]

(a)

2500

"" 2000 ., -£. 1500 :::: 0

~ 1000 .... "' Ql u u 500 o:l

0

0.015 0

*, .... ~ .. ~,. \ ,. ..

•' .. ·:.--frontal .. ..

0.005 0.01 0.015

time [s]

(b)

Figure 3.3: (a) Pressure-time histories; (b) Linear skull acceleration of nodes at frontal and occipital bone.(---) numerical response;(-) experimental results by Nahum (1977)

The calculated and measured resultant linear accelerations of the skull are also presented in Figure 3.3. The overall shape and amplitude of the acceleration-time histories show similar trends. However, the calculated accelerations show a phase shift and a greater width of the curves with respect to the measured accelerations. Striking with respect to the phase shift is the suddenness with which the measured accelerations start off.

Various causes for the differences in pressures and accelerations can be distinguished. These causes may be grouped. Firstly, concerning the model used. The overall geometry of the head in the numerical model is not the same as the heads of the cadavers used in the ex­periments. Further, the numerical model is relatively coarsely meshed, and the implications of the element size are studied in Section 3.6. Additionally the reference model contains no representations for the various structures that constitute the intracranial contents, nor for the skulL Interactions between structures inside the head and between the head and the neck have also been left out. In particular the representation of the skull-brain interface at the contrecoup site may form a too strong constraint on the brain. The negative pressures generated by tensile forces exerted by the skull on the lagging brain, create a pressure level much higher than the experimentally measured pressures. An interface that represents the skull-brain interaction more realistically, for instance by explicitly modeling the CSF-layer or by providing the possibility of relative motion, might generate a more realistic pressure level at the contrecoup site.

Secondly, as to the material properties, the measurements were performed on ca­davers, for which the mechanical behavior of the various tissues is unknown. The material properties used for the simulations were taken as an average from literature.

Last but not least the measured pressures in the subdural space may not be at the same locations as taken in the numerical model. Moreover, the numerical model does not contain a representation of the CSF layer, instead the pressures were calculated in corresponding locations in the brain. Concerning the phase-shift in acceleration seen in

58 Development of a 3-D human head model

Figure 3.3, other results reported by Nahum and Smith (1976), that showed a similar impact, present a more gradual increase of the acceleration.

Due to a shortage of experimental results, it can be dangerous to rely on only one source, in this case the results by Nahum et al. (1977). A number of the presented causes here will be further investigated in Chapter 4.

3.4.2 Shear response

To visualize the deformation of the brain in the mid-sagittal plane, Figure 3.4 shows a

-

L L L

t = 0.0 ms t = 2.0 ms t = 3.0 ms -

L L

t = 4.0 ms t = 5.0 ms t = 6.0 ms - - -

t = 8.0 ms t = 10.0 ms t = 12.0 ms

0.0 5.0 10.0 15.0 20.0 25.0 30.0 [kPa]

Figure 3.4: Von Mises stress distributions in the brain in the mid-sagittal cross-section at various points in time. Note that the time intervals between subsequent plots are not always the same.

Simulation of impact and comparison with experiments 59

time-sequence of plots of the von Mises stress distributions. In Appendix D it is shown, by calculation of the internal energy that the von Mises stress is solely related to the deformation.

Between t = 2.0 ms and t = 6.0 ms the brain is subjected to larger levels of von Mises stress, and especially the outer perimeter of the brain is affected with the largest stresses occurring at the top of the cerebrum.

3.4.3 Dilatational response

- - -

L L L

t = 0.0 ms t = 2.0 ms t = 3.0 ms - -

L L L

t = 4.0 ms t = 5.0 ms t = 6.0 ms

t = 8.0 ms t = 10.0 ms t = 12.0 ms

-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0.0 0.005 [-]

Figure 3.5: Minimum principal total strain in the brain in the mid-sagittal cross-section at various points in time. Note that the time intervals between subsequent plots are different.

60 Development of a 3-D human head model

The dilatational response may be represented by the minimum and maximum principal total strains. The minimum total strain is presented in Figure 3.5. The minimum total strain is largest at t = 4 ms in the frontal lobe of the brain.

The maximum principal total strain is shown in Figure 3.6

- - -

L L L

t = 0.0 ms t = 2.0 ms t = 3.0 ms - - -

L L L

t = 4.0 ms t = 5.0 ms t = 6.0 ms - -

t = 8.0 ms t = 10.0 ms t = 12.0 ms

-0.005 0.0 0.005 0.01 0.015 0.02 0.025 [- ]

Figure 3.6: Maximum principal total strain in the brain in the mid-sagittal cross-section at various points in time. Note that the time intervals between subsequent plots are different.

The results show maximum strains occurring at the outer perimeter in the posterior region of the cerebrum in the t ime interval from 3 ms to 5 ms. Since the minimum and maximum principal strain represent measures for compression and tension, respectively,

Time refinement 61

the coup-contrecoup phenomenon, i.e. positive pressures occurring at the impact site and negative pressures seen at the diametrically opposite location of the head, a..<> seen in Fig­ure 3.3 is also seen in Figures 3.5 and 3.6. This phenomenon can be observed in particular at 4.0 ms after the onset of impact.

For both the strain and the von Mises stress contour plots low levels of strain and stress are seen in the brain, when the impact has terminated after t = 8 ms. After the impact force has subsided the model continues to travel as a free body, where the head is rotating backwards over time. Next to the results presented in the mid-sagittal plane, results were also visualized for a coronal and transversal cross-section. The von Mises stress showed the same trend in these cross-sections, namely that the maximum stress was observed at the outer perimeter of the brain in the top half of the head.

3.5 Time step refinement

Smaller time steps are often used in transient analyses to check if the response has converged to a particular solution. If a large difference is found the initial time step was probably too large. Two simulations were performed to assess the influence of the time step. In the first, the time step was reduced from 2.0 . w-s .'3 to 2.0 . w-6 .'3 and the second simulation was done with a time step of 2.0 · w-4 s. Further the simulation time interval was reduced from 15.0 ms to 10.0 ms for two reasons. Firstly, as can be seen in the results presented in section 3.4.1 the largest amplitudes of the various parameters occur during the first 10.0 ms, and secondly by shortening the time interval the computational costs are reduced. Figure 3.7 depicts the results of the calculated frontal and occipital pressures, and linear

2500 150

100 ;:;-' 2000

"' \j ...._ Q, 50 £ 1500 ::::.. ;:::

"' 0 0 .... :;:; ;:l 00 ol 1000 Ul -50

.... "' "' .... Qi

"" u -100 u 500 <e

-150 0

0 0.005 0.01 0 0.005 0.01

time [.5] time [s]

(a) (b)

Figure 3.7: Frontal and occipital pressures (a) and linear resultant skull accelerations (b) calcu­lated at the frontal bone, using different time steps. (-) reference analysis, and (- -) results for the larger time step.

resultant skull acceleration of a node at the frontal bone, for the analyses with the different time steps and the reference simulation. For both the calculated pressures and acceleration

62 of a 3-D human head model

similar trends are seen. In that the simulation with the larger time step underestimates the pressures in the coup and the contre-coup region, whereas the pressures of the reference simulation and the results with smaller time step fully overlap. The acceleration also shows the overlapping for the smaller time step, and smaller skull accelerations for the larger time step. In the simulation with the larger time step an oscillation in the linear resultant acceleration is present after the maximum amplitude has been reached. From these results it may be concluded that the initially chosen time step was sufficiently small, to accurately calculate the response.

3.6 Mesh refinement

Large gradients of field parameters over the elements are often better estimated by using more and smaller elements. This is in particular true for linear elements, since these give only a linear transition of the stresses and displacements over an element. The mesh of the 3-D model was refined by subdividing each existing element in eight new elements (each edge is divided in two) giving a refined mesh with a total of 14,048 elements and 15,999 nodes. Using the initially chosen time step of 2.0 · 10-5 s the same impact force was prescribed as for the reference model, and for the same reasons as for the time step refinement only a time interval of 10 ms was simulated.

The pressure in the frontal and occipital region for the reference simulation and the simulation with mesh refinement are shown in Figure 3.8 The pressures were calculated

200

150

03 100 Q, ~ 50

"' "" :;l 0 (ll

'53 -50 ... c.

-100

-150

0 0.005 0.01

time [s)

Figure 3.8: Pressure-time histories.(---) reference simulation,(···) simulation with mesh refine­ment, and (-) the measured pressures.

for the same regions as in the reference simulation, that is instead of the 4 elements now 32 elements were used for each of the regions. The results show that the simulation done with the refined mesh gives lower pressure values in the frontal and occipital regions. The maximum pressure amplitude in the frontal region agrees better with the measured value by Nahum et al. (1977) (Figure 3.8). The contrecoup pressure is, like in the reference model, also overestimated in this simulation.

Mesh refinement 63

Figure 3.9 shows the von Mises stress in the brain for the mid-sagittal plane at various points in time for both the reference model and the model with refined mesh.

t = 3.0 ms

t = 4.0 ms

-t = 5.0 ms

lft·--0.0 5.0 10.0 15.0 20.0 25.0 30.0 [kPa]

Figure 3.9: Von Mises stress distributions in the brain in the mid-sagittal plane at various points in time with the reference model (left) and the refined mesh (right) .

The von Mises stress distributions show similar patterns for the reference and refined models , for t = 3.0 ms and t = 4.0 ms. The von Mises stress for a particular region was calculated using the same averaging method used to calculate the pressures in the frontal and occipital regions. The differences between the reference simulation and the model with mesh refinement become more apparent when time history plots for various locations are compared for the reference and refined model. Figure 3.10 shows the von Mises stresses at various locations in the brain for both simulations.

64 of a 3-D human head model

frontal region occipital region

25 25

~ 0..,

20 20

~ 15 15 "' "" <1> <1>

.;!l "' ;'?; 10 ::§ 10

;::: ;:::

~ 5 0 5 >

0 0

0 0.005 0.01 0 0.005 O.Ql

time [s] time [s]

foramen region top region

25 25

~ 20 20

~ 15 15

"' Kl <1> ,;!l "' ;'?; 10 ::§ 10

g ;:: 5 ~ 5

0 0 -------- -------···-·

0 0.005 0.01 0 0.005 0.01

time [s] time [s]

Figure 3.10: Von Mises stress-time histories. (-) reference simulation, and (- -) model with refined mesh.

The von Mises stress time-history plots for the various locations show similar trends as for the pressure distribution, in that the results obtained with the refined mesh show a smaller amplitude than for the reference model. This is in particular true for the maxi­mum amplitude, the overall pattern in time of the von Mises stress during the rest of the simulation is the same.

3. 7 Discussion and conclusions

The human head is a three-dimensional structure, but with the proper assumptions 2-D plane strain models of the human head may offer an attractive alternative to reconstruct the deformations in the head under impact conditions, because they have lower computational costs and may provide quick insight into the problem. However, 2-D models also have a number of drawbacks. Firstly, due to the nature of their numerical formulation, plane-strain models may especially under large deformations provide unrealistic results, because under all circumstances they will maintain the plane character of the strain distribution whereas these in reality are likely to be of a three-dimensional nature. Second, although the Young's

Discussion and conclusions 65

modulus of the skull is chosen the same in 2-D and 3-D models, a 2-D model lacks a so­called geometrical stiffness, because the 2-D model describes the skull as a cross-section of a cylinder, which has a considerable lower stiffness than the dome shaped container, that the skull actually is. Because of the lower load resistance of the skull in a 2-D model, the brain will be subjected to larger deformations than for a similar load applied to a 3-D model. Lastly, there is a functional drawback, in that a 3-D model can easily be subjected to loads applied in various directions, whereas for each new direction a new 2-D model has to be generated representing a particular cross-section of the head. Further, because of the development of hardware and software the necessity of 2-D modeling is becoming less compelling.

The comparison presented in this chapter between the numerical results with the gener­ated 3-D model and the experiments by no means represents a full validation of the model, but describes more to what extent the obtained results agree with the limited literature data. The calculated coup pressure in the frontal lobe and accelerations of the skull agreed quite reasonably with the data measured by Nahum et al. (1977). The comparison between numerical results is only performed on the basis of kinematic quantities, like skull accel­eration, and global dynamic quantities, like fluid pressures measured in the CSF layer. As described in Chapter 1, a full validation of an arbitrary parameter is by no means possible at this moment, due to the lack of experimental data and the fact that not every param­eter can be measured in vivo or in vitro. Whether the calculated stresses and strains are realistic, in the sense that they are comparable with injury levels, can therefore not be concluded.

The frequencies calculated in the modal analyses of the separate skull and brain and the complete head, correspond with frequencies calculated using other 3-D models and with the results obtained in experiments with live human subjects or in vitro with the heads of cadavers.

The results obtained with the reference model compare quite well with the mea­surements presented by Nahum et al. (1977). The variation of the time step size showed the initial time to be sufficiently small, since the various time-history plots fully over­lapped for the two simulations with the initial and smaller time step. The overall effect of mesh refinement was that the maximum amplitude of the various results was lower; for the coup pressure approximately 12% of the maximum amplitude of the pressure in the reference simulation. The maximum von Mises stress in the top region of the head was lower by approximately 33% of the maximum von Mises stress calculated in the reference simulation.

Comparing the developed model with other 3-D human head models presented in literature, that also used Nahum's cadaveric tests for the purpose of validation, some differ­ences and similarities become apparent. Ruan et al. (1994) simulated Nahum's experiments by using a similar method of prescribing impact as used in this thesis. Their results also show an overestimation of both the coup and contrecoup pressures. In an extended version of this model (Ruan et al., 1993) with more substructures included, and more elements used for describing the model, an impact was modeled using an impactor with a specific

66 Development of a 3-D human head model

initial velocity. The results of this model were in closer agreement with Nahum's results. The model developed by Zhou et al. (1995a,b) again showed an overestimation of both the coup and contrecoup pressures.

Since in this study, we are at first interested which parameters influence the results of various quantities, the reference model will be further used in this study. The much higher computational costs for the model with refined mesh, make this model for the moment impractical for usage in a parametric study.

Summarizing; • a 3-D model of the human head was developed using MRI and CT data. • the results obtained from modal analyses and transient dynamic simulations showed

trends similar to those experimentally measured. • parametric variation of the time-step size, showed the initial time step, based on a

modal analysis, to be sufficiently small. Mesh refinement gave slightly better results, but has for the time being been set aside from a computational cost point of view.

Chapter 4

Parametric study

4.1 Introduction

In this chapter the results of a parametric study using the 3-D model, described in Chap­ter 3, are presented. The study was conducted to obtain better insight in the importance of a number of parameters in the model. The model, presented in Chapter 3, will be denoted as reference model and using the same applied loads the following topics will be dealt with; constitutive properties:

• variation of the Young's modulus in the elastic brain model • variation of the time constant in the linear viscoelastic brain model

boundary and interface conditions: • relative motion between skull and brain • the influence of a foramen magnum • the influence of a kinematic constraint for the head

anatomical detail of the model: • substructures of the intracranial contents • frontal sinus in the skull

The choice of topics is based on both the literature survey presented in Chapter 1, and the possible causes mentioned in Chapter 3 for the differences observed between the calcu­lated and measured results. The results of the parametric study will indicate which of the assumptions made in the development of the reference model were justified, and addition-

67

68 Parametric study

ally provide indications and pointers for the setup of experiments and the development of numerical models in future simulations.

In section 4.2 the parameters relating to the constitutive properties are discussed, while the boundary and interface conditions are introduced in section 4.3 and the influence of modeling various substructures is investigated in section 4.4. The chapter is concluded with a discussion and summary of conclusions of the results obtained in this chapter.

Analogous to the previous chapter, pressure and von Mises stress will be evaluated for the frontal, top, occipital and foramen regions. In each region four adjacent elements were chosen. For each element the calculated results in all eight integration points were averaged. The results for a region were obtained by averaging the element results in each region.

4.2 Material properties

To calculate realistic responses with finite element models of the human head, the need for accurate constitutive properties is now the problem at the forefront. Experimentally it has been determined that brain tissue consists of materials with varying stiffness and directional rigidity (Arbogast and Meaney, 1995). In the finite element models typically homogeneous and isotropic material properties are used to describe the entire brain. For lack of adequate data on brain material properties a common procedure in the modeling of head impact is to represent the brain firstly by a linear elastic model and subsequently by using a linear viscoelastic model, whereas soft biological tissues are known to possess non-linear viscoelastic properties (Fung, 1993).

The material properties of brain tissue presented in experimental studies vary over a wide range and so do the linear elastic properties used in published 3-D numerical models of the human head. Table 4.1 gives an overview of the material properties used in recent years, from which in particular the Young's modulus can be seen to vary over a considerable range. Before comparing the results obtained with the models, it seemed therefore worthwhile to investigate the effect of varying the Young's modulus within this range in our model.

The 60% higher Young's modulus for the white matter with regard to the gray matter in the study by Zhou et al. (1995b) has been determined empirically in simulations of porcine brain models (Zhou et al., 1994). The justification for the higher Young's modulus for the white matter was that the white matter should be stronger and tougher because it is composed of axonal fibers, whereas the gray matter consists of nerve cell bodies. The reason as to why the modulus for white matter is only 20% higher than the modulus for gray matter in their other study (Zhou et al., 1996) is not clear from the publication. Further the responses obtained with the two different elastic moduli in the study by Zhou et al. (1996), form the upper and lower bounds of an envelope, in which the results of their parametric study for the viscoelastic material should lie. Surprising is the fact that the material data (Table 4.1) used in the two publications (Zhou et al., 1995b) and (Zhou et al., 1996) show such a large difference, whereas the used finite element model is the same, and the only difference seems to be that in the former the model was subjected to an impact whereas in

69

Table 4.1: Linear elastic material properties used for the brain in 3-D human head models.

Author Tissue E p v [N/m2

] [kgjm3

reference model (this study) brain 1.0· Ruan et al. (1994) brain 5.04-106 1.04·103

Bandak et al. (1995) brain 6.8-107 1.22-103 Zhou et al. (1995b) gray matter 5.0-105 1.04-103

white matter 8.0-105 1.04·103

Kumaresan and brain 6.67-104 1.04·103 0.48 Radhakrishnan (1996) Ruan and Prasad (1996) brain 5.58-105 1.04-103 0.499 Zhou et al. (1996)1 gray matter2 1.88-104

white matter2 2.27-104

gray matter3 1.01-105

white matter3 1.22-105

1: p, v not reported in publication, probably values used as in Zhou et al. (1995b) 2: lower bound study 3: upper bound study

the latter a rotational acceleration was prescribed. A similar difference in Young's modulus for brain tissue is also seen between the models presented by Ruan et al. (1994) and Ruan and Prasad (1996). Although the used head models were similar, comparison of the results is not possible, because in one publication (Ruan et al., 1994) an impact on the frontal bone was prescribed, whereas in the second publication (Ruan and Prasad, 1996) a modal analysis was performed to characterize the vibrational response of the head.

The brain material properties, in the various publications, are in majority derived from the same experiments, performed in the late sixties and early seventies by Ommaya (1968); Fallenstein et al. (1969); Galford and McElhaney (1970}; McElhaney et al. (1972) and Shuck and Advani (1972).

An often utilized approach in the simulation of head impact is to consider the skull as a homogeneous and isotropic structure with linear elastic material behavior. The properties used in the various finite element models are summarized in Table 4.2. The material properties for the skull do not show the large range as seen for the brain. The properties are in majority based on the experiments performed by McElhaney et al. {1970) and Wood (1971).

A linear viscoelastic model was introduced to investigate the influence on the head's res­ponse of a time-dependent constitutive model for the brain. Like the linear elastic prop­erties, the viscoelastic properties presented in literature (Table 4.3), also vary over a con­siderable range, in particular concerning the time constants used. The deviatoric behavior is described by the standard linear solid model G(t) =Goo+ (Go G00)e-~, with Goo the long-term shear modulus, G0 the short-term modulus, and T the time constant. To enable

70 Parametric

Table 4.2: Linear elastic material properties used for the skull in 3-D human head models.

Author Tissue E p v [N/m2

] [kg/m3] [-] reference model (this study) skull 6.5·109 2.07-103 0.2 DiMasi et al. (1991a) skull rigid Bandak and Eppinger (1994) Ruan et al. {1994) inner and outer table 1.22·1010 3.0·103 0.22

diploe 5.66·109 1.75·103 0.22 Bandak et al. (1995} skull 6.8·109 2.5·103 0.25 Zhou et al. (1995b) skull 8.0·10!1 2.1-103 0.22 Kumaresan and skull 6.5·109 1.41·103 0.21 Radhakrishnan (1996) Ruan and Prasad (1996) inner and outer table 5.465·109 3.0·103 0.22

diploe 2.684·109 1.75·103 0.22 Zhou et al. (1996) 1 skull 8.0·109

1: p, v not reported in publication, probably values used as in Zhou et al. (1995b)

Table 4.3: Linear viscoelastic material properties used in literature in 3-D human head models.

Author tissue Go Goo T K [N/m2] [N/m2] [s] [N/m2]

Ruan et al. (1993) brain 5.28·105 1.68·105 0.029 3.07·105

DiMasi et al. (199la) brain 3.44·104 1. 72-104 0.01 6.8·107

Bandak and Eppinger (1994) brain 3.44·104 1. 72·104 0.01 1.84·109

Zhou et al. (1996}1 gray matter 3.40·104 6.3·104 0.0014 2.19·109

white matter 4.10·104 7.6·104 0.0014 2.19·109

1:

a comparison of the results obtained with the viscoelastic model with the reference model, the short-term modulus was derived from the reference Young's modulus (106 [N/m2)). To study the influence of time-dependence two time constants were utilized.

4.2.1 Variation of Young's modulus in elastic brain model

The reference value 106 Nfm2 of the Young's modulus was varied by a factor of 10.0, both larger and smaller, thus approximately covering the range of data as presented in Table 4.1. Apart from the Young's modulus for the brain tissue the models were the same as the reference model. The simulation time was shortened from 15.0 to 10.0 ms, since the reference simulation has shown this to be the most relevant period. The pressure in the frontal and occipital regions, and the von Mises stresses for the various regions in the brain are depicted in Figures 4.1 and 4.2, respectively.

Figure 4.1 shows that the higher Young's modulus results in a lower pressure for both the coup and the contrecoup region. Where the other two models show an oscillation

Material properties

200

150

100

50

-100

-150

0 0.005

time [sj

71

O.Ql

Figure 4.1: Coup and contrecoup pressure. (-) E 106 Nfm2 (reference model); (···) E = 105 Nfm2; (---) E = 107 Nfm2•

for both the coup and the contrecoup pressure, the model with the higher Young's modulus has no oscillation. Additionally the first pressure peaks in the results for the lower Young's modulus lag behind those in the other two responses.

The results for the von Mises stresses (Figure 4.2) in the model with the higher Young's modulus show a similar pattern as in the reference model for all regions, except that the result for the reference model tends to oscillate after t :::::: 6.0 ms, which is best seen in the top region. The first peaks in the results for the model with the lower Young's modulus show a similar time lag with respect to the results obtained with the other Young's moduli, as was seen also for the pressure. Further the maximum amplitude is significantly higher than for the other simulations in the frontal and foramen regions. For the top region the reference model shows an oscillation with a period of approximately 1.66 ms. The simulation with the lower Young's modulus shows a larger period, in accordance with the expectation that a lower Young's modulus will result in lower eigenfrequencies, or larger periods. For the frontal region the lower modulus has a higher frequency. The reason for these high-frequency oscillations occurring in the frontal region for the low modulus, seen in Figure 4.1 for the pressure (coup-site) and in Figure 4.2 for the von Mises stress, could either be of a numerical or physical nature. Analysis of the deformation pattern in the brain showed elements to be subjected to large deformations particularly in the frontal region. For the closer analysis to establish whether these large deformations are the cause of both the oscillations and the negative pressure peak seen for the coup region (Figure 4.1) at t:::::: 8.0 ms, for the lower Young's modulus, impact experiments with physical models could provide more insight.

Discussion Examples of linear elastic material properties used by other authors in 3-D numerical models of the human head have been presented in Table 4.1. Comparison of the results between studies is yet only possible on the basis of pressures calculated in the frontal and

35

30

25

f!l 20

.:!l 15 ~ § 10 > 5

0

0

35

<r 30

0., 25 ~

~ 20

.!!l 15 ~

= 10 ~

5

0

0

frontal region

/',

j .... ~··· .... / ·.... r·· .. : ..... ··\.!

0.005

time [s]

foramen region

0.01

...... ···········.\................ . .... ··· .......... ··

0.005

time (s] O.ol

Figure 4.2: Von Mises stress-time histories. (-) E 105 N/m2 ; (---) E = 107 N/m2 .

35

30

25

20

15

10

5

0 -!----"""''·

35

30

25

20

15

10

0

5

0-1--0"'·

Parametric

occipital region

0.005

time [s]

top region

0.01

0 0.005 0.01

time [s]

106 N/m2 (reference model); (· .. ) E =

occipital regions, since these are the only quantities that are available for several other 3-D models as well. Ruan et al. (1994) and Zhou et al. (1995b) used a similar Young's modulus as in the reference model in this study. Their pressure-time histories for the frontal region are similar to the results obtained with the reference model in this study (Figure 4.3). For the contre-coup region larger differences for the pressure levels, between the studies in literature and the reference model are seen. The other authors have included a compliant layer between brain and skull in their numerical models, that represents the CSF layer. The presence of this layer may be a cause for the differences seen.

The simulations discussed in this section have shown the magnitude of the Young's modulus to be of particular great significance. The variations of the Young's modulus, which not even fully covered the range of data presented in Table 4.1, had a large influ­ence on the pressure and von Mises stresses. Therefore the conclusions as presented by various authors over the years may at least be questioned concerning the reliability of the calculated responses. So long as no additional experimental data become available for the description of the constitutive behavior of the tissues inside the head, the results obtained with numerical models may only be used to characterize trends.

Material 73

200

150

01 100 c., ~ 50

Q) .... ;:I 0 "' "' Q)

-50 .... 0.

-100

-150

0 0.005 0.01

time [s]

Figure 4.3: Coup and contrecoup pressures.(-) Nahum's experiments;(---) Zhou et al. (1995b); (···) Ruan et al. (1994); (-) reference model.

4.2.2 Variation of linear viscoelastic brain properties

Simulations with the viscoelastic model for two different values of the time constant were compared with each other and with the elastic reference modeL The short-term behavior, governed by G0 , was derived from the Young's modulus used for the reference modeL Two values of the time constant were chosen that form upper and lower bounds for the time constant values used in various publications (see Table 4.3). The values used for the parametric variation are summarized in Table 4.4. The case with 7 2 = 10-4 represents a fast

Table 4.4: Linear viscoelastic material properties used in parametric study.

Description

3.38·105

3.38·105 1.69·105

1.69·105

T

[s) 2.0 ·10 2

10-4 1.04 . 103 0.48 1.04 . 103 0.48

decay of the material stiffness whereas the model with r 1 = 2.0·10-2 lies between the former value and the elastic material model used in the reference model. The elastic material model may be represented by the linear viscoelastic model with an infinite time constant. Only the deviatoric behavior has been altered by the introduction of viscoelasticity, the dilatational behavior remains unchanged.

For the time interval analyzed (10 ms) it is expected that for the time constant 72

full relaxation of the material will have taken place, before any significant pressure build up occurs. Therefore the response will be largely governed by G00 • In the case of T1 as time constant, the process of relaxation will not have subsided at the end of the simulation interval.

The pressure-time history for the frontal and occipital regions in the brain are presented in Figure 4.4. The pressure histories are only slightly affected by the different material models used, which complies with the expectations, for pressure is determined by

74 Parametric

200

150

100

50 Q) ... ;:l 0 gj Q)

-50 ... ~

-100

-150

0 0.005 0.01

time [s]

Figure 4.4: Coup and contrecoup pressure for different time constants. (-) reference model; ( ···) 7t = 2.0 ·10-2s; {---) 72 = 1.0 ·10-4 s.

the dilatational behavior of the material, and the viscoelastic model only influenced the deviatoric behavior of the material.

The von Mises stress is presented for various regions in the brain in Figure 4.5. The decay in material stiffness becomes apparent after about 2 ms, and is best seen in the results for the von Mises stress for the 12 value. The oscillations seen for the top region clearly indicate the drop in frequency for the simulation with 12, indicating that the material model is already at its long-term modulus, which is lower and thus also has a lower eigenfrequency. The influence of r 1 will become apparent once a longer time interval is analyzed.

Discussion

Publications on the properties of brain tissue (Ommaya, 1968; Fallenstein et al., 1969; Gal­ford and McElhaney, 1970; McElhaney et al., 1972; Shuck and Advani, 1972) indicate that most authors have fitted their experimental data with a so-called Maxwell-Kelvin model also known as four-parameter fluid model. Most finite element codes however do not have this constitutive model implemented, but use the standard-linear solid model, which is also used in this thesis and by the authors summarized in Table 4.3. The necessary parameters for this material model may be obtained by fitting the model to the experimental data published by the various authors of the experimental studies. Which methods were used to obtain the necessary data for the constitutive models in the finite element studies, is not mentioned in the various publications. Direct comparison of the time constants as used by other authors and in this thesis with the experimental data on the properties of brain tissue is therefore not possible.

Of the recent publications on 3-D finite element models of the human head only Zhou et al. (1996) have performed a parametric study, where the parameters of the linear viscoelastic material model were varied, specifically the values of the time constant. How­ever, a direct comparison of the results is difficult, because Zhou et al. (1996) simulated

Material properties 75

frontal region occipital region

35 35

30 30 ~

25 tl., 25 ::::=..

"' 20 "' 20 0.) 0.)

"' til

~ 15 ~ 15

§ 10 l:: 10 0 ;.

5 ;.

5

0 0

0 0.005 O.oi 0 0.005 O.Ql

time [sJ time [s]

foramen region top region

35 35

~ 30 30

~ tl., 25 tl., 25 ::::=.. ::::=..

"' 20 "' 20 0.) ~ .~ ~

15 ~ 15

;:: 10 d 10 §;

5 §;

5

0 0

0 0.005 0.01 0 0.005 O.Ql

time [s] time [s] Figure 4.5: Von Mises stress-time histories for different time constants.(-) reference model;(···) T} 2.0 ·10-2s; (---) Tz = 1.0 ·10-4 s.

a time interval of 55.0 ms and applied a rotational acceleration on the head model. The trend seen was that the larger time constant of 1.43 ·10-2 s lead to higher stress levels than the smaller time constant (1.43 ·10-3 s). This result is in agreement with the findings of the parametric study in this thesis. Whether brain tissue really shows time dependent behavior for the analyzed simulation intervals can only be determined by additional experimental studies, where the constitutive properties of brain tissue are determined. As the results in this study have indicated, variation of the time constant leads to significant changes of the various responses. However, if the time constant is of the same order as the simulation time interval, the material behavior may just as well be represented as linear elastic, with the Young's modulus calculated on basis of the short-term shear modulus.

76 Parametric

4.3 Boundary and interface conditions

4.3.1 Relative motion between skull and brain

In Section 1.3.3 the importance of boundary and interface conditions has been elaborated. Specifically the way of modeling the skull-brain interface has been shown by various authors to greatly influence the response. A previously developed 2-D model by Kuijpers et al. (1995) that included the possibility of relative motion between brain and skull, gave results for the coup region that compared well with the data presented by Nahum et al. (1977).

A new model was constructed by decoupling the brain from the skull in the reference model. With this decoupling the number of elements in the model remained the same (1,756) but the number of nodes increased from 2,257 to 2,859 nodes, due to the double set of nodes defined at the interface between brain and skull. In the reference model the nodes at the interface are both coupled to the elements of the brain and the skull, whereas in the decoupled model one set of nodes is used for the brain and the other set for the skull. Between the brain and the skull the corrected contact algorithm, presented in Chapter 2, was applied.

To allow for a comparison with other analyses and to reduce computational costs a simulation time interval of 10.0 ms was chosen. For ease of reference the model with the contact algorithm at the skull-brain interface will be denoted free interface model, in contrast with the reference model that had a coupled interface.

The results for the pressure in the frontal and occipital regions are presented in Figure 4.6. From the results it may be directly concluded that the free skull-brain interface has a large

200

150

~ 100 i:l.., ~ 50 <ll .... ;::! 0 "' g)

-50 .... Cl.

-100

-150

0 0.005 0.01

time [s]

Figure 4.6: Pressure-time history. (~) reference model; (···) free-interface model; by Nahum et al. (1977).

pressures

influence on the results. The pressure-time histories for both the frontal (coup) and occipital (contrecoup) region show a much lower amplitude than the calculated pressures with the reference model and the pressures measured by Nahum et al. (1977). The low amplitude of the pressure in the contrecoup region is related to the applied interface condition. As opposed to the model with the coupled interface, in the model with the free interface the

Boundary and interface conditions 77

skull exerts no restraint on the brain. Due to the application of the contact algorithm only compressive forces can be transferred across the interface. In the free-interface model the brain lags behind the skull and recedes from the interface, creating a gap between brain and skull. A major difference between the measurements presented by Nahum et al. (1977) and the simulations performed here, is the fact that Nahum et al. (1977) measured the pressure in the CSF layer, and in the simulations the pressure is calculated in elements that represent brain tissue. In the real human head the gap between brain and skull is expected to be filled with CSF flowing away from the coup site to the ventricles or the spinal canal. Striking is also the fact that the coup and contre-coup pressures are built up much slower than in the reference model.

The CSF layer shows two mechanical functions, firstly it allows for an increase in pressure, and secondly it provides the possibility of relative motion between skull and brain. By using a contact algorithm to describe relative motion, only the second phenomenon has been included. How the modeling of the skull-brain interface causes the slow build up of pressure and the low pressure level in the coup region is discussed at the end of the next section.

The von Mises stress distributions for the various regions are presented in Sec­tion 4.3.2, where the foramen magnum is included in the model presented in this section.

4.3.2 Influence of foramen magnum

The foramen magnum is the largest hole in the skull and forms the transition area be­tween the brain and the rest of the human body. The location is in particular interesting for two reasons. Firstly, the brainstem which is located close to the foramen, is a vital and vulnerable organ, that connects the spinal cord to the other parts of the intracranial contents. Secondly, due to the incompressible nature of brain tissue, the hole may act as a pressure-release mechanism. When the head is impacted the intracranial contents will deform and due to the incompressible nature of brain tissue a small volume change may lead to a large increase in pressure. Through the outflow of fluids through the foramen, like CSF and blood, this pressure build up may be released.

Other authors (Ruan et al., 1993; Chu et al., 1994; Zhou et al., 1995b) have modeled the foramen magnum in combination with a coupled skull-brain interface, including the upper part of the spinal cord that fits inside the foramen magnum. However the influence of modeling this hole has not received much attention in these publications. Therefore in this thesis the effect of the presence of the foramen magnum was studied in the free-interface model. The hole was modeled by allocating the skull elements to the brain in the foramen region (Figure 4. 7). The four elements transferred from skull to brain roughly represent the upper end of the spinal cord. These elements are also separated from the skull by the contact algorithm. The foramen magnum is represented as a stress-free opening, and due to the presence of the contact algorithm the brain has the capability to move through the foramen magnum. This is in contrast to the previous model where, although relative motion was possible, the movement of the brain outwards was restricted by the skull. The created model still contains 1,756 elements, except that now the distribution between brain and skull is 1,004 elements for the brain and 596 elements for the skull. The face remains

78 Parametric study

(a) (b)

Figure 4.7: Mid-sagittal view of the foramen magnum region in the reference model (a), where no foramen magnum is present, and in the newly developed model (b).

unchanged. The total number of nodes in the model is 2,866. The same settings with regard to contact, damping and time step were used as in the free-interface model.

The pressure-time histories in the frontal and occipital regions are shown in Fig­ure 4.8 and are compared with the simulation with only the free-interface and the pressures measured by I'<ahum et al. (1977). The pressure-time history for the new model overlaps

150 - /-, 1/ \,_____--coup

I \ 100-I \

I \

1,' ~-- ......... p.,;-- '),~~-~ .... ... .

/ .. .....,.:- ...... __ ______ _ 50-

/ / / ' 0- ... .;;:~-:.::::~~ ... , ~ -- _ :~: -- ____ -_:::-:-~--~~rw

' I ' ' - 50 -

', __ /~ontrecoup -100-~------------r-----------~

I 0 0.005 0.01

time [s]

Figure 4.8: Pressure-time history. (- - -) free-interface model; ( · · ·) free-interface model with foramen magnum; (--)pressures by Nahum el al. (1977) .

with the results of the free skull-brain interface, indicating that modeling the foramen magnum does not influence the frontal and occipital regions, with regard to the pressure.

The von Mises stress distributions in the brain for the model with and without the foramen magnum modeled in the skull are depicted in Figure 4.9, and compared with the results obtained in the reference (coupled interface) simulation. The results for the free­interface models show in comparison with those for the reference model a large difference

and interface conditions 79

for the point in time the maximum stress level is attained. The maximum levels are attained later in the simulation interval for the free-interface models and their maximum amplitude of the von Mises stress are, except for the foramen region, approximately of the same magnitude. The von Mises stresses for the frontal, occipital and top regions of the brain

35

30

25

20

15

10

5

35

30

25

20

15

10

5

0

0

frontal region

0.005

time [s]

foramen region

0.005

time [s]

0.01

0.01

35

30

25

20

15

lO

5

o-+--~K··--.

35

30

25

20

15

10

5

0

0

0

occipital region

0.005

time [s]

top region

0.005

time [s]

0.01

O.Ql

Figure 4.9: Von Mises stress-time histories.(-) reference model;(---) free-interface model;(···) free-interface model with foramen magnum.

are roughly the same for both free-interface models. The stress for the foramen region shows quite a different pattern, here the von Mises stress is significantly higher for the model with the foramen magnum included. The pressures in the foramen region, for the reference model and the two free-interface models, are plotted in Figure 4.10. The pressure for the model with foramen magnum shows a maximum negative value and increases to a similar positive value in the observed simulation interval, whereas the model with only the free interface modeled gave higher pressure levels. The reference model oscillates around a pressure level of zero, with a decreasing amplitude over time.

Discussion

The results obtained over the years in the numerical and experimental studies of head im­pact have indicated that the CSF-layer plays an important role in two physical phenomena.

80

0 0.. .::.s "' ... ;:l til g) .. c.

20

15

10

5

0

-5

-10

-15

-20

0 0.005

time [s]

Parametric

O.Gl

Figure 4.10: Pressure-time history in the foramen region. (-)reference model; (---)free-interface model; ( · · ·) free-interface model with foramen magnum.

Firstly by providing the possibility of relative motion occurring between brain and skull when the head is impacted. Secondly by allowing for an increase in pressure in the fluid layer, either positive or negative, resulting in possible cavitation at the contre-coup site. In experimental studies of the impact of physical models (Nusholtz et al., 1996) the pos­sibility of cavitation occurring in the contre-coup region has been investigated. The voids that appear are believed to be responsible for injuries that occur in the occipital region.

The method of modeling the skull-brain interface used in this thesis only incor­porates the first phenomenon. The working of the contact algorithm only permits the transmission of compressive forces between structures. Once a tensile force becomes active, the two structures separate and a gap is created in between. In the real human head no gap can occur due to the presence of the CSF layer, instead the tensile forces would cause a pressure decrease in the CSF, possibly resulting in cavitation. To model the possibility of cavitation in combination with relative motion between brain and skull in a finite el­ement model of the human head two methods come to view. The first utilizes a coupled finite element code, permitting the simulation of both fluid and solid structures. However, these types of finite element codes are only just becoming available. The second method forms an extension of the contact algorithm by prescribing the way two structures should release, for instance by increasing the threshold tensile force, above which the structures are permitted to separate. In this study, because of the lack of knowledge on the behavior of the CSF layer under impact conditions, a threshold force level of zero has been assumed. By increasing the threshold level the motion along the interface is unrestricted but motion perpendicular to the interface (separation) is restricted or even prevented. As an extension to the sliding interface, that allows for only relative motion between structures, used in a 3-D model by DiMasi et al. (199la),· Bandak and Eppinger (1994) modeled the interface between a rigid skull and deformable brain in this manner.

The.slow increase in pressure in the frontal and occipital regions is caused by the free interface. Because the brain and skull are uncoupled pressure can only be built up

Boundary and interface conditions 81

there where the brain movement is restricted, whereas in the coupled model the brain is always restricted by the skull, giving an immediate increase in pressure. With regard to the low level of pressure in the coup region , further analysis of the results indicates that the left and right sides of the frontal lobe and the location where the frontal lobe interacts with the skull-base are subjected to high pressure levels (100.0- 250.0 kPa). In Figure 4.11 coronal cross-sections of the brain at the frontal lobe depicting the pressure distribution are shown for the reference model and the free-interface model. The pressure distributions show the

(a) (b)

-100.0 -30.0 40.0 110.0 180.0 250.0 [kPa]

Figure 4.11: Pressure distribution in a coronal cross-section of the brain at the frontal lobe at point in timet= 4.6 ms. (a) reference model; (b) free-interface model.

outer perimeter of the brain in the free-interface model to be subjected to high levels of pressure and a larger variation of the pressure level is seen, whereas in the reference model the pressure has an almost constant value over the entire cross-section. Results for the skull in the free-interface model indicate higher levels of pressure, by one magnitude, with respect to the reference model. Both in the reference model and in the free-interface model the maximum pressure occurs at the frontal bone, at the location where the impact load is applied. In addition in the free-interface model maximum levels of pressure are also found more irregularly spread throughout the whole skull. The von Mises stresses indicate for both the coupled and free-interface model the maximum level to occur at the skull-base. The maximum level in the free-interface model is roughly two orders of magnitude larger than in the coupled model.

In the modeling of only a free-interface, in particular the fact that both coup and contre-coup pressures were lower with respect to the pressures measured by Nahum et al. (1977) is remarkable. For using a similar method of modeling the skull-brain interface in a 2-D model representing a para-sagittal cross-section of the human head, Kuijpers et al. (1995) found coup-pressures in good agreement with Nahum's results, and like in the uncoupled model presented in this thesis, Kuijpers et al. (1995) found a pressure level of about zero for the contre-coup region. Using the same 2-D model but with a coupled

82 Parametric

interface the coup pressure was lower by a factor of two with respect to the measurements by Nahum et al. (1977). In conclusion the large differences in results found between the 2-D and 3-D head models show that extrapolation of the results obtained with a 2-D model to a 3-D situation, with regard to the modeling of the free interface, is virtually impossible.

As discussed in Section 1.3.3 various authors have described relative motion at the skull-brain interface. In most studies no comparison was made of a free-interface model with a coupled interface version of the model, making the influence of modeling the free-interface in these studies unclear. To compare the results obtained with our model with those of other authors that explicitly simulated the skull-brain interface, their results will first be briefly summarized. As mentioned above, using a 2-D model of a para-sagittal cross-section of the head the coup-pressures calculated by Kuijpers et al. (1995) compared favorably with experimental data. Bandak and Eppinger (1994) simulated rotational impact of the human head, by prescribing a rotational acceleration in the coronal plane. The stylized 3-D model consisted of a rigid skull, brain and falx cerebri. Their developed injury measure, the so­called cumulative strain damage, showed concentrations in the brain to occur near the skull surface. Ueno and Melvin (1995) used so-called frictionless gap elements to simulate the skull-brain interface in a 3-D model of the ferret, to simulate the cortical impact experiments performed by Lighthall et al. (1989). The model represented the brain and the skull was represented by a rigid surface model. Ueno and Melvin (1995) concluded that because of the incompressibility of the brain, a separation of the brain from the skull is very unlikely in a closed-head impact situation. In numerical simulations of half­cylinder physical models Galbraith and Tong (1988) and Tong et al. (1989) utilized in a 3-D model different interface conditions ranging from pure slip via various levels of friction to a no-slip condition. The model was subjected to a rotational acceleration in accordance with the experiments performed by Margulies (1987). Results showed that for the higher friction levels higher shear strains were induced and that for the no-slip condition maximum principal strains were located near the boundary, whereas for the slip conditions more spatial variations were found. Including a falx made the shear strains occur at the tip of the falx, relatively insensitive to the level of friction.

Concluding, only the models by Kuijpers et al. (1995), Ueno and Melvin (1995) and in our own study an impact in combination with a free interface was simulated, in all the other studies a rotational acceleration was prescribed. As mentioned above, the study by Kuijpers et al. (1995) used the same methodology as in this study, however the results by Kuijpers et al. (1995) were in better agreement with the experimental data by Nahum et al. (1977).

The modeling of a foramen magnum in combination with the free interface gave results similar to those of the model with only the free interface present. Only in the foramen region the pressure and the von Mises stress showed a different response. The lower pressures in the foramen region (Figure 4.10) for the model with free interface and foramen indicate that in our the model the foramen magnum may act as a pressure-release mechanism. In the real human head the presence of the spinal cord will restrict the movement of the brain outwards of the skull and decrease the effect of the pressure-release mechanism. The high

and interface conditions 83

von Mises stresses in the foramen region were also found by Ruan et al. (1993), where contour plots of the shear-stress distribution in a para-sagittal cross section of the head indicated the maximum levels also to occur in the foramen region. The high levels of shear are possibly caused by relative motion between various regions of the brain. Although a free interface is present, the brain material located in the foramen magnum opening is restricted in its movement due to the geometry of the skull at the foramen magnum wherea.c; the brain itself tends to move forward with respect to the skull, especially with the free skull-brain interface.

For the model in this thesis no dynamic boundary conditions were prescribed at the foramen magnum opening. In reality the spinal cord forms a continuum with the intracranial contents, and an extension of the model could therefore be to implement some kind of impedance to simulate the presence of the spinal cord. Data on the stiffness and damping factors for this impedance are unknown. Alternatively explicitly modeling the various structures of the neck in combination with the head model also gives the interaction between intracranial contents and spinal cord. This method was employed by Liu (1986) and Kumaresan and Radhakrishnan (1996). Their main reason for modeling the neck was to study the influence of the neck on the global motion of the head. An analysis of the results in the foramen region was not given. In a 2-D study by Kuijpers et al. (1995) where various models for the foramen magnum were presented in combination with either a free or coupled skull-brain interface, similar conclusions were obtained as in this thesis. The kind of model used to simulate the skull opening influenced especially the results in the foramen region. Chu et al. (1994) found the coup and contre-coup pressures to be hardly affected in a 2-D model of a para-sagittal section of the human head, where the influence of the foramen was studied in combination with a coupled skull-brain interface. The consequences for the field parameters in the foramen region were not analyzed. As mentioned in the introduction of section 4.3 other authors of 3-D models have modeled the foramen magnum, but no explicit analyses on the influence of the presence of the foramen magnum have been performed. Further possible applied boundary conditions were not reported on.

In conclusion, the way of modeling the skull-brain interface has been shown to have major implications for the field parameter distributions in the various regions of the intracranial contents. The two types of interfaces presented in this thesis, the coupled and free interface, represent the extreme cases. Looking at the anatomy, the real interface will be somewhere in between these two cases, but closer to the free-interface than to the fully coupled, because the presence of the layer, constituted by fluid and meninges, decouples the intracranial contents from the skull.

4.3.3 Modeling of a kinematic boundary condition

The neck interacts with the head in two ways. Firstly the neck restricts the motion of the head. In numerical head-impact simulations however it is often assumed that the impact takes place on such a small time scale, that the influence of the neck may be neglected and as a result no kinematic boundary conditions are prescribed. Secondly as discussed in

84 Parametric study

the previous section and in the anatomy description in Chapter 1, the spinal cord forms a continuum with the intracranial contents. When the head is subjected to an impact the spinal cord will interact with the intracranial contents in a complex manner.

In this section the influence of restraining the head is assessee! , by forcing the hearl to only rotate. The time sequence of plots depicted in Figure 3.4 shows the motion of the reference model to consist of a combination of translation and rotation. To understand the influence of restraining the head a kinematic constraint is defined. using the reference modeL that forces the head model to only rotate. The const raint consisted of rigid beams that connected the head model with a rigid node located 150.0 mm below the center of gravity. The beams were attached to nodes of the frontal and occipital bone respectively. The head model with the rigid beams attached is depicted in Figure 4.12. The impact was

ace

Rigid node

Figure 4.12: Mid-sagittal view of the left-half of the 3-D head model with the rigid beams.

simulated by prescribing the same pn~ssure-time history as in the previous simulations. The material properties , time st ep and simulation period were a lso the same.

The results of the simulation showed the head to rotate' approximately 12.0° around the defined constraint during the simulation interval. To compare the motions of the head for the reference simulation and the simu!ation with the kim~matic constraint, the transla­tional velocities of the center of gravity and rotational velocity around the center of gravity were calculated. The results are shown in Figure 4.13. The velocities were calculated using two nodes of the skull and assuming the skull may in approximation be seen as a rigid body. The directions are defined in Figure 3.2, and the rotation around the y-axis was taken positive when the head tilts backwards. Results show the head model in the simula­tion with the kinematic constraint to be subjected to a larger angular velocity and a larger

8

.,. 6 ..._

..§_ ;.., 4 ..., ·o ..9 "' 2 ;>

0

0

and interface conditions

velocity in x-direction of e.g. velocity in y-direction of e.g.

0.005

time [s]

50 .,. ..._ "tl 40 <:3 ~

.?;> 30 ·o 0

OJ 20 ;> ....

-3 10 bl)

Iii 0

0

"' -0.5-..._ £.. ;.., -1-..., ·o 0

OJ ;> -1.5-

-2-

0.01 0

angular velocity around e.g.

0 0.005 0.01

time [s]

I 0.005

time [s]

85

O.ol

Figure 4.13: Velocity of the center of gravity (e.g.) and the angular velocity of the head. (-) reference model; (---) model with kinematic constraint.

velocity in z-direction, with respect to the reference simulation. The results for the pressure in the frontal and occipital regions are depicted in

Figure 4.14 together with the results of the reference model. The pressures in the two regions are only to a small degree affected by the kinematic constraint. Both the coup and contrecoup pressures are lower for the model with kinematic constraint, in comparison to the pressures calculated using the reference model.

The von Mises stress distributions for the various regions in the brain are shown in Figure 4.15. The von Mises stress for the model with the kinematic constraint, shows higher values for all regions, but overall for the whole simulation interval the von Mises stresses are higher for the top, occipital and frontal regions. This would be directly related to the fact that the model is forced to rotate around the constraint node, instead of being able to move freely. The velocity in the head will be greatest at those locations farthest away from the center of rotation. Due to the difference in inertia and Young's modulus the brain will show the tendency to lag behind the skull. This effect will result in a maximum lag of the brain with respect to the skull for those locations where the velocity has the maximum amplitude, or that are farthest away from the center of rotation. Figure 4.16

86 Parametric study

200

150

'""ij"' 100 0.. ~ 50

Q) .... ;:l 0 "' "' "' -50 .... 0.

-100

-150

0 0.005 0.01

time [s]

Figure 4.14: Pressure-time history. Reference model: (-) line, and model including kinematic constraint: (---) line.

frontal region occipital region

35 35

Cl 30

ri; 30

0.. 25 25 ~ ~

"' 20 ~ 20

Q)

.:s 15 ~ 15 ;::;;; § 10 >:: 10

§: . ' .,. 5 5

0 0

0 0.005 0.01 0 0.005 0.01

time [s] time [s]

foramen region top region

35 35

Cl 30

'""ij"' 30

0.. 25 0.. 25 ~ ~

"' 20 "' 20 Q) Q)

"' 15 -~ ~ ;::;;; 15

>:: 0

10 § 10 > 5

.,. 5

0 0

0 0.005 0.01 0 0.005 0.01

time [s] time [s]

Figure 4.15: Von Mises stress-time histories. Reference model: (-) line, and model including kinematic constraint: (---) line.

Boundary and interface conditions 87

shows the velocity distribution in x-direction at t imet = 4.0 ms. The maximum velocities

0.0 1.0 2.0 3.0 4.0 5.0 6.0 [m/s]

Figure 4.16: Velocity distribution in x-direction in the mid-sagittal plane at timet= 4.0 ms.

are found for the top, frontal and occipital regions of the brain, for which the von Mises stresses as shown in Figure 4.15 a lso show higher levels than in the reference model. At the skull-brain interface, no actual lag can occur, because the two structures are coupled .

Discussion

Few authors have considered the influence of kinematic boundary conditions on the motion of the head and thus also on the field parameter distributions of the intracranial contents. using a 2-D pla in-strain model of a para-sagittal cross-section of the human head Chu et al. (1994) modeled a kinematic restraint as presented in this section. Results of frontal impact simulations showed high shear stresses to occur first in the top region and la ter also in the foramen region. Their results are similar to the result obtained in this thesis, except for the foramen region. T he reason for their high stress level for the foramen region might be caused by the fact that their model included a representation for the foramen magnum in combination with a coupled skull-bra in interface. Other authors (Liu, 1986; Kumaresan ami Radhakrishnan , 1996) have explicitly modeled the neck. Comparison with these models is of no use, because the global motion of the head has not been reported.

In conclusion, the results of modeling a kinematic constraint lead to an angular velocity of the head twice as high (Figure 4.13) as for the reference model. This more severe rotational motion resulted in a different shear behavior for the intracranial contents, resulting in significant higher von Mises stresses for the top and occipital regions. When the skull-brain interface is modeled as a free interface it is expected that these higher von ).1ises stresses might not occur, because the skull can mm·e freely with respect to the brain without inducing the applied rotation directly to the brain.

88 Parametric

4.4 Anatomical detail of the model

4.4.1 Modeling of substructures

The falx cerebri and tentorium have a structural function in the human head. The hypothe­sis regarding their mechanical function is that when the head is impacted, they support the cerebrum and prevent it from compressing underlying structures, such as the brain stem and cerebellum. In the reference model with the homogeneous intracranial contents the as­sessment of this possible supportive function is impossible. Therefore the reference model, introduced in Chapter 3 was used as a basis and various substructures were modeled.

The structures of the intracranial contents were included manually using an anatom­ical atlas and the CT and MRI images of the Visible human dataset. Included in the model were; falx cerebri, tentorium, cerebrum, cerebellum, brainstem, the skull, built up of one layer and the face. It was decided to first use the model with all interfaces between struc­tures coupled. When modeling the falx cerebri and tentorium, other structures such as cerebrum and cerebellum automatically come forward. To connect various structures with each other it was chosen not to use kinematic constraint equations ( tyings), that permit the coupling of the displacements of nodes of two bodies. The reason for this choice is that the usage of these constraints, when constructing and further manipulating the model, requires a lot of manual labor. Instead the geometry of the newly created structures was extended into the connecting structures, resulting for example in an additional row of elements in the skull at the mid-sagittal plane, to connect the falx cerebri to the skull. Further the usage of hexahedral elements was maintained, and only at those locations where no hexahedral element could be included, a tetrahedral element was used.

Figure 4.17 shows, for ease of visualization, only the left half of the model with the various substructures present. The intracranial contents were attached to the skull, and all additional substructures were also rigidly connected to each other. No relative motion was therefore possible between brain and skull and between the separate structures.

All the structures were represented as homogeneous linear elastic material. The values for the material properties were obtained from literature. The total model consisted of 12,126 elements and 13,300 nodes. The number of elements representing the various structures together with the used material properties are given in Table 4.5. The head model was subjected to the same pressure-time history as described for the previous models. The simulation time interval was 10.0 ms and the time step was 2.0 · 10-5 s, as used for the reference model.

Figure 4.18 depicts the calculated pressures in the frontal and occipital regions of the brain in comparison with the results calculated using the reference model. Both the coup and contrecoup pressures for the new model show a decrease of the maximum attained pressure level in comparison with the reference model. The decrease in the contrecoup region (29.5 kPa) is larger than the decrease in the coup region (14.6 kPa), indicating that for frontal impact the tentorium and falx cerebri have a supportive function for the cerebrum. As was shown in Section 4.3.3 the model without kinematic boundary condition defined, exhibits a translational and rotational motion. The functions of the falx cerebri and tentorium would probably emerge more explicitly if the head were only subjected to

Anatomical detail of the model 89

Falx Cerebri

Brainstem

Cerebellum

Figure 4.17: Mid-sagittal view of the left half of the 3-D model with the various substructures included. To ease visualization the intracranial contents are plotted separate from the skull and face.

Table 4.5: Linear elastic material properties used in the simulation with the 3-D model with various substructures.

Structure elements E p 1/

# [N/m2] [kg/m3] H skull1 2834 6.5·109 2.07·103 0.22 face1 188 6.5·109 5.0·103 0.22 cerebrum1 7568 1.0·106 1.04·103 0.48 cerebellum1 730 1.0·106 1.04·103 0.48 falx cerebri2 314 3.15·107 1.13·103 0.45 tentorium2 240 3.15·107 1.13·103 0.45 brainstem1 252 1.0·106 1.04·103 0.4 1: data averaged from literature 2: data obtained from Ruan et al. (1991)

rotational motions in respectively the sagittal plane for the function of the tentorium and the coronal plane for the function of the falx cerebri. In the discussion further attention is paid to the influence of the falx when the head is subjected to rotation. Due to the coupling of the structures, the stiff falx cerebri will restrain the brain material from moving, and thus induce shear stresses in the cerebrum. In Figure 4.19 this is depicted.

90 Parametric

200

150

100

50 i:: = 0 "' ., .,

-50 .... 0.

-100

-150

0 0.005 O.Dl time [s]

Figure 4.18: Pressure-time histories.(-) reference model;(---) model with substructures.

To further investigate this restraining of the cerebrum by the tentorium, contour plots of the mid-sagittal plane of only the brain, consisting of cerebrum, cerebellum and brainstem, have been made for various points in time. The pressure distributions are de­picted in Figure 4.20, in Figure 4.21 the von Mises stress distributions are shown. The results have been presented for three points in time. For t = 3.0 ms and 4.0 ms the pres­sure contours in the model with substructures indicate most clearly that the pressures just above and just below the tentorium are slightly different, again indicating the possible sup­porting function the tentorium has in frontal impact. In comparison to the reference model, the pressures in the model with substructures are lower, but the pressure distributions in the reference model in the mid-sagittal plane are more evenly spread.

In the von Mises stress distributions (Figure 4.21) high stress levels are seen in the cerebrum, whereas the cerebellum is more or less stress-free. The higher stresses are located at the interface where the falx is connected to the cerebrum. Due to the modeling of the falx cerebri and the coupling of all the structures, the stiffer falx will restrain the cerebrum and thus induce shear stresses in the cerebrum, see also Figure 4.19. Comparing the von Mises stress distributions with the distributions calculated with the reference model, shows the maximum stresses for the model with substructures to occur in the cerebrum, and for the reference model the von Mises stress is more evenly spread out over the entire intracranial contents.

Discussion

The modeling of substructures influenced both pressure responses and von Mises stress distributions. The tentorium supports the cerebrum in frontal impact, and the falx cerebri further restrains the brain, because of the coupling of all the structures.

In the numerical modeling of the human head it is common practice nowadays to include some of the structures that constitute the intracranial contents (Ruan et al., 1993; Bandak and Eppinger, 1994; Zhou et al., 1995b). Except for the model presented by Bandak and Eppinger (1994), the various structures of the intracranial contents modeled in the

Anatomical detail of the model 91

(a) (b)

0.0 8.0 16.0 24.0 32.0 40.0 [kPa]

Figure 4.19: Restraining effect of the falx cerebri, von Mises stress distribution in a transversal cross-section of the brain for t = 4.0 ms. The skull and falx cerebri are not shown. (a) reference model; (b) model with substructures.

various head models were fully coupled. As an extension of the model developed by DiMasi et al. (199 1a) , Bandak and Eppinger (1994) included a contact prescription between head and skulL Although most authors have included various structures of the human head, their relevance and/or function is almost never assessed. Results by Bandak and Eppinger (1994) of coronal rotational impact simulations indicate deformation concentrations to occur near the surface of the falx st ructure . Rather surprising was the conclusion by the authors with regard to the behavior of the falx , based on the result that the velocity of t he brain lagged behind the imposed velocity of the skulL They concluded that the brain was resisting the rotational motion of the falx, instead of the expected situation where the falx resists the motion of the brain. Although no specific remarks were made by Zhou et al. (1996) concerning the importance of modeling the falx cerebri , rotational accelerations performed in the coronal and sagittal plane indicated maximum shear stresses to occur at the tip of the falx cerebri .

In numerical simulations of the rotational accelerations of physical models the im­portance of the falx cerebri in combination with various friction levels at the skull-brain interface has been studied by various authors (Galbraith and Tong, 1988; Tong et al. , 1989; Cheng et al. , 1990). Galbraith and Tong (1988) concluded with regard to the influence of the falx that in particular higher shear levels were seen in the regions near the tip of the falx , compared to the model without the falx. Tong et al. (1989) and Cheng et al. (1990) concluded that both boundary and anatomic feature effec ts are important but it also ap­pears that the influence of the anatomic discontinuity (falx) is far more significant to both the magnitude and pattern of induced strains. The data in literature shows the effect of a partition such as a falx in combination with rotational motion. To fully understand the relevance of the partitions in the head when the head describes a translational motion, additional studies will need to be performed.

92 Parametric study

t = 3.0 ms

t = 4.0 ms

t = 5.0 ms

-150.0 -100.0 -50.0 0.0 50.0 100.0 150.0 200.0 [kPa]

Figure 4.20: Pressure distribution in the brain in the mid-sagittal plane at various points in time with the reference model (left) and the model with substructures (right). Full head is shown, but the results are shown only for the brain.

Simulations in this thesis have indicated the tentorium to significantly influence the field parameter dist ributions throughout the brain in impact situations. The falx cerebri restrained the cerebrum and induced additional shear stresses which will probably be absent when a free interface is utilized between the structures of the intracranial contents. Taking into account the results obtained by other researchers when a free skull-brain boundary interface was combined with a partition . higher levels of von Mises stresses are expected in the region near the tentorium, when for the model presented in this section a free interface is utilized between the structures of the intracranial contents and the skull.

Anatomical detail of the model 93

t = 3.0 ms

t = 4.0 ms

t = 5.0 ms

0.0 8.0 16.0 24.0 32.0 40.0 [kPa]

Figure 4.21: Von Mises stress distribution in the brain in the mid-sagittal plane at various points in time with the reference model (left) and the model with substructures (right). Full head is shown, but the results are shown only for the brain.

4.4.2 Modeling of frontal sinus

The frontal sinus is an empty cavity in the frontal bone. The CT-slice depicted in Fig­ure 4.22 shows the size of this cavity in the frontal bone. As mentioned in Chapter 3, in the reference model this cavity was not included, because of the method used for meshing the skull. Since the frontal sinus takes up a large volume in the actual head, an approxima­tion of the sinus was included in the model. The objective of modeling the frontal sinus is to see if the impact , using the model with frontal sinus, results in different field parameter distributions in the brain. Possible differences of field parameter distributions occurring in the skull have been neglected here.

The model of the skull with sinus was built by meshing the skull using two layers

94 Parametric study

of elements. In the frontal region these two layers were separated, such that the shape of the sinal cavity was roughly approximated. The new model contained 1,178 elements and 1,552 nodes. The model is also shown in Figure 4.22. The other properties and the loading of the model were equal to those of the reference model. The simulation interval was 10.0 ms.

The pressure results of the frontal and occipital regions of the brain are shown in Figure 4.23. The pressure is only slightly affected for both the frontal and occipital regions. The von Mises stress distributions for various regions in the brain are shown in Figure 4.24. Only two regions are shown, because the results were, like the pressures, only slightly affected by including the frontal sinal cavity.

Discussion

Including an approximation of the frontal sinus in the skull hardly affected the responses calculated for various regions in the brain. Both the pressures and von Mises stresses show a small increase of the maximum attained levels. Whether the modeling of the sinal cavity influences attained stress and strain levels in the skull might be the subject of a future study. The focus in this parametric study was on the variation of responses seen for various regions in the brain.

(a) (b)

Figure 4.22: Mid-sagittal view of the left-half of the 3-D head model with the frontal sinus included (a), and (b) mid-sagittal CT-image of the head, with only the skull shown in black (U.S. National Library of Medicine, 1996).

Anatomical detail of the model 95

200

150

100

50

"' .... :::l 0 "' "' Q)

-50 .... Cl.

-100

-150

0 0.005 0.01

time [s]

Figure 4.23: Pressure-time history. (-) reference simulation; (---) model with frontal sinus.

frontal region

35

30

25

"' 20 "' (J)

~ 15

§ 10 > 5

0

0 0.005 0.01

time [s]

35

<r 30

0... 25 ~

"' 20 Q) 00

~ J,5

,:: 10 §;

5

0

0

occipital region

0.005

time [s] 0.01

Figure 4.24: Von Mises stress-time histories. (-) reference model; (---) model with frontal sinus.

96 Parametric study

4.5 Discussion and conclusions

The data available in literature concerning the knowledge about material properties and geometric properties, like boundary and interface conditions, display large variations. To enlarge the insight into the effect of this data on the response, the parametric study pre­sented in this chapter was performed. The demonstrated response changes due to the application and variation of various parameters are considerable, but because of the lack of experimental data, the calculated results have been compared with similar numerical studies, available in literature, where the same parameters were studied. The focus has only been on the responses of various regions in the brain, because the interest in this study is primarily in closed-head injuries, where no skull fracture occurs.

4.5.1 Discussion

Two major conclusions may be drawn from the results obtained in this chapter. Firstly, the variation of the Young's modulus over a range, representative of the range found in literature, had significant influence on the pressure and von Mises stress responses. Since the results of the reference simulation were comparable to results found in literature, the conclusions drawn by the various authors, based on all these different material properties, become at least questionable. Second the modeling of relative motion at the skull-brain interface showed to have large implications for especially the pressures in the frontal and occipital region, with respect to the reference simulation. The von Mises stresses attained similar maximum levels, only the pattern was different. The correct modeling of the skull­brain interface lies probably somewhere in between the fully coupled interface, as used in the reference model, and the free interface as introduced in this thesis. However the correct model will be closer to the free interface than to the coupled interface model, because of the presence of the layer of CSF and meninges, which decouples the intracranial contents from the skull.

In the next section the conclusions for all parameters are summarized.

4.5.2 Conclusions

• variation of the Young's modulus of brain tissue has a large influence on the response. On the other hand describing the brain tissue as being viscoelastic had hardly any effect on the pressure response. Variation of the time constant showed in particular the shorter time constant to alter the von Mises stress response in comparison with the elastic reference model.

• modeling of the free skull-brain interface allowing relative motion between brain and skull resulted in lower coup and contrecoup pressures, whereas the von Mises stress remained at a similar level as seen in the reference model.

• incorporation of the foramen magnum only influenced the results in the foramen region.

• a kinematic condition, which forced the head to solely rotate, lead to lower coup and contrecoup pressures with respect to the reference model. For the von Mises stress

Discussion and conclusions 97 -----------------------------------------------------

distribution in particular those areas were affected that lie farthest away from the center of rotation.

• subdividing the intracranial contents by including the various partitions, such as falx cerebri and tentorium, and thus also modeling the cerebrum, cerebellum and brain stem resulted in lower pressures at the contrecoup site, possibly indicating the supporting function the tentorium has. The von Mises stress distributions displayed higher stresses with regard to the reference model, concentrated in the cerebrum. These were caused by the fact that the structures of the intracranial contents were coupled, thus the stiffer falx restrained the brain tissue from moving, thereby induc­ing higher levels of von Mises stresses.

• modeling the frontal sinus in the frontal bone had hardly a noticeable effect on the calculated responses for the brain, neither for the pressures nor for the von Mises stress distributions.

98 Parametric

5.1 Discussion

Chapter 5

Discussion, Conclusions and Recommendations

For head injuries that occur as a result of an impact, such as those that occur in car acci­dents, knowledge of injury mechanisms can aid in the development of new injury prevention measures. An injury mechanism describes the relationship between the biomechanical re­sponse of a biological system and the sustained injury. The process of the identification of injury mechanisms comprises two tasks; firstly the calculation of the field parameter distributions sustained by the tissues in the head under impact conditions and secondly providing the link between calculated deformations and observed injuries seen in patients. The focus in this thesis has been on the first task. The strategy has been to start off using simple finite element models, to calculate the deformations in the head and to assess the importance of both the degree of geometrical detail to which the components are modeled and the conditions at their interfaces. As a result a three-dimensional finite element model of the human head has been developed and because of the many uncertainties concerning material properties, and boundary and interface conditions, a parametric study was per­formed. This chapter discusses and summarizes the major findings of this study. Further, recommendations are given for future research. In many of the recommendations physical models are brought forward since they represent a tool with which various mechanisms that

99

100 Conclusions and Recommendations

play a role in the development of injuries, can be further studied without the drawbacks encountered when using biological tissues.

5.1.1 Contact algorithm

The numerical simulations in this study were performed with the finite element package MARC, because of the availability of this code. Both explicit and implicit time integration schemes can be chosen, but to avoid possible problems with the numerical stability of the solution when using an explicit time integration scheme, an implicit scheme was used in all the analyses in this study. The greater computational costs, due to the usage of the implicit scheme, were taken for granted.

To model interactions between structures, both inside and outside the head, contact algorithms have been used in this study. Since these algorithms fully determine how energy is transferred from one structure to the next, an in-depth study on the working of these algorithms was performed. Contact phenomena between deformable structures that impact are mathematically described, next to the equations of motion for each of the structures, by three constraint conditions. This set of kinematic and dynamic constraints consists of; (1) the no-penetration condition, (2) the requirement that in the point of contact, the contact forces are equal but opposite in sign, and (3) only compressive forces may be transmitted. The contact algorithm in the MARC finite element code, known as the solver-constraint method, in combination with the Newmark-,8 time integration scheme only used the first of the contact conditions, and as a result gave errors for the velocity and the acceleration of the nodes at the contact interface. This deficiency was corrected for the velocity by solving the momentum balance equation, derived from the second condition, for the nodes that contact, so as to obtain a new correct velocity for the moment directly after impact. Correct accelerations for the contacting nodes were obtained by solving the equilibrium equation, derived from the third condition. The corrected contact algorithm was shown to produce results, that apart from an oscillation, were in good agreement with an analytical solution, that was based on wave propagation theory through beams. The resulting oscillation in the solution, known as Gibb's phenomena, appeared to stem from the spatial1discretization, as was also shown by Chin (1975).

5.1.2 3-D head model

The Visible Human dataset, consisting of digital CT and MRI data, has been utilized to construct a finite element model of the human head. The reference model contained only the skull filled with a homogeneous continuum, representing the brain. An approximation of the facial structure was added to the model and the mass density of the face was tuned, such that the mass and moment of inertia of the head model were in accordance with data presented in literature. The results of modal analyses of models of the skull and the complete head proved to be in good agreement with in vivo and in vitro measurements. For the validation of human head models the only experimental data available, are the impact experiments performed on cadavers by Nahum et al. (1977). They measured the pressures at various locations in the heads of cadavers that were impacted by an impactor traveling at a

Discussion 101

particular initial velocity. Calculated and measured pressures in the frontal region showed the same course. The measured occipital pressure was overestimated by the simulation, which was attributed to the particular method of modeling the skull-brain interface. The interface was coupled, making relative motion between brain and skull impossible. This method of modeling the interface might form a too strong restraint on the brain.

The experiments by Nahum et al. (1977) only provide pressure data at various loca­tions in the head. For a correct interpretation of the occurrence of injuries seen in patients and to allow for a full validation of the numerical model, field parameter distributions throughout the head are necessary. Here the introduction of physical models might prove to be a useful intermediate step in validating the calculated deformation patterns in the numerical model. In physical models the various input parameters, such as applied load­ing conditions, material properties and geometry are known, and using digital non-contact measurement systems the field parameter distributions throughout the surrogate intracra­nial contents of the physical model can be measured. Once the numerical model has been validated using the physical model, additional experimental data for the real human head will have to be assessed using post-mortem human subjects.

5.1.3 Parametric study

In the reference model various approximations were made concerning the material proper­ties, boundary and interface conditions and the geometry. To assess the importance of these approximations a parametric study involving a limited number of topics was performed; material properties, boundary and interface conditions, and the anatomical detail of the model.

Material properties

Data presented in literature on constitutive properties of brain tissue, indicate that brain tissue is viscoelastic and nearly incompressible. The values for the material properties avail­able in literature are spread over a wide range. Due to the lack of conclusive experimental data, as a first approximation in this study the brain tissue was represented as linear elastic, with the Young's modulus taken as an average of the data presented in literature. Variation of the Young's modulus over a range representative for the range seen in literature, had a large influence on the pressure and von Mises stress responses. Lowering of the Young's modulus showed higher pressure levels for the frontal and occipital regions. The higher modulus resulted in responses similar to the original configuration. Comparing the results obtained by various authors is difficult due to the differences in used geometries, material properties and boundary and interface conditions. The results obtained in this study with the variatio~ of the Young's modulus show that the comparison of results between studies is actually impossible if the used material properties vary over a range of two orders of magnitude.

To further understand the obtained results, for instance concerning the oscillations seen for the lower Young's modulus, physical models are a valuable tool. Experiments with physical models where the Young's modulus of the gel, used for mimicking the brain, is

102 Conclusions and Recommendations

varied could answer the question whether the calculated oscillations are of a physical or numerical nature.

To study the influence of time-dependence, the standard linear viscoelastic model was introduced. In this model the compressibility, governed by the bulk modulus, was kept constant, whereas the shear behavior was varied by variation of the time constant. Results indicated the larger time constant to give almost the same von Mises stress distributions as the reference model. The smaller time constant showed a clear decrease of the von Mises stress. Due to the unchanged bulk modulus of the brain tissue, the pressure levels in the frontal and occipital regions were hardly affected by the introduction of a viscoelastic material model, in comparison with the elastic brain model.

In view of the findings presented in the parametric study on the variation of material properties, the results presented in literature become at least questionable. Comparison of the results of numerical studies appears to become particularly difficult when different material properties have been used. In the future additional experimental data on the constitutive properties of brain tissue should be gathered. Next to paying attention to the aspects of viscoelasticity, anisotropy and inhomogeneity the properties should preferably be characterized for the deformation rates found in head impact.

Boundary and interface conditions Specific attention has been paid in this study to the modeling of the interaction between structures within the head. The skull-brain interface consists of various meninges separated by fluid. The relative motion between brain and skull was modeled by including a contact algorithm between brain and skull. Perpendicular to the interface the contact algorithm can only transmit compressive forces, once a tensile force larger than a certain threshold level is present the two structures release. The way of modeling the interface has shown to affect the pressure responses to a large extent and the von Mises stress distributions for various regions in the brain to a lesser extent. Inclusion of the foramen magnum in the skull in combination with the free skull-brain interface influenced especially the pressure and the von Mises stresses in the foramen region. The zero-pressure level in the occipital region was attributed to the fact that a tensile threshold force of 0.0 N has been used for the contact algorithm. The pattern of release between two structures can be modified by altering the threshold level.

Forcing the reference model with coupled interface to only rotate, instead of allowing it to perform the combined translational and rotational motion affected in particular the von :Mises stress at those locations where the velocity was highest, in the upper part of the head. In these location the brain is probably subjected to much larger stress levels than when a free skull-brain interface would be used.

Anatomical detail of the model

The anatomical detail of the model was studied by including the two partitions, falx cerebri and tentorium. As a result the intracranial contents was also subdivided into the various structures. The frontal and occipital pressures indicated that the tentorium has a support­ive function in the head in frontal impact situations. However, the stiff falx cerebri also

Conclusions 103

restrained the brain as a result of the coupling of all the structures. Using a free interface between each of the structures will probably reinforce the supportive function of the ten­torium, because the skull and falx cerebri would not be able to restrain the brain anymore as is the case with the coupled interface.

The used CT data indicated that the person had a large frontal sinus, leading to a relatively thick frontal bone, due to the meshing method used. To investigate the possible influence of this air-filled space on the impact response of the head, a model of the cavity was included in the head model. Results indicated that the pressure and von Mises stress for regions in the brain were not affected. Implications of modeling the sinus for the response of the skull were not investigated.

In conclusion, various parameters have been varied and the calculated results have been compared with each other and the original configuration given in the reference model. The results indicate that in particular the interaction between brain and skull, modeled us­ing the contact algorithm, and the material parameters for brain tissue are of importance when reconstructing the field parameter distributions inside the head. The CT and MRI data provide detailed information on the geometry of the human head. However, the en­hancement of the geometrical representation of the numerical models is of lower priority compared to the high priority the proper characterization of the material properties has. The second task of confronting calculated field parameter distributions, obtained for in­stance in the process of accident reconstruction, with injuries seen in patients to identify injury mechanisms has not be pursued yet.

5.2 Conclusions

• The modeling of contact in combination with Newmark-f9 time-integration lead to erroneous results in the finite element code MARC for the velocity and accelerations of the nodes at the contact interface. These errors were caused by the incorporation of only one of the three contact constraints, necessary for the correct modeling of contact between two structures.

• The remaining contact constraints were reformulated into a momentum balance equation and an equilibrium equation for the nodes at the contact interface, respec­tively. Succesfull correction of the errors in velocity and acceleration was possible by solving the momentum balance and the equilibrium equation at the time of contact.

• A geometrically realistic 3-D finite element model of the human head was developed based on CT and MRI data. Results of modal analyses of the various parts proved to be in good agreement with experimental data. Comparison of the transient response of the model with experimental data, showed similar trends for the pressure levels in the frontal and occipital regions of the brain.

104 Conclusions and Recommendations

• A parametric study using the 3-D head model was performed. The material proper­ties and method of modeling the skull-brain interface showed to significantly change the results for pressures and von ~1ises stress for various regions in the brain. The modeling of the partitions, falx cerebri and tentorium influenced the responses of various regions to a lesser degree. Only a small difference in pressure was visible in the brain just below and above the model of the tentorium, indicating a possible supportive function of the tentorium.

5.3 Recommendations

• Constitutive properties: The constitutive properties of the various tissues of the human head should be characterized adequately, that is for their response under impact conditions. No new experimental results on the constitutive properties of the intracranial contents in impact conditions have been published during the last two decades. Various parts of the brain show a fiber orientation, for example the brain stem. Attention should therefore be paid on the aspects of anisotropy. Exper­imental research will further have to indicate whether the behavior of brain tissue is viscoelastic or may in good approximation be seen as elastic, when subjected to high strain rates. Mixed numerical-experimental methods may aid in this character­ization of properties.

• Validation: The process of validation comprises two tasks. The first task is con­cerned with the checking of the numerical tools. Simple physical models could be utilized to check if the numerical simulations accurately calculate deformation pat­terns. Under impact conditions wave propagation phenomena may play an important role in the human head. Their role can only be assessed if they can be described and measured in respectively numerical simulations and experiments. Additionally vari­ous kinds of interface conditions between the gel and the cylinder could be simulated in the physical models together with their numerical counterparts to understand the mechanical function of the fluid layer between brain and skull. In the second task controlled impact experiments using animals in combination with numerical models of these animals allow for a validation on the basis of measured and calculated kinematic trajectories and forces. Apart from the experimental prob­lems encountered when using animals as test subjects, the ethical aspects should also be considered. After this process of validation has been accomplished, injuries seen in the animal can be compared with calculated deformations in the numerical model permitting the identification of injury mechanisms. This second task has been schematically depicted in Figure 5.1. Once injury mechanisms have been identified in animals, they have to be converted for the human head. This conversion not only involves the scaling of the geometry but also of, among other parameters, the ma­terial properties and the injury tolerances. This is not a trivial process.

Recommendations 105

I Identification

1 injury mechanisms I_ --- ---- __ I

Figure 5.1: Flow diagram of using animal and numerical models in the identification of injury mechanisms.

• Numerical analysis: For the numerical analysis the usage of explicit time integra­tion schemes, instead of the implicit scheme used in this study, should be considered. For instance an estimate of the computational advantage of the explicit method can be obtained together with the numerical accuracy and stability of the solution using the impact of two beams. To model the human head as realistically as possible, more features have to be in­cluded. These features include in particular the modeling of interactions between structures other than only between skull and brain. Further to simulate the impact of the human head with other structures, the 3-D head model has to be extended by incorporating a representation of the scalp. The inclusion of additional geometrical details has the lowest priority. Both head and neck injury research may benefit from combined head and neck mod­els including the interactions between the two structures. The incorporation of the neck model will probably not lead to the same large changes in motion as were seen with the modeling of the kinematic constraint. For the simulation of the head interaction with the rest of the human body hybrid numerical methods, where finite element models and multi-body models are combined, could be a promising method.

106 Recommendations

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Appendix A

Derivation of contact conditions for velocity and acceleration

A.l Introduction

This appendix describes the kinematic conditions for velocity and acceleration, and how they are derived from the kinematic contact condition for the displacement.

A.2 Derivation of velocity constraint

Contact condition (2.19) describes the constraint of no-penetration, that is two points of continua A and B can never penetrate the other continuum (Figure A.l).

Figure A.l: Contact condition, gap vector g and normal vector n 8 .

117

118Appendix A. Derivation of contact conditions for velocity and acceleration

(A.l)

where the first part is the gap vector g. g connects the points that are closest together, in the example (Figure A.l) points P and Q. The direction of g is by definition the same as nB, normal to the surface of B. Time differentiation of this constraint equation gives the following constraint equation for the velocity

([u(~P,t) it(~Q,t)]) · nB(~Q,t) + ([x(~P)+u((,t)] [x(~Q)+u(~Q,t)])·nB(~Q,t) 0 (A.2)

The derivative of nB, nB, in the second term on the left hand side is by definition perpen­dicular to nB. The gap vector g is parallel to the vector nB and therefore also perpendicular to nB. Using these relations and the fact that the dot product of two perpendicular vectors equals zero, the velocity constraint reduces to

(A.3)

A.3 Derivation of acceleration constraint

The kinematic constraint for the acceleration is determined from the velocity contact con­straint by a time differentiation of equation (A.3)"

([u((, t)- u(~Q, t)]) · nB(~Q, t) + ([u(~P, t)- u(~q, t)]). nB(~q, t) = o (A.4)

The second term on the left hand side represents the effect of relative motion between the contacting points in the directions perpendicular to nB. This term is negligible when such relative motions are small or contact occurs between flat surfaces. Using this assumption the acceleration constraint reduces to

(A.5)

Appendix B

Application of contact conditions

B.l Introduction An example is presented in which the direct constraint method using only displacement conditions in combination with Newmark-P time integration leads to differences in the velocity and acceleration of a contacting set of nodes. In the finite element method, contact between continua is either established on a node-to-node basis or a node-to-edge basis. The used finite element package MARC K6.1 (MARC Analysis Research Corporation, 1994) uses the latter, the 2-D example presented here will also be based on this method. Since the primary interest here is the kinematic behavior of the contact algorithm and not the resulting distribution of field parameters, i.e. stress and strain, in the area of contact, the simplest possible configuration will be used in the example; the coming into contact of a single node with an element edge (Figure B.l). where node 1 has an initial velocity denoted by vector v and the accelerations of all nodes are initially zero.

As long as the two bodies are not in contact their equation of motion

(B.l)

may be split and solved independently for each of the bodies. The moment the two bodies contact the following three contact conditions have to be satisfied;

1. No-penetration condition,

119

120 Appendix B. Application of contact conditions

y,v

t x,u

B

( = -1 2

Figure B.l: The coming into contact of a single node with an element edge.

2. Contact forces in the point of contact are equal but opposite in sign for each of the bodies,

o.A(~P, t) · nA(~P, t) = t:T8 (~Q, t) · n 8 (~Q, t)

3. Only compressive forces in the point of contact,

(B.3)

(B.4)

Similar kinematic conditions for the velocity and acceleration can be derived from the first contact condition (B.2) as described in Appendix A.

Often instead of using all three contact conditions, just the kinematic conditions and its derivatives are used. In the next section the example is elaborated when at the moment of contact only contact condition (B.2) is considered. In section B.3 a correct modeling of impact is presented, using a momentum balance for the velocity, derived from the second contact condition, and weighting the accelerations using the masses of the contacting points. To include the momentum balance and the correction for the acceleration efficiently in the finite element program some assumptions were made which will also be briefly discussed. Section B.4 describes the implementation of the corrections when contact between two 3-D bodies occurs. The 2-D example will be extended by considering contact between again a single node and a so-called patch. A patch is the 3-D equivalent of an edge, and is formed by the four nodes of the face of a 3-D element (see also Figure B.2).

B.2 Coupling of only kinematic conditions

Suppose that at a certain point in time t in the analysis contact is detected between the two bodies. As long as the bodies are in contact, the no-penetration condition and its derivatives (Appendix A) have to be complied with, for the velocity

(B.5)

and for the acceleration

(B.6)

kinematic conditions 121

The last equation implies the assumption that relative motions between the contacting surfaces are small and contact occurs between flat surfaces. Condition (B.2) states that as long as the bodies are in contact the positions of the two contacting points coincide. The displacements since the time of contact of the two contacting points P and Q, during the time contact is established, are therefore also the same. When using an incremental formulation (2.18), the incremental displacements are the same from the moment of contact iluf+t:.t = ilu~t:.t·

For the case where contact is considered between a node and an edge the incremental displacements between three nodes are coupled, for the example in Figure B.l

(B.7)

where ( is a coordinate along the segment 2 -+ 3, defined as ( = -1 in node 2 and ( 1 in node 3, and ( denotes the location of node 1 on edge 2 -+ 3. The node that comes into contact with the edge is often called the tied node, whereas the nodes of the edge are called the retained nodes.

Equation (B. 7) describes the coupling of the displacements in x and y direction, no sliding of node 1 along edge 2 -+ 3 is possible anymore. To enforce the no-penetration condition (B.2), coupling the displacement normal to the edge 2 -+ 3 is sufficient. For this a local displacement vector uL is introduced, with component uL defined as the displacement normal to the edge 2-+ 3 and component vL the displacement along the edge (Figure B.1). Using the local incremental displacement a similar tying relation as equation (B.7) can be derived,

(B.8)

where only the displacement perpendicular to edge 2 -+ 3 is coupled. The local displace­ment is obtained from the global displacement by considering a transformation yL T.. ·1J with the transformation matrix T.. defined as

T = [ cos(/3) sin(/3) ] - -sin(/3) cos(/3)

(B.9)

with the angle f3 defined as the angle between the global and local x-axes. The direct constraint method imposes the no-penetration condition by eliminating

the equations and degrees of freedom of the tied node, in this case node 1, from the system matrices and columns. As equation (B.1) only contains global displacements, equation (B.8) has to be transformed to the global system using transformation matrix T

ilu(1 )t+t:.tcos(,B) + ilv(1)t+t:.tsin(f3)

~(1 () (flu(2)t+t:.tCOS(/3) + ~v(2)t+t:.tSin(f3)] +

~(1 + () [~u(3)t+t:.tCos(,8) + ~v(3)t+t:.tSin(f3)] (B.10)

122 of contact conditions

Depending on the magnitude of the angle {:J either L\u(l) or L\v(l) is eliminated. In the example L\u(l) will be coupled and eliminated. The process of elimination is done for system matrices M and and columns y, 'f!:, y and [, and leads to a fYStem with a reduced number of equations and degrees of freedom.

For each time step L\yt+Llt is solved from

(B.ll)

where the matrices and columns are in their reduced form. The incremental displacement for node 1 in x-direction (L\u(l)t+L:lt) is determined using kinematic relation (B.lO).

Solving the Newmark equations

L\yt+Llt = iJAt + [ ( ~ - o: )'!!t + a'!Jt+Llt] L\tz

'i!:t+Llt = 'i!:t +[(I o)'!!t + o'!Jt+Lltl L\t

(B.l2)

(B.13)

with o: and o the Newmark parameters, in combination with kinematic relation (B.lO) leads to a solution for the velocity and acceleration at timet+ L\t. Substitution of (B.12) in (B.lO) yields

(u(I)tl\t + [(~- o:)ii(l)t + o:ii(l)t+L:lt]L\t2) · cos({:J)

-(v(l)tl\t + [(~- o:)v(l)t + o:v(l)t+Llt]L\t2) · sin(,B) +

~(1- (){ (u(2)tl\t + [(~- o:)ii(2)t + o:ii(2)t+Llt]L\t2) · cos({:J) + (B.14)

(v(2)tl\t + [(~- a)v(2)t + o:v(2)t+Llt]L\t2) • sin({:J)} +

~(1 + (){ (u(3)tl\t + [(~ o:)ii(3)t + o:ii(3)t+L:lt]L\t2 ) · cos({:J)) +

(v(3)tl\t + [(~- a)v(3)t + o:v(3)t+Llt]L\t2) · sin(f:J))}

For the example it is assumed that prior to contact, the velocities are v(1)t it(2)t = v(2)t = u(3) = v(3)t 0 and it(l)t = Vo and the accelerations are ii(l)t = v(l)t ii(2)t v(2)t = ii(3) v(3)t = 0. For the sake of simplicity of the example it is assumed that node 1 contacts halfway between node 2 and 3, making ( = 0. Further it is assumed that edge 2 -t 3 is parallel to the global y-axis ({:J = 0), and that after contact the accelerations for node 2 and 3 are the same ( ii(2)t+L:lt ii(3)t+L:lt)· Using these assumptions relation (B.14) reduces to

ii(l)t+Llt- ii(2)t+Llt = vo

(B.l5)

The result is a difference in acceleration at the contact interface (Jiang and Rogers, 1988). In a similar way, a difference in velocity for the nodes at the contact interface can be

calculated using the previously obtained result for the acceleration and equation (B.13). For

Correct modeling of impact 123

the simplicity of the example the previous assumptions are used and it is further assumed that the newly calculated velocities for nodes 2 and 3 are equal (u(2)t+L>t = u(3)t+t>t)· The result for the difference in velocity is

u(2)t+L>t = Vo(l - _i.) a

(B.16)

When the Newmark time integration scheme is used such that it introduces no numerical damping (a:= h 8 == ~) the difference in velocity between the two nodes is v0 .

The differences for the velocity and acceleration are caused by the fact that only the first contact condition (B.2) has been used in the establishing of contact. An often used technique in numerical analyses for improving convergence of the results is the reduction of the time step size. This would be of no use here, since the difference in acceleration is inversely proportional to the time step size, and the difference in velocity is independent of the time step size.

B.3 Correct modeling of impact

To model the impact between continua correctly, extra contact conditions are required for the velocity and the acceleration. The extra contact condition for the velocity can be derived from contact condition (B.3). As described in section 2.3.3, using the balance of momentum before and after the coming into contact, a velocity can be calculated for the contacting points (Hughes et al., 1976; Jiang and Rogers, 1988). The accelerations are corrected by weighting the accelerations by the masses of the contacting points, so that the equation of motion is satisfied after the establishment of contact.

Similar to implementing the contact condition for the displacement, the velocity and acceleration only have to be corrected for their component perpendicular to edge 2 -t 3 in the example in figure B.l. This causes the tangential velocity (or relative motion) between the two bodies to remain unchanged.

The algorithm that calculates the new velocity based on the momentum balance will be explained step by step. Prior to contact the known variables are the global velocity y for the nodes that come into contact and the untied mass matrix for the nodes that contact. To ensure that only the velocity perpendicular to edge 2 -t 3 is corrected, the velocity of the node to be tied is determined using the local coordinate system defined in Figure B.l. In the case contact between more nodes and edges occurs in a larger system the Assembly part, of the following algorithm, is repeated for every set of tied nodes and edges and the tied mass matrix is assembled in the new system matrices. Once the assembly is finished the new velocity after contact is solved for the whole system, and using these new velocities, velocities for the tied nodes are determined using the algorithm described under Transformation.

124 Appendix B. Application of contact conditions

Assembly: 1. lump mass matrix:1 M -t ML 2. calculate momentum p before contact: p = ML · 1! 3. transform p for the node to be tied to local system. In the example this would be for

node 1. -4. substitute tying relation (B.8), as described in section B.2, for mass matrix ML--+ Mi

and p--+ pt. 5. lump- mass matrix Mi using the absolute value of the off-diagonal terms.

Solving: 1. solve new velocity for tied system from: Mi·'!Z~ P_t, where y~ is the new tied velocity.

Transformation: 1. calculate new velocity for the tied node in perpendicular direction in local coordinate

system using relation (B.8). 2. transform this velocity back to the global system.

The lumped mass matrix is used as opposed to the consistent mass matrix used in the rest of the analysis from an efficiency point of view. The new velocity can now be calculated without using matrix operations, like triangulizations. The lumped mass matrix using absolute values shows in comparison to the usage of a consistent matrix an error in the newly calculated velocity. The norm of the velocity vector has an error deviation with a maximum of 5%, depending on the direction of the normal vector of the edge and the direction of the initial velocity vector before contact.

The new acceleration is calculated using the same algorithms as described above, except where it says velocity, acceleration should be read. Further the product of lumped mass matrix with the velocity would now be some kind of force instead of the momentum.

B.4 Correct modeling of 3-D contact

The correct modeling of 3-D contact differs only in a few points from the modeling of 2-D contact presented in the previous section. For the detection of contact, contact is now considered between a node and the face of an element (Figure B.2). The kinematic tying relation between the four retained nodes of the element face and the single tied node has the same structure as equation (B.8), but has been extended for the additional nodes. Further next to the local variable ( an additional variable 7J is introduced to define the position of node 1 on the face of the element in the local coordinate system ( (, 71) as shown in figure B.3.

The tying relation for the displacement perpendicular to the element face is now defined as

wf (1 ()(1 ry)w~ + (1 + ()(1-rt)wf + (1 + ()(1 + rt)w~ + (1- ()(1 +rt)w~ (B.l7)

1. subscript L means a matrix is lumped and superscript t means a matrix or vector is tied. The latter denotes the fact that the tying relation has been substituted

Correct of 3-D contact 125

--1

3

Figure B.2: The coming into contact of a single node with an element face.

4 5 (-l,l)i._ _______ (l.l)

(-1,-1)--------e(l,-1) 2 3

Figure B.3: Definition of local coordinates ((, TJ).

The algorithm described in the previous section remains the same for all three parts (As­sembly, Solving and Transformation), except that the matrices involved in the various calcu­lations are larger due to the larger number of degrees of freedom per node (three instead of two) and the larger number of retained nodes.

126 Appendix B. Application of contact conditions

Appendix C

Mesh generation

C.l Introduction

This appendix describes the method of generation of the finite element mesh of the human head. The generation of the skull mesh and brain mesh is performed using two different methods. The brain mesh is determined with the so-called projection method and as outside surface of the brain the inside surface of the skull is taken. The skull mesh is generated by expanding the faces of the elements of the brain, that lie on the inside surface of the skull, to the outside surface of the skull.

C.2 Generation of elements

The projection method is actually a transformation from logical space to physical space. The logical domain, spanned by (ry,(,/), represents the initial mesh discretization and the physical domain (x,y,z) describes in the 3-D model the actual volume enclosed by the surface. In this case the physical domain consists of the volume enclosed by the inner skull surface.

For the generation of a 2-D mesh the transformation may be visualized as shown in Figure C.l. The boundary of the physical domain is known and the locations of the interior coordinates on the physical domain are the solution of a boundary-value problem. One way to describe these problems is by making use of potential theory, and a classic method often used, is to describe the boundary-value problem by means of elliptic partial differential

127

128

equations (Knupp and Steinberg, 1993). An advantage of using elliptic equations is that the calculated interior grid is very smooth.

Transformation

p Final mesh

y

U:-x _ __/ Figure C.l: Example of the generation of the mesh using the Laplace equation to describe the transformation. P denotes the physical domain and L the logical domain.

The principles of the transformation method will be explained for a two-dimensional case, to ease visualization. The simplest elliptic mesh generator is the length generator or AH (Amsden and Hirt, 1973) generator. This generator requires each component of the grid to satisfy the Laplace equation for each of the coordinates

(C. I)

(C.2)

The equations for determining x and y are uncoupled and linear and formulated on a square logical domain spanned by (ry,(). The boundary conditions for the domain are given by four maps for each coordinate, for example for the x-coordinate

x(17,0) = xa(1J) x(O, () = x1(()

x( 1], 1) Xt( 1J) x(1, () = Xr(()

(C.3)

(C.4)

where b, t, l and r denote bottom, top, left and right boundary of the logical domain. The Laplace equation is solved for each coordinate by a central difference discretization.

Generation of elements 129

Figure' C.1 demonstrates an example of how a logical10 x 10 elemPnt domain is transformed into a regular mesh using an ellipt ic mesh generator. The example shows a particular problem of the length generator in that the mapping is not one-to-one. For non-convex physical domains this results in folded regions of the final grid. In the example the top of the final mesh shows folded elements. Despite t his disadvantage , the length generator produces smooth grids and is computationally relatively fast.

In grnrrating a model of t he human head t he boundary conditions for the logical domain arc determined by finding t he intersection points of rays extending from the outside

Figure C.2: Oblique view of the right half of t he 3-D model that has been cut along the mid­sagittal plane. The part in light gray denotes the brain.

of the initial grid with t he inside surface of the skull. The direction of t he rays is governed by the vector from the center of the initial grid to that particular point for which the intersection is sought. l'sing the algorithm described above for 3-D situations, written in FORTRAN 77, a relatively coarse mesh was generated of t he brain and. as described before, by expanding the outside faces of the brain elements, the elements of t he skull were generated.

An initial grid was used with 10 elements in each direction , giving a mesh with 1,000 elements for the brain and 600 for the skull , 6 faces with 100 elements per face that are expanded. A. representat ion of the facial bones consisting of 156 elements, was constructed and added to the previously generated skull mesh. T he mesh was divided in two along the mid-sagittal plane, and half of the model was used, to obtain a completely symmetrical model along the mid-sagittal plane, by mirroring one half of t he model relative to this midsagittal plane. T he resulting model is shown in Figure C.2.

130 Appendix C. Mesh generation

Appendix D

Von Mises stress

D.l Introduction

This appendix explains how the von Mises stress is only related to the deformation of a particular volume of material. The change in volume is not incorporated by the von ~Hses stress. The method of proof starts of by calculating the internal energy of an arbitrary volume of material, during a particular state of deformation.

D.2 Von Mises stress related to distortion

The internal energy (W) is calculated as

w 1 - tr(u ·e) 2

(D.1)

Assuming linear elastic behavior according to Hooke's law and separating the stress in a hydrostatic (O"m) and a deviatoric part ( ud) according to ud = u- O"m I with O"m = !tr( u ), Hooke's law may be written as

(D.2)

131

132 D. Von Mises stress

Substituting this relation in the equation for the deformation energy, the following equation is obtained

W 1 ( 1 d d ( 1 1 ) m d 1 ( m)2 I) (D ) = 2tr 2Go- . o- + 2G + 3K a o- + 3K a ·3

Using the fact that tr(o-d) = 0 this equation reduces to

(D.4)

The deformation energy is now presented as a part that solely depends on the change in volume and the second part that represents the deformation. This last term may be represented in terms of the second invariant (J~ = -~tr(o-d · o-d)) of the deviatoric stress as

(D.5)

The von Mises stress is defined as o- 11 J -3Jj and therefore apart from a factor and the square root the von Mises stress represents only the deformation.

Samenvatting

Het menselijk hoofd is een van de meest kwetsbare delen van het menselijk lichaam, met name wanneer de mens betrokken is bij een ongeval, bijvoorbeeld een verkeersongeluk. De mechanism en via welke bepaalde belastingen tot bepaalde letsels leiden, zijn nog steeds niet volledig begrepen. Numerieke modellen bieden bier de mogelijkheid om deformatiepatronen binnen het hoofd te berekenen, die experimenteel niet te bepalen zijn. Verondersteld wordt dat deze deformatiepatronen een van de hoofdoorzaken vormen voor bet ontstaan van letsel. De eindige elementen methode is voor de simulatie van impact op bet menselijk hoofd uitermate gescbikt, vanwege de mogelijkbeid tot representatie van complexe geometrieen en de beschrijving van fysische en geometriscbe niet-lineariteiten.

Dit proefschrift beschrijft bet belang van de randvoorwaarden en de interactie tussen struc­turen in bet hoofd, de mate van detaillering van die structuren en de variatie van materi­aalgrootbeden, voor de simulatieresultaten.

Als eerste is er met behulp van MRI en CT data een basismodel ontwikkeld. Dit model bestaat uit een schedel gevuld met een homogeen continuum dat de hersenen repre­senteert. De berekende eigenfrequenties van het complete model en van de afzonderlijke schedel- en hersenmodellen komen goed overeen met experimentele resultaten uit de litera­tuur. De transiente respons van bet model is getoetst door berekende en gemeten drukken te vergelijken. De gemeten drukken zijn afkomstig van impact experimenten met mense­lijke lijken uitgevoerd door Nahum et al. (1977). Simulatie van deze impact experimenten resulteerde in drukken in de bersenen die redelijk goed overeenkomen met de gemeten resultaten.

In het basismodel is in tegenstelling tot de werkelijkheid geen relatieve beweging mogelijk tussen schedel en hersenen. De schedel-herseninterface is via een parameterstudie nader onderzocht, door de interface te simuleren met behulp van een contact algoritme. Testen aan de hand van de simulatie van de impact van twee balken hebben aangetoond dat het belangrijk is dat op het eerste moment van in contact komen van twee structuren, de drie contact voorwaarden correct in rekening worden gebracht. Simulatie van de relatieve beweging tussen schedel en hersenen onder dezelfde impactbelastingen als het basismodel

133

134 Samenvatting

resulteerde in drukken in het frontale gebied van de hersenen, die een factor drie lager waren dan die in het basismodeL Het grote verschil in drukken geeft het belang aan van de simulatie van relatieve beweging tussen structuren. Het modelleren van het foramen mag­num in de schedel in combinatie met relatieve beweging tussen schedel en hersenen heeft vergeleken met het vorige model met aileen relatieve beweging, aileen consequenties voor de drukken in het foramen-gebied. Variatie van de Young's modulus van hersenweefsel over een range representatief voor data uit de literatuur, heeft een grote invloed op de drukken en von Mises spanningen voor verschillende gebieden in de hersenen. Aan de hand van dit resultaat kan al geconcludeerd worden dat het vergelijken van resultaten gepresenteerd door verschillende auteurs, ieder met hun eigen geometrie, materiaaleigenschappen en be­lastingen, al dubieus is wanneer alleen de materiaaleigenschappen beschouwd worden. Het modelleren van verschillende substructuren van de hersenen, zoals cerebrum, cerebellum en hersenstam en van de vliesachtige structuren, zoals falx cerebri en tentorium, heeft ver­geleken met het basismodel, met name lagere drukken in het occipitale gebied tot gevolg. Het verschil in drukken net hoven en onder het tentorium vormt een indicatie voor de ondersteunende functie van het tentorium.

Concluderend kan gesteld worden dat de numerieke simulatie van impact op het hoofd met behulp van de eindige elementenmethode tot resultaten leidt die in overeenstemming zijn met experimentele gegevens uit de literatuur. De simulatie van relatieve beweging tussen verschillende structuren binnen het hoofd is belangrijk en moet in toekomstige studies een hoge prioriteit hebben. Naast de relatieve beweging lijken de materiaaleigenschappen net zo kritisch te zijn voor de simulatieresultaten. Implementatie van materiaaleigenschappen die bepaald zijn onder goed gedefinieerde impact condities, zullen het vertrouwen geven dat de numerieke resultaten correct zijn en er voor zorgen dat in de toekomst de nu­merieke resultaten van verschillende studies beter onderling te vergelijken zijn. Pas dan zullen de berekende grootheden, zoals rek-, reksnelheids-, spannings- en drukverdelingen aanknopingspunten leveren bij het onderzoek naar de identificatie van letselmechanismen.

Acknowledgments

I hereby would like to thank everyone who contributed to this research. In particular; • Fons Sauren for his support in this study and the pleasant cooperation over the

years. • Jan Janssen en Jac Wismans for their broader view of the research field. • Dave Brandts, Otwin Gunther, Peter Hempenius, Giovanni Henssen, Jack van Hoof,

Marc Huisman, John Janssen, Rob van Kessel, Peter Kessels, Carin de Kluijs, Ard Kuijpers, Thorvalt Lobel, Ron Peerlings, Marcel Radt and Paul Schwarte; the stu­dents who participated in this research.

• Colleagues from the Eindhoven University of Technology. • Robert Anderson from the Road Accident Research Unit (RARU) Adelaide, Aus­

tralia for his collaboration in the development of a 3-D finite element model of the human head.

• the TNO Crash Safety Research Institute for providing the funding to make the research possible.

• the people from Marc Analysis Research Corporation in Zoetermeer, especially Cees Gelten for his valuable assistance in the analysis and modification of the contact algorithm.

Finally I would like to thank my family and friends for their support and encouragement over the years.

Maurice Claessens

135

July 3, 1969

1981-1983

1983-1987

1987-1992

1992-1997

March, 1997-

Curriculum Vitae

Born in Eindhoven, the Netherlands

United World College of South East Asia, Singapore

Atheneum B, Anton van Duinkerken College, Veldhoven

Masters degree at the Department of Mechanical Engineering, Eindhoven University of Technology

Research assistant at the Department of Mechanical Engineering, Section of Computational and Experimental Mechanics, Eindhoven University of Technology

Engineer in the Noise and Vibration department, Audi AG, Ingolstadt, Germany

137

Stellingen behorende bij het proefschrift

Finite Element Modeling of the Human Head under Impact Conditions

1. Een volledige toetsing van numerieke simulaties van impact op het menselijk hoofd zal altijd onmogelijk blijven.

+ Dit proefschrift, hoofdstukken I en 3.

2. Ondanks de ontwikkelingen in CAD/CAE-software, blijft voor de ontwikkeling van een geometrisch realistisch 3-D eindige elementen model van het menselijk hoofd een grote mate van handwerk noodzakelijk.

+ Dit proefschrift, hoofdstuk 3.

3. Een realistische beschrijving van de interactie tussen structuren binnen bet hoofd, impliceert dat de relatieve beweging tussen deze structuren gemodelleerd dient te worden.

+ Dit proefschrift, hoofdstuk 4.

4. Modellering van contact is vanuit wiskundig oogpunt een eenduidige zaak. De imptementa­tie in eindige elementen software vindt op vele manieren plaats: eenduidigheid ontbreekt.

+Zhong, Z.-H. (1993). Finite element procedures for contact-impact problems, Oxford University Press, Oxford.

+ Dit proefschrift, hoofdstuk 2.

5. Literatuurgegevens over de materiaaleigenschappen van weefsels in bet menselijk hoofd vertonen een grote spreiding. Bovendien is vaak twijfelachtig of deze eigenschappen gelden voor de hoge deforrnatiesnelheden die optreden bij impact.

+Arbogast, K. B. and D. F. Meaney (1995). Biomechanical characterization of the constitutive relationship for the brainstem. In Proceedings of the Jrjh Stapp Car Crash Conference, SAE paper 952716, pages 153-159.

+ Thibault, K. L. and S. S. Margulies (1996). Material properties of the developing porcine brain. In Proceedings of the International IRCOBI Conference on the Biomechanics of Impact, pages 75-85.

+ Dit proefschrift, hoofdstuk 4.

6. Winstmaximalisatie op korte tennijn leidt tot technologieveranning in de toekomst. Ten gevolge van een op steeds kortere tennijn vereiste winstgevendheid van projecten binnen bet bedrijfsleven wordt op R&D budgetten steeds verder gekort, omdat de resultaten van technologisch onderzoek pas op langere tennijn zichtbaar zijn.

7. Akoestische optimalisatie van modeme personenwagens vindt voomamelijk plaats op basis van experimenteD en gevoel; numerieke technieken spelen ten onrechte nog een onderge­schikte rol.

8. Het beoefenen van sport onder werktijd kan de efficientie en samenhorigheid binnen een or­ganisatie bevorderen.

9. Aileen een rigoreuze arbeidstijdverkorting (b.v. 10 uur per week) zal zorgen voor meer ar­beidsplaatsen, daar een arbeidstijdverkorting van slechts enkele uren (b.v. 2 uur per week) snel wordt gecompenseerd door een arbeidsproduktiviteitsstijging.

lO.De regel bij squash dat de tegenstander niet op de lijn tussen de bal en de vqormuur mag staan is altijd goed voor pijnlijke discussies.

Maurice Claessens Ingolstadt, juli 1997