Simulation of dynamic crack growth using the generalized

Int J Fract (2007) 143:79–102DOI 10.1007/s10704-007-9051-z

ORIGINAL PAPER

Simulation of dynamic crack growth using the generalizedinterpolation material point (GIMP) method

Nitin P. Daphalapurkar · Hongbing Lu ·Demir Coker · Ranga Komanduri

Received: 4 August 2006 / Accepted: 5 January 2007 / Published online: 14 February 2007© Springer Science+Business Media B.V. 2007

Abstract Dynamic crack growth is simulated byimplementing a cohesive zone model in the gen-eralized interpolation material point (GIMP) met-hod. Multiple velocity fields are used in GIMP toenable handling of discrete discontinuity on eit-her side of the interface. Multilevel refinementis adopted in the region around the crack-tip toresolve higher strain gradients. Numerical simu-lations of crack growth in a homogeneous elas-tic solid under mode-II plane strain conditions areconducted with the crack propagating along a weakinterface. A parametric study is conducted withrespect to varying impact speeds ranging from 5 m/sto 60 m/s and cohesive strengths from 4 to 35 MPa.Numerical results are compared qualitatively withthe dynamic fracture experiments of Rosakis et al.[(1999) Science 284:1337–1340]. The simulationsare capable of handling crack growth with crack-tip velocities in both sub-Rayleigh and interson-ic regimes. Crack initiation and propagation arethe natural outcome of the simulations incorporat-ing the cohesive zone model. For various impactspeeds, the sustained crack-tip velocity falls eitherin the sub-Rayleigh regime or in the region be-tween

√2cS (cS is the shear wave speed) and cD (cD

N. P. Daphalapurkar · H. Lu · D. Coker ·R. Komanduri (B)School of Mechanical and Aerospace Engineering,Oklahoma State University, Stillwater, OK 74078,USAe-mail: [email protected]

is the dilatational wave speed) of the bulk mate-rial. The Burridge–Andrews mechanism for transi-tion of the crack-tip velocity from sub-Rayleigh tointersonic speed of the bulk material is observedfor impact speeds ranging from 9.5 to 60 m/s (fornormal and shear cohesive strengths of 24 MPa).Within the intersonic regime, sustained crack-tipvelocities between 1.66 cS (or 0.82 cD) and 1.94 cS(or 0.95 cD) were obtained. For the cases simulatedin this work, within the stable intersonic regime,the lowest intersonic crack-tip velocity obtainedwas 1.66 cS (or 0.82 cD).

Keywords Cohesive law · GIMP · Mode-II ·Dynamic crack propagation · Intersonic ·Burridge–Andrews mechanism

1 Introduction

Dynamic fracture has been an area of sustainedinterest with the aim of understanding various phe-nomena in engineering and geosciences. Linearelastic fracture mechanics (LEFM) has been app-lied extensively to crack dynamics. However,LEFM has been effective mainly for brittle fracturephenomena and not so in dealing with materialsundergoing ductile fracture. More particularly, itdoes not provide a physical description on how thecrack process zone is formed and how its state aff-ects the crack propagation phenomena (Yang and

80 N. P. Daphalapurkar et al.

Ravi-Chandar 1996). Dugdale [1960] and Barenb-latt [1962] introduced a methodology to explicitlymodel the crack process zone by considering crackopening displacements and tractions over the cracksurface. This approach is applicable for both duc-tile and quasi-brittle materials and independentof the microstructure at the crack-tip. As far asnumerical implementation is concerned, the intro-duction of the cohesive element methods (Nee-dleman 1987; Camacho and Ortiz 1996) provideda detailed force-separation relation as well as thecohesive strength in the process zone. Their appli-cations have resulted in satisfactory explanationsof numerous phenomena related to crackdynamics.

In this investigation, we use the generalized int-erpolation material point (GIMP) method, intro-duced by Bardenhagen and Kober [2000] whichis an enhancement of the material point method(MPM), originally developed by Sulsky et al. [1995].The GIMP method is a further development ofMPM to eliminate certain artificial numerical noiseassociated with material points just crossing theborders of background cells in the original MPM.As a result, GIMP gives essentially the same res-ults as finite element method (FEM) gives underrelatively small deformations but still works satis-factorily for large deformations (Ma et al. 2005).This numerical method uses the material points(Lagrangian description) for representing mate-rial continuum and the background fixed mesh(Eulerian description) for solving field equations.

Since GIMP uses combined Lagrangian andEulerian descriptions, it does not use body-fixedmesh for computation. Consequently it offers anadvantage in the simulation of some dynamic prob-lems over finite element and meshless methods(Nairn and Guo 2005). Multiple velocity-field tech-nique (Hu and Chen, 2003; Nairn and Guo, 2005)can be used to model discontinuities in the dis-placements and velocities across the propagatingcrack surfaces. In this paper, a methodology incor-porating cohesive zone model in GIMP is pre-sented to simulate the damage zone at the crack-tip. A parametric study is carried out by varyingthe impact speeds and the cohesive strengths. Theresults obtained for one of the cases are com-pared with the experimental results of Rosakis

et al. [1999] and good qualitative agreement wasfound.

2 Background

The past decade has seen advances in the fieldof dynamic fracture, including experimental andnumerical observations, determination of limitingvelocities for crack propagation in different modes,allowable speed regimes for crack propagation,intersonic crack propagation, and sub-Rayleigh tointersonic transition mechanism. The research hasbeen progressing towards the development of aphenomenological model as well as a unifying the-ory that incorporates the physical basis at all scalesfrom atomistic to continuum, and even extendingto geological scales.

Majority of studies involving size effects werecarried out using molecular dynamics (MD) sim-ulations. For example, Abraham and Gao [2000]performed MD simulations of mode-II crack prop-agation along a weak interface between two har-monic crystals. Their simulations showed that ashear dominated crack initially accelerates to theRayleigh wave speed, followed by the nucleationof a daughter microcrack ahead of the main crack,and finally coalescence of the mother and daugh-ter-cracks with the crack-tip velocity reaching avalue as high as the longitudinal wave speed. Aftercoalescence, when the far-field loading is relaxed,the crack decelerates and propagates at a steadyrate close to a speed of

√2cs, where cs is the shear

wave speed of the bulk material. Abraham andGao [2000] also reported that the essential fea-tures in the results obtained from MD simulationsmatched well with the experimental observationsreported by Rosakis et al. [1999] as well as withthe results of cohesive modeling (Needleman 1999)even though they were results obtained at differentscales.

Even though large scale simulations of up toone-billion atoms have been reported using mas-sive parallel processing [Abraham and Gao, 2000],most MD simulations reported in the literature arestill somewhat limited in their size (a few thou-sand atoms). As a result, multiscale simulationapproaches (see Lu et al. 2006) requiring seamless

Simulation of dynamic crack growth using GIMP method 81

coupling between atomistic and continuum havebeen emerging. The applicability of these meth-ods, in general, and the development of continuummethods containing atomistic information in par-ticular are of great interest to researchers. Bothmicroscopic scale and continuum scale simulationsare necessary to reflect appropriate behavior ofdynamic fracture since dominant physicalprocesses occur at both these scales. They includemechanisms seen in ductile materials, such as voidformation, their coalescence into microcracks andsubsequent growth into macrocracks, phase trans-formations, and friction at the asperity level involv-ing junction formation and collapse. In all thesecases, continuum simulations of dynamic fracturerequire a unifying law that will take into accountthe physics of deformation as well as the lengthscale phenomena.

A critical contribution to the computational fra-cture mechanics has been the development andapplication of cohesive element methods (Needle-man 1987; Tvergaard and Hutchinson 1993; Xuand Needleman 1994; Camacho and Ortiz 1996;Leonov and Panasyuk 1998). Modeling using thesemethods enables simulation of a nonlinear zone atthe crack-tip. The modeling of complex fracturephenomena, such as crack branching, kinking isstill under development. For instance, using a cohe-sive element network, there is an artificial soften-ing of material properties as the size of cohesiveelements decreases (Falk et al. 2001).

Belytschko and co-workers (Moes and Belyts-chko 2002) developed a new class of cohesiveelements which allowed the crack to propagatethrough an element instead of just at the elementboundaries and termed it “the extended finite ele-ment method (XFEM).” Thus, constraint in thedirection of crack motion is relieved. Furthermore,Belytschko et al. [2003] developed a methodologyto switch from a continuum to a discrete discon-tinuity based on loss of hyperbolicity of the gov-erning partial differential equations. They used theXFEM to deal with the discontinuity.

Gao and Klein [1998] used the virtual internalbond (VIB) method for crack growth simulation.In their model, randomized cohesive bond inter-actions are incorporated into the constitutive lawand cohesive interactions are assumed betweenthe material particles. Gao and Ji [2003] used VIB

method to model nanomaterials and demonstratedthat at a critical length scale of one nanometer,there is a change in the fracture mechanism fromthe classical Griffith fracture to one involving hom-ogenous failure close to the theoretical strength ofsolids. For this transition, they replaced the classi-cal singular deformation field near a crack-tip by auniform stress distribution with no stress concen-tration near the crack-tip.

Considerable amount of analytical, experimen-tal, and numerical work has been reported in theliterature on intersonic crack growth (Gao et al.2001; Huang and Gao 2001; Klein et al. 2001;Rosakis 2002; Guo et al. 2003; Hao et al. 2004; Xiaet al. 2004, 2005). The term “intersonic speed” isreferred to as the crack-tip velocity between shearand dilatational wave speeds of the material, whilethe term ‘sub-Rayleigh speed’ is referred to asthe crack-tip velocity less than the Rayleigh wavespeed of the material.

Washabaugh and Knauss [1994] observed thatthe sustained mode-I crack-tip velocity is alwaysless than the Rayleigh wave speed of the material.Ravi-Chandar and Knauss [1984a,b] observed mi-crocracks on the trailing crack surfaces and pointedout that this crack branching was responsible forlimiting the crack-tip velocities. They designed afracture specimen with a weak plane possessingvarying bond strengths so that the mode-I crackwas guided to propagate along the weak plane.They concluded that for mode-I, the crack-tip veloc-ity approaches asymptotically the Rayleigh wavespeed of the material in the limit of vanishing bondstrengths.

Andrews [1976] presented numerical work onplane strain, mode-II dominated shear cracks andconcluded that the crack would rapidly accelerateat first toward the Rayleigh wave speed and thenafter a short period of adjustment would start prop-agating at a speed close to but greater than

√2cS.

Following Andrews [1976] work, Burridge et al.[1979] studied the steady motion of a semi-infinitemode-II shear crack driven by a moving point loadwhich remains at a constant distance d from thecrack-tip. Based on their analysis, they reportedthat for very large values of d, the sub-Rayleighregime and the regime between shear wave speed,cS and the transition speed of

√2cS is unstable for

crack propagation.


Broberg [1989] used energy considerations andconcluded that the crack-tip velocity regime bet-ween cS and dilatational wave speed, cD is forbid-den for mode-I and the regime between cR and cS isforbidden for both mode-I and mode-II propagat-ing cracks due to negative energy release rate inthese regions. Freund [1979] used an asymptoticanalysis for the steady state mode-II intersoniccrack and determined that

√2cS is the only sta-

ble intersonic crack propagation velocity. Rosakiset al. [1999] conducted experiments on fracturespecimen with a weak plane successfully constrain-ing a shear dominated crack to propagate along thisweak plane under remote loading conditions. Theyobserved crack-tip velocities in intersonic regimewith crack pattern featuring shear shock waves anddiscussed the significance of the intersonic crackpropagation at preferred velocity of

√2cS which is

consistent with the analysis by Freund [1979].Although considerable work has been focused

on rate-independent cohesive zone models, Kna-uss [1993] presented a rate-dependent cohesivesurface relation incorporating viscoelasticity andstudied crazing in thermoplastic polymers. Sincethen, many researchers have been working on rate-dependent cohesive zone models for analytical(Samudrala et al. 2002) and numerical solutions(Nguyen et al. 2004). Cohesive zone models areanticipated to have a significant potential in dyn-amic fracture modeling including, fragmentation(Cirak et al. 2004), crack branching (Xu and Nee-dleman 1994; Belytschko et al. 2003), andkinking (Borst et al. 2006). At larger scales, geo-physicists model earthquakes as shear cracks prop-agating along tectonic plates with faults, or weakinterfaces (Xia et al. 2005). Depending on thelocal topography and the geological age of a fault,the wave speeds across a fault or inhomogeneitywould vary. Thus the fault is modeled as a weakplane along which mode-II cracks would dynami-cally propagate (Rosakis 2002). This is of particularinterest to geophysicists who investigate interson-ic fault rupture during shallow crustal earthquakeevents. Preferred weak planes exist even in earth’scrust in which the dominant fault motion is unsta-ble shear crack growth between tectonic platesduring earthquakes.

Recently a computational method, known asthe material point method (MPM), was developed

by Sulsky (Sulsky et al. 1995) for solving solidmechanics problems of dynamic nature. MPMevolved from the particle-in-cell method (Harlow1964; Brackbill et al. 1988), developed specificallyfor fluid mechanics. MPM has combined the advan-tages of both Eulerian scheme (provided by thegrid) and the Lagrangian scheme (provided by thematerial points). The grid is usually held fixed andused to determine spatial gradients and for solv-ing field equations. Irregular transitional grid canbe used to improve the accuracy in the case ofhighly localized strain gradients (Wang et al. 2005).The material points are convected by the deforma-tion of the solid throughout the background gridand are not subjected to mesh tangling. Addition-ally, incorporation of various constitutive modelsin MPM is relatively straightforward. In the casewhere the material points move in a single-val-ued velocity field, non-slip contact between thesurfaces can be handled automatically.

MPM has been applied to analyze a wide rangeof problems, including granular media (Barden-hagen et al. 2000), and dynamic material failure,such as spall failure in brittle materials using dec-ohesion constitutive model (Sulsky and Schreyer2004). Wieckowski [2004] demonstrated MPM forlarge strain engineering problems involving gran-ular flow in a silo, plastic deformation processes,such as extrusion and metal cutting. Nairn and Guo[2005] applied MPM for fracture problems in brit-tle materials. They used multiple velocity fields atnodes and allowed MPM to handle cracks by intro-ducing a discontinuity in the single-valued velocityfield. In their study, the crack propagation is basedon critical stress intensity factors and energy cri-teria. They reported that MPM is able to modelexplicit cracks very well and also passes the “crackpatch” test with good accuracy.

MPM, unfortunately, gives stress sign revers-ing problem (or numerical noise) when materialpoints just cross the border of cells. To resolve thisproblem, Bardenhagen and Kober [2000] intro-duced a new methodology involving generalizationof MPM using a variational form and incorporat-ing a Petrov–Galerkin discretization scheme. Theycalled this the generalized interpolation materialpoint (GIMP) method. The numerical noise pres-ent in MPM, as a result of material point cross-ing the cell boundaries, is eliminated by using C1


continuous shape functions. Ma et al. [2006]developed a structured mesh refinement techniquefor GIMP to model effectively regions with highstress gradients. GIMP has also been used to dem-onstrate large deformations using nanoindentation(Ma et al. 2005), mode-I and mode-II crack propa-gation problems in coupled MD/GIMP simulations(Ma et al. 2006), and foam densification problems(Brydon et al. 2005).

In this paper, we implement cohesive zonemodel in GIMP method to model crack propag-ation. This method can deal with cracks in bothductile and brittle materials. Additionally, the inc-orporation of a characteristic length scale in thecohesive zone model allows investigation of phy-sics-based (Klein et al. 2001) crack propagationbehavior.

3 Modeling crack propagation in GIMP usingcohesive zone model

3.1 Generalized interpolation material pointmethod (GIMP)

As stated earlier, generalized interpolation mate-rial point (GIMP) method (Bardenhagen andKober 2000) is a further development of the mate-rial point method (MPM) (Sulsky et al. 1995), inc-orporating C1 continuous shape function. For thepurpose of completeness, we present in somedetail their generalized interpolation materialpoint (GIMP) method.

In GIMP, material continuum is discretized intoa finite collection of N material areas (2D), or vol-umes (3D) as shown in Fig. 1. For convenience,each material volume is represented by a circleonly as a schematic representation. The actual

volume is rectangular (or square here) in thereference configuration and quadrilateral in thedeformed state with its center located at the centerof the circle. Each material volume is assigned amass (mp, where p = 1,…, N) consistent with thematerial density and volume (Vp) of the point, andall other variables, such as position, acceleration,velocity, strain and stress, and temperature. Phys-ical variables carried by the points are projectedonto the background grid, and motion and energyequations are solved at the background computa-tional grid.

The governing equations for such dynamic sim-ulations are the mass and momentum conservationequations:

dρ

dt+ ρ∇ · v = 0, (1)

and

ρa = ∇ · σ + b. (2)

where ρ is the material mass density, a is the accel-eration, σ and b are the Cauchy stress tensor andbody force density, respectively.

The displacement and traction boundary condi-tions are given as

u = u on ∂�u, (3)

τ = τ on ∂�τ , (4)

τ coh = τ coh on ∂�coh, (5)

where ∂�u ∩ ∂�τ = ∅, ∂�coh ∩ ∂�τ = ∅ and∂� = ∂�u ∪ ∂�τ ∪ ∂�coh. In variational form, the

Fig. 1 Schematic ofGIMP grid cells withmaterial points (adaptedfrom Tan and Nairn 2002)

uΩ∂

τΩ∂

Ω∂

Vp

BackgroundmeshMaterial points

Nodes


momentum conservation equation is written as

∫

�

ρa · δvdx =∫

�

(∇ · σ ) · δvdx +∫

�

b · δvdx, (6)

where δv is an admissible velocity field.Applying the chain rule,(∇ · σ ) · δv = ∇ · (σ ·

δv)−σ : ∇δv, and the divergence theorem, Eq. (6)can be written as∫

�

ρa · δvdx +∫

�

σ : ∇δvdx

=∫

�

b · δvdx +∫

∂�τ

τ · δvdS +∫

∂�coh

τ coh · δvdS

+∫

∂�u

τu · δvdS. (7)

Here, τu is the resultant traction due to the dis-placement boundary condition on ∂�u, τ is theexternal traction vector, and τ coh is the cohesivetraction at the interface ∂�coh. In GIMP, the do-main � is discretized into a collection of materialparticles, with �p as the domain of particle p. Phys-ical quantities, such as mass, stress, and momen-tum are defined for each particle. For example,the momentum for particle p can be expressed aspp = ∫

�p

ρ(x)v(x)χp(x)dx, where v(x) is the veloc-

ity and χp(x) is the particle characteristic func-tion. From this, the particle volume is calculatedby Vp = ∫

�∩�p

χp(x)dx.

The momentum conservation equation is thusdiscretized as∑

p

∫

�∩�p

ppχp

Vp· δvdx +

∑p

∫

�∩�p

σ pχp : δvdx

=∑

p

∫

�∩�p

mpχp

Vpb · δvdx

+∑

p

∫

∂�τ ∩�p

τ · δvdS +∑

p

∫

∂�coh∩�p

τ coh · δvdS

+∑

p

∫

∂�u∩�p

τu · δvdS (8)

where mp is the lumped mass of each materialparticle. In order to simplify computations, the

lumped mass is used instead of consistent massmatrix. Introducing a background grid and the gridshape function Si(x), the admissible velocity fieldcan be represented by the grid nodal data as δv =∑

iδviSi(x). Here, the grid shape function satisfies

partition of unity∑

iSi(x) = 1.

The momentum conservation, Eq. (8), can even-tually be written for each node i as

pi = finti + fb

i + fτi + fcoh

i . (9)

Here, the time rate of change of momentum is

pi =∑

p

Sippp/t (9a)

the nodal internal force vector

finti = −

∑p

σ p · ∇SipVp (9b)

the body force vector

fbi =

∑p

mpbSip (9c)

and the external force vector

fτi =

∑p

∫

∂�τ ∩�p

τSi(x)dS (9d)

and the cohesive force vector

fcohi =

∑p

∫

∂�coh∩�p

τ cohSi(x)dS (9e)

Sip is the weighting function between particle p andnode i, given as

Sip = 1Vp

∫

�∩�p

χp(x)Si(x)dx. (10)

The weighting function in GIMP is C1 continu-ous and satisfies partition of unity, as given byBardenhagen and Kober [2004]. The momentumconservation equation, Eqn. (8), is solved at each


node to update the nodal momentum, acceleration,and velocity. These updated nodal quantities areinterpolated to the material particles to update theparticle position, velocity, stress, and strain. Thematerial constitutive relation can be representedas

σ = C : ε and ε = 12

[(∇u) + (∇u)T

], (11)

where C is the fourth-order elasticity tensor, ε isthe infinitesimal strain tensor. This paper will con-sider only linear elastic, isotropic materials underinfinitesimally small deformations.

The mass of each material particle does notchange throughout the computation so that themass conservation equation is automatically sat-isfied. Using a background grid, the weak form ofmomentum conservation equation is discretized.However, the computation is independent of thegrid. In our computations, a spatially fixed struc-tured grid is chosen for convenience. The equationslisted above are applicable for both 2D and 3D. Fora uniform structural grid, the grid shape functionin 2D is defined as the product of two nodal tentfunctions (Bardenhagen and Kober 2004)

Si(x) = Sxi (x)Sy

i (y), (12)

in which the nodal tent functions are in the sameform, e.g.,

Sxi (x) =

⎧⎪⎪⎨⎪⎪⎩

0 x − xi ≤ −Lx

1 + (x − xi)/Lx −Lx ≤ x − xi ≤ 01 − (x − xi)/Lx 0 ≤ x − xi ≤ Lx

0 Lx ≤ x − xi

(13)

where Lx is the background mesh cell-length.Also, the particle characteristic function of the

material particle located at (xp, yp) is taken as

χp(x) = χxp(x)χ

yp(y), (14)

where χxp(x) = H[x−(xp − lx)]−H[x−(xp + lx)], H

denotes the Heavyside unit step function and 2lx isthe particle size along x.

3.2 Cohesive surface constitutive law

In this investigation, plane-strain conditions areassumed and the cohesive surface decohesionformulation of Needleman [1990] and Xu and Nee-dleman [1994] is used. The continuum is character-ized by two constitutive relations, namely, one thatrelates stress and deformation in the bulk mate-rial using the generalized Hooke’s law and theother that relates traction and displacement jumpacross a cohesive surface. Strength and work ofseparation per unit area are included as part ofthis latter characterization and thus a character-istic length enters the formulation. In addition tothe material models mentioned above, appropriatebalance laws together with the initial and boundaryconditions define completely the boundary valueproblem. Crack initiation and crack growth arethe natural outcome of the cohesive law of sep-aration. The cohesive surface constitutive relationused allows for both tangential and normal deco-hesion. The normal and tangential traction expres-sions are given as (Needleman 1999)

Tn = −φn

δnexp

(−n

δn

) {n

δnexp

(−2

t

δ2t

)

+1 − qr − 1

[1 − exp

(−2

t

δ2t

)] [r − n

δn

]},

Tt = −φn

δn

(2δn

δt

)t

δt

{q

+(

r − qr − 1

)n

δn

}exp

(−n

δn

)exp

(−2

t

δ2t

)

(15)

where, n = n · �, t = t · �, Tn = n · T, andTt = t · T with � = u+ − u− is the differencebetween the displacement at the upper surface andthe bottom surface; n and t are the unit normaland tangent vectors to the surface at a given pointin the reference configuration. Further, the ratioq = φt

/φn and r = ∗

n/δn, where ∗n is the value of

n after complete shear separation with Tn = 0.Figure 2 shows the normal and shear tractions

across a cohesive surface. The normal work (φn)

and shear work (φt) of separation are given as

φn = eσmaxδn , φt =√

e2τmaxδt, (16)


-1 0 4 7-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-3 -2 -1 0 1 2 3-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

1 2 3 5 6

(a)

(b)

Fig. 2 (a) Normalized normal traction, Tn, on the cohesivesurface as a function of n/δn with t = 0. (after Needle-man 1999) (b) Normalized tangential traction, Tt, on thecohesive surface as a function of t/δt with n = 0. (afterNeedleman 1999)

where, σmax and τmax are the cohesive surface nor-mal and shear strengths, respectively, e is 2.71828,and δn and δt are characteristic lengths correspond-ing to displacement jumps to the cohesive sur-face in the normal (n) and tangential directions(t), respectively. The traction components givenin Eqn. (15) are integrated over the cohesive seg-ments, or cohesive elements, which are part ofthe discretized interface in this case. It may benoted that we use four point Gaussian quadraturemethod to integrate the traction over the crack seg-ments and neglect the effect of crack segment rota-tion assuming that crack opening displacementsremain infinitesimal.

3.3 Cohesive zone in GIMP

Cohesive segments or elements are separate fromGIMP background mesh elements. Thus, GIMPcan allow extension of new fracture surfaces with-out any restrictions from the background mesh ele-ments. In this investigation, we focus on the crackpropagation along the interface in the horizontaldirection by defining a weak plane directly in frontof the initial crack-tip. To reduce computationaltime, we avoid additional calculations pertainingto the classification of nodes and particles acrossthe interface by letting the material points in theupper and lower parts of the plate (with respectto the interface) to convect in their own velocityfields. Thus, by adopting multiple velocity fields(Hu and Chen 2003; Nairn and Guo 2005), thereis no interaction between the material points oneither side of the interface and surfaces on bothsides of the interface are connected to each otheronly through the cohesive zone model.

The interface is discretized into cohesive seg-ments and, each segment is defined by two cohe-sive nodes, forming a line for 2D description of thecrack surface. The more the cohesive nodes arewithin a particular length, the more accurate is themodel of the crack surface. The length scales in-volve a macroscopic length scale characterized bythe size of the body, a length scale associated withthe computational background mesh size, lm, andthe cohesive zone length, lC. The elastic propertiesof the bulk material (E and ν ), the cohesive zonestrength (τmax), surface energy (γ ), and the crackpropagation velocity (vc) determine the cohesivezone length. In our simulations, we have consid-ered a simple finite traction separation potential atthe continuum level, proposed by Morrissey andRice [1998] as an approximate expression for thecohesive zone length at zero crack velocity given

by lC (for vc = 0) ≈ 9π32

(E

1−ν2

)2γ

τ 2max

. However, the

cohesive zone length decreases as the crack-tipvelocity increases. In our final simulations, we usedbackground mesh size of 62.5 μm for impact veloc-ities below 10 m/s, and background mesh size of31.25 μm for impact velocities higher than 10 m/s(based on the convergence study). We fixed thecohesive segment length to ∼2 μm (< lm, where lmis the background mesh size) for higher resolution


Materialpoints

Cracksurface

Cohesivesegment

Backgroundmesh

Fig. 3 Plate with crack showing cohesive segments(adapted from Tan and Nairn 2002)

of the cohesive zone within the grid element. Asa result, the grid size is ∼15 times or higher of thecohesive element size. This high resolution cohe-sive segment length along with smooth interpola-tion functions over the background mesh elementsresolves the cohesive zone very well in all our sim-ulations.

We next consider an interface defined bycohesive nodes represented by smaller solid blackcircles, as shown in Fig. 3. For simplicity, we shallconsider an example of a plate with separated inter-faces forming the crack surfaces. The bigger solidgrey circles are the material points convecting overthe background grid. While defining a vertical orslant weak plane with this methodology is also pos-sible, we restrict our investigation to horizontalinterfaces. Each of the surfaces, namely, upper andlower surfaces, has surface nodes associated withthem. These nodes overlap initially when thereis no displacement jump over the interface. Thematerial points in the upper and lower plates areconvected onto two different velocity fields. Thus,there is no interaction between the material pointsin the upper and lower plates. Both surfaces alongthe interface move in their respective velocity fields,defined by material points in the upper and lowerplates, respectively. The crack opening displace-ment (COD) measured at each crack nodecorresponds to the displacement jump betweentwo crack nodes on the opposite surfaces at theinterface when they were initially (time = 0)overlapping.

While carrying out the GIMP calculations ateach time step, we move the surfaces belonging

to the interface in their corresponding velocityfield. As the surfaces move, they contribute to dis-placement jump on nodes over the interface. Wenow consider a segment being formed betweentwo crack nodes, initially joined in the referencestate, which we will term as the cohesive segment.The cohesive tractions are integrated over cohe-sive segments forming the interface, using theprocedure described in Needleman [1987]. Thus,at the end of the time step, we have displacementjumps at the interface nodes along with the restor-ing forces acting on them at the interface. Thisrestoring force at crack nodes is interpolated tothe surrounding background mesh nodes based onthe weight functions and adds to the total nodalforce for solving the governing equations for thenext time step.

4 Simulation of mode-II crack propagationin a homogenous, isotropic, and linearelastic material

Rosakis et al. [1999] investigated experimentallythe intersonic crack growth in homogeneous elas-tic solids subjected to remote shear loading, usingHomalite plates with specimen geometry and load-ing conditions as shown in Fig. 4. They introduceda weak plane by bonding two Homalite plates withan adhesive having bonding modulus similar tobulk material. We use the same model geome-try and introduce a weak plane using prearrangedcohesive segments along a horizontal line and apredefined initial sharp crack of length 25 mm.Thus, the mode-II crack is constrained to propa-gate along this weak plane. Coker et al. [2003] con-ducted simulations of crack propagation in bima-terials, and provided an estimate of static strengthof the adhesive as σmax = τmax = 24 MPa (or0.4615% of the Young’s modulus of the bulk mate-rial) and we will use the same values in our simula-tions. The remaining parameters characterizing thecohesive surface used are, δn = 0.4μm, r = 0, andq = 1, which gives δt = 0.9327μm (Needleman1999). We assume elastic material behavior andplane strain conditions. Material properties usedfor Homalite are Young’s modulus E = 5.2 GPa,Poisson’s ratio ν = 0.34, and mass density ρ =1246 kg/m3. These material properties correspond


w = 125mm

Predefinedweak plane

mm

051y

a0 = 25 mm

x

Impact mm

05

velocity(Vimp)

mm

051

Fig. 4 Geometry of the specimen and loading conditions

to Homalite’s dynamic properties at a strain rateof ∼103 s−1. For this material, with plane strainidealization, the dilatational (cD), shear (cS), andRayleigh (cR) wave speeds are given by (see, e.g.,Freund 1998; Liu et al. 2005)

cD =(

κ + 1κ − 1

· μ

ρ

)1/2

, cS =(

μ

ρ

)1/2

,

cR = cS0.862 + 1.14ν

1 + ν, (17)

where, μ = E/(2 + 2ν) is the shear modulus, and

κ = 3 − 4ν for plane strain deformation and κ =(3 − ν)

/(1 + ν) for plane stress deformation.

Thus, for Homalite material and plane strainconditions, cD = 2, 534 m/s, cS = 1, 248 m/s, andcR = 1, 164 m/s. These wave speeds are representedin Fig. 5 along with the terminologies for eachregime.

At time t = 0, the body is stress-free and atrest. A normal velocity is prescribed at the edgex = 0, and −50 mm ≤ y ≤ 0. The remainingexternal surfaces are traction-free. The applied

Cra

cktpi

veticoly

/m(

s)

0

500

1000

1500

2000

2500

3000

c R = 1164 m/s

D

2cS

cS

FORBIDDEN GAP

SUPERSONIC

STABLE INTERSONIC

Dilatational wave speed

UNSTABLE INTERSONIC

SUB-RAYLEIGH

Rayleigh wave speed

Shear wave speed

c = 2534 m/s

= 1765 m/s

= 1248 m/s

Fig. 5 Representative plot showing wave speeds for Ho-malite-100 and nomenclature of various regions

velocity is given by

V(t)=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

V1t/tr, for 0 ≤ t < tr;

V1 for tr ≤ t ≤ (tp + tr

)V1

[1 − (

t − tp)

/ts] , for tp < t <(tp+ts+tr

)0, for t ≥ (

tp + ts + tr)

.

(18)

where tp is the pulse time, and tr, and ts are the risetime and step-down time, respectively. The velocityprofile selected is a representative of experimen-tal conditions. The pulse time in the experimentsdepended on the length of the projectile used. Inour simulations, the times used were adapted fromNeedleman [1999] and Coker et al. [2003].

Numerical simulations of dynamic crack prop-agation were carried out using the methodologypresented in Section 3 on an 8-node parallel com-puter cluster (processor speed of 2.8 GHz each)using the Linux operating system. Simulations werecarried out until 19.2 μs time and took ∼48 h ofcomputational time to complete. The parallel envi-ronment was provided by the Structured Adap-tive Mesh Refinement Application Infrastructure(SAMRAI) (Hornung and Kohn 2002) developedat the Lawrence Livermore National Laboratory(LLNL). The geometry of the plate is discretizedusing material points with different levels ofrefinement. The background mesh consists oftwo-dimensional, 4-node, square elements coveringthe entire region. Four material points are used


to represent the material in each cell. Here, eachmaterial point area equals to a quarter of the areaof each cell. Mesh convergence study is conductedusing the sustained crack-tip velocity and interfacecohesive forces. The smallest cell length used inthese simulations was 31.25 μm.

We make use of the structured mesh refinementscheme in GIMP to communicate between suc-cessive levels (Ma et al. 2005) with changing meshsize. Figure 6 is a schematic of the geometric modelused with six predefined levels of refinement. Inthis case, level 0 corresponds to the backgroundmesh of cell-length 2 mm while each immediatefiner level has a cell-length of half of the cell-lengthin the immediate coarser level. Thus, the finest level(level 6) has cell length of 31.25 μm. The pattern ofrefinement is chosen to accommodate the cohesivezone in the finest level so as to reveal the structureof the cohesive zone. The background mesh def-ined using the refinement pattern is fixed in all thesimulations. A total of 12,000 cohesive segmentsare pre-defined between x = 25 mm to x = 50 mm,which defines the interface in front of the initialnotch. The simulation time ends at 19.2 μs which is

Level 0

Level 1

Level 4 Level 2

Level 5 Interface

Level 6

Level 3

Fig. 6 Specimen geometry showing structured mesh refine-ment levels used for the simulation of asymmetric impactof a projectile

before the dilatational waves reach the end of thepredefined cohesive segments (at x = 50 mm) onthe right hand side.

The simulation results are presented in the formof shear stress contours, plots of crack propagationvelocity and position histories, interface profile,and cohesive forces along the interface. In Sect. 6,we also compare our results with the experimentalresults of Rosakis et al. [1999] in terms of photo-elastic stress patterns. In their experiments, Ros-akis et al. [1999] used dynamic transmission photo-elasticity to determine the stress field around theinterface. In this technique, isochromatic fringesare related to the contours of (σ1 − σ2) through thestress-optics relation, with σ1 and σ2 being the max-imum and minimum in-plane principle stresses.Thus

(σ1 − σ2) = NFσ

h(19)

where Fσ is the stress optical coefficient of Homa-lite, h is the specimen thickness, and N is the brightisochromatic fringe order.

5 Results

Dynamic fracture simulations were carried out un-der the loading and geometry conditions identicalto Rosakis et al. [1999] experiments (see Fig. 4for details). The results are presented in terms ofshear stress contours, crack-tip position and veloc-ity histories, and cohesive forces at the interface.Also, features that are characteristic of intersoniccrack growth, such as shear shock waves and tran-sition mechanism from sub-Rayleigh to intersoniccrack-tip velocities are presented. Next, a meshconvergence study is presented (Sect. 5.2) to dem-onstrate the convergence of results with decreasingmesh size. In the discussion part (Sect. 6), addi-tional results of shear dominated crack growthwith varying impact speeds and cohesive strengthsare discussed. Finally, the stress contours are com-pared with the experimental results of Rosakiset al. [1999].


Fig. 7 (Online color) Snapshots of τmax (MPa) for Vimp =30 m/s at various durations (a) Stress wavefront arrivingat the initial crack-tip (b) Stress wavefront loading theinitial crack in predominantly shear mode (c) Crack haspropagated along the interface after initiation at 10.4 μs

(d) Maximum shear stress pattern immediately after thetransition mechanism (e) Formation of shear shock waves(f) Crack propagation at a sustained crack-tip velocity inintersonic regime (cR = 1164 m/s)

5.1 Intersonic crack growth at an impact speedof 30 m/s

Numerical simulations were conducted for asym-metric impact of a projectile at Vimp = 30 m/susing the specimen geometry and loading condi-tions shown in Fig. 4 with normal and shear cohe-sive strengths of 24 MPa. Figure 6 shows the meshused in the simulations, where the smallest mesh

size used is 31.25μm. Isochromatic fringe patterns(contours of maximum shear stress, τmax) at differ-ent times are shown in Fig. 7. In each frame, thecorresponding time and crack-tip velocities areshown along with the crack-tip position indicatedby an arrow. In Fig. 7 (a), the traveling wavefrontarrives at the initial crack-tip, loading the crack-tip predominantly in the shear mode. The stresswave travels over the interface onto the upper half


of the workpiece and envelops the initial crack-tip(Fig. 7 (b)). In the next frame (Fig. 7 (c))(corresponding to time 10.56 μs), the crack haspropagated along the interface after initiating at10.4μs. The maximum shear stress pattern is typi-cal of a crack propagating at sub-Rayleigh speedsunder shear dominated loading. The crack propa-gates at sub-Rayleigh speeds until 11μs, after whichit propagates at intersonic speeds, with the stresspatterns shown in Fig. 7 (d–f). For frames after∼11μs time, (Fig. 7 (d–f)) there is a significantchange in the fringe pattern. There are two linesof discontinuity (called the shear shock waves)emanating from the crack-tip in all these frames.Fig. 7 (f) is taken downstream from the initialnotch. The inclination of these shear shock waveswith the horizontal gives the velocity of the crack-tip. At higher velocities, the angle becomes smaller.The expression relating the shear shock angle withthe crack-tip velocity is given by

β = sin−1(

cs

vc

)(20)

where β is the angle made by the shear shock wavewith the horizontal interface on the trailing side ofthe crack-tip and vc is the crack-tip velocity.

The crack-tip position as a function of time isshown in Fig. 8. The crack-tip is defined as the

Time (μs)

Cra

ck-

pitsop

oitin

(m

m)

10 12 14 16 18 20

25

30

35

40

45

Fig. 8 Plot of crack-tip position versus time for Vimp =30 m/s

farthest point from the initial crack-tip where thedisplacement jump is 5δt. This corresponds to ∼100%of cohesive energy dissipated in the resultingcohesive zone. The crack-tip velocity history isshown in Fig. 9 and is presented in terms of thenon-dimensional parameter vc

/cR, where cR is the

Rayleigh wave speed. The crack-tip velocity at aparticular time is obtained by the slope of thecrack-tip history using the least square fitting ofa straight line over 31 consecutive crack-tip posi-tions. These calculations of vc are conversant withthose obtained from the shear shock angle usingEqn. 20. The crack-tip starts accelerating from time∼10.75 μs to ∼ 11.5μs. Correspondingly, the calcu-lated crack-tip velocity increases transiently fromsub-Rayleigh velocity to a velocity above the dila-tational wave speed before settling down to a sus-tained intersonic speed of 2302 m/s (or 1.977cR).This is facilitated by the formation of a microcrackat some distance ahead of the main crack followedby coalescence of the main crack with the micro-crack (also called daughter-crack). This phenome-non, known as the Burridge–Andrews mechanism(after Burridge 1973; Andrews 1976), has been ob-served in the finite-element simulations by Nee-dleman [1999] and in the experiments by Xia et al.[2004] for homogeneous materials as well as in thesimulations in bimaterials by Coker et al. [2003].Through this mechanism, cracks transiently jumpfrom sub-Rayleigh to intersonic speed regime with-out going through the forbidden region betweenthe Rayleigh wave speed and the shear wave speed.This mechanism will be discussed in detail inSect. 6.

5.2 Convergence study

In this study, the governing equations of GIMP(Eqn. 7) are solved on the background mesh. Theeffect of mesh size on the sustained crack-tip veloc-ity and cohesive forces on interface nodes in bothtangential and normal directions are investigated.Mesh is refined by a factor of two where the small-est mesh cell-length is varied from 1 mm to31.25 μm. The crack-tip positions as a function oftime for different mesh sizes are shown in Fig. 10,for Vimp = 30 m/s and a cohesive strength of


Time (μs)

Cra

-kctip

velo

ctiy

/c

10 12 14 16 18 20

0.5

1

1.5

2

2.5

c D

Rc

S2c

Sc

R

500

1000

1500

2000

2500

3000

Cra

t-kcip

velo

city

(/

ms)

Fig. 9 Plot of normalized (with cR) crack-tip velocity withtime for Vimp = 30 m/s

24 MPa. The difference in the crack-tip positionhistory for different mesh sizes is negligible.

We next examine the mesh convergence by plot-ting the normalized (with cR) crack-tip velocitiesfor different mesh sizes as shown in Fig. 11. As themesh size is decreased from 1 mm to 62.5 μm, thetransition from sub-Rayleigh to intersonic speedsbecomes more abrupt. For the portion of the curvesprior to 12 μs in Fig. 11 and 13(a), convergencewas not attained for different mesh sizes becauseit is during this time interval that the transitionmechanism from sub-Rayleigh to intersonic speed

Time (μs)

Cra

ck-t

ipp

sot i

noi(

)m

m

10 12 14 16 18

25

30

35

40

45

1 mm0.5 mm0.25 mm0.125 mm62.50 μm31.25 μm

Mesh sizes

Fig. 10 (Online color) Variation of the crack-tip positionwith time for different mesh sizes

Time (µs)

Cra

ck-t

ipve

loci

ty/c

10 12 14 16 18

0.5

1

1.5

2

2.5

1 mm0.5 mm0.25 mm0.125 mm62.50 µm31.25 µm

R

Mesh sizes

500

1000

1500

2000

2500

3000

Cra

ck-t

ipve

loci

ty(m

/s)

Fig. 11 (Online color) Plot of normalized (with cR) crack-tip velocity with time for different mesh sizes for Vimp =30 m/s

regime occurs aided by the formation of a micro-rupture (daughter-crack tip) ahead of the mother-crack tip. Further, depending on the mesh size andhow accurately the daughter-crack tip can be re-solved, the crack-tip position and velocity historiesat or near this transition mechanism changes withlarger mesh sizes tending to average the positionand velocity, respectively. Even though the veloc-ity history plots show crack propagation throughthe forbidden zone, it is an artifact of poor gridresolution. At the smallest mesh size of 31.25 μm,an overshoot of the velocity to supersonic speedwas observed. It may be noted that this is an arti-fact, the result of calculation of the velocity fromthe crack-tip position. When a daughter-crack isformed ahead of the main-crack, the crack-tip algo-rithm identifies wrongly the new crack-tip as thelocation of the newly formed daughter-crack. Thisfeature is not captured by the coarser mesh sizes.However, in all cases, the same sustained crack-tipvelocity is reached at 2300 m/s (or ∼1.977cR ). Thecoarser mesh sizes result in oscillations around thisvalue. There is also a good agreement for all meshsizes for the time at which crack initiation occurs(∼10.4 μs).

For all mesh sizes, the shear and normal cohe-sive forces at the nodes along the interface areplotted in Fig. 12 at t = 17.6μs. The crack-tip islocated at ∼41 mm and both the shear and nor-mal forces increase as the crack-tip is approached


X (mm)

aT

ngen

tial

oche

isev

foecr

onin

terf

aec

node

s(N

)

36 38 40 42 44 46 48 50-10

0

10

20

30

40

50

60

1 mm0.5 mm0.25 mm0.125 mm62.50 μm31.25 μm

Mesh sizes

X (mm)

Nor

mal

coehsi

vefo

rce

oni

tner

faec

oneds

(N)

36 38 40 42 44 46 48 50-10

0

10

20

30

40

50

60

1 mm0.5 mm0.25 mm0.125 mm62.50 μm31.25 μm

Mesh sizes

X (m)

(Y

μ)

m

36 38 40 42 44 46 48

10

20

30

40

50

Cohesivetractionsabsent

Original crack interfaceLOWER PLATE

Bottom interface(penetrated into upper interface)

Upper interface

Cohesive tractions present

Interface starts separating

UPPER PLATE

(b)

(c)

(a)

Fig. 12 (a) (Online color) Tangential cohesive force (Ttintegrated over cohesive segments) on the upper interfacefor Vimp = 30 m/s at t = 17.6μs. (b) (Online color) Normalcohesive force (Tn integrated over cohesive segments) onthe upper interface for Vimp = 30 m/s at t = 17.6 μs. (c)(Online color) Interface surface profiles for Vimp = 30 m/swith a cell length of 31.25 μm and t = 17.6 μs

from the right and drops to zero when separationoccurs. In the case of Fig. 12 (a), the values ofthe forces over the interface have a tendency toclose the interface in the shear direction. In thecase of Fig. 12 (b), the nature of normal forcesis compressive. In other words, the values of theforces over the interface have a tendency of inter-face opening in the normal direction due to bot-tom surface having a tendency of pressing ontothe upper surface, due to Poisson’s effect. This isillustrated by plotting interface surface profiles inFig. 12 (c). In Fig. 12 (a) and (b), for the finermesh sizes, the cohesive region size is equal to∼7 mm, which is defined by 224 background meshelements and 3360 cohesive segments. From theseresults it is reasonable to conclude that mesh con-vergence is attained for element size of 62.5 μm;to resolve the highly transient Burridge–Andrewsmechanism, we investigated using a finer mesh size(31.25μm) for Vimp = 30 m/s.

6 Discussion

The effect of the impact speed on the crack-tipvelocities was studied by varying the impact speedsfrom 5 to 60 m/s. The normal and shear cohesivestrengths of 24 MPa were used at the interface.Normalized crack-tip velocity histories for all casesare shown in Fig. 13 (a). A mesh size of 31.25μmwas used for Vimp of 20 m/s and above. However, amesh size of 62.5μm was adopted for Vimp ≤ 10 m/sbecause of the occurrence of a numerical instabilityduring the formation of a microcrack, as a result ofwhich the simulation quits. A more detailed discus-sion pertaining to this issue is given in the latter partof this section. For Vimp < 9.4 m/s, the crack prop-agates at a velocity less than the Rayleigh wavespeed of the Homalite. As the impact speed are in-creased (starting from 5 m/s), the crack-tip veloc-ity approaches cR earlier in time. As the crack-tip velocity approaches cR with increasing impactspeed, the crack either crosses cR and enters theintersonic regime (Vimp > 9.4 m/s) or it oscillatesat values lower than cR (for Vimp < 9.4 m/s). Inthis latter case of sub-Rayleigh crack propagation,the crack-tip velocity approaches cR with time andthe oscillations (maximum amplitude of ∼40 m/s)decreases. The sustained crack-tip velocity is taken


cR

cS

9.2 m/s

9.5 m/s

Time (μs)

Cra

t-kcip

velo

ctiy

/c

10 12 14 16 18 20

0.5

1

1.5

2

2.5

cD

2cS

30 m/s

5 m/s

9.3 m/s

60 m/s20 m/s10 m/s

R

500

1000

1500

2000

2500

3000

500

1000

1500

2000

2500

3000

rC

cak-

tipv

leoc

ity(m

/s)

Impact speed (m/s)

uS

tsai

ned

carck

t-pi

evol

icty

/c

0 10 20 30 40 50 60

0.5

1

1.5

2

2.535 MPa24 MPa12 MPa4 MPa

c R

c D

1.66cS

cS

5

60

20102.25

12.5

9.2

30

S2c

9.3Numbers indicate value ofimpact speed at corresponding point

9.5

12.2

5.7

R

Cohesive strengths

uS

stai

ned

car

t-kcip

velo

cti y

(/

m)s

(a)

(b)

Fig. 13 (a) (Online color) Variation of normalized (withcR) crack-tip velocity with time for various impact speeds,considering cohesive strength of 24 MPa and plate Young’smodulus, E of 5.2 GPa. (b) Variation of normalized (with cR)sustained crack-tip velocity with impact speeds and cohesivestrengths, for plate modulus, E of 5.2 GPa

as the peak of the oscillating values. For the caseof Vimp > 9.4 m/s, the sustained crack-tip veloc-ity (at time ∼19.2 μs, which more or less remainsat a constant value with time) is found to be inthe region between

√2cS and cD. Further, as the

impact speed is increased, the transition Burridge–Andrews mechanism occurs earlier in time.

The transition of sustained crack-tip velocityfrom sub-Rayleigh to intersonic, with increasingimpact speed is abrupt and occurs at Vimp ≈ 9.5 m/s.This abrupt change can be seen in Fig. 13 (b) wherethe sustained crack-tip velocity is plotted as a func-tion of the impact speed. The corresponding lowestvalue of the sustained crack-tip velocity in the int-ersonic regime is called the transition crack-tip

velocity (Burridge et al. 1979) which, from oursimulations is ∼1.66cS. Thus, in all these cases,there was no steady-state crack-tip velocity obs-erved between cR and

√2cS which is consistent

with the finite energy arguments of Freund [1979]and Rosakis et al. [1999]. However, in our case,with the cohesive parameters selected (σmax = τmax

= 24 MPa), no sustained crack-tip velocity wasobserved between

√2cS and 1.66cS either. This

result appears to be consistent with the analysisof Burridge et al. [1979] on the crack propaga-tion under a step impact loading. In their anal-ysis, they reported that for constant shear load-ing at a distance d from the crack-tip, the transi-tion crack-tip velocity of

√2cS will occur only for

very high values (up to infinity) of distances, d, ofthe crack-tip from the point of load applicationand that the transition crack-tip velocity increasesas d decreases. However, Needleman [1999] con-ducted FEM simulations where cohesive strengthas well as impact speed pulse duration time werevaried and obtained crack propagation with veloc-ities close to

√2cS. Thus, the lowest attainable

stable intersonic crack-tip velocity of√

2cS canstill occur under such situations as reported byBurridge et al. [1979] (analytical work) and Ros-akis et al. [1999] (experiments). In our simulations,we used pulse duration time of 15 μs, and cohe-sive strength of 24 MPa, while in Needleman’s sim-ulations (1999) the time duration was 1 μs andthe cohesive strength was 30 MPa, in which caseNeedleman observed sustained intersonic crack-tip velocity of

√2cS. In our simulations the lowest

sustained crack-tip velocity could reach√

2cS ifthe pulse duration time were reduced, or if thecrack were allowed to propagate over a longerlength.

We further investigated the effects of cohesivestrength of the interface and Young’s modulus ofthe bulk material on the value of transition crack-tip velocity. Cohesive strength was varied from 4to 35 MPa. Fig 13 (b) shows the transition speed asa function of impact speed for cohesive strengthsof 4, 12, 24, and 35 MPa. The transition speed re-mains at a constant value of ∼1.66cS, independentof the cohesive strength. This transition speed isreached at impact speeds of 2.25, 5.7, 9.5, and12.5 m/s, and for cohesive strengths of 4, 12, 24, and35 MPa, respectively. Thus for the cases studied


cR

cD

1.66cS

cS

2 m/s

4.5 m/s

5 m/s

4 m/s

3 m/s

2.5 m/s

Time (μs)

Cra

t-kcip

velo

ctiy

/c

10 12 14 16 18 20

1

1.5

2

2.5

3

R

500

1000

1500

2000

2500

3000

Cra

t-kcip

velo

c ity

(/

ms)

Fig. 14 (Online color) Variation of normalized (with cR =883 m/s) crack-tip velocity with time for various impactspeeds, considering cohesive strength of 24 MPa and platemodulus, E of 3 GPa

in our simulations the lowest sustainable crack-tip velocity in intersonic region was 1.66cS. Lastlyfrom Fig. 13 (b) we can also see that as the impactspeed is increased, the sustained intersonic crack-tip velocity increases from 1.66cS (or 0.82cD) and1.94cS (or 0.95cD) and tends to approach cD at adecreasing rate with respect to the impact speed.Figure 14 shows the normalized crack-tip veloc-ity history plots for impact speeds correspondingto relatively lower Young’s modulus of the plate(E = 3 GPa, ν = 0.34) and cohesive strength of4 MPa. It can be seen that the transition speed re-mains more or less the same (∼1.66cS) correspond-ing to an impact speed of 4 m/s.

Next, simulations are conducted for an impactspeed, Vimp of 10 m/s with cohesive strengths,σmax = τmax = 24 MPa, and background meshsize of 62.5 μm, to observe the Burridge–Andrewsmechanism. Fig. 15(a) to (d) show sequential plotsof maximum shear stress contours showing thetransition mechanism. In Fig. 15 (a) only a singlecrack-tip can be seen (see black arrow) propagat-ing at a velocity below the Rayleigh wave speed.In Fig. 15 (b) a microrupture or a daughter-crackis formed ahead of the trailing mother-crack-tip asindicated by a white hollow arrow-head. The crite-rion used for the formation of the daughter-cracktip should be the same as that used for the maincrack-tip, namely, when the displacement reachesa critical value of 5δt (δt is the characteristic length

corresponding to the cohesive surface in thetangential direction) the crack-tip is formed. How-ever, in our simulations the daughter-crack wasnot fully developed at fine background mesh size(31.25 μm) as the simulation quits. We had to use acoarser mesh size (62.5 μm) so that the simulationcould continue to run. However, at this coarserbackground mesh size, the crack-tip displacementsin shear were unable to reach the value of 5δt due tothe effect of mesh averaging, so that the daughter-crack tip could not be recognized in this way. Asa result, the initiation of the daughter-crack wasdetermined approximately through the observa-tion of the stress contour at a small distance aheadof the major crack-tip and from the jump in thevelocity history plot (Fig 13 (a)). The mother-crack-tip then accelerates rapidly (Fig. 15 (c)) and joinsthe daughter-crack-tip as shown in Fig. 15 (d). Itis during this process that the apparent crack-tipvelocity increases abruptly to and above the dila-tational wave speed. From this point, shear shockwaves start developing from the crack-tip and af-ter some time, the crack-tip velocity reaches a sus-tained value in the intersonic region.

This mother-daughter-crack transition mecha-nism has been explained theoretically by Broberg[1989] using the steady state analysis. In that work,he pointed out that the crack-tip cannot travelsteadily between cR and cS due to violation of theenergy considerations. The accelerating crack-tipthus goes through a transition mechanism so asto pass from sub-Rayleigh to intersonic regime toavoid the forbidden region. This transition mecha-nism is called the Burridge–Andrews or the mother-daughter crack mechanism.

In all the simulations conducted in this inves-tigation of varying mesh sizes and impact speeds,we observed that even though mesh convergencemay have been attained (based on the above con-siderations) identifying the mother-daughter (Bur-ridge–Andrews) transition mechanism requiresfiner mesh sizes in order to observe an abrupt inc-rease in the crack position history. However, forthe case Vimp = 10 m/s, we observed that the fin-est mesh size (cell-length of 31.25 μm) is limitedby the instability in the simulation occurring dur-ing Burridge–Andrews transition mechanism afterwhich simulation would quit. Figure 16 shows theresults obtained with a cell-length of 31.25 μm and


Fig. 15 (Online color)Snapshots of τmax (MPa)for Vimp = 10 m/s. (a)Main crack-tippropagation at asub-Rayleigh wave speed.(b) Formation ofmicrorupture ahead of themain crack-tip. (c)Coalescence of maincrack-tip withmicrorupture. (d)Resultant crack-tippropagation at intersonicwave speed


Fig. 16 (Online color)Results for Vimp = 10 m/swith cell-length of31.25 μm, showinginstability

Time (μs)

Cra

ck-t

ippo

soiti

n(

)m

m

10 12 14 16 18 20

26

28

30

32

34

36

38

40

Cell-length: 31.25 μm

Instability occurs

Crack-tip position history

(a)

4037343128252219161310741

Snapshot of τmax (MPa) at time 16.96 μs

(b)

Vimp of 10 m/s. Figure 16 (a) shows the distinctjump in the crack position history which is alsothe point where instability occurs. Figure 16 (b)shows the corresponding plot of maximum shearstress contour at time 16.96μs, just before the sim-ulation quits. This instability, at the point wheremicrocrack (daughter-crack) nucleates, could beattributed to elastic snap-back instability (Gao andBower 2004) occurring when the stress reachesthe peak strength of the interface. The mesh sizerequired to resolve the daughter-crack is differ-ent (less) than that required to obtain convergedsustained crack-tip velocities, since daughter-crackformation occurs at a relatively smaller scale thanthe major (mother) crack with respect to the size oftheir cohesive zones. In our work, we have checkedfor convergence of the major crack-tip cohesivezone and not for daughter-crack tip cohesive zone.For potential intersonic cracks, depending on themesh size and impact speeds, the daughter-cracktip cohesive zone is resolved to a certain level. The

daughter-crack is better resolved at lower impactspeeds, if we keep the mesh size the same at all imp-act speeds. One such simulation at which we wereable to resolve the daughter-crack (resolve theactual distance between the daughter-crack andmajor crack in conjunction with crack openingdisplacements) is at 10 m/s impact speed and31.25 μm mesh size. Here, we observed sharp jump(discontinuity) in the crack-tip position history in-stead of continuous increase, as in other cases ofhigher impact speeds. It is for this unique case (inour work) that the nucleation of daughter-crackwas resolved. However, during nucleation, the sim-ulation ran for some time and quit due to cohesiveforces reaching very high values. Gao and Bower[2004] also had experienced a similar problem intheir simulations. In their work, they reported thatby introducing a viscosity term in the constitu-tive equations for the cohesive law they were ableto suppress the instability during crack nucleationand thus allow simulation to go further.


Fig. 17 Views of (a)Simulated result showingmaximum shear stress forVimp = 10 m/s andσmax = τmax = 24 MPa,compared with (b)experimental pattern, (c)theoretical predictionsbased on a Dugdale-typecohesive zone model(after Rosakis et al. 2000)

In our case, we simulated the results (Fig. 13 (a))for the case of Vimp = 10 m/s with the mesh size(cell-length of 62.5 μm) one level coarser than theone in which simulation becomes unstable (cell-length of 31.25 μm). However, this mesh size couldnot resolve the distinct jump near the crack-tip thatotherwise propagates at the same sustained crack-tip velocity. Thus, we found mesh size of 31.25 μmto be appropriate for Vimp of 20 m/s and above,while a mesh size of 62.5 μm for Vimp of 10 m/s andlower.

Our simulation results obtained for the caseof Vimp = 10m/s with cohesive strengths, σmax =τmax = 24 MPa, and background mesh size of62.5 μm are compared qualitatively with the iso-chromatic fringe patterns obtained from experi-ments using photoelasticity by Rosakis et al. [1999],in Fig. 17. The general shape of the isochromaticpatterns in front of the crack-tip matches with theexperimental results of Rosakis et al. [1999], asshown in Fig. 17 (b) (from Rosakis et al. 2000);and theoretical predictions based on a Dugdale

type cohesive zone model, as shown in Fig. 17 (c)(from Rosakis et al. 2000). It is somewhat difficultto compare the patterns quantitatively due to diffi-culty in extracting stress values from the fringe pat-terns obtained in the experiments. Furthermore,the velocity of the crack-tip is 1.66cS for the numer-ical pattern and 1.47cS for the experimental pat-tern. This can also be seen from the angles formedby the shear shock waves; in the simulation, thevalue is 37◦ and in the experiments, it is 43◦. Fin-ally, due to the absence of contact algorithm inthe numerical simulation model, we see complexstress patterns corresponding to waves travelingover the trailing crack surfaces, which are not ex-actly comparable with experiments. From this, itappears that in actual experiments the crack sur-face contact plays some role and it could even influ-ence the stress contours trailing the crack-tip.

It may be noted that numerical implementa-tion of the normal cohesive law has to take intoconsideration the possibility of interpenetrationof crack surfaces. Interpenetration in the cohe-


sive zone has been prevented by the use of steepslope of the traction-displacement cohesive law inthe case of negative (relative) displacements. Thecurrent simulation has taken this into consider-ation and prevented penetration of material pointsin the cohesive zone. In the wake of the crack-tip, where the cohesive law is turned off (due todifficulty associated with searching cohesive nodeswithin each cell, as a result of relatively large dis-placements at crack surfaces), algorithm to preventinterpenetration of the crack faces has not beenimplemented. Based on published results (moredetails to follow), it is seen that the stress field infront of the crack-tip should not be affected bythe interpenetration behind the crack-tip becausethe behavior of the crack-tip is governed by theasymptotic stress field around the crack-tip. It maybe noted that the present work has focused on thestress-field in front of the crack-tip. Comparisonsbetween experiments and finite element simula-tions (Needleman 1999; Rosakis et al. 1999, 2000;Coker et al. 2003; Yu et al. 2002) imply that thecrack behavior for shear-dominated crack growthis not affected by the particular behavior in thewake of the crack-tip. In the FEM work, citedby Needleman [1999] and Coker et al. [2003], theinterpenetration was allowed and frictional con-tact was not considered in the wake of the crack-tip. In the 3D FEM work by Yu et al. [2002], fric-tional contact was considered in the wake of thecrack-tip. In all these cited FEM works, simula-tion results were compared with experimental re-sults of Rosakis et al. [1999]; Coker and Rosakis[2001] and Coker et al. [2003]. It may be noted thatin the experiments, interpenetration did not oc-cur and frictional contact was present. Despite thedifferences in the conditions in the wake of crack-tip between FEM simulations, the global behavior(crack-tip speed) and the local behavior (fringepatterns in front of the crack-tip) showed goodagreement between experiments (where there iscontact and frictional sliding of the crack faces)and simulations. In this work, we have focused onimplementing the cohesive zone model and did notimplement the contact algorithm to prevent inter-penetration behind the crack-tip because we areinterested primarily in the crack-tip dynamics.

Some discrepancy can be seen between the sim-ulated crack initiation time and the experimental

values, with the experimental values being slightlyhigher. This can be attributed to the geometryof the initial crack-tip, which is mathematicallysharp in the case of simulations but not so in theexperiments. There is also a possibility that someof the impact energy may be consumed in mate-rial nonlinearities not considered in the numericalmodel. Further, since the sustained crack veloc-ities in the experiments are lower than that ofthe numerical simulations, for relatively higher im-pact speeds, it is possible that the actual bond-ing strength might be higher than the cohesivestrength used in the simulations. Overall, the com-bined GIMP and cohesive zone model methodol-ogy are able to capture the salient features of thedynamic crack growth.

7 Conclusions

Cohesive zone model was implemented in GIMPto simulate dynamic crack propagation. Dynamiccrack growth was investigated by numericalsimulations of shear dominated crack growth in anominally brittle, homogeneous, linear-elastic solidunder plane strain conditions with the crack con-strained to grow in the direction of the initial notch.Based on dynamic crack growth simulation results,the following conclusions are reached:

(1) Dynamic crack propagation occurs in bothsub-Rayleigh and intersonic crack-tip veloc-ity regimes. A potential intersonic crack,after initiating, accelerates and experiencescrack-tip velocities close to the Rayleigh wavespeed. In order for the crack-tip velocity toincrease from sub-Rayleigh to intersonicregime, it goes through a transition Burridge–Andrews mechanism, namely, formation ofa microcrack at some distance ahead of themain crack, and coalescence of the main crackwith the microcrack, after which it reaches astable state of crack propagation.

(2) Mesh convergence study was carried out byvarying the smallest element size from 1 mmto 31.25 μm. There was a good agreementfor the sustained crack-tip velocity and cohe-sive force distribution over the interface for62.5 μm and 31.25 μm mesh sizes. However,


the sub-Rayleigh to intersonic speed transi-tion mechanism is only captured by thesmallest mesh size of 31.25 μm used in theinvestigation.

(3) The effect of impact speed was investigatedfor speeds ranging from 5 to 60 m/s and cohe-sive strengths of 4 to 35 MPa [Fig. 13(b)]. Forthe case of cohesive strength of 24 MPa, thesustained crack-tip velocity within the sub-Rayleigh region was found to increase withimpact speed up to Rayleigh wave speed, cR.This was followed by a sudden increase in thesustained crack-tip velocity from cR to 1.66cS.With further increase in the impact speed, thesustained crack-tip velocity was found to in-crease up to 1.94cS (or 0.95cD) correspondingto an impact speed of 60 m/s. In our simula-tions, the lowest sustained crack-tip velocityobtained was 1.66cS and did not reach

√2cS

as reported in the analytical work by Bur-ridge et al. [1979], experiments by Rosakis etal. [1999], and FEM simulations by Needle-man [1999].

(4) A transition crack-tip velocity of 1.66cS wasobserved, which remained the same (for sim-ulations in this work) for different cohesivestrengths of 4, 12, 24, and 35 MPa (for bothshear and normal strengths) and correspond-ing impact speeds of 2.25, 5.7, 9.5, and 12.5 m/s.

(5) Good qualitative agreement was observedbetween the numerical results presented hereand the experimental results of Rosakis et al.[1999] in terms of the photoelastic stress pat-terns ahead of the crack-tip.

Acknowledgements The work was supported by the AirForce Office of Scientific Research (AFOSR) through aDEPSCoR grant (No. F49620-03-1-0281). The authors wouldlike to thank Drs. Craig S. Hartley, Dr. J. Tiley Jr., and Capt.Brett Conner, Program Managers (during the course ofthe project) for the Metallic Materials Program at AFOSRfor the support of and interest in this work. RK acknowl-edges the A.H. Nelson, Jr., Endowed Chair in Engineeringfor additional support. NPD acknowledges graduate assis-tantship of the Oklahoma NSF EPSCoR scholars program.The authors also acknowledge Dr. Richard Hornung andDr. Andy Wissink of the Lawrence Livermore NationalLaboratory for providing the SAMRAI code and Dr. JohnA. Nairn for providing the 2D MPM code. Lastly, the au-thors thank Mrs. Sharon Green for her assistance in thepreparation of this manuscript.

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