interlaminar fracture analysis of cross ply frp laminates using fea
TRANSCRIPT
Acknowledgement
We express deep sense of gratitude and profound regards to S Srilakshmi,
Asst. Professor, Dr. V Balakrishna Murthy, Professor, Department of Mechanical
Engineering, Prasad V Potluri Siddhratha Institute of Technology, Vijayawada, for
their keen interest, constant encouragement and invaluable guidance during the
course of the work.
We express our deep sense of gratitude to our beloved Principal, K
Srinivasu, for a kind patronage. We express our sincere thanks to Dr. K Sivaji Babu,
Head of the Department, Mechanical Engineering, for his encouragement
throughout the project.
We express our thanks to our programmer Mr G Rajendra Prasad in CAD lab,
Department of mechanical Engineering for providing the lab during the entire
course of our work.
Finally we thank others who assisted us directly or indirectly for successfully
completing this project.
Project Associates,
Prathyusha Achanta,
Kranthi Chand Musunuru,
Pranoy Yerraguntla,
Ashish Kalipatnapu.
Abstract One of the most important damage mechanisms in composite
materials is the delamination between plies of the laminate. In industrial
applications, composite plates are sensitive to impact and delamination
occurs. Many composite components have curved shapes, tapered
thickness and plies with different orientations, which make the
delamination grow depending on the extent of the crack. It is therefore
important to analyse the delamination characteristics of composite
structures.
The main objective of the present investigation is the
characterization of the delamination growth in four layered
unidirectional fibre reinforced composite laminates under variations of
crack length, stacking sequence and different load applications. The
analysis has been carried out using Virtual Crack Closure Technique
(VCCT) in combination with Finite Element Methods (FEM) theoretically
and numerically with the help of commercially available Finite Element
Software, ANSYS.
<Summary/Conclusions Briefing>
This work can be useful in analysing the effect of various factors such
as location of the crack and nature of the load on fracture response of
FRP structures, saving a great amount of time for further research in this
field.
CONTENTS
1. INTRODUCTION 1.1 INTRODUCTION 1.2. FIBER REINFORCED POLYMERS 1.3. INTRODUCTION TO FRACTURE
1.3.1 MODES OF FAILURES 1.3.2EXAMPLES OF STRUCTURAL FAILURES CAUSED BY FRACTURE• MECHANICAL, AERONAUTICAL, OR marine• CIVIL ENGINEERING
1.4. FRACTURE MECHANICS VS. STRENGTH OF MATERIALS
2. LITERATURE SURVEY 2.1 INTRODUCTION 2.2 TYPOLOGY OF FRP DELAMINATION
2.2.1 INTERLAMINAR CRACKS AND LINEAR ELASTIC FRACTURE MECHANICS 2.2.2 BASIC ANALYSIS OF INTERLAMINAR FRACTURE TOUGHNESS
2.3. FRACTURE MODES 2.3.1. MICROSCOPIC ASPECTS
2.4. MAJOR HISTORICAL DEVELOPMENTS IN FRACTURE MECHANICS 2.5. THE STUDY OF DELAMINATION USING NUMERICAL METHODS
3 PROBLEM STATEMENT AND METHODOLOGY
3.1. INTRODUCTIONNTRODUCTION 3.2 PROBLEM STATEMENT 3.3 METHODOLOGY 3.4 ASSUMPTIONS
4.ANALYSIS OF EDGE CRACK LAMINATES
4.1 INTRODUCTION 4.2 PROBLEM MODELLING
4.3 CASE ONE: PRESSURE LOADING ANALYSIS OF RESULTS: 4.3.1 ALL LONGITUDINAL FIBRES 4.3.2 ALL TRANSVERSE FIBRES 4.3.3 SYMMETRIC FIBRE ORIENTATION 4.3.4 ANTISYMMETRIC FIBRE ORIENTATION
4.4 CASE TWO: LINE LOADING ANALYSIS OF RESULTS:
5.4.1 ALL LONGITUDINAL FIBRES 5.4.2 ALL TRANSVERSE FIBRES 5.4.3 SYMMETRIC FIBRE ORIENTATION 5.4.4 ANTISYMMETRIC FIBRE ORIENTATION
5 ANALYSIS OF CENTRE CRACK LAMINATES
5.1 INTRODUCTION 5.2 PROBLEM MODELING 5.3 CASE ONE: UNIFORM PRESSURE
5.3.1 ALL LONGITUDINAL FIBRES 5.3.2 ALL TRANSVERSE FIBRES 5.3.3 SYMMETRIC FIBRE ORIENTATION 5.3.4 ANTISYMMETRIC FIBRE ORIENTATION
5.4 CASE TWO: LINE LOADING ANALYSIS OF RESULTS: 5.4.1 ALL LONGITUDINAL FIBRES 5.4.2 ALL TRANSVERSE FIBRES 5.4.3 SYMMETRIC FIBRE ORIENTATION 5.4.4 ANTISYMMETRIC FIBRE ORIENTATION
6 CONCLUSIONS AND SCOPE
ERROR! REFERENCE SOURCE NOT FOUND. SCOPE
7 REFERENCES
List of Figures
Fig.1.1 Cracked Cantilevered Beam
Fig 1.2 Failure Envelope for a Cracked Cantilevered Beam
Fig.2.1.Replica of a cross-ply laminate with an inner delamination in the
0/90 ply interface
Fig.2.2 Internal delamination: (a) disposition across the laminate and (b)
effect on the overall stability
Fig.2.3.Near-surface delamination: (a) open in tension; (b) closed in tension;
(c) open buckled; (d) closed buckled; (e) edge buckled and (f) edge buckled
with secondary crack. [5]
Fig.2.4.Elastic variation of P versus and change in energy
Fig.2.5.Load-displacement curve for stable crack growth in (a) brittle matrix
and (b) tough matrix
Fig.2.6.Load-displacement curve for unstable crack growth
Fig.2.7.Crack propagation modes: (a) mode I; (b) mode II and (c) mode III
Fig.2.8.Stress field of a resin rich area point and matrix micro crack
formation ahead of the crack tip
Fig.2.9 Crack growth at (a) θ/0 and (b) 0/0 ply interfaces illustrating the
crack plane migration mechanism
Fig.2.10. Fibre bridging in a mode I inter-laminar crack. The vertical arrow
indicates a point 20 mm behind the crack tip
Fig.2.11.Formation and growth of a mode II delamination at the ply
interface: (a)micro crack formation ahead of the crack tip; (b) micro crack
growth and opening and (c) micro crack coalescence accompanied by shear
cusps
Fig.2.12.Development of shear micro cracks in mode II non-reversed and
reversed Delamination
Fig. 4.1 Geometry and Loaded model for edge crack at centre interface.
Fig. 4.2 Geometry of a 20 Node SOLID95 Element
Fig 4.3 Deformed model after cylindrical bending due to uniform pressure
Fig 4.4: Variation of ‘G’ with respect to ‘a’ from an edge crack with 0-0-0-0
laminate.
Fig 4.5: Variation of ‘G’ with respect to ‘a’ from an edge crack with 90-90-90-
90 laminate.
Fig 4.6: Variation of ‘G’ with respect to ‘a’ from an edge crack with 0-90-90-0
laminate.
Fig 4.7: Variation of ‘G’ with respect to ‘a’ from an edge crack with 90-0-0-90
laminate.
Fig 4.8: Variation of ‘G’ with respect to ‘a’ from an edge crack with 0-90-0-90
laminate.
Fig 4.9: Variation of ‘G’ with respect to ‘a’ from an edge crack with 90-0-90-0
laminate.
Fig: 4.10 Loaded model for edge crack opening at centre interface
Fig. 4.11 Deformed model for edge crack opening at centre interface
Fig 4.12 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 0-
0-0-0 laminate
Fig 4.13 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with
90-90-90-90 laminate
Fig 4.14 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 0-
90-90-0 laminate.
Fig 4.15 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with
90-0-0-90 laminate.
Fig 4.16 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 0-
90-0-90 laminate
Fig 4.17 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with
90-0-0-90 laminate.
Fig. 5.1 Geometric model for centre crack at centre interface.
Fig. 5.2 Geometry of a 20 Node SOLID95 Element
Fig. 5.3 Loaded model for centre crack at centre interface
Fig 5.4 Deformed model after laminate bending due to uniform pressure
load
Fig 5.5: Variation of ‘G’ with respect to ‘a’ from an edge crack with 0-0-0-0
laminate.
Fig 5.6: Variation of ‘G’ with respect to ‘a’ from an edge crack with 90-90-90-
90 laminate.
Fig 5.7: Variation of ‘G’ with respect to ‘a’ from an edge crack with 0-90-90-0
laminate.
Fig 5.8: Variation of ‘G’ with respect to ‘a’ from an edge crack with 90-0-0-90
laminate.
Fig 5.9: Variation of ‘G’ with respect to ‘a’ from an edge crack with 0-90-0-90
laminate
Fig 5.10: Variation of ‘G’ with respect to ‘a’ from an edge crack with 90-0-90-
0 laminate.
Fig. 5.11 Loaded model for edge crack at centre interface
Fig. 5.12 Deformed model for centre crack opening at centre interface
Fig 5.13 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 0-
0-0-0 laminate.
Fig 5.14 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with
90-90-90-90 laminate.
Fig 5.15 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 0-
90-90-0 laminate.
Fig 5.16 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with
90-0-0-90 laminate.
Fig 5.17 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 0-
90-0-90 laminate.
Fig 5.18 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with90-
0-90-0 laminate
NOMENCLATURE
E1 Young`s Modulus In longitudinal direction of the lamina
E2 Young`s Modulus in the in-plane transverse direction of the lamina
E3 Young`s Modulus in the out-of-plane transverse direction of the lamina
G12 Shear Modulus in 1-2 longitudinal plane of the lamina
G23 Shear Modulus in 1-3longitudinal plane of the lamina
G13 Shear Modulus in transverse plane of the lamina
Ѵ12 Major in- plane Poisson’s ratio
Ѵ23 Out- of- plane Poisson’s ratio
Ѵ13 Out-of-plane Poisson’s ratio
KI Stress intensity factor
KIc Critical stress intensity factor
VCCT Virtual crack closure technique
SERR Strain energy release rate
GI Strain energy release rate
GIC Critical strain energy release rate
1
INTRODUCTION
1.1 INTRODUCTION
The use of advanced composite materials has increased considerably in
the fabrication of structural elements. Advanced composite materials
progressively substitute traditional materials, such as steel, aluminium or
wood, due to their better specific properties. The excellent stiffness to weight
and strength to weight ratios of polymeric matrix composite materials, particularly
those reinforced with glass or carbon fibres make them very attractive for certain
manufacturing sectors. Initially, this type of materials was exclusively used in
technologically advanced applications, such as in aeronautical and aerospace
industries. Nowadays, due to technological development and reduction of
manufacturing costs, these materials are being used more and more in different
applications. These applications range from sportive items, biomedical
implants, car parts, fluid containers and pipes, small boats and road bridges
to advanced aircraft and space vehicles. It can be seen the important
increment in the use of this type of material by the general industrial sector.
Nevertheless, the especial characteristics of the design of a composite
part have limited a wider generalization of these materials. Designing a new
composite element not only requires the design of the element geometry, but the
design of the material itself. Traditionally, due to the reduced knowledge of the
behaviour of composite materials, this process was accomplished using methods
based on available empirical data. However, this methodology is limited to the
characterization of definite materials and stacking sequences, meanwhile the
number of material combinations is nearly unlimited. The experimental
characterization of the material is expensive and difficult to be extended to other
material configurations. Until a better knowledge about composite materials
behaviour and properties was achieved, the dependence on experimentation
limited, in part, a higher use of composites in more common applications.
1.2. FIBER REINFORCED POLYMERS
The oldest composite materials appeared long time ago in the nature.
Wood can be seen as a lignin matrix reinforced by cellulose fibres. Human and
animal bones can be described as fiber-like osteons embedded in an interstitial
bone matrix. The first man made composite was straw-reinforced clay for bricks
and pottery. Present composite materials use metal, ceramic or polymer binders
reinforced with different fibres or particles. Then, composite materials can be
defined as those materials resulting from the combination of two or more
materials (known as components or constituents), different in composition, form
or function at a macroscopic scale. In the resulting composite material, the
components conserve their initial identity without dissolving or mixing
completely. Usually, the components can be physically distinguished and it is
possible to identify the interface between components. Taking into account their
structural properties, composite materials can be defined as those materials having
a reinforcement component (fibre or particles) in an agglutinating component
(matrix). Reinforcement is responsible for the composite high structural properties;
meanwhile matrix gives physical support and ambient integrity.
With the combination of different matrices (usually polymeric matrices
or light metals) with different fibres (glass, carbon, organic and polymeric
fibres, among others), it is possible to obtain composite materials with
different mechanical properties specially designed for certain applications.
Thus, the great number of combinations results in a great number of composites.
They can be distinguished in function of their typology (long o short fibres, random
or oriented, single or multiple plies, etc.) or in function of their components
(thermoplastic or thermo set polymer matrix, aluminium or titanium metal
matrix, carbon matrix, inorganic or organic fibres, metal whiskers, etc.).
Usually, polymeric matrix fibre reinforced composite materials are found in
the shape of unidirectional laminates (all the reinforcement fibres in one
direction) or as bidirectional laminates (many unidirectional plies with different
fibre orientations). Different types of reinforcement can be used to improve the
properties of the resulting composite, which is then known as hybrid. This is the
case of reinforced concrete, a particle-reinforced composite (concrete) further
reinforced with steel rods. When a light core material is sandwiched between two
faces of stiff and strong materials, the result is an improved material called
sandwich.
1.3. INTRODUCTION TO FRACTURE
In this introductory stage of fracture, we shall start by reviewing the
various modes of structural failure and highlight the importance of fracture induced
failure and contrast it with the limited coverage given to fracture mechanics in
Engineering Education. In the next section we will discuss some examples of well
known failures/accidents attributed to cracking. Then, using a simple example we
shall compare the failure load predicted from linear elastic fracture mechanics with
the one predicted by “classical” strength of materials.
1.3.1 MODES OF FAILURES
The fundamental requirement of any structure is that it should be designed
to resist mechanical failure through any (or a combination of) the following modes:
1. Elastic instability (buckling)
2. Large elastic deformation (jamming)
3. Gross plastic deformation (yielding)
4. Tensile instability (necking)
5. Fracture
Most of these failure modes are relatively well understood, and proper
design procedures have been developed to resist them. However, fractures
occurring after earthquakes constitute the major source of structural damage, and
are the least well understood. In fact, fracture often has been overlooked as a
potential mode of failure at the expense of an overemphasis on strength. Such a
simplification is not new, and finds a very similar analogy in the critical load of a
column. If column strength is based entirely on a strength criterion, an unsafe
design may result as instability (or buckling) is overlooked for slender members.
Thus failure curves for columns show a smooth transition in the failure mode from
columns based on gross section yielding to columns based on instability. By
analogy, a cracked structure can be designed on the sole basis of strength as long as
the crack size does not exceed a critical value. Should the crack size exceed this
critical value, then a fracture-based failure results. Again, on the basis of those two
theories (strength of materials and fracture mechanics), one could draw a failure
curve that exhibits a smooth transition between those two modes.
1.3.2EXAMPLES OF STRUCTURAL FAILURES CAUSED BY FRACTURE
Some well-known, and classical, examples of fracture failures include:
• Mechanical, aeronautical, or marine
1. Fracture of train wheels, axles, and rails
2. Fracture of the Liberty ships during and after World War II
3. Fracture of airplanes, such as the Comet airliners, which exploded in mid-
air during the fifties, or more recently fatigue fracture of bulkhead in a Japan
Air Line Boeing 747
4. Fatigue fractures found in the Grumman buses in New York City, which
resulted in the recall of 637 of them
5. Fracture of the Glomar Java sea boat in 1984
6. Fatigue crack that triggered the sudden loss of the upper cockpit in the Air
Aloha plane in Hawaii in 1988
• Civil engineering
1. Fractures of bridge girders (Silver Bridge in Ohio)
2. Fracture of Stratford A platform concrete off-shore structure
3. Cracks in nuclear reactor piping systems
4. Fractures found in dams (usually unpublicized)
Despite the usually well-known detrimental effects of fractures, in many
cases fractures are man-made and induced for beneficial purposes examples
include:
1. Rock cutting in mining.
2. Hydro-fracturing for oil, gas, and geothermal energy recovery.
3. “Biting” of candies.
1.4. FRACTURE MECHANICS VS. STRENGTH OF MATERIALS
In order to highlight the fundamental differences between strength of
materials and fracture mechanics approaches, we consider a simple problem, a
cantilevered beam of length L, width B, height H, and subjected to a point load P at
its free end is given by
Fig.1.1 Cracked Cantilevered Beam
We will seek to determine its safe load-carrying capacity using the two
approaches.
1. Based on classical strength of materials the maximum flexural stress should
not exceed the yield stress σy, or
2. In applying a different approach, one based on fracture mechanics, the
structure cannot be assumed to be defect free. Rather, an initial crack must
be assumed. Governed failure; for the strength of materials approach in the
linear elastic fracture mechanics approach (as discussed in the next
chapter), failure is governed by:
KI≤ KIc
Where K Iis a measure of the stress singularity at the tip of the crack and KIc is the
critical value of K I. K I is related to σmax through:
The two equations governing the load capacity of the beam according to
Two different approaches, call for the following remarks:
1. Both equations are in terms of BH2 /6L
2. The strength of materials approach equation is a function of a material
property that is not size dependent.
3. The fracture mechanics approach is not only a function of an intrinsic
material property but also of crack size a.
On the basis of the above, we can schematically represent the failure envelope of
this beam in fig.where failure stress is clearly a function of the crack length.
Fig 1.2 Failure Envelope for a Cracked Cantilevered Beam
On the basis of this simple example, we can generalize our preliminary finding by
the curve shown in Fig. We thus identify four corners: on the lower left we have our
usual engineering design zone, where factors of safety are relatively high; on the
bottom right we have failure governed by yielding, or plasticity; on the upper left
failure is governed by linear elastic fracture mechanics; and on the upper right
failure is triggered by a combination of fracture mechanics and plasticity. This last
zone has been called elasto-plastic in metals, and nonlinear fracture in concrete.
2
LITERATURE SURVEY
2.1 INTRODUCTION
In the previous chapter, it has been stated that the main objective of the
present work is the study of inter-laminar fracture behaviour of four layered cross-
ply unidirectional continuous fibre reinforced composite laminates However, before
analysing the behaviour of inter-laminar cracks, it is necessary to address the
principles of the delamination mechanics, the onset and propagation of inter-
laminar cracks under static conditions, the interaction of delamination with
other micromechanics of composite laminates, etc. In this way, a better
understanding of the phenomenon can be achieved.
This chapter includes a sort of review on basic aspects of
composite eliminations. The review starts taking into account a classification of
inter-laminar cracks and continues with the application of fracture mechanics basic
concepts. Next, an overview on microscopic aspects of composite delamination is
considered. The historical approaches to the characterisation of this phenomenon
under static conditions are also presented and briefly discussed
2.2 TYPOLOGY OF FRP DELAMINATION
Crack formation between two adjacent plies, or delamination, is a damage
mechanism of composite laminates that can form during any moment of the life of
the structure: manufacturing, transport, mounting and service. According to [1]
and [2] the technological causes of the delamination can be grouped in two
categories. The first category includes delamination due to curved sections,
such as curved segments, tubular sections, cylinders and spheres, and
pressurised containers. In all these cases, the normal stresses in the interface of
two adjacent plies can originate the loss of adhesion and the initiation of the inter-
laminar crack. The second category includes abrupt changes of section, such as ply
drop-offs, unions between stiffeners and thin plates, free edges, and other
bonded and bolted joints. A third category related to temperature and moisture
effects can be added. The difference between the thermal coefficients of matrix
and reinforcement results in differential contractions between plies during the
curing process. The residual stresses originated by these differential contractions
may originate delamination [3]. Similarly, the differential inflation of the plies
during the absorption of moisture might be the cause of delamination [4].
Delamination can be also originated during the manufacturing stage
due to the shrinkage of the matrix, formation of resin-rich areas due to poor
quality in lying the plies, etc. [5-6]. Impact is an important source of delamination
in composite structures. Inter-laminar cracks can be originated by internal damage
in the interface between adjacent plies as a consequence of an impact in the
laminate, due to the drop of a tool during production, mounting or repairing, or
ballistics impacts in military planes or structures.
Location within the stacking sequence of the laminate has an important
effect on the growth of delamination [7]. According to [5-6], two types of
delamination can be considered: internal delamination and near-surface
delamination. Internal delamination originate in the inner ply interfaces of the
laminate and can be due to the interaction of matrix cracks and ply interfaces.
Delamination originated in the 0/90 interfaces by transversal matrix cracks in
the 90º plies of cross-ply laminates are common examples of this type of
delamination. Figure 2.1 shows a replica with an inner delamination growing from a
transverse crack to the left in a 0/90 interface of a carbon/epoxy cross-ply
laminate subjected to axial load [8]. In the replica, some fibre breaks in the 0º ply
can be seen due to the stress concentration near the transverse crack.
Fig.2.1.Replica of a cross-ply laminate with an inner delamination in the
0/90 ply interface [8]
Inner delaminations considerably reduce the load-capacity of composite
elements. In particular, when compression loads are applied, the overall flexural
behaviour of the laminate is significantly affected (as shown schematically in Figure
2.2). Although the delamination separates the laminate in two parts, there is an
interaction between the deformation of the one part of the laminate and the
other. Due to this interaction, both parts of the laminate deflect in a similar way.
Fig.2.2 Internal delamination: (a) disposition across the laminate and (b)
effect on the overall stability
Near-surface delaminations, as its name indicates, originate near the surface
of the laminate and represent a more complex scenario than internal
delaminations. The deformation of the delaminated part is less influenced by the
deformation of the rest of the laminate. Therefore, the deformation of the near-
surface delaminated part does not necessary follow the deformation of the rest
of the laminate. Consequently, not only the growth of the near-surface
delamination has to be taken into account but also its local stability. [5-6] classified
the different types of near-surface delaminations than can originate in plate
composite components in different load conditions. Figure 2.3 shows different
types of near-surface delaminations.
Fig.2.3.Near-surface delaminations: (a) open in tension; (b) closed in
tension; (c) open buckled; (d) closed buckled; (e) edge buckled and (f)
edge buckled with secondary crack. [5]
After initiation, both types of delaminations can propagate either under
static loads either under fatigue conditions. In both cases, the reduction in strength
and stability of the composite part to flexural loading is considerable.
2.2.1 INTERLAMINAR CRACKS AND LINEAR ELASTIC FRACTURE MECHANICS
Fracture mechanics is concerned with crack-dominated failures and
delamination is a fracture mechanism of composite laminates. Therefore, fracture
mechanics is a suitable methodology to approach the onset and propagation
of composite delaminations problem. In addition, usual composite laminates
are very stiff in the laminate plane and behave as linear elastic materials in
their gross deformation. Thus, it is reasonable to base the analysis of interlaminar
toughness on linear elastic-fracture mechanics (LEFM).
2.2.2 BASIC ANALYSIS OF INTERLAMINAR FRACTURE TOUGHNESS
Nowadays, composite materials are tailored in order to profit their
high in-plane tensile strength. However, the through-thickness properties of such
materials are in most cases very low compared to the in-plane tensile strength.
Therefore, the through-thickness stresses in laminated composite may initiate
delamination, especially if particular geometries (free edges, holes, ply drops
etc.) or previous damage (matrix cracks or micro-delaminations as a consequence
of impacts, fabrication problems, etc.) are present in the material [9]. After
delamination onset, the consequent propagation is not controlled by the through-
thickness strength any more but by the interlaminar fracture toughness.
If the interlaminar toughness is expressed in terms of energy
release rate, the delamination will propagate when the energy release rate
achieves a critical value, Gc. According to [10], for any form of elastic behaviour,
the energy release rate can be expressed as a function of the increment of
external work Ue, strain energy Us (kinetic energy is ignored in this case) and
crack increment ∆a. Therefore, for a crack of width b and length a, the
energy release rate can be expressed as
Figure 2.4 presents an elastic variation of the load P versus the
displacement for an interlaminar crack growing from an initial length a to a final
length a+∆a. In point A1 the applied load is P1, the displacement is 1 and
delamination length a. In point A2 the applied load and displacement are P2 and 2,
respectively, and the crack length is a+∆a.
Fig.2.4.Elastic variation of P versus and change in energy
Therefore, the external work and strain energy of the linear variation
shown in Figure 2.4 can be expressed, respectively, as
The change in energy is determined by the area OA1A2 (dashed area in the
figure). If linear deformation behaviour is assumed, the straight lines showed in the
figure are to be used and the change in energy becomes
For the considered crack increment and width, the increment in crack area
would be b∆a. Thus, as the critical energy release rate can be defined as the change
in energy per unit of new crack surface and denoting P1 as P, P2 as P+∆P, δ1 as δ
and δ2 as δ+∆δ, the expression for Gc can be written as
The compliance of the system depends on the crack length and is defined
as
Taking into account the increments of load and displacement and equation,
the increment in displacement can be expressed as
and combining equations a final expression for the critical energy
release rate can be found after mathematical manipulation as:
or in differential form as
For the experimental study of interlaminar crack propagation in composite
materials, the variation of the applied load with respect the obtained displacement,
as shown in Figure 2.4, is basic. This experimental data, together with the crack
length, is the basis for the calculation of G and the generation of the R-
curve. However, the experimental determination of the onset and propagation
values of G for an interlaminar crack is complicated and different methods
can be used. The first method is based in the determination of Gc by visual
observation of the crack onset.
Nevertheless, this method is imprecise and highly dependent on the
observer. The second method is based on the calculation of Gc at the point
of non-linearity of the load-displacement curve. According to [9], for brittle
matrix composites the non-linearity point coincides with the point at which
the initiation of the crack can be observed (see Figure 2.5(a)). However, for tough
matrices a region of non-linear behaviour may precede the observation of the crack
initiation (see Figure 2.5(b)). In the third method, Gc is determined as the
intersection between the load-displacement curve and the line that corresponds to
an increase by the 5 % to the original compliance of the system. If the maximum
load occurs before intersection, then the maximum load and corresponding
displacement are used to compute Gc.
Fig.2.5.Load-displacement curve for stable crack growth in (a) brittle
matrix and (b) tough matrix
The load-displacement curves represented in Figure 2.5 are for stable
crack growth cases. Unstable crack growth is characterised by one or more periods
without crack propagation (or very slow) followed by rapid propagations,
which results in sharp drops in the load-displacement curve. These rapid
propagations are normally followed by arrest and a reloading, which results in
a local peak load when delamination growth restarts. This behaviour is usually
known as stick-slip growth and results in typical saw-teeth load-displacement
curves [11]. Figure 2.6 shows a typical load-displacement curve for the case of
unstable interlaminar crack growth
Fig.2.6.Load-displacement curve for unstable crack growth
2.3. FRACTURE MODES
According to fracture mechanics, the growth or propagation of an
interlaminar crack, or delamination, may occur in mode I (opening), mode II
(shearing), mode III (tearing) and in any combination of these (see Figure 2.7). Every
mode has a fracture toughness value and an R-curve associated which are intrinsic
material characteristics. In the case of isotropic materials, only mode I toughness is
considered. For these materials, the fracture toughness is lowest in this mode
and even if the crack starts to grow under a different mode, the crack will
deviate and grow in mode I [9].
Fig.2.7.Crack propagation modes: (a) mode I; (b) mode II and (c) mode III
The propagation of delaminations in laminated composite materials is
mainly limited to lie between the strong fibre reinforced layers. In this way, it is
possible for a delamination to propagate in any combination of the three
propagation modes. A clear example is the case of transverse matrix cracks growing
in the 90º plies of cross-ply laminates loaded in tension (see Figure 2.1). Once the
crack reaches the strong fibres at the (0/90) interface, the crack is forced to deviate
and change direction in order to remain in the interface. Then, the propagation
mode is changed. In fact, composite delaminations are mostly studied under pure
mode I, pure mode II and mixed-mode I/II. It is generally accepted that the mode III
contribution in delamination growth is negligible. In fact, the mode III
contribution is typically quite small for composite structures as a consequence
of the constraints of adjacent plies, as shown by [12] for a layered structure and
by [13] in laminated lap-joints. In addition, the fracture toughness values for
delamination in composite laminates are higher in mode III than in the other
modes [9]. In the foregoing the term mixed-mode will stand for the mixed-
mode I/II condition.
In isotropic materials, toughness values are commonly expressed in
terms of the critical stress intensity factor. However, interlaminar fracture
toughness of laminated composites is normally expressed in terms of the
critical energy release rate. The stress intensity factor is governed by the local
crack-tip field and is extremely sensitive. It is difficult to obtain true values of
K at the crack tip due to the inhomogeneous composition of composite
laminates complicates. [2] state that the use of G for composite materials is
certainly more consistent with the analytical models in use than K, even
though the K-mix can be defined rigorously, in contrast to G. Therefore, the
majority of the studies about delaminations in composites use the critical
energy release rate, Gc, instead of the critical stress intensity factor, Kc, to
predict the initiation of the crack.
2.3.1. MICROSCOPIC ASPECTS
At the microscopic level, the growth of an interlaminar crack is
preceded by the formation of a damage zone ahead of the crack tip. This damage
zone is characterised by the formation of microcracks in the resin rich areas that
exist between the plies. According to [14], at this microscopic level, the matrix can
be seen as an isotropic and homogeneous material, which in general, like metals,
will only crack under tensile load conditions (local mode I). Therefore, matrix
microcracks will form and grow in the plane subjected to maximum tensile
stress. Figure 2.8 shows schematically a point of a resin rich area in the ply
interface subjected to mode I (opening) and mode II (shearing) loading and the
formation of a matrix microcrack in the plane subjected to the resulting maximum
tensile stress, σm.
Fig.2.8.Stress field of a resin rich area point and matrix microcrack
formation ahead of the crack tip
Under mixed-mode load condition, microcracks ahead of the crack tip form
at an angle from the plane of the plies and grow in this direction. According to [15],
when such a microcrack is located at 0/0 ply interface, where 0 stands for an off-
axis ply, fibres on the off-axis ply allow the propagation of the microcrack through
the ply. Consequently, the crack tip of the delamination migrates through the off-
axis ply. A change in the crack plane can be achieved if the crack tip
encounters the next ply interface. In this case, the study and characterisation
of the delamination become complicated. The crack plane migration mechanism is
represented in Figure 2.9(a). Oppositely, when the microcrack is located at 0/0
ply interface, the fibres at both 0º plies prevent the propagation of the crack
through the plies. The interlaminar crack is forced to remain adjacent to the fibres
of the ply. The mechanism is represented in Figure 2.9(b). In this case, no change in
the crack plane is present and the study and characterisation of the delamination
become easier. Actually, in order to avoid the crack plane migration, the study of
delaminations in composite laminates is usually carried out using unidirectional
laminates with the fibres parallel to the crack growth.
Fig.2.9 Crack growth at (a) θ/0 and (b) 0/0 ply interfaces illustrating the
crack plane migration mechanism [15]
As mentioned, since further growing would require fibre fracture, at 0/0 ply
interfaces the growth of the microcracks is arrested when they reach the
fibres of one of the boundaries of the interlaminar zone. In the general case of
mixed-mode loading, the propagation of the interlaminar cracks results from
the coalescence of these matrix microcracks [16]. For a greater contribution of
mode I, the matrix microcracks grow relatively parallel to the plane of the
plies but progressively displacing to one of the boundaries of the interply
zone. Consequently, the interlaminar crack progressively grows to one of the
interply boundaries, where the presence of the fibres modifies the damage zone
ahead of the crack tip and increases the stress concentration. This results in the
growth of the delamination by the peeling of the matrix from the fibres. According
to [14], this process justifies the presence of fibres in one of the fracture surfaces
while on the other only the fibre imprints are present.
However, the general scenario in delamination test specimens is different.
The presence of fibres bridging both fracture surfaces near the crack tip is
commonly observed. This phenomenon is known as fibre bridging and tends to
arrest or reduce the propagation of the delamination. In fact, the growth of the
crack involves pulling these bridging fibres from the resin under a tensile stress
state until they finally break. Accordingly, an artificial increment of the material
fracture toughness that depends on the crack extension is observed. For longer
crack lengths, more fibres from both fracture surfaces are bridging the crack. It has
been experimentally found that this effect is more important for higher mode I
contributions and less important for higher mode II dominated fractures [15],[17].
In this case, fibre breakage, broken pullout fibres, behind the crack tip can
be observed. According to [18], fibre bridging is a characteristic
micromechanism of unidirectional ply testing that will not occur in real
structures. Figure 2.10 shows an interlaminar crack with the presence of fibre
bridging and fibre breakage in a glass-fibre/vinyl ester composite [19].
Fig.2.10. Fibre bridging in a mode I interlaminar crack. The vertical arrow indicates a
point 20 mm behind the crack tip [19]
For a greater contribution of mode II the size of the damage zone
increases and matrix microcracks start to form at a relatively considerable
distance ahead of the crack tip. In addition, the angle between the direction
of the microcracks and the plane of the plies increases up to 45º. The coalescence
of the microcracks results in the growth of the interlaminar crack but with uneven
surfaces. These uneven surfaces are due to the formation of shear cusps or hackles.
For a greater contribution of mode II, more shear cusps form and deeper they
are. In addition, less influence of fibre bridging is observed [20]. The increased
area of the uneven fracture surfaces at microscopic level in mode II justifies
the increases of the measured fracture toughness for this mode, since more
atomic bonds have to be broken [14]. Figure 2.11 shows schematically the
formation and coalescence of mode II microcracks that result in the formation and
growth of a mode II delamination accompanied by some shears cusps. If these
surfaces are subsequently subjected to a fatigue process in mode II, the shear
cusps will degrade into matrix rollers due to the effect of the friction between
shear cusps of both surfaces.
Fig.2.11.Formation and growth of a mode II delamination at the ply
interface: (a)microcrack formation ahead of the crack tip; (b) microcrack
growth and opening and (c) microcrack coalescence accompanied by shear
cusps
The previous figure shows the formation of the damage zone ahead of
the crack tip for mode II delamination for a non-reversed loading condition. If a
reversed loading condition is considered, a second block of microcracks appear in
the normal direction to the previous, this is at -45º. Therefore, two sets of
microcracks form at approximately 90º [21]. Figure 2.12 shows the schema of the
microcrack formation ahead of the crack tip for mode II reversed and non-
reversed loading conditions.
Fig.2.12.Development of shear microcracks in mode II non-reversed and
reversed Delaminations
2.4. MAJOR HISTORICAL DEVELOPMENTS IN FRACTURE MECHANICS
As with any engineering discipline approached for the first time, it is helpful
to put fracture mechanics into perspective by first listing its major developments:
1. In 1898, a German Engineer by the name of Kirsch showed that a stress
concentration factor of 3 was found to exist around a circular hole in an infinite
plate subjected to uniform tensile stresses [22].
2. While investigating the unexpected failure of naval ships in 1913, [23] extended
the solution for stresses around a circular hole in an infinite plate to the more
general case of an ellipse. It should be noted that this problem was solved 3 years
earlier by Kolosoff (who was the mentor of Muschelisvili) in St Petersbourg,
however history remembers only Inglis who showed that a stress concentration
factor of
Prevails around the ellipse (where a is the half length of the major axis, and ρ is the
radius of curvature) 5.
3. Inglis’s early work was followed by the classical studies of Griffith, who was not
originally interested in the strength of cracked structures (fracture mechanics was
not yet a discipline), but rather in the tensile strength of crystalline solids and its
relation to the theory based on their lattice properties, which is approximately
equal to E/10 where E is the Young’s Modulus *24+. With his assistant Lockspeiser,
Griffith was then working at the Royal Aircraft Establishment (RAE) at Farnborough,
England (which had a tradition of tolerance for original and eccentric young
researchers), and was testing the strength of glass rods of different diameters at
different temperatures [25]. They found that the strength increased rapidly as the
size decreased. Asymptotic values of 1,600 and 25 Ksi were found for infinitesimally
small and bulk size specimens, respectively. On the basis of those two observations,
Griffith’s first major contribution to fracture mechanics was to suggest that internal
minute flaws acted as stress raisers in solids, thus strongly affecting their tensile
strengths. Thus, in reviewing Inglis’s early work, Griffith determined that the
presence of minute elliptical flaws were responsible in dramatically reducing the
glass strength from the theoretical value to the actually measured value.
4. The second major contribution made by Griffith was in deriving a thermo
dynamical criterion
for fracture by considering the total change in energy taking place during cracking.
During crack extension, potential energy (both external work and internal strain
energy) is released and “transferred” to form surface energy. Unfortunately, one
night Lockspeiser forgot to turn off the gas torch used for glass melting, resulting in
a fire. Following an investigation, (RAE) decided that Griffith should stop wasting his
time, and he was transferred to the engine department.
5. After Griffith’s work, the subject of fracture mechanics was relatively dormant for
about 20 years until 1939 when Westergaard [26] derived an expression for the
stress field near a sharp crack tip.
6. Up to this point, fracture mechanics was still a relatively obscure and esoteric
science. However, more than any other single factor, the large number of sudden
and catastrophic fractures that occurred in ships during and following World War II
gave the impetus for the development of fracture mechanics. Of approximately
5,000 welded ships constructed during the war, over 1,000 suffered structural
damage, with 150 of these being seriously damaged, and 10 fractured into two
parts. After the war, George Irwin, who was at the U.S. Naval Research Laboratory,
made use of Griffith’s idea, and thus set the foundations of fracture mechanics. He
made three major contributions: (a) He (and independently Orowan) extended the
Griffith’s original theory to metals by accounting for yielding at the crack tip. This
resulted in what is sometimes called the modified Griffith’s theory. (b) He altered
Westergaard’s general solution by introducing the concept of the stress intensity
factor (SIF). (c) He introduced the concept of energy release rate G
7. Subcritical crack growth was subsequently studied. This form of crack
propagation is driven by either applying repeated loading (fatigue) to a crack, or
surround it by a corrosive environment. In either case the original crack length, and
loading condition, taken separately, are below their critical value. Paris in 1961
proposed the first empirical equation relating the range of the stress intensity
factor to the rate of crack growth.
8. Non-linear considerations were further addressed by Wells, who around 1963
utilized the crack opening displacement (COD) as the parameter to characterize the
strength of a crack in an elasto-plastic solid, and by Rice, who introduce his J
integral in 1968 in probably the second most referenced paper in the field (after
Griffith); it introduced a path independent contour line integral that is the rate of
change of the potential energy for an elastic non-linear solid during a unit crack
extension.
9. Another major contribution was made by Erdogan and Sih in the mid ’60s when
they introduced the first model for mixed-mode crack propagation.
10. Other major advances have been made subsequently in a number of
subdisciplines of fracture mechanics: (i) dynamic crack growth; (ii) fracture of
laminates and composites; (iii) numerical techniques; (iv) design philosophies; and
others.
11. In 1976, [27] introduced the fictitious crack model in which residual tensile
stresses can be transmitted across a portion of the crack. Thus a new meaning was
given to cracks in cementitious materials.
12. In 1979 [28] showed that for the objective analysis of cracked concrete
structure, fracture mechanics concepts must be used, and that classical strength of
materials models would yield results that are mesh sensitive.
2.5. THE STUDY OF DELAMINATION USING NUMERICAL METHODS
Nowadays, many practical problems in delamination of composite
materials are solved using finite element methods (FEM) and other numerical
methods. Numerical methods were applied to the study of interlaminar cracks
shortly after the first studies about delamination in composites. [29] used a
numerical elasticity solution based on finite differences for the analysis of free
edge delaminations. [30] studied the same problem using a 3-D finite
element analysis combined with Maxwell stress functions and minimisation of
complementary energy. [31] developed a 2-D finite element model in which
the distribution of the axial displacements was taken into account. Until this work,
FEM solutions were only available under plane stress or strain conditions. [32]
were the first to obtain sufficiently accurate results of the free edge problem using
a 2-D finite element model. [33] were the first to report the calculation of the
stress singularities in the free edge of the laminate. This work, based on
Lekhnitskii’s stress potentials, was an important contribution and the basis for
other analytical models in the field of composite delaminations. This work was
followed by [34] with a solution to the free edge problem for a ±45 angle ply
laminate.
One of the numerical approaches more commonly used nowadays is the
virtual crack closure technique (VCCT). This technique is based on Irwin’s crack
closure integral [35-37] and assumes that the energy ∆E released when the crack
is extended by an increment ∆a, from a to a + ∆a, coincides with the energy
required to close the crack to its original condition, from a + ∆a to a. At present,
this technique is considered one of the most rigorous techniques for the analysis of
the propagation of interlaminar cracks [3], [38-40].
A current approach to model composite delaminations is based in
homogenisation methods [41-42]. These methods, also known as double scale
methods, are based on finite element methods and solve a submodel for
every integration point of the model. It is in the submodel where the mechanical
behaviour and damage characteristics of the material are defined.
3
PROBLEM STATEMENT AND METHODOLOGY
3.1. INTRODUCTION
In the previous chapter, the relevant literature available has been reviewed
and scope for the present work has been identified. In this chapter, statement of
the problem of present work and method used for solution of the problem has
been explained.
3.2 PROBLEM STATEMENT
The objective of the present work is to analyse the inter-laminar fracture
behaviour of four layered cross-ply unidirectional continuous fibre reinforced
composite laminates with delamination at two different locations (edge crack and
centre crack) using virtual crack closure technique (VCCT) in combination with finite
element method.
3.3 METHODOLOGY
Three dimensional finite element models are generated in ANSYS software
to represent four layered symmetric cross-ply laminate. The inter-laminar crack is
modelled as a longitudinal discontinuity with different nodes attached to the top
and bottom crack surfaces. The nodes at both discontinuity (crack) sides have the
same coordinates and are coupled through multi-point constraints. This multi-
point constraint type ties all the degrees of freedom at the tied node to the
corresponding degrees of freedom at the retained node.
The virtual crack closure technique, assumes that the energy ∆E
released when the crack is extended by an increment ∆a, from a to a + ∆a,
coincides with the energy required to close the crack to its original condition, from
a + ∆a to a. The energy release rate is determined as the work done by the
reaction forces developed due to the multi-point constraints at the crack tip per
unit area of virtual cracked surface for the considered loading condition.
The details of the finite element modelling, loading, boundary conditions,
material properties and crack length are explained in forth coming chapters.
3.4 ASSUMPTIONS
The assumptions in the present work are as follows
1. Possibility of branch cracks is ignored
2. State of plane strain is assumed.
3. Each layer of the laminate is a unidirectional continuous fibre reinforced
lamina
4. Each lamina behaves as a transversely isotropic material
5. The behaviour of the structure is geometrically linear
4
ANALYSIS OF EDGE CRACK LAMINATES
4.1 INTRODUCTION This chapter presents the analysis of edge crack laminates supported along two
opposite edges and subjected to two different load cases. In first case, a uniform pressure
is applied on the top surface of the laminate. And in second case, a uniform line load is
applied along the top edge of the laminate parallel to the supported end.
4.2 PROBLEM MODELLING In this section a four layered laminate structure is modelled with the longitudinal
dimension of the plate taken as 100 mm span (L) with a span/depth ratio of 10. Four layers
of equal thickness (10/4=2.5mm) are considered. The width of the plate is considered to be
infinite. The crack location is at one end of the laminate. The length of the crack is varied
from 15mm to 85mm. By trial and error method, comparison of theoretical and analytical
Energy release rate for isotropic material, the crack extension is found to be 0.11mm. . The
same value is extended to orthotropic material.
Fig. 4.1 Geometry and Loaded model for edge crack at centre interface.
Finite element mesh is generated using 20 node quadratic solid element SOLID95 in
ANSYS software. This element is defined by 20 nodes having three degrees of freedom per
node: translations in the nodal x, y and z directions. The element may have any spatial
orientation. SOLID95 has plasticity, creep, stress stiffening, large deflection, and large strain
capabilities. It has the capability to inherit orthotropic material properties and hence, best
suited for analysing FRP composites.
Fig. 4.2 Geometry of a 20 Node SOLID95 Element
The surfaces between the laminates, apart from those at the crack, are bonded
together using a contact pair. The coincident nodes at the surface of the crack extension
length are coupled. Simply supported boundary condition is applied on the bottom edges
of the x-z plane, i.e.; degrees of freedom along the z-axis are constrained. The plate is also
constrained in the y-direction to imply infinite length, and plane strain condition.
Unidirectional fibre reinforced layers of Carbon-Epoxy with the following properties
are used in the laminate. [45]
E1=147GPa; E2=E3=10.3GPa
v12= v13=0.27; v23=0.54
G12=G13=7GPa; G23=3.7GPa
4.3 CASE ONE: PRESSURE LOADING The transverse uniform pressure of 1 MPa is applied on the surface along the top
edge of the laminate, parallel to the supported end.
Fig 4.3 Deformed model after cylindrical bending due to uniform pressure
The position of the crack is varied between different laminate interfaces, along
with the increase in the length of the crack. The nature of deflection in the Z-axis direction
is studied and the Strain Energy Release Rate in the plate is calculated from the element
tables.
ANALYSIS OF RESULTS:
With respect to Fig 4.4 to 4-9, the total amount of Energy is first used to deform
the crack region transversely and then for the extension of the crack longitudinally. Initially,
the energy required to deform is less. But, as the crack length increases, the energy
required to deform the crack region increases. Hence, more amount of energy has to be
supplied to shear the crack. After a certain crack length, the material resistance of the
laminate decreases. As a result of this the amount energy required to shear the crack
decreases.
4.3.1 ALL LONGITUDINAL FIBRES
Fig 4.4: Variation of ‘G’ with respect to ‘a’ from an edge crack with 0-0-0-0 laminate.
4.3.2 ALL TRANSVERSE FIBRES
Fig 4.5: Variation of ‘G’ with respect to ‘a’ from an edge crack with 90-90-90-90 laminate.
The amount of energy required shear the crack is very much less when all the fibres are
oriented longitudinally than transversely, as the fibre and crack extension are in the same
direction.
0
20
40
60
80
100
120
0 20 40 60 80 100
G (
J/m
2)
a (mm)
Edge Crack 0-0-0-0
Top InterfaceBottom InterfaceCentre Interface
0
200
400
600
800
1000
1200
1400
1600
0 20 40 60 80 100
G (
J/m
2)
a (mm)
Edge Crack 90-90-90-90
Top InterfaceBottom InterfaceCentre Interface
4.3.3 SYMMETRIC FIBRE ORIENTATION
Fig 4.6: Variation of ‘G’ with respect to ‘a’ from an edge crack with 0-90-90-0 laminate.
Fig 4.7: Variation of ‘G’ with respect to ‘a’ from an edge crack with 90-0-0-90 laminate.
In symmetric fibres there is a very slight difference in the Strain energy release rate at the
top and bottom interfaces. The symmetric nature of the Energy distribution is disrupted by
the proximity of the boundary conditions applied on the bottom layer of the laminate.
0
100
200
300
400
500
600
700
0 20 40 60 80 100
G (
J/m
2)
a (mm)
Edge Crack 0-90-90-0
Top InterfaceBottom InterfaceCentre Interface
0
50
100
150
200
250
300
350
400
450
0 20 40 60 80 100
G (
J/m
2)
a (mm)
Edge Crack 90-0-0-90
Top InterfaceBottom InterfaceCentre Interface
4.3.4 ANTISYMMETRIC FIBRE ORIENTATION
Fig 4.8: Variation of ‘G’ with respect to ‘a’ from an edge crack with 0-90-0-90 laminate.
Fig 4.9: Variation of ‘G’ with respect to ‘a’ from an edge crack with 90-0-90-0 laminate.
The structural resistance when the top layer is oriented 00 is greater than fibres oriented
with 900 at the top layer.
0
100
200
300
400
500
600
0 20 40 60 80 100
G (
J/m
2)
a (mm)
Edge Crack 0-90-0-90
Top InterfaceBottom InterfaceCentre Interface
0
100
200
300
400
500
600
0 20 40 60 80 100
G (
J/m
2)
a (mm)
Edge Crack 90-0-90-0
Top InterfaceBottom InterfaceCentre Interface
4.4 CASE TWO: LINE LOADING A line load of 10 N is applied on the line along the top edge of the laminate, parallel
to the supported end.
Fig: 4.10 Loaded model for edge crack opening at centre interface
Fig. 4.11 Deformed model for edge crack opening at centre interface
Analysis of Results:
Fig 4-12 to 4-17 show that the opening up of a crack requires more amount of energy, as
the energy required for deformation of the crack dominates more than the energy required
for crack propagation.
4.4.1 ALL LONGITUDINAL FIBRES
Fig 4.12 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 0-0-0-0 laminate
4.4.2 ALL TRANSVERSE FIBRES
Fig 4.13 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 90-90-90-90
laminate
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 10 20 30 40 50 60 70 80 90
G (
J/m
2)
a (mm)
Edge Crack Opening 0-0-0-0
0
1
2
3
4
5
6
7
8
9
0 10 20 30 40 50 60 70 80 90
G (
J/m
2 )
a (mm)
Edge Crack Opening 90-90-90-90
5.4.3 SYMMETRIC FIBRE ORIENTATION
Fig 4.14 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 0-90-90-0
laminate.
Fig 4.15 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 90-0-0-90
laminate.
0
0.5
1
1.5
2
2.5
3
0 10 20 30 40 50 60 70 80 90
G (
J/m
2 )
a (mm)
Edge Crack Opening 0-90-90-0
0
0.5
1
1.5
2
2.5
3
0 10 20 30 40 50 60 70 80 90
G (
J/m
2)
a (mm)
Edge Crack Opening 90-0-0-90
4.4.4 ANTISYMMETRIC FIBRE ORIENTATION
Fig 4.16 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 0-90-0-90
laminate
Fig 4.17 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 90-0-0-90
laminate.
0
0.5
1
1.5
2
2.5
3
3.5
0 10 20 30 40 50 60 70 80 90
G (
J/m
2)
a (mm)
Edge Crack Opening 0-90-0-90
0
0.5
1
1.5
2
2.5
3
0 10 20 30 40 50 60 70 80 90
G (
J/m
2 )
a (mm)
Edge Crack Opening 90-0-90-0
5
ANALYSIS OF CENTRE CRACK LAMINATES
5.1 INTRODUCTION This chapter presents the analysis of centre crack laminates supported along two
opposite edges and subjected to two different load cases. In first case, a uniform pressure
is applied on the top surface of the laminate. And in second case, a uniform line load is
applied along the top edge of the laminate parallel to the supported end.
5.2 PROBLEM MODELING
In this model the cross-section of the rectangular dimension plate are taken as 100
mm span (L) with a span/depth ratio of 10. Four layers of equal thickness (10/4=2.5mm)
are considered. The width of the plate is considered to be infinite. The crack location is at
centre of the laminate. The length of the crack is varied from 15mm to 75mm. The crack
extension length is taken to be 0.11mm on both sides of the crack.
Fig. 5.1 Geometric model for centre crack at centre interface.
Finite element mesh is generated using 20 node quadratic solid element SOLID95 in
ANSYS software. This element is defined by 20 nodes having three degrees of freedom per
node; translations in the nodal x, y and z directions. The element may have any spatial
orientation. SOLID95 has plasticity, creep, stress stiffening, large deflection, and large strain
capabilities. It has the capability to inherit orthotropic material properties and hence, best
suited for analysing FRP composites.
Fig. 5.2 Geometry of a 20 Node SOLID95 Element
The surfaces between the laminates, apart from those at the crack, are bonded
together using a contact pair. The coincident nodes at the surface of the crack extension
length are coupled. Simply supported boundary condition is applied on the bottom edges
of the x-z plane, i.e.; degrees of freedom along the z-axis are constrained. The plate is also
constrained in the y-direction to imply infinite length, and plane strain condition.
Unidirectional fibre reinforced layers of Carbon-Epoxy with the following properties
are used in the laminate. [45]
E1=147GPa; E2=E3=10.3GPa
v12= v13=0.27; v23=0.54
G12=G13=7GPa; G23=3.7GPa
5.3 CASE ONE: UNIFORM PRESSURE The transverse uniform pressure of 1 MPa is applied on the surface along the top
edge of the laminate, parallel to the supported end.
Fig. 5.3 Loaded model for centre crack at centre interface
The position of the crack is varied between different laminate interfaces, along
with the increase in the length of the crack. The nature of deflection is studied and the
Strain Energy Release Rate in the plate is calculated from the element tables.
Fig 5.4 Deformed model after laminate bending due to uniform pressure load
5.3.1 ALL LONGITUDINAL FIBRES
Fig 5.5: Variation of ‘G’ with respect to ‘a’ from an edge crack with 0-0-0-0 laminate.
5.3.2 ALL TRANSVERSE FIBRES
Fig 5.6: Variation of ‘G’ with respect to ‘a’ from an edge crack with 90-90-90-90 laminate.
Due to the symmetric nature of crack when at centre interface, the amount of resistance to
deformation is more. When the crack is at the top or bottom interface, the required energy
to propagate the crack is lesser. Due to the constraints on the bottom layer, forces develop
in opposite direction, hence propelling easier crack growth.
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40 50 60 70 80
G (
J/m
2)
a (mm)
Centre Crack 0-0-0-0 Top Interface
Bottom Interface
Centre Interface
0
50
100
150
200
250
300
350
400
0 10 20 30 40 50 60 70 80
G (
J/m
2)
a (mm)
Centre Crack 90-90-90-90 Top Interface
Bottom Interface
Centre Interface
5.3.3 SYMMETRIC FIBRE ORIENTATION
Fig 5.7: Variation of ‘G’ with respect to ‘a’ from an edge crack with 0-90-90-0 laminate.
Fig 5.8: Variation of ‘G’ with respect to ‘a’ from an edge crack with 90-0-0-90 laminate.
The symmetric nature of the orientation, the energy required at bottom and top surfaces is
almost the same amount. Minor variations in the value of G are only a result of constraints
applied. And also, it is easier to open up a crack at the interface of 0-0 fibres than 90-90.
0
20
40
60
80
100
120
140
160
180
0 10 20 30 40 50 60 70 80
G (
J/m
2)
a (mm)
Centre Crack 0-90-90-0 Top Interface
Bottom Interface
Centre Interface
-20
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50 60 70 80
G (
J/m
2)
a (mm)
Centre Crack 90-0-0-90
Top Interface
Bottom Interface
Centre Interface
5.3.4 ANTISYMMETRIC FIBRE ORIENTATION
Fig 5.9: Variation of ‘G’ with respect to ‘a’ from an edge crack with 0-90-0-90 laminate
Fig 5.10: Variation of ‘G’ with respect to ‘a’ from an edge crack with 90-0-90-0 laminate.
The resistance when the top layer is oriented 00 is greater than when oriented at 900.
-20
0
20
40
60
80
100
120
140
160
180
0 10 20 30 40 50 60 70 80
G (
J/m
2)
a (mm)
Centre Crack 90-0-90-0
Top Interface
Bottom Interface
Centre Interface
0
20
40
60
80
100
120
140
160
180
0 10 20 30 40 50 60 70 80
G (
J/m
2)
a (mm)
Centre Crack 0-90-0-90 Top Interface
Bottom Interface
Centre Interface
5.4 CASE TWO: LINE LOADING A line load of 1000N is applied on the line along the centre of the laminate, parallel
to the supported end.
Fig. 5.11 Loaded model for edge crack at centre interface
Fig. 5.12 Deformed model for centre crack opening at centre interface
Analysis of Results:
The graphs 5.13-5.18 show the constant trend of the amount of energy required to open
up the crack to be a continuous increasing curve with respect to the crack length.
5.4.1 All Longitudinal Fibres
Fig 5.13 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 0-0-0-0 laminate.
5.4.2 ALL TRANSVERSE FIBRES
Fig 5.14 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 90-90-90-90
laminate.
0
50
100
150
200
250
300
350
0 10 20 30 40 50 60 70 80
G (
J/m
2 )
a (mm)
Centre Crack Opening 0-0-0-0
0
500
1000
1500
2000
2500
3000
0 10 20 30 40 50 60 70 80
G (
J/m
2)
a (mm)
Centre Crack Opening 90-90-90-90
5.4.3 SYMMETRIC FIBRE ORIENTATION
Fig 5.15 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 0-90-90-0
laminate.
Fig 5.16 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 90-0-0-90
laminate.
0
200
400
600
800
1000
1200
0 10 20 30 40 50 60 70 80
G (
J/m
2 )
a (mm)
Centre Crack Opening 0-90-90-0
0
200
400
600
800
1000
1200
0 10 20 30 40 50 60 70 80
G (
J/m
2 )
a (mm)
Centre Crack Opening 90-0-0-90
5.4.4 ANTISYMMETRIC FIBRE ORIENTATION
Fig 5.17 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 0-90-0-90
laminate.
Fig 5.18 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with 90-0-0-90
laminate.
Fig 5.18 Variation of ‘G’ with respect to ‘a’ of opening a centre crack with90-0-90-0
laminate
0
100
200
300
400
500
600
700
800
900
0 10 20 30 40 50 60 70 80
G (
J/m
2 )
a (mm)
Centre Crack Opening 0-90-0-90
0
200
400
600
800
1000
1200
1400
0 10 20 30 40 50 60 70 80
G (
J/m
2)
a (mm)
Centre Crack Opening 90-0-90-0
6
CONCLUSIONS AND SCOPE
INTRODUCTION
The energy required for the growth of a crack under shear and opening conditions is
analysed for a four layered cross ply unidirectional fibre reinforced composite laminate
with delamination at two different locations (edge crack and centre crack) using
commercially available Finite Element Software, ANSYS.
Trend Pattern
Edge Crack
The energy required to shear the crack in the laminate under shear follows an increasing
trend until a certain crack length and then decreases as the length of the crack increases.
When in opening mode, the energy required to open up the crack almost follows a linear
increasing trend until a definitive crack length and then shows a parabolic growth.
Centre Crack
The energy required to shear and also open up the crack are in a progressively in a
parabolic increasing pattern
CONCLUSIONS
It is much safer to have centre crack structures than those with an edge crack.
The presence of a crack at the centre interface is less harmful compared to
structures with a crack closer to the bottom or top layers.
Transverse fibres are safer than longitudinal fibres in the case of presence of a
crack in a laminate structure.
SCOPE The above analysis can be extend at various levels
The energy release during delamination in angle ply laminates
To study the effect of number of layers
To study the effect of stacking sequence
The effect of other boundary conditions can be studied
The effect of thickness ratio of the laminate on the strain energy
To separate individual energy modes
7
REFERENCES
[1]. Kedward, K.T., 1995. Mechanical Design Handbook. (ed.) Rothbart, H.A., Harold, A.
McGraw-Hill, New York (USA), pp. 15.01-15.29.
[2]. Pagano, N.J., Schoeppner, G.A., 2000. Delamination of polymer matrix composites:
problems and assessment. Comprensive Composite Materials, 2. (ed.) Kelly, A.,
Zweben, C. Elsevier Science Ltd., Oxford (UK).
[3]. Tay, T.E., Shen, F., 2002. Analysis of delamination growth in laminated composites with
consideration for residual thermal stress effects. Journal of Composite Materials 36
(11), pp. 1299-1320.
[4]. Crasto, A.S., Kim, R.Y., 1997. Hygrothermal influence on the free-edge delamination of
composites under compressive loading. Composite Materials: Fatigue and Fracture 6,
ASTM STP 1285. (ed.) Armanios, E.A. American Society for Testing and Materials,
Philadelphia (USA), pp. 381-393.
[5]. Bolotin, V.V., 1996. Delaminations in composite structures: Its origin, buckling,
growth and stability. Composites Part B-Engineering 27 (2), pp. 129-145.
[6]. Bolotin, V.V., 2001. Mechanics of delaminations in laminate composite structures.
Mechanics of Composite Materials 37 (5-6), pp. 367-380
[7]. Greenhalgh, E., Singh, S., 1999. Investigation of the failure mechanisms for
delamination growth from embedded defects. Proceedings of the 12th
International Conference on Composite Materials, Paris (France).
[8]. Gamstedt, E.K., Sjogren, B.A., 2002. An experimental investigation of the
sequence effect in block amplitude loading of cross-ply composite laminates.
International Journal of Fatigue 24 (2-4), pp. 437-446.
[9]. Robinson, P., Hodgkinson, J.M., 2000. Interlaminar fracture toughness. Mechanical
Testing of Advanced Fibre Composites. (ed.) Hodgkinson, J.M. Woodhead Publishing,
Cambridge (UK), pp. 170-210.
[10]. Hashemi, S., Kinloch, A.J., Williams, J.G., 1990a. The analysis of interlaminar
fracture in uniaxial fiber-polymer composites. Proceedings of the Royal Society of
London Series A-Mathematical Physical and Engineering Sciences 427 (1872), pp.
173-199.
[11]. Kusaka, T., Hojo, M., Mai, Y.W., et al., 1998. Rate dependence of mode I
fracture behaviour in carbon-fibre/epoxy composite laminates. Composites Science
and Technology 58 (3-4), pp. 591-602.
[12]. Jensen, H.M., Sheinman, I., 2001. Straight-sided, buckling-driven delamination of
thin films at high stress levels. International Journal of Fracture 110 (4), pp. 371-385.
[13]. Glaessgen, E.H., Raju, I.S., Poe, C.C., 2002. Analytical and experimental studies
of the debonding of stitched and unstitched composite joints. Journal of
Composite Materials 36 (23), pp. 2599-2622.
[14]. Singh, S., Greenhalgh, E., 1998. Micromechanics of interlaminar fracture in
carbon fibre reinforced plastics at multidirectional ply interfaces under static and
cyclic loading. Plastics Rubber and Composites Processing and Applications 27 (5), pp.
220-226.
[15]. Greenhalgh, E.S., 1998. Characterisation of mixed-mode delamination growth
in carbon-fibre composites. PhD Thesis. Imperial College of Science, Technology
and Medicine, London (UK).
[16]. Purslow, D., 1986. Matrix fractography of fiber-reinforced epoxy composites.
Composites 17 (4), pp. 289-303.
[17]. Tanaka, H., Tanaka, K., 1995. Mixed-mode growth of interlaminar cracks in
carbon/epoxy laminates under cyclic loading. Proceedings of the 10th
International Conference on Composite Materials, Whistler B.C. (Canada), pp. 181-189.
[18]. Olsson, R., Thesken, J.C., Brandt, F., Jonsson, N., Nilsson, S., 1996. Investigations of
delamination criticality and the transferability of growth criteria. Composite
Structures 36 (3-4), pp. 221-247.
[19]. Compston, P., Jar, P.Y.B., 1999. The influence of fibre volume fraction on the mode
I interlaminar fracture toughness of a glass-fibre/vinyl ester composite. Applied
Composite Materials 6 (6), pp. 353-368.
[20]. Tanaka, K., Tanaka, H., 1997. Stress-ratio effect on mode II propagation of
interlaminar fatigue cracks in graphite/epoxy composites. Composite Materials:
Fatigue and Fracture 6, ASTM STP 1285. (ed.) Armanios, E.A. American Society for
Testing and Materials, Philadelphia (USA), pp. 126-142.
[21]. Dahlen, C., Springer, G.S., 1994. Delamination growth in composites under
cyclic loads. Journal of Composite Materials 28 (8), pp. 732-781.
[22]. Timoshenko, S. and Goodier, J.: 1970, Theory of Elasticity, McGraw Hill.
[23]. Inglis, C.: 1913, Stresses in a plate due to the presence of cracks and sharp corners,
Trans. Inst. Naval Architects 55, 219–241.
[24]. Kelly, A.: 1974, Strong Solids, second edn, Oxford University Press.
[25]. Gordon, J.: 1988, The Science of Structures and Materials, Scientific American
Library.
[26]. Westergaard, H.: 1939a, Bearing pressures and cracks, J. Appl. Mech.
[27]. Hillerborg, A. and Mod´eer, M. and Petersson, P.E.: 1976, Analysis of crack
formation and crack growth in concrete by means of fracture mechanics and finite
elements, Cement and Concrete Research 6(6), 773–782.
[28]. Baˇzant, Z. and Cedolin, L.: 1991, Stability of Structures, Oxford University Press.
[29]. Pipes, R.B., Pagano, N.J., 1970. Interlaminar stresses in composite laminates under
uniform axial extension. Journal of Composite Materials 4 , pp. 538-548.
[30]. Rybicki, E.F., 1971. Approximate 3-dimensional solutions for symmetric
laminates under inplane loading. Journal of Composite Materials 5 (JUL), pp. 354-360.
[31]. Herakovich, C.T., Renieri, G.D., Brinson, H.F., 1976. Finite element analysis of
mechanical and thermal edge effects in composite laminates. Army Symposium
on Solid Mechanics, Composite Materials: The Influence of Mechanics of Failure
on Design, Cape Cod (USA), pp. 237-248.
[32]. Wang, A.S.D., Crossman, F.W., 1977. Some new results on edge effect in symmetric
composite laminates. Journal of Composite Materials 11 (JAN), pp. 92-106.
[33]. Wang, S.S., Choi, I., 1982. Boundary-layer effects in composite laminates: I Free-
edge stress singularities. Journal of Applied Mechanics-Transactions of the Asme
49 (3), pp. 541-548.
[34]. Wang, S.S., Choi, I., 1983. The interface crack between dissimilar anisotropic
composite-materials. Journal of Applied Mechanics-Transactions of the Asme 50 (1),
pp. 169-178.
[35]. Irwin, G.R., 1958. Fracture I. Handbuch der Physik, VI. (ed.) Flüge, S.
Springer-Verlag, New York (USA), pp. 558-590.
[36]. Rybicki, E.F., Kanninen, M.F., 1977. Finite-element calculation of stress
intensity factors by a modified crack closure integral. Engineering Fracture
Mechanics 9 (4), pp. 931-938.
[37]. Broek, D., 1986. Elementary engineering fracture mechanics. Noordhoff
International Publishing, Alphen aan den Rijn (The Nederlands).
[38]. Camanho, P.P., Dávila, C.G., 2002. Mixed-mode decohesion finite elements for
the simulation of delamination in composite materials. NASA-Technical Paper
211737, National Aeronautics and Space Agency, USA.
[39]. Kim, I.G., Kong, C.D., Uda, N., 2002. Generalized theoretical analysis method
for free-edge delaminations in composite laminates. Journal of Materials Science 37
(9), pp. 1875-1880.
[40]. Krueger, R., 2002. The virtual crack closure technique: history, approach and
applications. NASA/CR-2002-211628, National Aeronautics and Space Agency, USA.
[41]. Caiazzo, A.A., Costanzo, F., 2001. Modeling the constitutive behavior of
layered composites with evolving cracks. International Journal of Solids and
Structures 38 (20), pp. 3469-3485.
[42]. Rand, O., 2001. A multilevel analysis of solid laminated composite beams.
International Journal of Solids and Structures 38 (22-23), pp. 4017-4043.
[43] Villaverde, N.B., 2004. Variable mix mode delamination in composite laminate
under fatigue condition: Testing and Analysis
[44] Saouma, Victor E., 2000. Fracture Mechanics
[45] Daniel, Isaac M., Ishai, O. 2006. Engineering Mechanics of Composite Materials,
Oxford University Press.
[46] Prashant Kumar, 1999. Elements of Fracture Mechanics, Wheeler Publications