-
Pergamon
J. Mech. Pl~.w. Solid.~, Vol. 42, No. 9, pp. 1397-1434, 1994
Copyright X; 1994 Elsevier Science Ltd
Printed in Great Britain. All mhts reserved
00225096(94)E0030--8 0022-5096194 $7.00 + 0.00
NUMERICAL SIMULATIONS OF FAST CRACK GROWTH IN BRITTLE SOLIDS
X.-P. XU and A. NEEDLEMAN
Division of Engineering, Brown University, Providence, RI 02912,
U.S.A
(Receked 20 December 1993; in recised,form 29 March 1994)
ABSTRACT
Dynamic crack growth is analysed numerically for a plane strain
block with an initial central crack subject to tensile loading. The
continuum is characterized by a material constitutive law that
relates stress and strain, and by a relation between the tractions
and displacement jumps across a specified set of cohesive surfaces.
The material constitutive relation is that of an isotropic
hyperelastic solid. The cohesive surface constitutive relation
allows for the creation of new free surface and dimensional
considerations introduce a characteristic length into the
formulation. Full transient analyses are carried out. Crack
branching emerges as a natural outcome of the initial-boundary
value problem solution. without any ad hoc assump- tion regarding
branching criteria. Coarse mesh calculations are used to explore
various qualitative features such as the effect of impact velocity
on crack branching, and the effect of an inhomogeneity in strength,
as in crack growth along or up to an interface. The effect of
cohesive surface orientation on crack path is also explored, and
for a range of orientations zigzag crack growth precedes crack
branching. Finer mesh calculations are carried out where crack
growth is confined to the initial crack plane. The crack
accelerates and then grows at a constant speed that. for high
impact velocities, can exceed the Rayleigh wave speed. This is due
to the finite strength of the cohesive surfaces. A fine mesh
calculation is also carried out where the path of crack growth is
not constrained. The crack speed reaches about 45% of the Rayleigh
wave speed. then the crack speed begins to oscillate and crack
branching at an angle of about 29 from the initial crack plane
occurs. The numerical results are at least qualitatively in accord
with a wide variety of experimental observations on fast crack
growth in brittle solids.
1. INTRODUCTION
Much is known about crack initiation and linear elastic singular
fields for straight cracks growing dynamically (Freund, 1990).
There are, however, a number of obser- vations that are not
adequately accounted for by current theory. For example, obser-
vations of fast crack growth in brittle solids typically reveal
complex patterns of crack branching [see, e.g. Field (1971) and
McClintock and Argon (1966), p. 5011, but a predictive theory for
such crack branching has not yet been developed. Also, although
fracture mechanics theory predicts that the limiting crack speed is
the Rayleigh wave speed [see, e.g. Freund (1990)], observed crack
speeds are rarely greater than half this value.
The issues of crack path and limiting crack speed may very well
be related. Based on their experiments, Ravi-Chandar and Knauss
(1984b,c) argued that the occurrence of micro-cracks in front of
the main crack controls the crack speed and plays an important role
in the branching process. More recently, Gao (1993) has put
forward
1397
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13% X.-P. XIJ and A. Nlz.EDI.liMAN
a wavy-crack model in which the tendency of fast moving cracks
to deviate from their plane limits the apparent crack velocity to
about half the Rayleigh wave speed. while Slepyan (1993) has
proposed a principle of maximum energy dissipation to explain
limiting crack speeds. However, analyses relevant to these issues
have relied on highly idealized models. What has been lacking are
full field solutions for fast moving cracks. where the cracks are
free to propagate away from the current crack plane.
In this investigation. we carry out simulations of dynamic crack
growth in isotropic elastic solids. The theoretical framework is
the cohesive surface decohesion for- mulation of Needleman (1987).
In previous work, attention has been confined to problems with a
single cohesive surface. Here, to allow for a variety of possible
crack growth paths. potential surfaces of decohesion are
interspersed throughout the material. The material failure
characteristics are embodied in the geometrical and constitutive
characterization of the cohesive surfaces. The discretization is
based on a finite element formulation. with volume finite elements
(or, in two dimensions. area finite elements) bordered by cohesive
surface elements. Although the creation of neM free surface must be
along finite element boundaries. the location and path are
otherwise unrestricted. Furthermore. new free surface is not
required to emanate continuously from a pre-existing crack.
Crack initiation and crack growth are calculated directly in
terms of the elastic properties of the material and of the
parameters characterizing the cohesive surface separation law.
which include a strength and the work of separation per unit area.
Hence, a characteristic length enters the formulation. This
framework has been used to address issues regarding void nucleation
(Needleman. 1987 : Tvergaard. 1990 ; Povirk (It rd., 1991 ; Xu and
Needleman, 1993), quasi-static crack growth (Needleman. 1990a,b;
Tvergaard and Hutchinson 1992. 1993). stability of the separation
process (Suo ct al., 1992; Levy, 1994). and reinforcement cracking
in metal matrix composites (Finot ct rd., 1994). The results
obtained here reproduce. at least qualitatively. a variety of
observed phenomena on fast crack growth in brittle solids,
including, for example, crack branching. the dependence of crack
speed on impact velocity and abrupt crack arrest. There is no
unified description of these phenomena within a traditional
fracture mechanics framework.
The specific problem analysed is a block with an initial central
crack. Plane strain conditions are assumed to prevail. The loading
is tensile. with a constant imposed velocity after a small rise
time. The material is characterized as an isotropic hypcr- elastic
solid and full finite strain transient analyses are carried out.
Although the strains generally remain small. the finite strain
formulation properly accounts for the local large strains and
rotations accompanying separation of cohesive surfaces. In the
plane of deformation, the cohesive surfaces are lines parallel to
the coordinate axes and at 45 to them. Some calculations exploring
the effect of varying the cohesive surface orientation are also
carried out. The cohesive surface constitutive relation is that
given by Xu and Needleman (1993). and allows for tangential as well
as normal separation.
Although attention is focused on brittle crack growth.
computational modelling of the creation of new free surface along
arbitrary paths is important in a wide variety of applications; for
example. in other branches of fracture mechanics and in the
analysis of manufacturing processes such as machining. For ductile
fracture of metals
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Simulations of fast crack growth 1399
due to void nucleation, growth and coalescence, the element
vanish technique of Tvergaard (1982) has been successfully employed
in a number of analyses, e.g. Needle- man and Tvergaard (1987),
Tvergaard and Needleman (1993). However, a suitable computational
framework for general separation processes has been lacking and
there has been interest in developing numerical approaches for
general fracture paths, e.g. Belytschko et al. (1994). Provided
that an appropriate decohesion relation is known or can be
developed, the formulation here provides an attractive
alternative.
3 _. PROBLEM FORMULATION
The continuum is characterized by two constitutive relations ; a
volumetric consti- tutive law that relates stress and strain, and a
cohesive surface constitutive relation between the tractions and
displacement jumps across a specified set of cohesive surfaces,
that are interspersed throughout the continuum.
A convected coordinate Lagrangian formulation is employed with
the initial unde- formed configuration taken as reference, so that
all field quantities are considered to be functions of convected
coordinates, J, which serve as particle labels, and time t.
Relative to a fixed Cartesian frame, the position of a material
point in the initial configuration is denoted by x. In the current
configuration the material point initially at x is at X. The
displacement vector u and the deformation gradient F are defined
by
u=x-x, F_%, 8X
The undeformed base vectors in the reference configuration
are
where y is the inverse of the metric tensor
Y,, = g,*g,.
The principle of virtual work is written in the form (Needleman,
1987; Xu and Needleman, 1994),
where s is the nonsymmetric nominal stress tensor, A is the
displacement jump across the cohesive surface, A : B denotes AB,,,
V, Se,, and S,,, are the volume, external surface area and internal
cohesive surface area, respectively, of the body in the reference
configuration. The density of the material in the reference
configuration is p and the traction vector T on a surface in the
reference configuration with normal v is given by
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I400 X.-P. XI and A. NEEDLEMAN
T = V.S. (5)
The volumetric constitutive law is that for an isotropic
hyperelastic solid so that
with MJ, the strain energy density, taken as
IZ = jE : I> : E. (7)
Here. L is the tensor of elastic moduli. and the second Piola
Kirchholf stress. S. and the Lagrangian strain. E. are given by
S=s.F , (8)
E zz i(F. F-1). (9)
where I is the identity tensor, ( ) denotes the inverse. and ( )
denotes the transpose. In component form the moduli are taken to
be
(10)
with E being Youngs modulus and 1 Poissons ratio. Although a
full finite deformation formulation is employed, tinite strain
and
rotation effects are negligible in the circumstances considered.
except very locally where new free surface is being created. The
general features of the overall response are accurately described
by linear isotropic elasticity. For example. the speeds oi
dilatational, shear and Rayleigh surface waves are [see, e.g.
Freund (1990)]
The material parameters are taken to he representati\,c of PMMA
with E = 3.24 GPa, 1 = 0.35 and 11 = 0.001 19 MPa (m s ) . From (I
I ). the dilatational and shear wave speeds are c
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Simulations of fast crack growth 1401
traction across the surface is given by
The specific form used for the potential C#I is one given by Xu
and Needleman (1993) that allows for tangential, as well as normal,
decohesion. Restricting attention to two dimensions,
where n and tare the normal and tangent, respectively, to the
surface at a given point in the reference configuration, and A,, =
n * A and A, = t-A.
In (13).
(14)
where 4,Z is the work of normal separation, 4, is the work of
tangential separation, and AZ is the value of A,, after complete
shear separation with T,, = 0. The normal work of separation, c$!,,
and the shear work of separation, I$,, can be written as
Here, e = exp (I), and gmax and TV,, are the cohesive surface
normal strength and tangential strength, respectively, and 6,, and
6, are corresponding characteristic lengths.
The cohesive surface tractions are obtained from (12) and ( 13)
as
T,,= -?exp(-?)kexp(-$)+z[l-exp(-$)][r-$11. (16)
T,= -f(2~~)${q+(~)$~exp(-$)exp(-$). (17)
Figure l(a) shows the normal traction across the surface, T,,,
as a function of A,, with A, z 0. The maximum value of - T,, is
crmilr and occurs when A,, = 6,,. The variation of T, with A,,
given by (I 7) when A, = 0, is shown in Fig. 1 (b). The maximum
value of 1 T, 1 = z,,,~~ is attained when 1 AI / = @6,/2.
In most of the computations all cohesive surfaces are taken to
have identical cohesive properties. Unless specified otherwise,
these are crrndl = E/10 = 324.0 MPa,
= 755.4 MPa and a,, = 6, = 4.0 x lo- m, so that (15) gives q = 1
with 2:: 4, = 352.3 J rnd2. The remaining parameter in (13), (16)
and (17), r, is taken to be zero. In order to give some indication
of the implications of the cohesive surface characterization for
fracture toughness, we note that for Mode I crack-like behavior
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1402 X.-P. XI and A. NEEl)I.EMAh
1.5~
in plane strain .I,, = &,, (Rice, 196X). Using the relation
K,, = b EJ,_ ( I - 11). the material and cohesive surface
parameters correspond to K,, == 1.14 MPa ,\: 111.
The c~~~t~put~~t~ons are carried out for a center cracked
r~ct~llt~uI~~r block as shown in Fig. 2. Plane strain conditions
are assumed to prevail and ;I Cartesian coordinate system is used
as reference, with the ,I. ,I. plane being the plane oldeformation.
The length ofthe specimen is 3L and the width is I?II,. The tensile
axis is aligned with the I,-direction and a crack of initial length
2~1, lies along the line ,\. = 0. At / = 0. the body is stress free
and at rest. u(J~..I., 0) = 0 and h(~, .j-. 0) = 0. Attention is
restricted to de~orI~~~tioIls that remain symmetric about 3. = 0.
with the region analyscd numcri- ally being J 2 0.
The boundary conditions on .I. = + L are
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Simulations of fast crack growth
t t t t t f
1403
and
where
Fig. 2. Geometry of the center cracked block
uz= s
V+(t)dt, T=O on y*=L (18)
uz = c
V_(t)dt, T =0 on y2 = -L, (19)
V+(f) = i
VI tit,, for I d t, ; v
ir for t > t,,
V-(t) = i
v,t/t,, for t 4 t, ; v
2, for t > t,. (21)
Here, either V, -L - V, or Vz = 0 and the rise time, t,, is
taken to be 0.1 ps. Symmetry about y = 0 requires
rJ=O, 7nz,,O
and the side +v = w remains traction free
(22)
T = 0, 1 = 0. (23)
The initial crack is specified by having J? = 0 as a cohesive
surface for which G,,,, = r max =OforOd~~ Gui.
In order to facilitate interpretation of the results,
dimensional values are used for the material, cohesive and
geometric parameters. Key dimensionless groups include VI/cd,
amirn/E, a,/6,, and L/q,. In linear elasticity, the stress carried
by the loading wave is proportional to V,, so that V,/c,, is a
measure of this stress magnitude relative to E. The ratio of Vi/cd
and amaxlE is a measure of the ratio of the stress carried by the
loading wave to the cohesive surface strength. Since the
formulation contains a characteristic length, the behavior depends
on the ratio of the specimen size to this characteristic length.
Crack-like behavior is obtained when all specimen dimensions
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1404 X:P.XU and A.Nk:EDLEMAN
are large compared to the cohesive surface characteristic
lengths [a factor of IO- IO is needed, depending on the material
constitutive behavior; Needleman (I>OOa,b)]. Various specimen
sizes are considered, but the dimensions are of the order of I mm
or larger ; the smallest specimen dimension is either (/, or L and
+2,, (or I>:ii,,) is in the range 10~lOJ. A measure of the
specimen size divided by a characteristic wa\e speed, say L/cd,
gives a characteristic time for a wave to travel over the specimen.
The time over which crack growth occurs. relative to this
characteristic time, gives an indication of the significance of
wave reflections.
Since both the volumetric and surface constitutive relations are
elastic. no dih- sipative mechanism is incorporated into the model.
Balance of energy then requires
/,.\,T*adS = 1, KdC+[, Wdr+/\ ,,,, ~/~d.S. (24)
where
The work done by the imposed loading is partitioned into kinetic
energy. strain energy stored in the material volume and elastic
energy stored in the cohesive surfaces. Over the course of the
deformation history the relative proportions of the three terms on
the right-hand side of (34) will vary substantially.
3. NUMERICAL IMPLEMENTATION
The finite element discretization is based on linear
displacement triangular elements that are arranged in a
crossed-triangle quadrilateral pattern. Some calculations are
carried out where there is a single cohesive surface. which is
along the line J. = 0 111 front of the initial crack. In most
calculations. however, the cohesive surfaces are all the lines in
the _Y_\. plane defined by the clement boundaries. In the latter
cast, displacement continuity is not required across any element
boundaries. so that. in two dimensions. the number of unknowns is
six times the number of triangular elements. In a standard
displacement finite clement formulation. the number ot unknowns
would be approximately twice the number of elements. Thus. the
present formulation leads to a greatly increased number of
unknowns. However. in the explicit solution of a dynamic problem.
the computer time basically scales with the nunbet of elements.
rather than with the number of unknowns. Numerical experiment\
showed that the increase in computational time over a conventional
displacement finite element formulation was about a factor of two,
with the additional time being mainly due to computing the cohesive
surface contributions to the nodal forces.
Figure 3 sketches how the nodes are connected, once the
discretization has been carried out. The two types of element
intersection are circled in Fig. 3(a). In one case. four
quadrilaterals meet, while the other intersection is where the four
triangles in each quadrilateral meet. Figure 3(b) sketches the
connections that result between nodes at an intersection of
quadrilaterals and Fig. 3(c) illustrates the connections at
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Simulations of fast crack growth 1405
(a)
lb)
(cl
Fig. 3. Schematic showing element arrangements and node
connections m the finite element discretization. (a) Triangular
elements arranged in a crossed triangle quadrilateral pattern. (b)
Node connections at an
intersection of quadrilaterals. (c) Node connections at a
quadrilateral center.
a point where four triangles meet. The separation between
elements in Fig. 3 is for illustrative purposes only. The initial
coordinates of each node at an intersection are identical.
Interpenetration is discouraged by the strong stiffening response
of the normal cohesive traction in compression (16).
When the finite element discretization of the displacement field
is substituted into the principle of virtual work (4) and the
integrations are carried out, the discretized equations of motion
are obtained as
where U is the vector of nodal displacements, M is the mass
matrix and R is the nodal force vector consisting of contributions
from the area elements and the cohesive surfaces. Along each
element boundary the surface integral on the left-hand side of (4)
is computed as described by Needleman (1987).
A lumped mass matrix is used in (26) instead of the consistent
mass matrix, since
-
I406 X.-P XU and A. NEEDLEMAN
this has been found preferable for explicit time integration
procedures. from the point of view of accuracy as well as
computational etficiency (Krieg and Key. 1973). An explicit time
integration scheme that is based on the Newmark /i-method with /i =
0 is used to integrate (26) to obtain the nodal velocities and the
nodal displacements (Belytschko 01 trl.. 1976) via
(-[; /,, I
_ , = M -It,, ,. ( I-
iu,,_, iI:,, I i/
- + -At,, i/ 3 i
(2X)
(29)
Here. the subscripts II and II + I refer to quantities evaluated
at I,, and I,, , , . respectivei!. At each time step, displacements
LJ,,, , are obtained from (27). The volumetric
constitutive updating LISCS (U,,., , - U,,):A/,, to represent
the displacement rate com- ponents. The stress components. 5. and
strain components. I?,,. at I,, , are obtained from
is i I:.,, .s:,i., = s::+ i, A/,,. tic,,),,, , = (fc,),,+- i,
A/t,. (.30)
where CS) i/ and iE,,, i/ are related by the moduli (IO). The
cohesive surface tractions T,, and T, are calculated from ( 16) and
( 17). The force vector K,, , is determined from the left-hand side
of (4) (there are no prescribed tractions on the external surlhcc).
Accelerations and velocities at I,, , arc then obtained from (28)
and (29).
The three meshes shown in Fig. 4 are used in the calculations.
In each case. the mesh consists of a uniform region around the
initial crack tip surrounded by ;I graduated mesh out to the block
boundaries. Rather coarse ~mxl~~s. ax shown in Fig. 4(a) with 40 x
40 quadrilaterals and 12 x 40 squares of side length 0.075 mm, are
used to explore some qualitative features of the behavior of the
model. This mesh has 6400 triangular elements and 38.400 degrees of
freedom. The mesh in Fig. 4(b) consists of 100 quadrilaterals in
the .I,-direction and 40 quadrilaterals in the I,-direction giving
16.000 triangular elements and 96,000 degrees of freedom. The
uniform mesh in front of the initial crack tip is comprised of X0 x
15 rectangles. In most calculations using this resolution. these
are 0.01875 mm x 0.0 I X75 mm squares and so extend I .S mm in
front of the initial crack tip. A 700 x 120 quadrilateral mesh
(336.000 triangular elements and 2,016.OOO degrees of freedom when
aII element boundaries are cohesive surfaces) is shown in Figs 4(c)
and 4(d). In this case. the uniform region also extends 1 .S mm in
front of the initial crack tip. but has 600 x 40 square elements.
with side length 0.0025 mm.
Initially, there is a well-defined crack tip location, namely
the terminus of the interval over which CJ,, ,,,, = T,,,,,, = 0.
Once crack growth initiates, this is no longer the case because of
the continuous dependence of the cohesive surface tractions on the
displacement jump A. For presentation of the results, the largest
value 0f.j. for which
-
a:
(a)
(d)
Fig.
4. M
esh
es use
d fo
r th
e regio
n a
naly
sed in
the c
alc
ula
tions
(~3 >
0).
(a)
A 4
0 x
40 q
uadri
late
ral
mesh
. (b
) A
100 x
40 q
uadri
late
ral
mesh
wit
h 8
0 x
16 u
niform
sq
uare
ele
ments
ahead o
f th
e in
itia
l cr
ack
tip
. (c
) A
700 x
120 q
uadri
late
ral m
esh
wit
h 6
00 x
40 u
niform
square
ele
ments
ahead o
f th
e in
itia
l cr
ack
tip
. (d
) M
esh
near th
e in
itia
l cr
ack
tip
of(
c).
-
Simulations of fast crack growth I409
A,, 3 6,, is recorded together with the current time. This value
of J is denoted by cl and is identified with the current crack tip
position. A quadratic polynomial is fit through three points of the
a versus t curve, say (I,,_, , u,, and a,,, , , and the slope of
this quadratic at t,, is taken as the crack speed at t,,, ci,,.
Note that this procedure does not guarantee that the location so
recorded is continuously connected to the current main crack.
Presuming that there is such a continuous connection (the numerical
results indicate that this is generally the case), the value
recorded numerically cor- responds to the projection of the current
crack tip location on the initial crack line. Some numerical
experiments were carried out using other values of A,, to define
the crack location, e.g. 26,, or 5d,,, and the predictions of crack
location and crack speed were not sensitive to the precise
choice.
4. QUALITATIVE BEHAVIOR
First, we consider plane strain wave propagation in a block
without an initial crack, i.e. N, = 0. Figure 5 shows results from
two calculations using a 40 x 40 quadrilateral mesh. One
calculation is based on an ordinary displacement finite element
formu- lation, with displacement continuity across element
boundaries. In the other case, the cohesive surface formulation is
used with all area elements surrounded by cohesive surfaces. The
calculations are carried out for a block with L = 5 mm and II = 10
mm, undergoing one-sided impact, V, = 0 in (21), with the impact
velocity. V, in (20), equal to 10 m s-. The wave is essentially one
of uniaxial strain, with wave speed 2090 m s- and the stress
carried by this wave is 25 MPa, which is 7.7% of Go,:,\. Curves of
cz2. where cr12 is the axial physical component of Cauchy stress,
versus ~3 along _t. = 2 mm at 1.5, 2.5 and 3.5 ps are shown in Fig.
5. These are taken from contour plots.? The contouring program uses
nodal values of field quantities, which are obtained by
extrapolation from the element integration points to the nodal
points. The extrapolated values associated with all elements
connected to a node are then averaged. The time step used in these
calculations is At = 0.01/1/~.~. where h is the minimum mesh
spacing. This time step is used because accuracy requires small
changes in A,,/6,, and A,/fi, in each time step. Of course, for the
ordinary displacement finite element calculation in Fig. 5(b) much
larger time steps can be taken. In subsequent calculations where a
finer mesh is used, the time steps are taken as AZ = 0. l/?/cd.
Note that regardless of the formulation, there is a numerically
induced smoothing of the leading edge of the wave. However, the
wave speed as calculated from the progression of a stress level of
z 5 MPa or greater is in good agreement with the theoretical value.
Figure 5 shows that as long as the stress levels remain small
compared to the cohesive strength, the predictions using the
cohesive surface model are in excellent agreement with those of a
conventional formulation.
The remaining calculations discussed in this section are for a
block with L = 1.5 mm. II = 3 mm, and an initial crack length a, =
0.3 mm. Deformed finite element meshes for the region analysed
numerically, y 3 0, are shown in Figs 668. These calculations were
carried out using a 40 x 40 quadrilateral mesh, with the
element
t Using the commercial plotting program Tecplot from Amtec
Engineering Inc.. Bellewe. WA
-
1310
diagonals at 45 to the coordiwte :tscs and with each plonc
strain i~rcil clement surrounded by cohesive surface elements.
Clearly. the mesh ib not fine enough to resolve detailed fields
around the crack tip. Nevertheless. the analyses illustrate quali-
tative features of observed failure behavior. Note also that
although crack branching can only take place parallel to the
coordinate axes or at 1-45 to them, the o\er~ll branching angle is
noticeably less than 45 from the .1,-axis.
Figure 6 illustrates the elect of varying the impact velocity.
Symmetric loading ib imposed; in Fig. 6(a). V, = - J., == I.0 m s .
lvhile in Fig. h(b) J, .= ..- 1~: = 15.0 n-l s . The main feature
of the results is that the higher the impact velocity the lea crack
growth ther-c is before branchins. With J., = I .O III s. crack
br~tllcll~i~~ occurs
-
Simulations of fast crack growth
0-3
Fig. 6. Deformed finite element meshes for blocks with L = I.5
mm, IV = 3 mm and N, = 0.3 mm.
Syn nmetric loading with V, = - V2 = 1 m SC. (b) Symmetric
loading with V, = - V, = ISms~ .
-
ta)
tb)
-
Simulations of fast crack growth
fa)
tb)
Fig. 8. Deformed finite element meshes for the blocks with t =
1.5 mm. tt = 3 mm and n, = 0.3 mm. (a) The cohesive strength is
reduced by 10% in y2 > 0 and increased by 10% in J, < 0,
while the cohesive strength along J. = 0 has the standard values
CJ,,, = 324.0 MPa and r,,,;,, = 755.4 MPa. (b) The cohesive
strength has the standard values for 0 < ,I. < I mm and is
increased by a factor of 3 for _I, 3 I mm.
-
1414 X -P. XU and A. KCEDLEMAN
when (I = I .5 mm and the crack speed is 757 m SK (0.81~~) ; the
corresponding values with I, = 15.0 m s- are a = 0.6 mm and 674 m
SK (0.72~~). The crack branching angles, which are taken to be the
angles between the initial crack tip and the point at which growth
on a multi-element segment parallel to the Is-axis begins. are 31
in Figs 6(a) and 6(b). The failure patterns are symmetric about 1.
= 0, although this is not explicitly enforced by the solution
procedure. This shows that the symmetric pattern is stable with
respect to the perturbations induced by the round offcrrors that
occur in a numerical solution. It will be seen that this is not
necessarily the case with the much finer meshes used in the next
section. Additional branching occurs near the free surface, ~3 =
II, particularly with the higher impact velocity in Fig. 6(b). The
triangular elements on the crack surface in Fig. 6(b) are the
result of the start of branches that were not taken. The
qualitative features in Fig. 6 agree well with those seen in Field
(1971) and McClintock and Argon (1966, p. 501).
Calculations with symmetric and asymmetric impact are shown in
Fig. 7. In Fig. 7(a). V, = 5.0 m s- and V, = -5.0 m s- ; while in
Fig. 7(b), V, = IO m s and 1,: = 0. All the parameters in Fig. 7
are the same as in Fig. 6. except for the impact velocity.
Symmetric crack branching occurs at (I = 1.28 mm. which is
intermediate between the two values in Fig. 6. In the calculations,
when growth initiates along several branches, continued growth
takes place on one or more of these branches and the remaining
branch or branches heal. This process occurs very locally, at the
current crack tip. and also on a larger scale. For example. at the
stage shown in Fig. 7(a) crack growth is occurring along four
branches. Subsequently. two of the branches dominate (one for which
_I > 0 and its symmetric counterpart with J. < 0) and the
other two close. The crack branching angle in Fig. 7(a) is about 34
Figure 7(b) illustrates that asymmetric loading can lead to a
strongly asymmetric cracking pattern. There is also some cracking
in Fig. 7(b) that is not connected with the main crack.
Some elects of a spatial variation in cohesive surface strength
are illustrated in Fig. 8. In Fig. 8(a). the cohesive surface along
J. = 0 in front of the initial crack tip has the standard values
listed in Section 2, while cohesive strength is reduced by 10% in
1% > 0 and increased by 10% in _I. < 0. In Fig. 8(b), G,,,.!,
= 324 MPa for 0 < .I. < I .O ~IIN and (T,,,,, = 972 MPa for
J. > 1.0 mm (T,,,:,, is also increased by a factor of 3). The
loading is symmetric in both cases with V, = - C:, = 5 m s . In
Fig. X(a). the amount of crack growth prior to branching is about
the same as in Fig. 7(a). although evidence of an earlier abortive
attempt at branching off the interface at (I = 0.83 mm can be seen.
In Fig. 8(b) the crack grows straight to the interface and then
comes to an abrupt stop.
Curves of crack speed. ir, versus time are shown in Fig. 9 for
the cases shown in Figs 7(a) and 8(b). For the calculation in Fig.
7(a), opening (A,, 3 (j,,) in the earl) stages of crack growth
occurs only for the cohesive surface directly in front of the
current crack tip. Eventually, a stage is reached where A,, 3 ij,,
occurs on the cohesive surfaces at +45 to the current crack plane
as well as directly in front of the crack. The crack continues to
grow straight ahead when this first occurs and the openings at +4S
heal. The first drop in crack speed is associated with this event.
At the next element, the opening directly in front of the current
crack tip heals and growth takes place at ,45 Two curves of crack
speed versus time are shown for this case. One
-
1200.0
T 600.0 E .rn
400.0
200.0
Simulations of fast crack growth
0.0
4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5
t (WI
_
IO
1415
.O
Fig. 9. Curves of crack speed, d. versus time, I, for the cases
shown in Fig. 7(a) and Fig. 8(b). For comparison purposes, a curve
of crack speed versus time is shown for a calculation where crack
growth is confined to 1. = 0. For the case in Fig. 7(a), crack
speeds based on two definitions of crack position are
shown; A,, = 6,, and A,, = 56,,. The dashed line shows the
Rayleigh wave speed.
curve is based on using A,z 3 6,, to define the crack position
(as is the case for all the other curves in Fig. 9) and the other
curve is based on A,, 3 56,, defining the crack position. As is
evident in Fig. 9, the crack speed versus time curve is not
sensitive to this choice. Oscillations in crack speed versus time,
as occur for the case in Fig. 7(a), have been observed by Fineberg
et al. (1992).
For comparison purposes, a curve of crack speed versus time is
shown in Fig. 9 for a calculation where cmax = 324 MPa along y = 0
in front of the initial crack and
~max = 972 MPa for $ # 0, so that the crack is constrained to
grow along the initial crack line. In this case, the crack speed
increases rapidly initially, then rather gradually to about 900 m
ss, and then falls off somewhat when the crack is very near the
free surface ~9 = iv. There are some slight oscillations in crack
speed prior to that, probably due to wave reflections. Abrupt
arrest when the crack in Fig. 8(b) reaches the stronger region is
clearly seen in Fig. 9. The crack then remains stationary until the
stress level becomes high enough for it to propagate through the
stronger material, which occurs at t = 10.4 ps. Abrupt crack
arrest, although in different circumstances from what is modelled
here, was observed by Ravi-Chandar and Knauss (1984a).
Figure 10 shows how the energy of each term in (24) varies as a
function of time for the calculation in Fig. 7(a), which is a case
having uniform properties and symmetric loading with V, = - Vz = 5
m s-. In the very early stages of loading, there is a relatively
equal partition of the work done by the imposed loading into
kinetic energy and strain energy stored in the material. As the
stresses build up, the strain energy increases much more than the
kinetic energy. Energy stored in the cohesive surfaces also
increases and crack growth eventually initiates. The kinetic energy
and cohesive surface energy increase at the expense of the strain
energy after crack branching. The strain energy reaches a plateau
and then decreases with the stress relaxation accompanying the
final stages of failure. As a check, the kinetic
-
1316 X.-F. XU and A. NEEDLEMAK
24o i -..- .- .-. -.-.. Work of imposed loading
energy, the strain energy and the elastic energy stored in the
cohesive surfaces wcr~ computed and, to a very good
~~pl.oxinl~tioll. their sum was equal to the work done by the
imposed loading throughout the deformation history.
5 _. CRACK GROWTH
Figure I1 (a) shows curves of crack speed. ir, versus time for
three c~~lc~ll~iti~~lls where crack growth is confined to the
initial crack plane. The fine mesh in Figs 4(c) and 4(d) is used
for these calculations. but symmetry about ,I. = 0 is imposed so
that only the region ,? 3 0 is analysed numerically. The only
cohesive surface is along j* = 0 and symmetry about j = 0 is
imposed by setting I = (/ = 0. with 6, 7t 0. so that T = 0 on J =
0. Two block sizes are considered. In both cases. 11~ = 10 mm and
(I, = 4.25 mm. For one block L_ = I mm, while f. = 3 mm for the
other. With f_ = I mm. the first loading wave arrives at the crack
plane at / = 0.48 ,us, the second zt t = I .44 J~S and the third at
t = 2.39 ps. When L = 3 mm, the loading wave arrives at I = 1.44
/IS and there are no further wave arrivals over the time interval
considered. The impact velocities are V, = 1Om sag with L = I mm
and both I/, = I5 and 30 m s 1 with L = 3 mm. Calculations for five
impact velocities for the block with II = 10 mm and L = 3 mm are
shown in Fig. I l(b) using the mesh in Fig. 4(b).
In Fig. 1 l(a). when there are no wave reflections (L = 3 mm).
the crack speed increases smoothly to a limiting speed. With V, =
I5 m s . a constant speed. just below the Rayleigh wave speed. is
reached when the crack has grown through 340 elements and the
plateau corresponds to growth through 250 elements (the uniform
region of the mesh extends 600 elements in front of the initial
crack tip). With I, = 30 m s , ;i constant crack speed is reached
after crack growth through 260 elements and this is maintained as
crack growth extends through 330 elements. In this case. however.
the crack speed is 1010 m s-. which is 7.7% above the Rayleigh
W;L\Y
-
Stmulations of fast crack growth 1417
speed. Similar trends are seen in Fig. 1 I(b), with the limiting
crack speeds being slightly slower with the coarser mesh. The
calculations in Fig. 11 (b) show that as the impact velocity
increases, the crack speeds attained get closer together and appear
to be approaching a limiting value. The small amplitude, high
frequency fluctuations in Fig. I l(a) are a consequence of the
discretization and the frequency of these fluc- tuations is lower
in Fig. 11 (b), where the mesh spacing is larger.
The good agreement between the limiting crack speeds in Figs 11
(a) and 1 l(b),
L=Smm, 1000.0 - V,&Omfs 4
L=3mm, V,=lSm/s
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
t (WI
L
1000.0 -
800.0 -
F 5600.0 -
.tU
400.0 -
200.0 -
p v =30&s /,,,,,..,,,,,,,,,,/.,,,,,/.,,
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
t (Id
Fig. Il. Results for calculations where crack growth is confined
to the initial crack plane, _I. = 0. and symmetry about x2 = 0 is
assumed. (a) Crack speed. 2, versus time. 1. with a 700 x 60
quadrilateral mesh for a LU = 10 mmxL = I mm block with V, = IO m
s- and for a IZ = 10 mm XL = 3 mm block with P, = I5 m SK and with
V, = 30 m s-. (b) Crack speed, ii, versus time, 1. with a 100x 20
quadrilateral mesh for a w = 10 mm x L = 3 mm block with I, = 10,
IS, 20, 25 and 30 m s-. (c) J versus time, i, for the cases in (a).
The dashed line in (a) and (b) shows the Ravleigh wave speed, while
the dashed line in (c)
-
141x X.-P. XU and A. NEEDLEMAT\;
600.0 ,,,, ,.,,/I,,,, ,.,,,,, , ,,,/,,,I
L=Smm 500.0 -
400.0
N~300.0 - 2 7 V,=l5m/s
200.0 -
/ i
100.0 -
0.0 "" """'1"""""' "I "4 "' 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
40
t (lls)
(c)
t.q! I I (~~oil//Jr/wr/)
where the node spacings in the miilorm mesh region ditter by 2~
factor of 7.5. show\ that the crack speed exceeding the Rayleigh
wave speed is not an artifact of the spatial discretization.
Furthermore, with I,, = 30 m s . a numerical experiment in\,olving
increasing the time step by a factor of 5 did not result in a
significant change in the crack speed. although the amplitude of
the high frequency fluctuations about the 111eat1 speed did
increase substantially.
That the crack speed can exceed the Rayleigh wa\c speed IS a
consquence of the cohesive surface model. Under quasi-static
loading conditions and for non-vanishing ci,,. decohesion involves
a combination of crack-like propagation and ;I lifting-of mode of
separation. Needleman (1990a). To see the implications of this in
the prcscnt circumstances. consider ;I plane wave impinging on a
cohesi\c surface capable of purely normal separation. q = I. = 0 in
( 16). If the stress cnrricd by the plane \va\c exceeds (T ,,,,,,.
decohesion can occur uniformly along the surface (cdpe clttcts and
stability considerations aside) in a lifting-off mode. so that the
apparent speed 01 propagation along the surface is infinite. With a
crack-like defect. the amount of lif- off increases with the
amplitude of the loading wave and this can act to increase the
speed of propagation when the crack tip position is defined by an
opening displace- ment. At I., = 30 m s . the stress carried by the
loading wave is 231~0 otrr,,,,,,. so that the effects of the finite
strength are not negligible. Evidence for this is seen in the crack
speed versus time curve for the block with L = I mm and I , = IO
111 s . The rclati\cl\, abrupt increase in crack speed at I 2 2.4
jts is associated with the art-i\21 of ;I retlected loading
wave.
Figure 1 l(c) shows curves of Rices (1968) ./-integral \crsus
time. where under dynamic loading conditions the computation of J
involves an area integral as well as a line integral. as in
Nakamura CI N/. (I 985). Here, ./was computed on several contours
ignoring any contribution of the cohesive surface. Suficiently far
away from the current crack tip path independent J values arc
obtained. within 2.5!~1 over the time interval in Fig. I I (c) and
with even less variation in the early stages of crack gro\vth
-
Simulations at fast crack growth 1419
-
X.-P. XEJ and A. NEERL~M~N
Fig. 11. Contours of the axial physical component of Cuuchy
stress. (rll. The extent ol the region s1wu.n is 0. I mm in the
j*-direction and 0.073 mm in the ,v-direction. (a) For the wse in
Fig. 1 I (a) with L = 3 mm and iV, = 30 m s- at f = 3.06 p (b) The
asymptotic field for a crack speed of 910 m \i plotted on the
deformed configuration of Fig. 13(a).
-
Simulations of fast crack growth 1421
Fig. 15. Deformed finite element meshes at three stages of crack
growth for a w = IO mm x L = I mm black with V, = - Vz = IO m s-
using a 700 x I20 ~~adrj~~t~ra~ mesh. The extent of the region
shoxvn is 0.767 mm in the y-direction and 0.4 mm in the
,y-dire&on and the left edge is the initial crack tip
posirian.
(a) At t = 1.50 jts. (b) At t = 1.75 ps. fcf At I = 2.25
#is.
-
1422 X.-P. XU and A. N~~~LE~A~
225 200
17s
150
$25
toa 75
50
25
0
Fg. 17. Contours of the axial physical component of Cauchy
stress. (T:~, over the region in Fig. 15. (a) At f = 1.50 its. (b)
At f = 1.75 ps. (c) At f = 2.25 /IS.
-
Simulations of fast crack growth 1423
200 175
150
125
loo
75
50
25
0
Fig. 18. Contours of the hoop physical component of Cauchy
stress, ~Oo. over the region in Fig. 16. (a) At
r=l.25ps.(b)Att=1.50ps.(c)Att=1.75/~~.
-
I424 X.-P. XU and A. NEEDLEMAN
(4
Fig. 20. Deformed meshes for a II = 10 mm x L = 1 mm block with
V, = - Vz = 10 m s- using a 100 x 40 quadrilateral mesh
illustrating the modes of crack growth with different cohesive
surface orientations. (a) 15 . at / = 1.71 DS, u = 4.58 mm. (b) 30
. at f = 1.6Ops. a = 4.44mm. (c) 45 , at t = 1.64 ps. CI =
4.66mm.
(d) 60 , at t = 2.86 ps, a = 5.62 mm.
-
Simulations of fast crack growth 1425
(within 1% at initiation). With V, = 10 and 15 m SK, crack
growth begins when J has increased to $,,. With V, = 30 m SK, crack
growth begins when J = 346 J me2. which is 1.8% below 4,,. This may
be an effect of the lift-off that occurs at high stress levels. In
any case, J is nearly constant during the early stages of crack
growth and then exceeds 4,,. For the two cases with L = 3 mm, J is
varying while there is a clearly constant value of the crack speed.
Values of J were also calculated for the computations in Fig. 1
l(b), but high frequency numerical oscillations in J were obtained
with the coarser mesh and these masked any trends.
Contours of the axial physical component of Cauchy stress, g2?,
are shown in Figs 12 and 13. Figure 12(a) is for the calculation
with V, = 15 m s- and b = 928 m s- . while V, = 30 m ss and ir =
1010 m SK in Fig. 13(a) (recall that the Rayleigh wave speed is 938
m s-l). For comparison purposes, corresponding contours from the
asymptotic linear elastic field (Freund, 1990, p. 163), are plotted
in the current configuration in Figs 12(b) and 13(b) for crack
speeds of 870 and 910 m SK, respec- tively. These values of crack
speed, with the amplitude of the singular field computed from the
current value of J, were chosen to provide a qualitative, visual
match to the location, shape and orientation of the stress contours
in the range between 100 and 150 MPa. In both cases, the high
stresses very near the crack tip that are predicted by the
asymptotic field are reduced by the lift-off. This is particularly
evident in Fig. 13.
The contour plots in Figs 12 and 13 are taken from the last time
step of each calculation and the fit in Fig. 12 is substantially
better than in Fig. 13. One possible explanation is that at higher
speeds the region of dominance of the singular fields is smaller so
that the role of non-singular terms is greater. Another explanation
for the greater discrepancy in Fig. 13 is that at the higher crack
speed, the time for the singular field to develop is greater.
Although the crack speeds are constant, the stress fields are
evolving and appear to be tending to the linear elastic singular
fields. Approximately at least, there does appear to be a ring
where the stress field is reasonably described by the linear
elastic singular field, but corresponding to a crack speed lower
than the actual crack speed. In conventional linear elastic
fracture mechanics the energy supplied to the crack tip vanishes at
the Rayleigh wave speed (Freund, 1990). If the crack tip stress
field is described by a singular linear elastic crack tip field
with a positive energy release rate, that field must correspond to
one for a crack speed less than the Rayleigh wave speed.
Figure 14 shows a curve of crack speed versus time for a
calculation for the block with L = 1 mm and where all element
boundaries are cohesive surfaces. Symmetric loading with V, = - VZ
= 10 m SK and the fine mesh in Figs 4(c) and 4(d) is used. For
comparison purposes. the crack speed versus time curve for the same
block size and loading conditions is repeated from Fig. 11. Crack
initiation is somewhat delayed because the additional compliance of
the cohesive surfaces somewhat reduces the stress concentration at
the initial crack tip. The crack speed at first increases smoothly.
then an oscillating crack speed versus time curve is obtained. The
crack speed reaches about 420 m s- (0.45~~) before the first large
oscillation in crack speed, which occurs between t = 1.40 and 1.42
ps. This is shortly before the arrival of the reflected stress wave
at t = 1.44 ps. There are then some high frequency, relatively low
amplitude oscillations in crack speed followed by an attempted
branching that gives rise to the large drop in crack speed at t =
1.57 ps. Crack growth then resumes on the initial
-
1426
z600.0 .!% .a
400.0
X.-P. XU and A. NEEDLEMAN
I / I I I c
Constrained crack path
crack plane and the crack speed increases. The crack bifurcates
into two branches at about I = 1.75 ~_ts. In the latter stages of
crack growth the mean crack speed is 2350 m s (~0.37c,). The
oscillations in crack speed for the calculation with unconstrained
crack growth are of larger amplitude and lower frequency than the
discretization induced fluctuations. Typically. from peak to
trough. the crack has grown through two quadrilateral elements.
Three stages of crack growth are shown in Fig. 15. Crack growth
is initially straight and the stage shown in Fig. 15(a) is after
the first large oscillation in the crack speed versus time curve in
Fig. 14. There is an initial attempt at branching off the crack
plane that results in a slight asymmetry about J. = 0. even though
the block con- figuration and loading are symmetric about this
axis. This unsuccessful attempt at branching results in the
protuberance on the lower crack surface in Fig. 15(b). Crack
branching is not quite symmetric, but for both branches the angle
of crack branching is about 29 in Fig. 15(c). After branching. the
crack path changes direction to become more or less parallel to the
j,-axis. as in Figs 6 and 7(a). The initiation of additional
branching can be seen in Fig. 15(c), particularly in the lower
branch.
Figure 16 shows the current crack tip configuration at three
stages of deformation. In Figs 16(a) and 16(b), the crack growth
continues straight ahead. although openings at k45 to the current
crack plane are evident. Figure 16(b) shows that the crack
continued to grow along its initial line and that some of the
micro-cracks in Fig. 16(a) have healed. Figure 16(c) is near the
beginning of crack branching. Some micro- cracking not directly
connected to the current crack tip can be seen in Fig. 16(c).
In Fig. 17. contour plots of the axial physical component of
Cauchy stress, (T:?. are shown at the same three stages of crack
growth and to the same scale as in Fig. 15. The first stage is
before crack branching, and the expected shape of the elastic
singular field is evident. Even though the amount of crack
branching is small in Fig. 17(b).
-
Fig. 16. Deformed meshes near the current crack tip at three
stages cfcrack growth for 8 M = t 0 mm x t = I mm block with I/, -I
- tl; = 10 m s-l using a 700 x IX! mesh. The extent of the region
shawn is 0.07h7 mm in the ,v-directian and 0.04 mm in the
)--dire&cm. (aa) At 2 = 125 $s, (b) At I = 1.50 jw. Ic) At
f = 1,75 ps.
-
14% X.-P. XU and A. IcEEDLEMAI\;
the surrounding stress field is affected. After substantial
branching. Fig. 17(c). local high stress fields have developed near
the tip of each branch. However. the beginning of further branching
can be seen at this stage.
The contour plots of (TV),,, the hoop stress component in a
polar coordinate system centered at the current crack tip position
with 0 measured from the J-axis. in Fig. IX show the stress
distribution near the current crack tip (the same region as shown
in Fig. 16). At the stage shown in Fig. IS(a), the crack speed
VU-sus time curve is still smooth and crack growth is straight
ahead. Figure 1 X(b) is the same stage of crack growth as Fig.
17(a) and there is crack growth along several directions lrom the
current crack tip. It appears that each branch is essentially
growing in a Mode I fashion. However. the very local details are
not resolved by the discretization used here. For example, the
extensions at the tip of the main crack extend only one or two
elements and with the constant strain triangles. the detailed
stress distribution around the new branches is not resolved. In any
case, crack growth immediately following the stage in Fig. 18(b) is
straight ahead and the additional branches close. Figure IX(c) is
the same stage as Fig. 17(b) ; crack growth now continues along the
branches off the initial crack line and the opening directly in
front of the main crack closes. Some opening of the cohesive
surfaces can be seen in the region around the main crack tip as a
consequence of the relatively high stresses that occur away from
the main crack tip.
In all the calculations so far. the cohesive surfaces not along
the coordinate axe\ have been taken to be at + 45 to them. Figures
19 and 20 show the effects of varying this angle. Curves of crack
speed versus time for cohesive surfaces at IS . 45 and 60 to the
initial crack line are shown in Fig. 19. Figure 20 shows the modes
of crack growth for four cases; 15 , 30 . 45 and 60 cohesive
surfaces (for clarity, curbus of crack speed versus time are only
shown for three of these cases in Fig. 19). In these
800.0 -
T600.0 - g .CU
-
Simulations of fast crack growth 1429
calculations L = 1 mm, the loading is symmetric with V, = - VZ =
10 m s-r and a 100 x 40 quadrilateral mesh was used. For the cases
with 15 and 30 cohesive surfaces, the crack grows in a zigzag mode
from the point of initial crack growth. All three cohesive surfaces
at the initial crack tip open, but crack growth eventually occurs
on only one of the cohesive surfaces off the initial crack plane,
and healing takes place on the other two. Initial crack growth is
straight ahead for the 45 and 60 cases. Since the geometric
configuration and loading are symmetric, the asymmetry in the mode
of crack growth is due to the very small perturbations that are
inevitable in a numerical solution. For the cases with 15, 30 and
45 cohesive surfaces, the onset of crack bifurcation occurs at a =
4.58, 4.44 and 4.66 mm, respectively, and the corresponding crack
speeds are ci = 725, 553 and 654 m s-. Note that the crack
branching point does not vary monotonically with cohesive surface
orientation. Also, it appears that crack bifurcation is imminent in
the 60 case as well, but the stage of deformation shown in Fig.
20(d), at a = 5.62 mm, is near the end of the uniform mesh region.
The crack speed versus time curve in Fig. 19 for the case with
cohesive surfaces at 15 shows lower amplitude oscillations with
crack zigzagging and higher amplitude oscillations associated with
crack branching. A similar curve was obtained for the calculation
with cohesive surfaces at 30.
Additional calculations were carried out to explore the effect
of varying various parameters. For the case with cohesive surfaces
at 30 a calculation was carried out with V, = 5 m sP . Crack growth
in a zigzag mode occurred as seen in Fig. 20, but crack branching
was delayed until LZ= 4.60 mm. A calculation was also carried out
for the 30 case and V, = IO m s-l with the cohesive surface
parameter P = 0.5, as opposed to r = 0, which was used in all other
calculations. There was no change in the qualitative response ;
branching was slightly delayed from a = 4.44 to 4.48 mm. Finally,
one calculation with 45 cohesive surfaces was carried out with
T,,,~ halved, so that q = 0.5. Again, the qualitative response was
the same, but crack branching occurred somewhat sooner, at CI =
4.60 mm, instead of at CI = 4.66 mm as in Fig. 20.
6. DISCUSSION
When crack growth is confined to the initial crack plane, and
over a time scale before wave reflections from the block boundaries
reach the crack, the crack speed reaches a plateau that increases
with increasing impact velocity and that appears to be reaching a
limiting value for large impact velocities [Fig. 11 (b)]. What is
surprising is that this plateau can exceed the Rayleigh wave speed.
This can be understood as a consequence of the finite strength of
the cohesive surface; for V, = 30 m s-, the stress carried by the
loading wave is 23% of urnax and the separation mode is a
combination ofcrack-like propagation and lift-off. It seems
plausible that the Rayleigh wave speed limit would be obtained as
crmax ---f cc, because for Mode I crack-like behavior conventional
fracture mechanics corresponds to the limit cmax -+ co and 6,, -+ 0
(with & remaining finite). It is interesting to note that crack
speeds inferred from linear elastic singular fields, as in Figs 12
and 13, are less than the Rayleigh wave speed. Thus, the present
results suggest the possibility that crack speeds determined
-
I430 X.-P. XU and A. NEEDLEMAN
experimentally from a measure of crack position versus time and
those determined from a measure of the crack tip stress field could
differ.
The calculations provide a rationale for the result that very
high crack speeds. approaching 90% of the Rayleigh wave speed. have
been measured in -anisotropic solids, but that limiting crack
speeds in isotropic solids are closer to 50% of the Rayleigh wave
speed (Field, 1971). The cohesive surface formulation indicates
that what matters is the orientation dependence of strength, with
the initial crack plane being weaker than alternative cleavage
planes, so that crack growth off the initial crack plane is
suppressed. or at least delayed. Similarly, the observation that
very fast crack speeds can be obtained in isotropic solids when the
initial crack plane is intentionally weakened (Lee and Knauss,
19X9) is consistent with the present results.
For fast cracks very large stresses occur over some distance
around the crack. Some microcracking unconnected to the main crack
is seen in the computations even though the cohesive properties
have been taken to be uniform. lf some statistical distribution of
defects were included in the problem formulation, more extensive
microcrackins would be expected.
Yoffe (1951) observed that the hoop stress maximum shifts to
about 60 from the initial crack plane when the crack speed exceeds
O.~C,. This has been identitied as the speed for crack branching
given by linear elastic fracture mechanics. so that the general
observation that crack branching and crack surface roughening are
associated with crack speeds 20.4~, has been unexplained. The
picture of crack growth that emerges from the calculations here is
as if the crack were performing a stability test at each step. Some
crack growth occurs along each cohesive surface emanating from the
current crack tip. A point is reached at which one or more of the
incipient cracks continue to grow and the remaining unload and
close. The oscillations in the crack speed versus time curves occur
shortly before crack branching and are not associated with any
dramatic change in the mode of crack growth; crack extension
appears to slow down as growth occurs along several alternative
branches and then speed up when the crack extends along one or more
of these branches and the others heal. This scenario suggests that
the crack speed oscillations are associated with the resistance to
crack growth being nearly the same for straight ahead growth as for
growth along the inclined branches. Indeed, Rice ct (I/. (1994)
have recently observed that the analyses of Eshelby (1970) and
Freund (1972) indicate that there is enough energy available to
create two crack surfaces as was available to create one when the
crack speed is about 0.45~,,.
Once the point of incipient branching is reached, the branching
process can be sensitive to small perturbations. For example, the
very small asymmetries induced by the numerics leads to the
somewhat asymmetric mode of crack growth in Fig. 15. The
sensitivity to numerical perturbations is more pronounced in Fig.
20. For the cases with cohesive surfaces at + 15 and + 30 to the
coordinate axes, crack growth initiates in a zigzag mode, even
though the block configuration and the loading are symmetric.
Within the context of the cohesive surface model. the tendency to
zigzag depends on the orientation of the cohesive surfaces relative
to the initial crack line. The crack growth mode prior to branching
is much like the wavy crack mode discussed by Gao (1993).
The oscillations in the crack speed versus time curve in Fig. 14
are very much like
-
Simulations of fast crack growth 1431
those observed by Fineberg et al. (1992). The correlation found
here between these oscillations and incipient crack branching is
consistent with the relation between crack speed oscillations and
fracture surface roughness found by Fineberg et al. (1992),
although the calculations are two dimensional while the roughness
in the experiments was fully three dimensional. It is expected that
a corresponding three dimensional analysis would lead to
non-uniform growth in the crack plane as well as out of it. Rice et
al. (1993) carried out a perturbation solution for a three
dimensional model problem with a single displacement variable that
satisfies the wave equation. Oscil- lations in response arise from
pre-existing heterogeneities in fracture resistance on the crack
plane. This is different from the situation here where oscillations
in crack speed occur with spatially uniform properties (but
separation is limited to discrete surfaces).
When discretized, the present formulation leads to a set of
equations that can be regarded as representing a collection of
nodal points connected by springs. Viewed this way, it bears some
relation to discrete models of macroscopic cracks. Discrete lattice
models have been used to address issues of fast fracture in brittle
solids, e.g. Kulakhmetova et al. (1984) and Marder and Liu (1993).
Marder and Liu (1993) have shown that straight ahead steady state
crack growth in a triangular lattice becomes unstable at a critical
crack speed. Above this speed, the solution involves the breaking
of bonds off the initial crack line. Although such discrete models
give a range of possible paths for defect growth, the extent to
which they provide an accurate rep- resentation of macroscopic
cracks is questionable. The cohesive surface formulation includes a
description of the separation process within a framework that
allows the continuum stress and deformation fields to be resolved.
In this regard, it is interesting to note that some qualitative
features of fast crack growth in brittle solids are repro- duced by
analyses (Figs 6-8) that clearly do not resolve the details of the
crack tip fields. However, there are important features, e.g. the
crack speed at which branching occurs and the average crack speed
after branching, that do require more detailed resolution.
Continuum formulations are available, such as the cleavage grain
formulation in Tvergaard and Needleman (1993) or the cell model of
Broberg (1979), where brittle fracture involves the loss of stress
carrying capacity over a volume (or area). In particular, Johnson
(1992, 1993) has carried out finite element calculations of rapid
crack growth in brittle solids using a cell damage model where the
linear elastic stiffness tensor is multiplied by a factor that is a
function of the relative density and that decreases from unity to
zero with decreasing density. The physical basis for this damage
model is not clear. A limiting crack speed is found in a number of
the calculations, but interestingly not when softening is allowed
in only one row of elements. Crack branching is also found to occur
for certain parameter values. However, in the implementation in
Johnson (1992, 1993) the cell size is tied to the finite element
mesh size, so that there is no way to distinguish between cell size
dependence and mesh size dependence. In contrast, in the cohesive
surface framework, the characteristic length associated with the
cohesive surfaces is independent of the discretization, and the
finite element size and the cohesive surface spacing can be varied
independently.
In principle, the extension of the present formulation to three
dimensions is straight- forward. although the increase in the
computational resources needed would be
-
I431 X-P. XII and A. NEEDLEMAN
substantial. The two dimensional calculation in Figs 14-I X
using the 700 x 120 quadri- lateral mesh and having 2016,000
degrees of freedom took 30.914 time steps and required about 22.5 h
of CPU time on a Gray C90 computer with the code running at 265
MFlops. A similar calculation using a mesh with 100 x 40
quadrilateral elements and having 96.000 degrees of freedom took
4200 time steps and required about X min of CPU time for the same
amount of crack growth. The computer time could be reduced somewhat
by having some, rather than all. element boundaries as cohesive
surfaces. Also, additional code optimizations are most likely
possible.
For quantitative predictions, an accurate characterization of
the cohesive properties of the material is required. While
difficult in practice, what is needed is conceptually clear. What
is more difficult to quantify. except for crystalline solids or for
interfacial fracture, is the geometry of the cohesive surfaces. The
computations indicate that some features of the response. e.g.
crack branching. are not particularly sensitive to this
characterization, but others, such as crack zigzagging. are. Also.
while a systematic mesh refinement study was not carried out. the
agreement between calculations using various degrees of mesh
resolution suggests that the finer meshes do accurately represent
the initial-boundary value problem solutions. For example. even
though the cohesive surfaces in Fig. 15 are parallel to the
coordinate axes or at i45 to them. the crack branching angles are
about 29 Furthermore, the crack branching angle in this fine mesh
calculation is not very different from those in the coarse mesh
cal- culations in Figs 6 and 7(a).
The computational framework involves the explicit solution of
the equations of motion. The increase in the number of equations
associated with the additional finite element nodal points in the
present formulation would make a quasi-static solution based on a
direct solution of a system of linear equations prohibitively
expensive. However, quasi-static crack growth phenomena could be
investigated using dynamic relaxation or other iterative
methods.
In the computations. both the material stress ~strain response
and the cohesive surface separation law have been taken to be
elastic. The results show that many characteristic features of fast
crack growth in brittle solids occur in such a conservative system.
In the calculations, the microcracking and attempted branching
prior to the actual onset of branching are suggestive of the
mirror. mist, hackle fracture surface evolution often seen in fast
brittle fracture. e.g. Field (1971). Ravi-Chandar and Knauss
(1984b). However, due to the elastic characterization of the
cohesive surfaces much complete crack healing occurs. A more
realistic model would account for time dependence of the separation
process, which would preclude complete crack healing after some
opening time. Also. in the computations the only interaction
between material elements is through the initial cohesive surface.
Large separations can occur and, effectively, free bodies can be
created as fracture progresses. Here, the motion of such free
bodies is unimpeded by the surrounding material. Computation of
extensive fracturing and. in particular. of the transition from
cracking to frag- mentation would need to account for such
interactions.
In the cohesive surface formulation, the fracture resistance
ofthe material is charac- terized by the geometry and properties of
a set of cohesive surfaces. Calculations carried out within this
framework provide a unified description of a broad range of
observed phenomena characteristic of fast crack growth in brittle
solids. They exhibit.
-
Simulations of fast crack growth 1433
in detail, the relation between the mode of crack growth and
crack speed. The present formulation, perhaps involving other
cohesive surface and material constitutive relations, can be used
to investigate a variety of fracture processes that have not, at
least readily, been amenable to treatment within a conventional
fracture mechanics framework.
ACKNOWLEDGEMENTS
This work was supported by the National Science Foundation
through grant DDM-9016568. We are grateful to Professor L. B.
Freund of Brown University for helpful discussions. A.N.s work on
this topic was stimulated by participation in the program on
Spatially Extended Nonequilibrium Systems at the Institute for
Theoretical Physics, University of California. Santa Barbara during
October-November 1992. The computations were carried out on the
Cray C90 computer at the Pittsburgh Supercomputing Center.
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