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Finite element analysis of dynamic crack propagation using remeshing technique

A.R. Shahani *, M.R. Amini Fasakhodi

Faculty of Mechanical Engineering, K.N.T. University of Technology, Pardis Street, Mollasadra Avenue, Vanak Square, P.O. Box 19395-1999, Tehran, Iran

a r t i c l e i n f o

Article history:

Received 29 April 2008

Accepted 21 June 2008

Available online 2 July 2008

Keywords:

Dynamic fracture toughness

Remeshing

Arrest

RDCB

a b s t r a c t

This paper presents a finite element analysis based on the remeshing technique to predict the dynamic

crack propagation and crack arrest in a brittle material, namely Araldite-B. The dynamic fracture tough-

ness criterion is used to form the crack tip equation of motion. Plane stress condition is invoked in the

present two-dimensional fracture analysis of the RDCB (rectangular double cantilever beam) specimens

and the unknown crack tip position, and velocity is computed during the analysis by the dynamic fracture

criterion. According to the new crack tip position, a remeshing algorithm has been used to simulate the

dynamic crack growth and arrest. The obtained results including kinetic energy and strain energy, crack

tip velocity, dynamic stress intensity factor during crack growth and also crack arrest length have been

presented. Comparison of the results with those cited in the literature has shown a good agreement.

Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Dynamic fracture deals with fracture under conditions where

inertia forces must be included in the problem formulation. Thisoccurs either under dynamic loading or in the case of static loading

as a rapidly propagating crack runs through a structure. In this pa-

per, the latter one is investigated in the RDCB (rectangular double

cantilever beam) specimens.

The double cantilever beam specimen is one of the most com-

monly used test configurations for determining crack growth

speed and fracture toughness of materials. Some theoretical analy-

sis of the RDCB specimen, via either simple beam or shear beam

theories, can be found in previous literature [1–6]. The solutions

have been given based on the energy balance for rapid crack prop-

agation and arrest. The work implemented by Shahani and Forqani

[6] includes the beam theory considering shear deformation ef-

fects, and the obtained final intricate equations of motion for the

case of dynamic analysis were simplified by eliminating the inertia

terms and were solved as a quasi-static state. The application of

beam theory to dynamic crack propagation is particularly attrac-

tive because it is one-dimensional analysis. Although the beam

analysis cannot predict the details of crack tip stresses or strains,

it does provide an accurate account of energy quantities which

form the basis of the fracture criterion.

Parallel to the analytical research, the finite element method

provides a powerful alternative of analyzing most real crack prop-

agation in specimen configurations. In general, analytic solutions

for the prediction problems of dynamic crack propagation are

rarely available. Therefore, numerical methods are necessary to ob-

tain solutions to predict these problems. Among the numerical

methods, finite element method is the most popular numerical

method used for the analysis and predictions of dynamic fracture,and when it is employed accuracy of the results heavily depends on

the simulation method of crack growth. Consequently, a reliable

numerical analysis method for dynamic fracture is needed.

The earliest finite element methods for crack propagation use a

node decoupling technique with a simple nodal force release

mechanism [7–9]. As discussed by Kanninen [10], these early

methods generally produce inaccurate results, and are inappropri-

ate for modeling crack propagation. Node release technique has the

drawback of requiring a priori the knowledge of the path followed

by the crack, and also singular elements could not be employed

around the crack tip in this method.

Nishioka and Atluri [11] introduced a moving singular element

procedure for dynamic crack propagation analysis. In their method,

a special singular element that follows the moving crack tip is

used, and during the simulation of crack propagation only the con-

ventional elements immediately surrounding the singular element

are distorted. Dynamic fracture analysis of RDCB specimen using

the moving finite element method was carried out by Nishioka

and Atluri [12].

A version of the mixed Eulerian–Lagrangian kinematics descrip-

tion (ELD) was developed by Koh and Haber [13] and Koh etal. [14].

In this method, a single mesh pattern changes continuously and

independently from the material motion to model the crack propa-

gation, and auxiliary regional mapping is used to relate the changes

of the nodal coordinates to the crack tip motion. A major disadvan-

tage of this method is that thefinalgeneralized coefficient matrix for

the finite element equation results in a full-sized nonsymmetrical

0261-3069/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.matdes.2008.06.049

* Corresponding author. Tel.: +98 21 84063221; fax: +98 21 88674748.

E-mail address: [email protected] (A.R. Shahani).

Materials and Design 30 (2009) 1032–1041

Contents lists available at ScienceDirect

Materials and Design

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matrix, and solving these equations greatly increases the required

time. Koh et al. [15] applied ELD approach in finite element method

to predict the dynamic fracture in RDCB specimens.

The element-free Galerkin (EFG) method [16,17] is another

technique which, based on the moving least-square interpolants,

aimed at the simulation of crack growth problems. It requires only

nodal data, and no element connectivity is needed. Imposition of

the geometrical boundary conditions by the EFG technique is morecomplicated than for the FEM.

In elastic–plastic materials, Beissel et al. [18] introduced an ele-

ment failure algorithm for crack growth in general direction. This

algorithm does not require remeshing technique, and is achieved

by tracking the path of the crack tip and failing the elements

crossed by the path such that they can no longer sustain deviatoric

or tensile volumetric stresses. Singular elements cannot be em-

ployed in this algorithm because the model has not been re-

meshed, and analysis is performed via the single and fixed mesh

pattern during the entire process.

One of the unique features of any discrete fracture propagation

finite element code is its remeshing algorithm. Crack propagation

in elastic and elastic–plastic materials can be analyzed via this

method. Also this technique is not restricted to the special

mode in fracture mechanics, and due to this ability crack propaga-

tion in problems which have no symmetry, neither in geometry nor

in loading conditions, can be simulated.

Another important consideration in any such remeshing algo-

rithm is the application of elements to adequately model the crack

tip singularity [19]. In principle, two remeshing approaches have

been reported in the literature. The first is to remesh the entire

model in every step as advocated by Shephard et al. [20] and Sumi

[21]. The advantage of this approach is that well shaped elements

can usually be generated. However, its disadvantage is that a large

number of state variables must be transferred from the old to the

new mesh, particularly when an elasto-plastic material is involved.

The second approach employs a local remeshing technique where-

by only the region of elements within a certain distance of the

crack tip is modified. This approach has the advantage of requiringonly the transference of state variables from a small region which

is to be remeshed. However, the generated elements may not be as

well-shaped as in the first approach since there tend to be greater

constraints on the placement of the new elements due to the exist-

ing mesh surrounding the region to be remeshed.

Some crack propagation analysis based on the remeshing tech-

nique can be found in the previous works. In the LEFM conditions,

the local approach was used to investigate the fatigue crack growth

by Wawrzynek and Ingraffea [22,23] and simulation of dynamic

crack propagation in an infinite medium by Swenson and Ingraffea

[24]. Bittencourt et al. [25] presented an algorithm named quasi-

automatic simulation of stable crack propagation for two-dimen-

sional LEFM problems in order to predict crack trajectory that is

very similar to the work done by Swenson and Ingraffea [24] inremeshing algorithm. The analysis in [25] was said to be quasi-

automatic only because the user still needs to provide a desired

crack length increment at the beginning of each simulation, and

this is for the sake of being a stable crack growth. Tradegard

et al. [26] employed a combination of entire remeshing and nodal

relaxation to study mode I stable crack propagation in an elastic–

plastic material by ABAQUS software. Rethore et al. [27] employed

a stable numerical scheme for the analysis of dynamic crack

growth with remeshing but their study focused on the stability

subject of dynamic calculations. In principle, the aim of their work

is not to develop an efficient remeshing procedure. Dynamic frac-

ture mechanics has been used as a basis for their study, and the

crack growth speed has been calculated via the criterion based

on the energy release rate considering that the material toughnessis independent of the crack speed. Recently, Shahani and Seyyedian

[28] employed an entire remeshing approach in order to simulate

glass cutting with the impinging hot air jet which could be inter-

preted as a controlled crack growth due to thermal stresses caused

by the hot air jet.

In this paper, the attempt is made toward gaining a finite ele-

ment simulation based on the remeshing technique compatible

with the actual dynamic fracture process. In this regard, mode I dy-

namic crack propagation and arrest phenomena are investigated inthe RDCB specimens under fixed displacement loading condition.

The finite element modeling is accomplished with the standard FE

code ANSYS 7.0. Since the dynamic crack growth is intrinsically

an unstable phenomenon, crack growth increment cannot be de-

fined by the user, and a dynamic fracture criterion should be em-

ployed to find the crack tip velocity and crack extension every

time, automatically. In this study, dynamic fracture toughness cri-

terion is used to predict the crack tip velocity.The quarter point sin-

gular elements are used around the crack tip in order to model the

singularity. Since the crack is to propagate, a remeshing algorithm

is used in each crack extension step. For this to be achieved, ANSYS

Parametric Design Language, APDL, is employed. Owing to time-

dependent nature of the problem and inertia effects, nodal data

should be transferred from the old mesh to the new mesh in each

step of the remeshing. Finally, the problem is analyzed for two dif-

ferent crack tip bluntings to investigate its effects on crack arrest

length, crack arrest time, crack growth velocity and the quantity

of strain energy and kinetic energy during the crack advancement.

Thepredicted results show good agreement with those obtainedvia

the experimental study and the other numerical techniques.

2. Dynamic crack propagation analysis

2.1. Dynamic stress intensity factor

In general, the crack tip stress intensity factor represents the ef-

fect of the applied loading, the geometrical configuration of the

body and the bulk material parameters in the crack tip region for

any motion of the crack tip. For a dynamically propagating crack

under a steady-state or under an unsteady-state condition, these

parameters depend on time, the crack length and the crack tip

velocity. For the model problem considered here, the instanta-

neous value of the crack tip stress intensity factor for an arbitrary

motion of the crack tip depends on crack motion only through the

instantaneous value of the crack tip speed, _a, and time, t . Moreover,

this dependence is of separable form [29,30]:

K dI ðt ; _aÞ ¼ kIð _aÞK ÃI ðt Þ ð1Þ

where K dI is the instantaneous dynamic stress intensity factor for

crack propagation, K ÃI is the equilibrium stress intensity factor that

depends on the current length of the crack, the applied load, and the

history of crack extension, but not on the crack velocity. It has the

dimensions of a static stress intensity factor, but in general it isnot equal to the static stress intensity factor for a stationary crack

of the same length as the moving crack. kIð _aÞ is a function of crack

speed, and can be approximated in the form:

kIð _aÞ %1 À _a=C R

1 À 0:5 _a=C R

ð2Þ

or

kIð _aÞ %1 À _a=C R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 À _a=C Dp ð3Þ

where

C 2

D

¼

2lq

1Àv

1À2vÀ Áplane strain

2lq

11Àv

À Áplane stress

8<: ð4Þ

A.R. Shahani, M.R. Amini Fasakhodi/ Materials and Design 30 (2009) 1032–1041 1033



Here C R and C D are the Raleigh wave speed and the dilatational

wave speed, respectively. Rayleigh wave speed is a root of

equation

4aSaD À ð1 þ a2S Þ2 ¼ 0 ð5Þ

where

a2D ¼ 1 À

_

aC D

2

ð6Þ

a2S ¼ 1 À

_a

C S

2

ð7Þ

where C S is the shear wave speed given by

C 2S ¼

l

qð8Þ

in which l is the shear modulus and q is the density of the material.

The crack-velocity function kI is a universal function for all elas-

todynamically propagating cracks. When the crack is not propagat-

ing ( _a ¼ 0), this function takes a unit value (kI(0) = 1). The function

kIð _aÞ decreases monotonically with increasing crack velocity _awhile kI reduces to zero for _a ¼ C R . Since kIð _aÞ is the decreasing

function as explained above, the stress intensity factor drops dras-

tically for a sudden increase of the crack velocity. The details of this

discussion can be found in [30].

2.2. Dynamic fracture toughness

During unstable crack propagation, the fracture toughness of

the material is introduced as dynamic fracture toughness, and is

denoted by K dIC, which is not a constant value. The quantity K dIC

represents the resistance of the material to crack growth. The

magnitude of K dIC in a special temperature is expected to depend

on the crack speed and on the properties of the material. All iner-

tial, plasticity and rate effects are lumped into the material prop-

erty K dIC.

In general, the available experimental results for dynamic crack

propagation are very limited. Results of several dynamic fracture

experiments on materials by Kobayashi and Dally [31], Rosakis

et al. [32], Zehnder and Rosakis [33] for metals and by Paxton

and Lucas [34] for polymers indicate that the material’s level of

resistance to crack advance may depend on the instantaneous

crack tip speed. These are typical of most of the data that have been

reported which suggest that K dIC is roughly speed independent at

low crack speeds. But the most significant feature of the speed-

dependency is the increasing sensitivity of dynamic fracture

toughness to crack tip speed with increasing speed.

Fig. 1 exhibits dynamic fracture toughness vs. crack tip speed

for Araldite-B polymer. In this investigation, the extrapolated dataof dynamic fracture toughness for the velocity greater than 340 m/s

are used in the analysis of fast fracture if needed.

Experimental data presented in [32], relating the dynamic frac-

ture toughness to the crack tip velocity, can be correlated by the

heuristic experimental relation suggested by Kanninen and Popelar

[35]

K dIC ¼

K IA

1 À _a

V l

m ð9Þ

where K IA, V l and m are the material constants that must be deter-

mined empirically. These constants have a clear physical interpreta-

tion: V l corresponds to a limiting crack speed, K IA corresponds to the

nearly constant value in the low crack speed regime, while m is adimensionless shape factor.

2.3. Dynamic fracture criterion and crack tip equation of motion

In order to specify an acceptable crack tip equation of motion,

dynamic crack growth criterion is required. For crack growth pro-

cesses in materials which fail in a predominantly brittle manner, or

in which any inelastic crack tip zone is completely contained with-

in the surrounding elastic crack tip zone, the most common crack

growth criteria are the generalization of Griffith’s critical energy

release rate criterion and Irwin’s critical stress intensity factor cri-

terion. According to Irwin’s criterion, a crack must grow in such a

way that the crack tip dynamic stress intensity factor is always

equal to the dynamic fracture toughness of the material which

characterizes the resistance of the material to crack advance andmust be specified:

K dI ½aðt Þ; _aðt Þ; t ; load� ¼ K

dICð _aÞ ð10Þ

where K dI is the dynamic stress intensity factor, and as mentioned

earlier it depends on crack length, crack speed, time and applied

load, and K dIC is the dynamic fracture toughness. In principle, K dImay be determined by a pure elastodynamic analysis. In practice,

K dI cannot generally be determined analytically. Thus, numerical

and optical techniques are necessary to interpret dynamic fracture

experiments.

Through the introduced equation of motion (10), crack tip posi-

tion and crack tip velocity could be found during the unstable frac-

ture process, and also crack arrest could be predicted by this

equation. According to the criterion given by (10), crack arrest oc-curs when the stress intensity factor at the crack tip becomes smal-

ler than or equal to a critical value. This can be expressed as

K I 6 K dICð0Þ K

dynIa ð11Þ

where K dynIa denotes the dynamic crack arrest toughness.

3. Physical model for dynamic crack propagation

Fig. 2 depicts a RDCB specimen geometry used in experimental

study performed by Kalthoff et al. [36], and is employed in this

study. The dimensions of the specimens are as follows: length

L = 321 mm, initial crack length a0 = 67.8 mm, beam height

h = 63.5 mm, pin diameter d = 25 mm, distance from beam end to

pin e = 16 mm, distance from crack plane to pin b = 20 mm andthickness B = 10 mm. They were able to deduce the dynamic stress

0

0.5

1

1.5

2

2.5

0 100 200 300 400

velocity (m/s)

d y n a m i c f r a c t u r e t o u g h n e s s ( M P a . m

^ 0 . 5

) Kalthoff et al [36]

Extrapolated data

Fig. 1. dynamic fracture toughness vs. crack tip speed for Araldite-B.

1034 A.R. Shahani, M.R. Amini Fasakhodi/ Materials and Design 30 (2009) 1032–1041



intensity factor for a running crack in the polymer, Araldite-B, by

using the method of shadow patterns or caustics.

In their experiments, different values of the crack initiation

stress intensity factor K IQ were obtained according to the degree

of blunting of the initial crack tip. This means that the cracks were

initiated from blunted notches at initiation stress intensity factors

K IQ which are larger than the fracture toughness K IC.

Consequently, crack growth is initiated from an artificially

blunted initial crack tip so that the running crack, which propa-

gates as a sharp crack, is driven at higher speeds. But because

the load pin displacement does not change appreciably, the crackdriving force diminishes and crack arrest can occur.

In the experimental study [36], the RDCB specimens with K IQ

values 2.32 and 1.76 MPa m1/2 were identified as specimens 4

and 8, respectively. For convenience, the same identification is

used in the present numerical simulation. Plane stress condition

is postulated in the analysis because of the relatively thin nature

of the specimen, and also was assumed in the experimental study

[36]. The dimensions and material properties of the wedge-loaded

RDCB specimen used in the experimental study [36] and also in

this investigation are summarized in Table 1.

Araldite-B exhibits small differences between the dynamic and

static material properties. The dynamic material properties are

used in this study, and sometimes have been compared with the

results obtained with the static ones.

As shown in Fig. 2, the RDCB specimen has geometrical symme-

try, and by applying symmetrical loading with respect to the crack

plane the conditions of mode I crack growth could be preserved in

the problem. Two common types of loading in this specimen are

fixed displacement and fixed load on the pins shown in Fig. 2.

The former causes the crack growth to be stopped after a specified

time, because the load pin displacement does not change and con-

sequently crack driving force diminishes, while in the latter crackpropagation will be continued until the full rupture of specimen.

4. Finite element modeling procedure for dynamic crack

propagation

This section presents a finite element analysis for modeling dy-

namic fracture problems using the remeshing technique. Quarter-

point singular isoparametric elements are used for modeling the

singular field near the crack tip. The procedure for simulation of

dynamic crack extension is outlined in the next two parts.

4.1. Singular field near the crack tip

Because of the symmetry, only one half of the RDCB specimen is

modeled by FEM. A finite element mesh layout of model and also

the mesh pattern generated around crack tip are illustrated in

Fig. 3. The FE standard code ANSYS 7.0 has been employed for

modeling the problem. Quadratic isoparametric triangular ele-

ments with two degrees of freedom for each node are employed

for the discretization of the model (plane 2, ANSYS 7.0). These ele-

ments are capable of constructing singular elements, which are

used for modeling the familiar square root singularity ðr À1=2Þ at

the crack tip in elastic fracture mechanics. This can be easily done

by shifting the mid-side nodes of the six-node element to the quar-

ter-point positions [37].

Another consideration for using the quadratic six-node triangu-

lar element is that singular form of these elements leads to far bet-

ter results (stress intensity factors) than rectangular elements for

elastic fracture [37].

4.2. Remeshing algorithm

Due to the fact that fast fracture is intrinsically a dynamic anal-

ysis and also rupture phenomenon causes the geometry of the

model to be time dependent, ANSYS Parametric Design Language,

APDL, is employed for creating the fully automatic program to sim-

ulate the problem without any user interaction through the

process.

Fig. 2. Geometry of RDCB specimen used by Kalthoff et al. [36].

Table 1

Elastic properties of material Araldite-B [36]

Static elastic modulus, E s 3380 MN/m2

Dynamic elastic modulus, E D 3660 MN/m2

Static Poisson’s ratio, mS 0.33

Dynamic Poisson’s ratio, mD 0.39

Density, q 1172 kg/m3

Dynamic bar wave speed, C 0 ¼ ffiffiffiffiffiffiffiffiffi E =q

p 1767 m/s

Fig. 3. Finite element model of the problem: (a) the entire model of half-RDCB; (b) the meshes generated around the crack tip.




Fig. 4 shows the Flow-chart of the prepared APDL code of the

problem based on the remeshing technique. Transient analysis of

the problem is carried out via the Newmark scheme by choosing

time integration parameters d = 0.5 and b = 0.25. According to the

algorithm, after initial geometrical and physical modeling of the

problem, mesh generation of the model is accomplished via the

combination of free and manual methods in the software, and then

the dynamic analysis of the problem is performed. Finally, at the

end of every time incrementDt , stress intensity factor is computed

and the crack tip velocity is predicted by the dynamic fracture cri-

terion which includes the material property K dIC: In order to find

new crack tip position, the magnitude of _aDt is computed, where_a is the current crack tip velocity. At this step, remeshing is accom-

plished according to the new crack tip position. In fact, remeshingstarts from a rosette of singular elements around the crack tip and

a new mesh generation with this start point is done. Subsequently,

the necessary data are transferred from the previous mesh to the

new mesh, and is interpolated to the nodal values of the present

mesh. Since the problem is fixed in time during the remeshing step

and only the mesh is being changed, the shape functions are used

to interpolate the nodal data. The algorithm is repeated through

the mentioned steps during the analysis.

5. Results and discussion

In this part, some numerical examples are considered. All the

specimens are of the same geometry and material except in the de-

gree of crack tip blunting, which have been used in the experimen-

tal study [36], numerical analysis such as moving finite elementmethod by Nishioka and Atluri [12] and moving finite element

Setting the geometrical and input material properties data of the problem

Start

Discretization of the model by Plane2 elements

Applying the load and boundary conditions

Structural analysis of the dynamic problem

Computing the Stress Intensity Factor, KI

K I > K ID (0)

Remeshing is accomplished to the base of the new crack tip position

Interpolation of the previous mesh nodal data for finding the corresponding

data of the present mesh

STOP

An arrest

phenomenon

has happened.

Yes

No

Computation of dynamic stress intensity factor and then finding the crack

tip velocity and amount of movement using the dynamic fracture criterion

Setting the dynamic parameters of transient analysis

Fig. 4. Flow-chart of the FEM algorithm of the problem.




0

0.4

0.8

1.2

1.6

2

0 100 200 300 400 500

Time (micro second)

d y n a m i c S I F ( M P a . m

^ 0 . 5

)

present study with dynamic properties

experimental by Kalthoff et al [36]

Nishioka & Atluri [12]

Koh et al [15]

Kobayashi [39]

Fig. 5. Variation of dynamic stress intensity factor vs. time for RDCB-4.

0

0.5

1

1.5

2

2.5

60 100 140 180 220

Crack length (mm)


^ 0 . 5

)

Present study with dynamic properties

Experimental by Kalthoff et al [36]

Gehlen et al [38] Without torsional springGehlen et al [38] With torsional spring

Koh et al [15]

Fig. 6. Variation of dynamic stress intensity factor vs. crack length for RDCB-4.

0

0.4

0.8

1.2

1.6

2

0 100 200 300 400 500

Time (micro second)


^ 0 . 5

)

Experimental Results by Kaltoff et al [36]


Present study with static properties

Fig. 7. Dynamic stress intensity factor vs. time for RDCB-4 based on the dynamicand static material properties.

0

100

200

300

400

0 100 200 300 400 500 600

Time (micro second)

v e l o c i t y ( m / s )


Experimental by Kalthoff et al [36]

Koh et al [15]Gehlen et al [38] without torsional spring

Nishioka & Atluri [12]

Fig. 9. History of crack tip velocity for RDCB-4.

0

0.1

0.2

0.3

0.4

0 100 200 300 400 500 600 700

time (micro second)

k i n e t i c & s t r a i n e n e r g y ( J o

u l e )

Strain energy with dynamic properties, Present study

Strain energy with static properties, Present study

Kinetic energy with static properties, Present study

Kinetic energy with dynamic properties, Present study

Strain energy by Gehlen et al [38]

Kinetic energy by Gehlen et al [38]

Kinetic energy by Nishioka & Atluri [12]

Strain energy by Nishioka & Atluri [12]

Fig. 10. History of energy quantities for RDCB-4.

60

100

140

180

220

0 100 200 300 400 500 600

time (micro second)

c r a c k l e n g t h ( m m )

Present by dynamic propertiesExperimental by Kalthoff et al [36]Koh et al [15]Nishioka & Atluri [12]

Fig. 8. History of crack extension for RDCB-4.




method based on the ELD by Koh et al. [15]. The results presented

in [12] and [15] were obtained based on the static and dynamic

material properties, respectively. Gehlen et al. [38] investigated

the rapid crack growth in these specimens, and obtained equations

of motion by theory implementation based on the one-dimen-

sional assumption. The final achieved equations by their approach

were solved via the finite difference method. In the following, the

obtained results of this study are compared with those obtained bythe above researchers showing good agreement.

5.1. Results for RDCB-4 specimen

This specimen has a degree of crack tip blunting correspond-

ing to the initiation stress intensity factor K Q = 2.32 MPa m1/2.

Fig. 5 shows the time variations of the dynamic stress intensity

factor for RDCB-4 specimen. The results have been compared

with the previous numerical results cited in the literature, Nish-

ioka and Atluri [12], Koh et al. [15] and Kobayashi [39] as well

as the experimental results of Kalthoff et al. [36]. In principle,

the dynamic stress intensity factor predicted by this study

agrees well with the experimental results particularly in the

range of time between 70–230 ls and 370 ls to the arrest time.

It is also seen that the dynamic stress intensity factor obtained

in this study, Nishioka and Atluri [12] and Kobayashi [39], shows

nearly the same trend in behavior where they have a maximum

value at the start of crack growth, and thereafter decreasing and

again reaching to a local maximum value at the instant of about290ls. The behavior of dynamic stress intensity factor curve at

the early stage could not be confirmed by the experimental re-

sults due to lack of experimental results in this interval. In addi-

tion, the arrest time predicted by this study shows a very good

agreement with the experimental study, and has a value about

504 ls. This value for the work done by Koh et al. [15], Nishioka

and Atluri [12] and Kobayashi [39] is 438 ls, 528 ls and 497 ls,

respectively.

Fig. 6 shows the variations of the dynamic stress intensity factor

with respect to crack length for this specimen. It should be noted

that the crack length is measured from the pin location. It is seen

Fig. 11. Variation of mesh layout during dynamic crack propagation in RDCB-4 specimen.




that the best estimation for the crack arrest with reference to the

experimental study corresponds to this study.

Fig. 7 shows the results for the dynamic stress intensity factor

vs. time obtained based on the dynamic and static material proper-

ties separately. Although the results are nearly similar the predic-

tions using the static elastic properties compare generally less

favorably than those based upon the dynamic properties in this

specimen.The crack length history and crack tip velocity history are

shown in Figs. 8 and 9, respectively. According to Fig. 9, the behav-

ior of crack tip velocity curve is similar to the dynamic stress inten-

sity factor curve, and has a local maximum at a time about 290 ls.

It is seen that prior to arrest the crack velocity decreases with an

abrupt slope from the local maximum value. It is also seen from

Fig. 9 that the predicted results via the different methods agree

well with each other with the exception of Gehlen et al.’s method

[38], which shows the maximum difference at the arrest time. In

general, the crack tip velocity predicted by this study and the

method of [12] is close to each other.

Energy quantities such as kinetic energy and strain energy dur-

ing dynamic fracture process with the fixed displacement loading

condition are depicted in Fig. 10. The figure shows the results ob-

tained by Nishioka and Atluri [12] which is in good agreement with

those of this study. Fig. 10 also contains the kinetic energy and

strain energy values obtained based on the dynamic material prop-

erties by Gehlen et al. formulation without using torsional spring

in the equations [38]. As shown, their results are in the higher level

of magnitude and time process among the methods. As shown in

Fig. 10, the value of kinetic energy is zero at the beginning of prop-

agation because the model is in the static conditions on the thresh-

old of crack propagation. The magnitude of kinetic energy starts to

increase parallel to the crack advancement and reaches a maxi-

mum value at the time about 140 ls, and thereafter begins to de-

crease in the remainder path until arrival at the zero value at the

arrest time. This is the time that the model again reaches a static

condition because of crack arrest. Owing to loading condition in

this problem, the maximum value of strain energy occurs on thethreshold of crack advancement, and thereafter decreases through-

out the analysis until the arrest time where its magnitude is about

0.0287 joule at this instant for RDCB-4. For this specimen, whether

using dynamic or using static material properties the maximum ki-

netic energy is 26.4% of the initial strain energy which does not

indicate that the event occurs statically. Decrease of both types

of energy, however, is because of energy release due to the crack

advancing.

Fig. 11 shows the mesh details and the variation of mesh layout

during the dynamic fracture simulation of RDCB-4 specimen. It is

observed that because of using remeshing technique, the finite ele-

ment analysis of the problem is accomplished with the well-

shaped elements, and the optimum mesh is generated each time

during the crack propagation. Moreover, the singular elementsare always at the crack tip position.

5.2. Results for RDCB-8 specimen

This specimen has a degree of crack tip bluntness that corre-

sponds to the initiation stress intensity factor K Q = 1.76 MPa m1/2.

This specimen is analyzed based on the dynamic material proper-

ties only. Fig. 12 shows the variation of dynamic stress intensity

factor vs. crack length obtained from the present method along

with the existent results for this specimen, which includes the

experimental results [36] and numerical results of [15]. The curve

obtained for this case shows a trend similar to that presented for

RDCB-4, and comprises a maximum value at the start of crack

growth. Despite the prediction of Koh et al. [15], the predictedcrack arrest length via the present method is nearly identical with

that measured in the experimental study [36], and is about

170 mm.

Figs. 13 and 14 show the crack length and energy history,

respectively. It is seen that there is a good agreement between

crack length histories. Also, the arrest time predicted by this study

and [15] is identical, and has a value equal to 369 ls.

The time duration obtained by the experimental study [36] is

larger with respect to the value computed by this investigation.According to Fig. 14, the maximum value of strain energy for

RDCB-8 is 0.16 joule and corresponds to the magnitude of critical

applying load (fixed displacement) for initial crack advancement.

This quantity was about 0.3 joule for RDCB-4 specimen. But the

amount of strain energy at the arrest time is 0.0276 joule which

is nearly equal to the similar value of RDCB-4 at the arrest time.

For this specimen, the maximum kinetic energy is 21.6% of the ini-

tial strain energy which is 4.8% less than the first studied type of

specimen, and does not indicate that the event occurs statically.

Fig. 15 shows the crack tip velocity vs. crack length for RDCB-8

which represents the comparison of the present results with the

experimental results.

0

0.4

0.8

1.2

1.6

60 90 120 150 180

crack length (mm)


^ 0 . 5

)


Experimental results by Kalthoff et al [36]

Koh et al [15]

Fig. 12. Variation of dynamic stress intensity factor vs. crack length for RDCB-8.

60

90

120

150

180

0 100 200 300 400 500 600

time (micro second)

c r a c k l e n g t h ( m m )



Koh et al [15]

Fig. 13. History of crack extension for RDCB-8.




5.3. Crack velocity–crack length relation

A further comparison could be implemented for more assurance

of the predicted crack arrest length using the hypothesis presented

by Hahn et al. [40] and Shmuely [41]. Their theoretical calculations

predict a unique relation between the length of the crack at arrest

and the maximum constant crack velocity which depends only on

the geometry of the specimen but not on the toughness of thematerial. Both theoretical curves in a properly normalized form

are shown in Fig. 16. The present results together with the exper-

imental results of Kalthoff et al. [36] for both RDCB-4 and RDCB-8

specimens are summarized in Table 2 in order to compare them

with the predicted results by Shmuely [41] and Hahn et al. [40]

arisen from Fig. 16. Here, c 0 is the bar wave speed in the material.

This study yields to the values of 0.1726 and 0.1613 for V max

c 0in

the RDCB-4 and RDCB-8 specimens, respectively, the crack arrest

lengths of which are presented in the table. The predicted crack ar-

rest lengths corresponding to the same values of V max

c 0have been ex-

tracted from Fig. 16 and presented in other rows of the same table.

It is seen that a relatively good agreement exists between the re-

sults. Nevertheless, the experimental results [36] assure the pre-

ciseness of the results of this investigation over those of Hahnet al. [40] and Shmuely [41].

6. Conclusion

The problem of dynamic crack propagation in RDCB specimen,

made of a brittle material, has been analyzed. A finite element

analysis based on the remeshing technique has been used to sim-

ulate the crack growth during the fracture process. The remeshing

technique has been preferred among the traditional methods, and

this is due to the fact that when it combines with the singular ele-

ments it could model the crack tip singularity and conserve it

through the crack advancement. Moreover, a dynamic fracture cri-

terion, which includes the dynamic fracture toughness, has been

employed in order to extract the crack growth velocity at each time

step. In general, fast fracture mechanics analysis deals with the

parameters such as dynamic stress intensity factor, history of crackpropagation and velocity, energy quantities and crack arrest

length. The aforementioned parameters in this study have shown

good agreement with the experimental results as compared with

the other numerical results published in the literature. The follow-

ing conclusions could be drawn from the above-mentioned unsta-

ble crack growth analysis:

With decreasing crack tip blunting, the initial stored strain

energy in the specimen decreases causing the crack arrest length

and time to decrease.

The ratio of the maximum kinetic energy to the initial strain

energy in the specimen is considerable, and this implies that

the dynamic effects are dominant in the unstable crack propaga-

tion phenomenon and should be taken into account in theanalysis.

Table 2

Summarized results of different predictions for RDCB-4 and RDCB-8 specimens

RDCB-4 RDCB-8

_amax;cons=c 0 aarrest=a0 _amax;cons=c 0 aarrest=a0

Present study 0.1726 3.038 0.1613 2.51917

Predicted by Shmuely [41] 0.1726 2.9516 0.1613 2.6926

Predicted by Hahn et al. [40] 0.1726 2.7716 0.1613 2.5726

Experimental by Kalthoff et al.

[36]

0.16978 2.9522 0.155 2.51917

0

50

100

150

200

250

300

350

67 87 107 127 147 167

crack length (mm)

v e l o c i t y ( m / s )



Fig. 15. Variation of crack tip velocity vs. crack length for RDCB-8.

0

0.05

0.1

0.15

0.2

0 2 4

relative crack length, a(arrest)/a0

r e l a t i v e c r a c k v e l o c i t y , ( V

m a x / c 0 )

Predicted by

Shmuely [41]

Predicted by Hahn

et al [40]

1 3

Fig. 16. Relation between maximum constant velocity and crack length at arrest.

0

0.04

0.08

0.12

0.16

0 50 100 150 200 250 300 350 400

time (micro second)

s t r a i n & k i n e

t i c e n e r g y ( j o u l e )

strain energy with dynamic properties, Present study

kinetic energy with dynamic properties, Present study

Fig. 14. Variation of energy quantities vs. time for RDCB-8.




The size of blunting at the crack tip influences the level of crack

growth velocity, the magnitude of crack arrest length and also

the arrest time in the fixed-displacement loading condition.

The strain energy related to the arrest point for both specimens

analyzed here, i.e., RDCB-4 and RDCB-8, is nearly identical, and

this implies that the arrest occurs when the strain energy

reaches a pronounced value related to the value of crack arrest

toughness. The accuracy of this investigation is more noticeable when it

concerns with the crack arrest length or arrest time predicted

in the specimens.

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