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Faculty Publications from the Department of
Engineering Mechanics
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2010
Studies of dynamic crack propagation and crack branching with Studies of dynamic crack propagation and crack branching with
peridynamics peridynamics
Youn Doh Ha Ph.D.
University of Nebraska at Lincoln
Florin Bobaru Ph.D.
University of Nebraska at Lincoln
, fbobaru2@unl.edu
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Ha, Youn Doh Ph.D. and Bobaru, Florin Ph.D., "Studies of dynamic crack propagation and crack branching
with peridynamics" (2010).
Faculty Publications from the Department of Engineering Mechanics
. 71.
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Abstract
In this paper we discuss the peridynamic analy-
sis of dynamic crack branching in brittle materi-
als and show results of convergence studies un-
der uniform grid renement (m-convergence)
and under decreasing the peridynamic horizon
(δ-convergence). Comparisons with experimen-
tally obtained values are made for the crack-tip
propagation speed with three different peridyn-
amic horizons. We also analyze the inuence of
the particular shape of the micro-modulus func-
tion and of different materials (Duran 50 glass and
soda-lime glass) on the crack propagation behav-
ior. We show that the peridynamic solution for this
problem captures all the main features, observed
experimentally, of dynamic crack propagation and
branching, as well as it obtains crack propagation
speeds that compare well, qualitatively and quan-
titatively, with experimental results published in
the literature. The branching patterns also corre-
late remarkably well with tests published in the
literature that show several branching levels at
higher stress levels reached when the initial notch
starts propagating. We notice the strong inuence
reecting stress waves from the boundaries have
on the shape and structure of the crack paths in
dynamic fracture. All these computational solu-
tions are obtained by using the minimum amount
of input information: density, elastic stiffness, and
constant fracture energy. No special criteria for
crack propagation, crack curving, or crack branch-
ing are used: dynamic crack propagation is ob-
tained here as part of the solution. We conclude
that peridynamics is a reliable formulation for
modeling dynamic crack propagation.
Keywords: dynamic fracture, crack branching,
brittle fracture, peridynamics, nonlocal methods,
meshfree methods
1 Introduction
1.1 Literature review of dynamic crack
propagation
In a brittle material, a propagating crack can de-
part from its original straight trajectory and curve
or split into two or more branches. Under very
high states of stress, the propagating crack will di-
vide into a river-delta crack pattern (Bowden et al.
1967; Ramulu and Kobayashi 1985). This fragmen-
tation of highly loaded, brittle materials is often a
succession of multiple branching of what was ini-
tially a single crack. Increases in the roughness of
the fracture surface prior to branching were con-
sistently observed in all reported investigations
(Ramulu and Kobayashi 1985; Döll 1975). In crack
branching of edge notch specimens of brittle mate-
rials it has also been observed that the crack tip ve-
locity drops by no more than 5–10% in the branch-
ing region (Döll 1975).
In atomistic models, under conditions that lead
to instability of the crack path, cracks can branch
without a specic criterion (see Zhou et al. 1996).
Published in International Journal of Fracture 162 (2010), pp. 229–244; doi: 10.1007/s10704-010-9442-4
Copyright © 2010 Springer Science+Business Media B.V. Used by permission.
Submitted August 31, 2009; accepted January 5, 2010; published online January 20, 2010.
Studies of dynamic crack propagation and crack
branching with peridynamics
Youn Doh Ha and Florin Bobaru
Department of Engineering Mechanics, University of Nebraska–Lincoln, Lincoln, NE 68588-0526, USA
Corresponding author — F. Bobaru, email fbobaru2@unl.edu
229
230 Y. D. Ha & F. Bo B a r u i n In t e r n a t I o n a l Jo u r n a l o f fr a c t u r e 162 (2010)
Particle-type models (see Bolander and Saito 1998)
are also capable of simulating crack branching.
However, none of these methods is able to cap-
ture the crack propagation speed or the angle of
crack branching correctly. For instance, MD sim-
ulations show instabilities that lead, shortly after
the bifurcation of a crack, to the propagation of
only one of the two branches, the other being ar-
rested. Moreover, the branching angle computed
with MD (see Zhou et al. 1996) is greater than 90°,
whereas experiments show much smaller crack
branching angles (Ramulu and Kobayashi 1985).
One may ask whether quantum mechanical calcu-
lations are needed to predict the phenomenon of
dynamic fracture in brittle materials (see Cox et al.
2005) which is one of the great challenges in dy-
namic fracture. One likely reason for MD simu-
lations’ failure to correctly predict dynamic frac-
ture is that, for example, crack branching events
are controlled by the interaction and wave reec-
tions from the boundaries (Ravi-Chandar 1998).
Because of this, one would have to either model
the entire structure with MD (not a viable op-
tion) or use a multiscale model that is capable of
transferring the waves between the scales cor-
rectly (still an open problem). Numerical simu-
lations based on continuum methods of dynamic
crack propagation behavior have been very dif-
cult to develop and, to this date, a reliable method
for simulating this complex problem has not been
found in spite of considerable efforts in this di-
rection (e.g. Xu and Needleman 1994; Cama-
cho and Ortiz 1996; Ortiz and Pandol 1999; Be-
lytschko et al. 2003; Rabczuk and Belytschko 2004;
Song et al. 2006). All these methods use some ver-
sion of cohesive-zone models. As such, they all
modify the local continuum mechanics equations
and introduce a nonlocal effect given by the pa-
rameters and length scales in the cohesive-zone
model. To reduce mesh dependency when the
grids are rened special methodologies have to
be used (Zhou and Molinari 2004). For the exist-
ing approaches, the difculties in modeling dy-
namic fracture processes like crack branching are
many. For example, continuum-type methods us-
ing the cohesive FEM or the XFEM require a dam-
age criterion and a tracking of the stresses around
the crack tip to decide when to branch the crack.
Decisions also have to be taken in terms of the an-
gle of propagation of the branches and about how
many branches will be allowed to form. In meth-
ods in which the crack advances along the element
sides by separating elements from one another,
the crack path becomes non-smooth (see Xu and
Needleman 1994; Camacho and Ortiz 1996; Ortiz
and Pandol 1999). Since the correct path (which
minimizes the strain energy) of the crack prop-
agation is not computed correctly, there are sig-
nicant departures from the true energy released
during the crack propagation event. In such cases,
reliable prediction of strength of brittle ceram-
ics under impact, for example, becomes difcult.
Mesh dependency is an additional problem in co-
hesive-zone FEM-based methods. Important prog-
ress has been recently made by using the XFEM
method which allows cracks to pass through the
nite elements (see e.g. Belytschko et al. 2003).
Subdivision of the cut elements for numerical in-
tegration purposes increases the complexity and
the cost of the method. This method requires phe-
nomenological damage models and branching
criteria, as well as tracking of the crack path us-
ing level sets, for example. It is not yet clear if the
method is applicable to problems that involve
fragmentation and/or multiple crack interactions,
branching, and coalescence. The method does not
predict the experimentally observed crack prop-
agation speeds (see Song et al. 2008). Cohesive-
zone based models need to modify the experimen-
tal values of the fracture energy by several factors
in order to get propagation velocities in the range
of measured ones (Song et al. 2008).
In the present contribution we try to answer
whether quantum, atomistic, or multiscale mod-
els are needed in dynamic fracture in order to cor-
rectly simulate the observed crack propagation ve-
locities and crack paths (Cox et al. 2005; Song et
al. 2008). We will show that peridynamics is able
to correctly model and simulate dynamic fracture,
in particular crack branching in brittle materials.
Peridynamics, which is a reformulation of contin-
uum mechanics (Silling 2000; Silling et al. 2007b),
does not require criteria for crack propagation or
crack branching: these happen spontaneously in
this method and are autonomously generated by
a simple bond-failure criterion that is correlated to
the material’s energy release rate. The name “peri-
dynamics” comes from the Greek “peri” which
means “nearby,” and dynamics. Peridynamics is a
nonlocal method in which material points interact
not only with their nearest neighbors but also with
points nearby, inside a horizon. This is what phys-
ically happens at the atomic scale, for example, but
peridynamics extends this idea to the continuum
scale. We will show convergence in terms of the
crack path and the crack propagation speed under
St u D i e S o F DY n a m i c c r a c k p r o p a g a t i o n a n D cr a c k B r a n c H i n g w i t H p e r i D Y n a m i c S 231
grid renement (the so-called δ-convergence, Bo-
baru et al. 2009) and under decreasing peridynamic
horizon (the so-called δ-convergence, Bobaru et al.
2009). The crack branching patterns obtained using
peridynamics follow remarkably close the exper-
imental results which show secondary branching
taking place when higher stress levels are reached
at the tip of the pre-notch prior to crack propaga-
tion. Moreover, the only input parameters in the
model are the Young’s modulus, the density, and
the fracture energy (which is kept constant, and
not a function of the propagation velocity or of the
incurred damage, in this rst study).
The paper is organized as follows: in the next
section we describe the sample problem setup. In
Section 2 we briey review the peridynamic for-
mulation and the connections between the parame-
ters in the formulation and the material properties
like the energy release rate. In Section 3 we present
the numerical results for the convergence studies.
We look at both the crack path and the propagation
speed of the crack, as measures of convergence. In
Section 4 we analyze the inuence of the micro-
modulus function on crack branching results as
well as the solutions for two different brittle ma-
terials under higher loading conditions that lead
to cascading branching. We also comment on the
roughening zones that take place in the branching
regions and on the effect of the reection waves on
the propagation paths of the dynamic cracks. The
conclusions are given in Section 5.
1.2 Problem setup
We consider the following setup as a bench-
mark problem for analyzing crack branching phe-
nomena: a prenotched thin rectangular plate with
0.1m by 0.04m as shown in Figure 1. All simula-
tions in this paper are 2D simulations. For some
3D results we refer the readers to Ha et al. (2010).
The materials chosen for this study are selected
because for these materials there are experimen-
tal results published on the crack propagation ve-
locity in the region of branching or the maximum
propagation velocity measured. The two materi-
als used here are a Duran 50 glass (taken form Döll
1975) and a soda-lime glass (taken from Bowden et
al. 1967). The material properties are summarized
in Table 1. Please note that in the bond-based peri-
dynamic implementation used here, the numerical
models will be limited to using a xed Poisson ra-
tio of 1/3 (for 2D plane stress problems). If other
Poisson ratios are desired, then the state-based
peridynamics formulation should be used (see
Silling et al. 2007b). For dynamic fracture prob-
lems, the Poisson ratio value does not have a sig-
nicant inuence on the propagation speed or the
crack path shapes (Silling et al. 2007a).
In the experimental settings the loading of the
sample may take tens of seconds or more. In ex-
plicit dynamic simulations that would be too ex-
pensive to compute. Instead, we choose to ap-
ply, along the upper and lower edges (see Figure
1), traction loadings σ suddenly at the initial time
step and maintain this loading constant after that.
The theoretical background for the peridynamics
analysis is based on Silling’s original peridynam-
ics paper (Silling 2000), the imposition of the trac-
tion boundary conditions is as in Ha and Bobaru
(2009), and the numerical implementation of fail-
ure is like in Silling and Askari (2005). The same
geometrical setup for studying crack branching
simulations has been used in other studies (Be-
lytschko et al. 2003; Rabczuk and Belytschko 2004;
Song et al. 2006).
While there is no analytical solution for the
crack branching problem, we can compare our
simulation results with experiments. Unfortu-
nately, the experimental papers we found do not
provide a complete description of the conducted
experiment on crack branching: some papers show
the crack paths but do not provide crack propaga-
tion speed data, others give the propagation speed
but do not show the crack paths, and most do not
describe in detail the loading conditions. We de-
cided to perform the peridynamic simulations
using a setup similar to that used in a few recent
simulation papers (Belytschko et al. 2003; Rabczuk
and Belytschko 2004; Song et al. 2006). The mate-
rial parameters, however, are like those used in
the experiments (Bowden et al. 1967; Döll 1975).
The maximum crack propagation speed, or the
Figure 1. Description of the problem setup for the
crack branching study.
232 Y. D. Ha & F. Bo B a r u i n In t e r n a t I o n a l Jo u r n a l o f fr a c t u r e 162 (2010)
crack propagation speed in the region of branch-
ing, is data that is fairly reproducible in experi-
ments and this is reported in Bowden et al. (1967)
and Döll (1975), for example. We are not aware of
any numerical method that can reproduce the ex-
perimentally measured dynamic crack propaga-
tion velocity. Note that in previous studies, for
certain methods, the fracture energy has to be sig-
nicantly modied (by several factors) in order to
bring the dynamic crack propagation speed closer
to the measured values (see, e.g. Belytschko et al.
2003; Rabczuk and Belytschko 2004).
2 The peridynamic formulation
The peridynamic formulation (Silling 2000) re-
lies on integration of forces acting on a material
point and thus it does not face any of the mathe-
matical inconsistencies seen in the classical con-
tinuum mechanics equations. The integration
takes place over a “horizon” (which, in princi-
ple, extends to innity but, for convenience is -
nite) within which the material points are interact-
ing with each other. In certain problems, the size
of the horizon can be correlated to an intrinsic ma-
terial length-scale. However, in many cases a ma-
terial length scale is not “visible” either because
the micro-structure and the specic loading and
boundary conditions do not lead to a measurable
effect of the length-scale. In such cases, the horizon
is selected by the user according to convenience
(see Bobaru et al. 2009). Allowing a variable hori-
zon (with a correspondingly scaled micromodulus
parameter) denes away of introducing adaptive
renement for this nonlocal method. It is impor-
tant to notice that peridynamics is a continuum
theory, not a particle-type method. This allows the
convergence results of the peridynamic solution to
the classical elasticity solutions in the limit of the
horizon going to zero (Bobaru et al. 2009; Silling
and Lehoucq 2008).
An important advantage of peridynamics is
the way damage is introduced: material points
are connected within the horizon via elastic (lin-
ear or nonlinear) bonds that have a critical relative
elongation, s
0
, at which they break (Silling 2000).
The critical relative elongation for brittle mate-
rials is computed from the experimentally mea-
sured value of the fracture energy for a specic
material (Silling and Askari 2005). Damage is im-
plemented as the fraction between the number
of broken bonds and the number of initial bonds
(Silling and Askari 2005). Cracks in peridynam-
ics form as surfaces between material points form,
as a consequence of sequential breaking of bonds.
Thus, there is no need to track the cracks like in
other continuum methods, or to impose criteria
for when cracks should branch, change direction,
turn, coalesce, etc. Moreover, peridynamics allows
for spontaneous generation of cracks where no
aws were present before. This is shown, for ex-
ample, in Silling et al. (2009) for the crack nucle-
ation and in simulation of spallation (see Xie 2005)
where spallation is treated as real fracture and not
modeled by void-growth formulations as in exist-
ing literature results.
We now briey review the peridynamic formu-
lation based on Silling’s original peridynamics pa-
per (Silling 2000). Also, we consider the summary
of the numerical implementation of the traction
boundary conditions in peridynamics (Ha and Bo-
baru 2009) and the formulation for the damage
model in peridynamics (Silling and Askari 2005).
The peridynamic equations of motion are given
by:
ρü (x, t) =
∫
H
f(u (xˆ , t) − u(x, t), xˆ − x) dxˆ + b(x, t) (1)
where f is the pairwise force function in the peri-
dynamic bond that connects node xˆ to x and u is
the displacement vector eld. ρ is the density and
b (x, t) is the body force. The integral is dened
over a region H called the “horizon,” which is the
compact supported domain of the pairwise force
function around point x.
A micro-elastic material (Silling 2000) is dened
as one for which the pairwise force derives from a
potential ω:
f (η, ξ ) =
∂ω (η, ξ )
(2)
∂η
where ξ = xˆ − x is the relative position in the refer-
ence conguration and η = uˆ − u is the relative dis-
Table 1. Material properties for Duran 50 and soda-lime glasses
Density (ρ) (kg/m
3
) Young’s modulus (E) (GPa) Poisson ratio (υ) Fracture energy (G
0
) (J/m
2
)
Duran 50 glass 2,235 65 0.2 204
Soda-lime glass 2,440 72 0.22 135
