10-Numerical simulations of fast crack growth in brittle solids.pdf

Pergamon

J. Mech. Pl~.w. Solid.~, Vol. 42, No. 9, pp. 1397-1434, 1994 Copyright X; 1994 Elsevier Science Ltd

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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 observations that are not adequately accounted for by current theory. For example, observations 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

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 formulation 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

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 constitutive 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 undeformed 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

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


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

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

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

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


(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 components. 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)

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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 corresponds 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 formulation, 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 qualitative 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


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)


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


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 fluctuations 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 displacement. 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).


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.


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.


(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, respectively. 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 configuration 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 microcracking 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


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


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 quadrilateral 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 calculations 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 characterized 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.


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.

REFERENCES

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