calculation of j-integral and stress intensity factors ... · pdf filecalculation of...

Copyright c© 2004 Tech Science Press CMES, vol.?, no.?, pp.1-14, 2004

Calculation of J-Integral and Stress Intensity Factors using the Material PointMethod

Y. GUO and J. A. NAIRNMaterial Science and Engineering, University of Utah, Salt Lake City, Utah 84112, USA

Abstract: The Material Point Method (MPM), whichis a particle-based, meshless method that discretizes ma-terial bodies into a collection of material points (the par-ticles), is a new method for numerical analysis of dy-namic solid mechanics problems. Recently, MPM hasbeen generalized to include dynamic stress analysis ofstructures with explicit cracks. This paper considers eval-uation of crack-tip parameters, such asJ-integral andstress intensity factors, from MPM calculations involv-ing explicit cracks. Examples for both static and dynamicproblems for pure modes I and II or mixed mode loadingshow that MPM works well for calculation of fractureparameters. The MPM results agree well with results ob-tained by other numerical methods and with analyticalsolutions.

keyword: Material point method (MPM), dynamicfracture,J integral, stress intensity factor.

1 Introduction

Many experimental methods are available for investi-gating the dynamic fracture properties of materials andstructures. Because of the very short time scales fordynamic fracture events, it is difficult to directly mea-sure physical fracture quantities such asJ-integral, en-ergy release rates, or stress intensity factors, particu-larly in opaque specimens or structures of practical in-terest. Computational calculations have the potential toovercome the difficulties associated with interpreting dy-namic fracture mechanics experiments. The approachwould be to use calculations to evaluate physical fracturequantities of a dynamic crack tip at any instant of timeduring the experiment. The advancement of dynamicfracture mechanics, therefore, relies heavily on advance-ments of numerical fracture methods.

The numerical analysis of dynamic fracture is often con-sidered as a package using various numerical methods,

but the problem actually partitions into three distinctproblems that can be solved independently:

1. Analysis of explicit cracks: The first problem isto develop numerical methods that can evaluatestresses and displacements around explicit cracks.

2. Calculation of fracture parameters: Once explicitcrack analysis is possible, each numerical methodneeds techniques to evaluate key crack-tip parame-ters such asJ integral, energy release rate, stress in-tensity factors, or various other local crack-tip prop-erties.

3. Prediction and inclusion of crack propagation:Once crack tip parameters are available, the nextissue is to predict what conditions are required forcrack propagation and in what direction the crackwill propagate. This problem is a material scienceproblem and not dependent on the particular numer-ical method chosen for analysis. For a particular nu-merical method to be effective, however, it shouldbe capable of modeling crack propagation in arbi-trary directions and continuing the analysis as thecrack grows.

One of the earliest applications of numerical methods todynamic fracture problems is the finite difference method(FDM) developed by Chen and Wilkens (1977). Later, fi-nite element analysis (FEA) became the preferred numer-ical tool [Nishioka and Atluri (1983), Nishioka (1995),Nishioka (1983), Nishioka (1997), Nishioka, Tokudome,and Kinoshita (2001), Nishioka and Stan (2003)]. Theanalysis of explicit cracks in FEA is easily handled byintroducing cracks in the mesh, but FEA can have diffi-culty dealing with crack surface contact. FEA can evalu-ate fracture parameters by methods such as crack closure[Rybicki and Kanninen (1977)], but encounters difficul-ties in coping with crack propagation, especially crack

2 Copyright c© 2004 Tech Science Press CMES, vol.?, no.?, pp.1-14, 2004

propagation in arbitrary directions [Nishioka, Tokudome,and Kinoshita (2001)]. Another problem of FEA is dif-ficulty with problems having large deformations or rota-tions, where the finite element mesh could become dis-torted requiring re-meshing methods and causing a greatdecrease in calculation efficiency. A dual boundary inte-gration method was developed to obtain dynamic stressintensity factors [Wen, Aliabadi, and Rooke (1998)].Because the crack opening displacements are used tocalculate stress intensity in this method, attention mustbe paid, and some techniques have to be used, to im-prove the accuracy of crack opening displacements nearcrack tips before stress intensity factors are reliable. Toavoid mesh problems of FEA and to better handle crackpropagation in arbitrary directions, some meshless meth-ods [Atluri and Shen (2002), Han and Atluri (2003b)]have been developed for dynamic fracture analysis [Be-lytschko, Lu, and Gu (1994), Organ, Fleming, and Be-lytschko (1996), Batra and Ching (2002), Han and Atluri(2003a)], as well as a cell method [Ferretti (2003)]. Forexample, Batra and Ching (2002) extended the MeshlessLocal Petrov-Galerkin (MLPG) method for inclusion ofcracks and evaluation of stress intensity factors (prob-lems 1 and 2 above). As in other meshless methods,however, the inclusion of cracks in MLPG required defi-nition of particle “influence” zones near crack surfaces tobe able to handle explicit cracks. These influence zoneshave to be evaluated by approximate node-visibility orstress-diffraction rules [Organ, Fleming, and Belytschko(1996), Batra and Ching (2002)]. Furthermore, it mightbe hard to incorporate contact algorithms for handlingcrack surface contact in MLPG.

A new particle-based method, called the material pointmethod (MPM), is developing into a new numerical toolfor solving dynamic solid mechanics problems [Sulsky,Chen, and Schreyer (1994), Sulsky, Zhou, and Schreyer(1995), Sulsky and Schreyer (1996), Nairn (2003), Bar-denhagen, Guilkey, Roessig, Brackbill, Witzel, and Fos-ter (2001), Bardenhagen and Kober (2004)]. In the mate-rial point method, the object being analyzed is discretizedinto a collection of particles or material points. As thedynamic analysis proceeds, the solution is tracked on theparticles by updating all required properties such as po-sition, velocity, acceleration, stress state,etc.. At eachtime step, the equations of motion for the particles aresolved on a background grid; this solution is used to up-date the particles and the background mesh can be dis-

carded or reused for the next time step in its initial, undis-torted form. This combination of Lagrangian and Eule-rian methods has proven useful for solving solid mechan-ics problems, particularly for problems with large defor-mations and rotations. Despite the use of a backgroundmesh, a recent derivation of a generalized MPM method[Bardenhagen and Kober (2004)] shows that MPM is aPetrov-Galerkin method having more in common withother meshless methods than with finite element meth-ods.

The standard derivation of MPM enforces continuousdisplacements. Although such an analysis can handlesome fracture properties with special symmetries [Tanand Nairn (2002)], it can not handle arbitrary explicitcracks. Recently we have developed a new MPM algo-rithm called CRAMP for cracks in material point calcula-tions [Nairn (2003)]. This new algorithm solves the firstproblem in numerical analysis of fracture on the analy-sis of explicit cracks. It handles cracks naturally withthe full accuracy of MPM. In other words, it is compa-rable to FEA in the ease of including explicit cracks andhas advantages over other meshless methods by not re-quiring crack approximations such as node-visibility orstress-diffraction rules [Organ, Fleming, and Belytschko(1996), Batra and Ching (2002)]. CRAMP may also haveadvantages over both FEA and other meshless methodsin its ability to fully model crack surface contact in-cluding frictionless sliding, sliding with friction, or stickconditions [Nairn (2003), Bardenhagen, Guilkey, Roes-sig, Brackbill, Witzel, and Foster (2001), Bardenhagen,Brackbill, and Sulsky (2000)].

Now that MPM/CRAMP can handle explicit cracks, thegoal of this paper was to address the second problem innumerical analysis of fracture or to develop MPM meth-ods for calculating fracture parameters and to verify thatMPM is suitable method for such calculations. We foundthat MPM is well-suited to efficient and accurate calcula-tion of J integral around cracks. By tracking crack open-ing displacements, it was further possible to convertJ-integral results into mode I and mode II stress intensityfactors. These results suggest that MPM is an excellentcandidate for the last fracture problem or the implemen-tation of failure criteria to predict both crack growth andgrowth direction. Because the MPM solution and thecrack definition are defined on particles, it is easy to im-plement crack propagation in arbitrary directions. Per-haps MPM will combine the algorithmic efficiencies of

MPM Calculations of J Integral 3

FEA for analysis of explicit cracks with the flexibility ofmeshless methods for describing arbitrary crack paths.

2 Numerical Methods

2.1 MPM Calculations with Explicit Cracks

The material point method (MPM), as originally derived,extrapolates particle information to a background gridfor solution of the equations of motion [Sulsky, Chen,and Schreyer (1994), Sulsky, Zhou, and Schreyer (1995),Sulsky and Schreyer (1996)]. Because this extrapolationenforces continuous velocities on the grid, the originalmethod could not accommodate cracks which are math-ematically described by velocity or displacement discon-tinuities. To extend MPM to crack problems, we de-veloped a new algorithm called CRAMP for cracks inmaterial point method calculations. This section gives abrief review of the CRAMP algorithm; the detailed al-gorithm is in Nairn (2003). The key difference in MPMwith cracks are:

1. To account for cracks, modify the MPM extrapo-lation of particle information to the grid to allownodes near cracks to have multiple velocity fields.The separate velocity fields contain information forthe solution on opposite sides of the cracks.

2. To determine which nodes need multiple velocityfields, cracks are introduced in the problem by ad-ditional mass-less particles. In 2D problems the ad-ditional particles are connected by line segments todefine the crack path. In 3D problems, crack def-inition requires defining a crack surface [Guo andNairn (2004)]. When extrapolating any particlep tonodei, a line is drawn from the particle to the nodeand each particle/node pair is assigned a field num-berν(p, i) which isν(p, i) = 0 if the line crosses nocrack (i.e., conventional MPM),ν(p, i) = 1 if theline crosses a crack and the particle is above thecrack, orν(p, i) = 2 if the line crosses the crackand the particle is below the crack. The concept of“above” or “below” the crack is relative to a coordi-nate system with the crack tip at the origin and thecrack direction in the negativex direction (or crackplane in the negativex-z plane in 3D problems).

3. Solve the equations of motion by conventionalMPM methods, but for cracks with multiple velocity

fields, solve the equations of motion for each veloc-ity field separately.

4. To prevent non-physical overlap at displacementdiscontinuities, implement contact rules at all nodeswith multiple velocity fields [Nairn (2003), Bar-denhagen, Guilkey, Roessig, Brackbill, Witzel, andFoster (2001)].

5. Using the nodal solutions, interpolate back to theparticles and update the solution on the particlessuch as particle stress, strain, velocity, and displace-ment. The interpolation back to the particles usesthe appropriate velocity field as determined in step2.

6. Finally, update the crack position within the body.In the previous paper [Nairn (2003)], this step up-dated the position of all crack particles by usingthe center-of-mass velocity field. In this work, weadded an extra update of crack information to trackcrack surfaces or crack opening displacements (seebelow).

2.2 CRAMP-Modified MPM Equations

Consider a solid body containing a crack subjected tobody force per unit volume~B and surface force~T whichhas been resolved into forces applied directly to particles,~Fp (see Fig. 1). The virtual work principle for the systemcan be expressed by [Cook, Malkus, and Plesha (1989)]:Z

V~B·δ~udV+∑

p

~Fp ·δ~u =Z

V~σ ·δ~εdV

+Z

Vρ

d2~udt2

·δ~udV+Z

Vdk

d~udt·δ~udV (1)

whereρ is density,dk is a damping coefficient,~u,~ε, and~σare the displacement, strain, and stress, respectively, andt is time. The termδ~u denotes an arbitrary virtual dis-placement that is allowed physically. In MPM, the bodyis divided into a collection of particles each assigned amass ofmp consistent with the material density. All thevariables needed to solve the problem (e.g., position, ve-locity, stress,etc.) are carried on the particles. To solvethe equations of motion for the particles, MPM uses abackground computational grid, as shown in Fig. 1. In-terpolating particle properties and virtual displacementto the grid and numerically evaluating the integrals using


Fp

Fp

Top

Plane

Bottom

Crack

Figure 1 : Discretization of a solid object with a crackinto material points or particles. The grid is the back-ground mesh for MPM calculations. The CRAMP algo-rithm tracks the crack as mass-less particles connectedby line segments; it also tracks the top and bottom cracksurfaces to provide information about crack opening dis-placement.

the particles as integration points, the virtual work equa-tion reduces to separate equations for nodal accelerationsof each velocity field (~ai,ν(p,i)):

~ai,ν(p,i) =~f tot

i,ν(p,i)

MLi,ν(p,i)

(2)

where ~f toti are the total nodal forces, andML

i are thelumped nodal masses. The second subscriptν(p, i) =0, 1, or 2 corresponds to each of the three possible ve-locity fields in the CRAMP algorithm. Although eachnode has three potential velocity fields, only nodes nearcracks will have more than the singleν(p, i) = 0 field.For the example, in Fig. 1 only the nodal points along thecrack plane will have multiple velocity fields. Further-more, even nodes near cracks will have at most two ve-locity fields that correspond to information extrapolatedfrom opposite sides of the crack near that node. (Note:although the crack plane in Fig. 1 is shown along gridlines, the crack particles will numerically reside on oneside or the other from the grid line. Although issues canarise due to numerical round off, they can be avoidedby careful selection of the line-crossing algorithm [Nairn(2003)].)

The remaining MPM equations [Sulsky, Chen, and

Schreyer (1994)] all need modification to account for thepossibility of multiple velocity fields. The lumped massmatrix at a nodei is

MLi,k = ∑

pmpNi,pδk,ν(p,i) (k = 0,1,2) (3)

The total nodal forces are the sum of the external forcesand the internal forces:

~f toti,k = ~f ext

i,k +~f inti,k (k = 0,1,2) (4)

where

~f exti,k = ∑

p

~FpNi,pδk,ν(p,i)−dk

ρ ∑p

mp~vpNi,pδk,ν(p,i) (5)

~f inti,k = ∑

pmp

(~Bs

pNi,p−~σsp ·∇Ni,p

)δk,ν(p,i) (6)

for k = 0,1,2 and where superscripts denotes a specificquantity,Ni,p = Ni(~xp) and∇Ni,p = ∇Ni(~xp) are the shapefunction and its’ gradient for nodei at the position ofparticle p (~xp), ~vp is the particle velocity, andδk,ν(p,i) isKronecker delta or 1 ifk = ν(p, i) and 0 otherwise.

Once nodal velocities have been updated, the nodal stateis interpolated back to the particles to update their ve-locity, position, stress, strain,etc.. In CRAMP, the inter-polation is done using the proper velocity field for eachparticle near a crack [Nairn (2003)]. Finally, the center-of-mass velocity for each particle in the definition of thecrack plane is used to update the crack position. A newfeature added to CRAMP in this work was to addition-ally use the separate velocity fields to track the positionsof the crack surfaces. Initially the crack is assumed tobe closed. During each MPM time step, the separatevelocity fields near cracks are used to update the posi-tions of the top and bottom surfaces of the crack. Figure1 shows a crack “Plane” which is defined by a collec-tion of massless particles connected by line segments.The “Top” and “Bottom” crack surfaces are tracked asdisplacements from that crack plane during the analysis.The crack opening displacement information is useful inpartitioning the stress state into mode I and mode II stressintensity factor.

An additional use for tracking crack opening displace-ments might be to implement numerical methods fordetection of crack contact. Prior MPM methods fordetecting two-body or crack contact have been basedon surface velocities or on nodal volumes or on both


[Nairn (2003), Bardenhagen, Guilkey, Roessig, Brack-bill, Witzel, and Foster (2001), Bardenhagen, Brackbill,and Sulsky (2000)]. Because contact detection is donefor grid nodes with multiple velocity fields while crackopening displacements are tracked on the massless crackparticles, it is more efficient to use grid methods ratherthan the new crack opening displacement information.The calculations in this paper have thus continued to usethe grid- and volume-based method described in Nairn(2003). By monitoring the crack opening displacementon the crack particles, we could confirm that the grid-based crack-contact detection algorithm worked well forall examples in this paper. We have experimented withdisplacement-based methods for crack-contact detectionthat extrapolates displacements within the two velocityfields to grid nodes and detects contact based on relativenormal displacements. This approach works well, but re-quires an extra extrapolation for nodal displacement in-formation.

In typical problems with cracks, only a few nodes nearcracks will actually have multiple velocity fields and thusthe CRAMP algorithm is very efficient. The computa-tional overhead for cracks in 2D problems is typicallyabout 10% [Nairn (2003)]. Additionally, MPM withcracks may have advantages over other particle-based ormeshless methods for inclusion of explicit cracks. TheCRAMP algorithm is anexactMPM representation ofcracks or displacement discontinuities analogous to thefact that the addition of cracks to an FEA mesh providesan exactFEA representation of discontinuities. In con-trast, other meshless methods can only include displace-ment discontinuities by approximate methods such asnode-visibility rules [Belytschko, Lu, and Gu (1994)] orstress-diffraction parameters [Organ, Fleming, and Be-lytschko (1996)].

3 J-integral and Stress Intensity Factor Calcula-tions in MPM

The CRAMP algorithm modification to MPM has ex-tended MPM to include explicit cracks and thereby de-termine stress fields, displacements,etc., near crack tips.To implement fracture mechanics models for crack prop-agation, the next step is to calculate fracture parametersfor crack tips such as energy release rate and stress in-tensity factors. This section describes MPM calculationsof J-integral around a crack tip and partitioning of theresults into mode I and mode II stress intensity factors.

The J-integral, as a key fracture parameter, was intro-duced by Cherepanov (1967) and Rice (1968). Althoughthe first derivation ofJ-integral was for quasi-static prob-lems with no kinetic energy, the concept can be ex-tended to dynamic problems by including kinetic energy.The definition of dynamic J-integral components (Jm form= 1,2) at a crack tip (see Fig. 2) is [Nishioka (1995),Cherepanov (1979)],

Jm = limε→0

ZΓε

[(W+K

)nm−σi j n j

∂ui

∂xm

]dΓ (7)

=Z

Γ

[(W+K

)nm−σi j n j

∂ui

∂xm

]dΓ

+Z

A(Γ)ρ[

∂2ui

∂t2

∂ui

∂xm− ∂ui

∂t∂2ui

∂t∂xm

]dA (8)

whereW andK are the stress-work density and kineticenergy density, respectively,σi j are stresses,ui are dis-placements (accordingly,∂ui/∂xm are the components ofdisplacement gradients,∂ui/∂t is velocity, and∂2ui/∂t2

is acceleration),nm are components of the unit normalvector to theJ-integral contour (Γ or Γε), ρ is density,and repeated indicesi and j are summed. The energyterms are

W = σi j dεi j and K =12

ρu̇i u̇i (9)

whereεi j are strains. TheΓ and Γε are line integralsalong contours around the crack tip. TheA(Γ) integralis an integral over the area enclosed by the contour.

For static problems, the area integral drops out and theresult is path independent for any crack tip contour (Γor Γε) [Rice (1968)]. This path independence is lost fordynamic problems because the transmission of energy todifferent paths depends on the time stress waves reach thepath. There are two solutions to the problem as expressedby the two results forJm above [Nishioka (1995)]. In thefirst solution (Eq. 7),Jm is calculated in the limit of asmall contourΓε. In this limit the components of theJ-integral are well defined because theA(Γ) integral be-comes negligible. The second solution (Eq. 8) can beused for an arbitrary contour,Γ, provided an extra term,which integrates the area enclosed byΓ or A(Γ), is in-cluded. The two combined terms are independent ofΓ[Nishioka (1995)], but the calculation ofJ is no longera single line integral. Most of the results in this paperused the first equation by numerical integration over apath close to the crack tip. The effect of path size, using


2'

1'

2

1

n1

n2

crack tip

Γε

θc

n̂

Figure 2 : Nomenclature and coordinates for terms usedin theJ-integral calculation. The 1 and 2 axes are globalx andy axes. The 1′ and 2′ axes are aligned with the cur-rent crack direction. The contourΓε starts on one cracksurface and proceeds counter-clockwise to the oppositecrack surface.

an example when the second term is needed, was alsoinvestigated.

Once the components of theJ-integral are evaluated, thetotal energy release rate for crack growth in elastic mate-rials (linear or non-linear) is given by [Nishioka (1983)]:

G = J1cosθc +J2sinθc (10)

whereθc is the crack propagation angle measured fromx-axis in the global coordinates (see Fig. 2).J1 andJ2 arecomponents ofJ integral, as evaluated in Eq. 7.

Figure 3 shows two potentialJ-integral contours arounda crack tip. The curveΓ1 is a circular path centered on thecrack tip. The curveΓ2 is a rectangular path that followsgrid lines in the background MPM mesh; it is centeredon the node nearest to the crack tip. BecauseJ-integral ispath-independent in the limit of small path length or byinclusion of the volume integral, theJ-integral contourcan be chosen arbitrarily. All calculations in this paperwere done with the rectangular path. By using a contourthat follows mesh lines in the background grid, the nu-merical integration to findJ integral was more efficient.Rectangular paths are denoted here asn×mpaths wheren andmare the distance from the central node to the con-

Γ1

Γ2

Figure 3 : Two possibleJ-integral contours around acrack tip in MPM calculations. The calculations in thispaper all used the rectangular path for computational ef-ficiency. This path is centered on the node nearest to thecrack tip and extends 2 cells in each direction from thatnode.

tour in thex andy directions. All paths where chosen tobe square (n = m). The path in Fig. 3 is a 2×2 contour.

The numericalJ-integral calculations proceeded as fol-lows. First, to get terms required for the contour integral,all needed terms from the current particle states were ex-trapolated to the nodes on the background mesh. Thenodal values were extrapolated by standard MPM meth-ods using

fi,k =∑pmp fpNi,pδk,ν(p,i)

MLi,k

(k = 0,1,2) (11)

wheref represents any property;fi,k are the nodal values,and fp are the values on the particles. The extrapolationto get displacement gradients used particle strain (e.g.,∂u1/∂x = εxx); to include cross-derivatives, the code sep-arately tracked∂u1/∂y and ∂u2/∂x on particles ratherthan just the sum or shear strainγxy. Importantly, theextrapolation must preserve the multiple velocity fields(k = 0,1,2) to provide information for the two sides ofthe crack. Next, the node closest to the crack tip was lo-cated and a rectangular path centered on that node wascalculated. For reasonably behaved cracks, this path will


cross the crack once. That crossing point was located,two nodes were placed at the cross-over point, and thepath was split to create a contour starting on one cracksurface and ending on the opposite crack surface. Thenodal values for stresses, energies, displacements,etc.,for the two new nodes were found by interpolation be-tween the two neighboring background mesh nodes onthe rectangular path. The neighboring nodes will alwayshave two velocity fields. The interpolation of these twovelocity fields gave two separate results resulting in onenew node that corresponded to the stress state on one sur-face of the crack while the other one corresponded to thestress state on the opposite side of the crack. Finally, as-suming there werenJ nodes (ornJ−1 segments) alongthe final J-integral contour, the components of theJ-integral were numerically integrated using the midpointrule:

Jm =nJ−1

∑i=1

(F(i)

m +F(i+1)m

) ∆i

2(m= 1,2) (12)

where∆i is the length of segmenti and the integrand ateach node,F(i)

m , is:

F(i)m =

(Wi,k +Ki,k

)nm−~σi,kn̂·

∂~ui,k

∂xm(13)

Here subscripti denotes nodal value while subscriptk de-notes the appropriate velocity field. The integration startson one crack surface with the velocity field appropriatefor that crack surface (velocity field for particles above orbelow the crack depending on crack orientation). When-ever nodes have multiple velocity fields, the initial veloc-ity field was used for the first half of the nodes while theopposite velocity field was used for the later half of thesegments. In practice, nodes near the middle on the con-tour had single velocity fields and there was never ambi-guity of the appropriate velocity field.

For calculations using Eq. 8, the second term was evalu-ated by numerical integration over the area enclosed bythe rectangular path. The particles enclosed by the pathwere used as equally-weighted integration points. Thecurrent particle properties for velocity, displacement gra-dient, and acceleration were used to calculate the inte-grand. The representation of the solution on the particlesdoes not have separate velocity fields, but, the particlestates automatically provide the proper result for the areaintegration.

The energy release rate,G, was calculated by Eq. 10;the angleθc for the calculation was found from the an-gle of the line segment at the crack tip. To partition totalenergy release rate,G, into mode I and mode II compo-nents, theG results were converted into mode I and modeII stress intensity factors —KI andKII . The calculationof G is valid for any elastic material (linear or nonlin-ear) and for heterogeneous materials. The conversion tostress intensity factors, however, required an assumptionof linear-elastic, homogeneous materials. The formulaeare given by [Nishioka (1995), Nishioka, Murakami, andTakemoto (1990)],

KI = δI

√2µGβII

AI(δ2

I βII +δ2II βI

) (14)

KII = δII

√2µGβI

AII(δ2

I βII +δ2II βI

) (15)

whereµ is the shear modulus,δI and δII denote crackopening and shearing displacements near the crack tip,andβI , βII , AI , andAII are parameters related to crackpropagation velocityC. They are given by,

βI =√

1−C/C2s and βII =

√1−C/C2

d (16)

and

AI =βI (1−β2

II )4βI βII − (1+β2

II )2(17)

AII =βII (1−β2

II )4βI βII − (1+β2

II )2(18)

whereCs andCd are the shear and dilational wave speeds

C2s =

µρ

and C2d =

(κ+1κ−1

)µρ

(19)

For a stationary crack withC = 0 (all examples in thispaper),βI = βII = 1 and

limC→0

AI = limC→0

AII =κ+1

4(20)

In the above equations,κ = (3− ν)/(1+ ν) for planestress andκ = 3− 4ν for plane strain, whereν is Pois-


son’s ratio. These limits lead to

KI =δI

δ√

GE (plane stress)

=δI

δ

√GE

1−ν2 (plane strain) (21)

KII =δII

δ√

GE (plane stress)

=δII

δ

√GE

1−ν2 (plane strain) (22)

whereE is tensile modulus andδ is the magnitude of thecrack opening displacement.

The new MPM results needed for stress intensity factorare the crack opening displacements,δI and δII . Thecrack opening displacements were calculated from theextension to CRAMP described in the previous sectionfor tracking positions of the crack surfaces. The crackis discretized into a series of massless points connectedby segments. The last point defines the crack tip. Thelast segment defines the crack orientation,θc. The crackopening displacements were calculated using the cracksurface positions of the next to last point. These openingdisplacements were resolved into displacement compo-nents normal and transverse to the last crack segment toprovideδI andδII , respectively; the partitioning only de-pends on the ratioδI/δII .

4 Results and Discussion

4.1 Static Results

TheJ-integral calculations and stress intensity factor par-titioning were first verified by comparison to static re-sults. Under static conditions, theJ-integral calculationis path independent and it is easy to find analytical re-sults for comparison. Figure 4 shows the analysis geom-etry for two static beam specimens. The specimens wereloaded at time zero with loads per unit thickness,PT andPB, applied to the top and bottom beams. This geometryresults in a pure mode I loading, double cantilever beamspecimen (DCB) whenPB =−PT or a pure mode II load-ing, end-notch flexure specimen (ENF) whenPB = PT .The material was assumed to be isotropic, linear elasticwith a modulus ofE = 2300MPa and a Poisson’s ratioof ν = 0.33. The specimens were 100mm long, 24mmhigh, and 1mm thick, and analyzed in plane-stress condi-tions. All the specimens were clamped on the left edge.

2h=24a=50

l = 100

PT

PB

P (N/mm)

t

P (N/mm)

t

PT = 4 X 10-4

PT = PB = 4 X 10-4

PB = -4 X 10-4

Mode I

Mode II

Figure 4 : The geometry of beam specimens used tocompare MPM results to static results. By varyingPT

andPB, this specimen can be pure mode I (DCB), puremode II (ENF), or mixed mode. The loads were appliedinstantaneously at time zero and then held constant.

The mass density wasρ = 1.5g/cm3. In order to com-pare the results of stress intensity evaluated by MPMwith static results, external damping was incorporated inthe computations and adjusted to make the dynamic vi-brations damp out after a few oscillations. The dampedout results should converge to the static solution. Thedamping coefficient in Eq. 1 for these specimens was setto dk = 1000sec−1. To test convergence, the dimensionsof the background mesh cells for both the DCB and ENFspecimens were varied from 4×4mm down to 1×1mm.For all meshes, there were 4 particles per cell. The ini-tial spacing between particles thus varied from 2mm to0.5mm or 4% to 1% of the crack length (50mm). TheJ-integral contour was 2× 2 cells. The distance of thecontour to the crack tip thus varied from 8mm to 2mmor 16% to 4% of the crack length.

Figures 5 and 6 compare the results of dynamic stress in-tensity evaluated by MPM to the static results. For theDCB specimen, the static stress intensity factors werefound using corrected beam theory [Kanninen (1973)]:

KIs = 2√

3PT(a+χh)

h3/2and KIIs = 0 (23)

whereχ is a correction factor to account for crack-rootrotation effects andχ = 2/3 for isotropic materials. Forthe ENF specimen, the static stress intensity factors werefound from [Carlsson, Gillespire, and Pipes (1986), Guoand Tang (1993)]:

KIs = 0 and KIIs =3PTa

h3/2

√1+

2(1+ν)5

(ha

)2

(24)


0 2 4 6 8 10 12-20

0

20

40

60

80

100

120 c=1*1mm c=2*2mm c=4*4mm

KIs

KId

KIId

Str

ess

Inte

nsity

(X

10-6 M

Pa

/m)

t (ms)

Figure 5 : Comparison of the stress intensity factors,KI

andKII , for the DCB beam specimen calculated by MPManalysis using three different cell sizes or by analyticalresults (horizontal line). The dynamic MPM results haddamping such that they should converge to the static so-lution.

For both the DCB and ENF specimens, the convergedMPM results oscillated around the analytical results andconverged to those results when damping was complete.The mode I results converged for 2×2mm cells, whilethe mode II results required smaller 1×1mm cells. Thecorrect partitioning of energy release rate intoKI andKII is shown by the calculation ofKII = 0 for the DCBspecimen (Fig. 5) andKI = 0 for the ENF specimen(Fig. 6). The partitioning was further verified by anal-ysis of mixed-mode specimens such as usingPT = P andPB = 0.

4.2 Dynamic Results

Three specimens were used to compare the results ofstress intensity factors evaluated by MPM throughJ-integral to the results computed by other approaches. Thefirst two specimens were a central cracked tension (CCT)specimen and a single edge notched tension (SENT)specimen. Figures 7a and 7b show the geometry and theaxial Heaviside load boundary condition for these speci-mens. They were analyzed under plane strain conditions.The crack-tip loading was pure mode I. The material waslinear elastic with a modulus ofE = 200GPa and Pois-son’s ratio ofν = 0.3. The mass density of the materialwas set toρ = 5.0g/cm3. The dimensions of the speci-

0 2 4 6 8 10 12-10

0

10

20

30

40

50

60

70

80 c=1*1mm c=2*2mm c=4*4mm

KIIs

KIId

KIdStr

ess

Inte

nsity

(X

10-6 M

Pa

/m)

t (ms)

Figure 6 : Comparison of the stress intensity factors,KI

andKII , for the ENF beam specimen calculated by MPManalysis using three different cell sizes or by analyticalresults (horizontal line). The dynamic MPM results haddamping such that they should converge to the static so-lution.

mens and the crack lengths are given in Fig. 7. The di-mensions of background mesh cells were 0.5× 0.5mmwith 4 particles per cell. The particle spacing was thus0.25mm which is 1.25% of the specimen width, 5.2% ofthe CCT crack length, and 5% of the SENT crack length.TheJ-integral contour was 2×2 cells.

Figures 8 and 9 compare the results of dynamic mode Istress intensity factors of the CCT and SENT specimenscalculated by MPM to prior results calculated by the fi-nite difference method [Chen and Wilkens (1977)]. Inthe two figures, the stress intensity factors were normal-ized by the static, infinite-sheet stress intensity factor ofKI0 = σ0

√πa whereσ0 = 400MPa is the value of the ap-

plied stress, anda is 2.4mm and 5mm for the CCT andSENT specimens, respectively. Figures 8 and 9 show thatthe new results calculated by MPM agree well with priornumerical results.

The third dynamic specimen analyzed was a double edgenotched plate (DENP) specimen dynamically loadedin impact compression between the two notches (seeFig. 7c). The DENP specimen has been used as an impactspecimen to study the dynamic fracture toughness undershear or mode II loading [Kalthoff and Winkler (1987)].Although analysis shows the loading is mostly mode II,


a=2.4mm

a=5mm

2a

a

20mm

20mm

40mm

a. CCT

c. DENP

32mm

b. SENT

200mm

a=50mm

100mm

25mmσ(t)

σ(t)

σ(t)

σ(t)

σ0

t

Figure 7 : Specimen geometries for dynamic stress in-tensity factor calculations. a. Center crack tension (CCT)specimen. b. Single edge notched tension (SENT) spec-imen. c. Double edge notched plate (DENP) specimen.All specimens where loaded (as indicated by arrows) in-stantaneously at time zero and then the stress was heldconstant.

0 2 4 6 8 10 12 14-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

by MPM by FDM [1]

KId/K

I0

Time (µs)

Figure 8 : Comparison of the mode I stress intensity fac-tor (KI ) for the CCT specimen calculated by MPM orby finite difference method (FDM in Chen and Wilkens(1977)).

0 2 4 6 8 10 12 14 16 18 20 22-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

by MPM by FDM [1]

KId/K

I0

Time (µs)

Figure 9 : Comparison of the mode I stress intensity fac-tor (KI ) for the SENT specimen calculated by MPM orby finite difference method (FDM in Chen and Wilkens(1977)).

the mode I component is non-zero or the loading is actu-ally mixed mode. The material was linear elastic with amodulus ofE = 210GPa and Poisson’s ratio ofν = 0.29and analyzed under plane-strain conditions. The impactcompression stress wasσ0 = 200MPa. The mass den-sity of the material was set toρ = 7.833g/cm3. Thedimensions of the specimens and the crack lengths aregiven in Fig. 7c. The dimensions of the background meshcells were 2×2mm with 4 particles per cell. The parti-cle spacing was thus 1mm which is 1% of the specimenwidth and 2% of the crack length. TheJ-integral contourwas 2× 2 cells. The MPM calculations used symme-try and analyzed only the top half of the specimen. Thenodal velocities in they direction in the midplane of thespecimen were constrained to zero to define the symme-try plane.

Figure 10 compares the results of mode I and mode IIstress intensity factors evaluated by MPM to prior re-sults calculated by the Meshless Local Petrov-Galerkin(MLPG) method [Batra and Ching (2002)]. The twomethods have similar results. The slight differences maybe a real difference because the MPM results and MLPGresults were for slightly different problems. The MLPGanalysis used highly refined particle density near thecrack tip in an attempt to resolve the effect of the ac-tual machined crack tip with a radius of 0.15mm [Batraand Ching (2002)]. The MPM analysis assumed a sharp


0 2 4 6 8 10 12 14 16 18 20 22 24-40

-35

-30

-25

-20

-15

-10

-5

0

5

KIId

KId

by MPM by MLPG [11]

KId &

KII

d (M

Pa

m )

Time (µs)

Figure 10 : Comparison of the mode I and mode II stressintensity factors (KI andKII ) for the DENP specimen cal-culated by MPM or by Meshless Local Petrov Galerkin(MLPG in Batra and Ching (2002)) method.

crack, although at the analyzed resolution, the MPM re-sults would not distinguish crack radii less than the halfthe inter-particle spacing (0.5mm). The MPM resultscould be extended to higher resolution, but the similar-ity of the results in Fig. 10 indicates the details on thecrack tip shape do not have a large effect provided thecrack is reasonably sharp (< 0.5mm).

4.3 Path Effects on DynamicJ Integral

The results in the previous section were for small rect-angular paths (see Fig. 3) that extended 2 cells in thexand y directions from the node nearest to the crack tip(2× 2 cell paths). The distance from the crack tip tothese contours was 8% of the crack length for most spec-imens, but was 20% of the crack length for the CCT andDENP specimens. These contours were judged to be suf-ficiently small (by comparison to other methods), that theJ-integral could be evaluated by Eq. 7. In other words,J-integral was evaluated by the first term in Eq. 8 andthe second term was assumed to be negligible. The ad-vantage of this approach is numerical efficiency becausethe extrapolations and calculations needed for the secondterm could be skipped. For calculations using larger con-tours or for different problems with different dynamic ef-fects, the second term might be needed. This section de-scribes some calculations for a problem where the secondterm is needed and shows it can be evaluated by MPM.

We repeated the DENP calculations for a rectangularpaths from 2×2 cells up to 10×10 cells. The later wasthe largest square path that could be accommodated inthe DENP specimen and remain within the material whenusing cells that were 2×2mm. For small contours (un-der 5×5 cells), the second term in Eq. 8 was negligibleand thus Eq. 7 was accurate for calculating theJ-integral.For larger paths, however, the differences became signifi-cant. Figure 11 compares mode II stress intensity factorscalculated by the smallest and largest paths either withor without the second term in Eq. 8. The lack of dif-ference for the two 2× 2 cells path results verifies thesecond term was negligible for that size path. In con-trast, the second term has a large effect on the 10× 10cells path results. When the second term was ignored,the large path give poor results. The oscillations corre-spond to stress waves passing through the contour. Whenthe second term was included, the large path calculationagreed with the small path calculations for most times.This result shows that MPM can evaluate the second termand that analysis using both terms in Eq. 8 gives a path-independent result.

There are discrepancies between the small path and thetwo-term, large path results for times less than about 9µs.This time corresponds to the time required for the stressinitiated by the impact loading to travel from the edgeof the specimen to the crack tip. Thus, the results fortimes less than 9µs correspond to times for which stresswaves have reached theJ-integral contour, but have notyet reached the crack tip. The results in this region areprobably more reliable with the smaller path. The rec-ommendation is that all MPM calculations ofJ integralshould use a small path. The second term may be in-cluded to insure path independence, but it will probablybe negligible for typical problems.

4.4 Effect of Background Mesh Size on Stress Inten-sity Factor

The stress intensity factors were computed here by con-vertingJ-integral results rather than by direct evaluationfrom local crack-tip stresses and displacements. BecausetheJ-integral calculations work well, even for relativelycrude meshes, it is reasonable to expect that the calcu-lation of stress intensity in MPM can be accurate andmore computationally effective than in direct methods,which might require finer meshes. In dynamic analysis,computational efficiency is extremely important because


0 5 10 15 20 25-35

-30

-25

-20

-15

-10

-5

0

5

2X2, 1 term 2X2, 2 terms 10X10, 1 term 10X10, 2 terms

KII

d (

MP

a /m

)

Time (ms)

Figure 11 : Calculation ofJ integral using a line integral(one term or Eq. 7) or using a line and and area integrals(two terms or Eq. 8). The calculations were for a smallcontour (2×2cells) or for a large contour (10×10cells).The results are for the mode II stress intensity factor (KII )in a DENP specimen

an enormous number of time steps are often required fortypical problems. The use of a relatively coarse back-ground mesh, while still obtaining goodJ-integral re-sults, would save a great deal of computational time. Thissection describes calculations to assess the computationalefficiency of dynamicJ-integral in MPM,i.e., to assessthe tolerance of the calculation to large background cellsizes.

We repeated the SENT calculations (see Fig. 9), whichpreviously had cell size ofc = 0.5mm or 0.5 by 0.5mmcells, with background meshes havingc = 0.25mm,c =1.0mm, andc = 2.0mm. For a crack length of 5mm,a specimen width of 20mm, and four particles per cell,these cell sizes correspond to inter-particle spacings (s=c/2) that are 2.5%, 5%, 10%, and 20% of the crack length(a) and 0.625%, 1.25%, 2.5%, and 5% of the specimenwidth (W). The J-integral contours were chosen alonggrid line segments two cells away from the crack tip inboth thex andy directions. The normalized mode I stressintensity factors for each cell size are plotted in Fig. 12.

The results in Fig. 12 show that theJ-integral results con-verge for the relatively coarse mesh ofc = 1mm, whichcorresponds to relative mesh size ofs/a = 10% ands/W = 2.5%. The results for all finer meshes were nearly

0 2 4 6 8 10 12 14 16 18 20 22-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

c=0.25mm c=0.5mm c=1mm c=2mm

KId/K

I0

Time (µs)

Figure 12 : Calculation of the mode I stress intensity fac-tor (KI ) in an SENT specimen as a function of the back-ground mesh density. All calculations used a rectangu-lar path that extend two cells from the crack tip in eachdirection. The background cells sizes, however, werevaried usingc = 0.25mm,c = 0.5mm, c = 1.0mm, orc = 2.0mm.

identical. The results for the coarsest mesh (c = 2mm)showed deviations or lost accuracy. The ability to obtainaccurateJ-integral results without the need for highly re-fined crack-tip meshes is especially important in dynamicproblems because each factor of 2 in cell size leads toa factor of 23 = 8 (in 2D or 24 = 16 in 3D) increasein calculation time. There is a factor of two for eachdimension and another factor of two because smallermeshes require proportionally smaller times steps in ex-plicit solvers. The ability to find accurateJ-integral whenc = 1.0mm instead of needingc = 0.25mm provides afactor 64 (in 2D) or 256 (in 3D) efficiency in crack prob-lems.

5 Conclusions

The material point method is a new method that has inter-esting potential for handling fracture problems involvingcrack propagation. MPM has previously been extendedto allow analysis of problems with explicit cracks [Nairn(2003)]. It can include explicit cracks with the ease offinite element analysis while retaining the advantages ofmeshless methods for the ease of handling crack propa-gation in arbitrary directions. This paper has described


new MPM algorithms for calculatingJ-integral and forpartitioning the total energy release rate into mode I andmode II stress intensity factors. Several examples wereconsidered and all show that MPM results agree withanalytical solutions or with prior numerical results. Al-though calculations of dynamicJ-integral, can usually becalculated by Eq. 7 using a small path around the cracktip, true path-independent results require the line inte-gral to be supplemented with an area integral inside thecontour as in Eq. 8. MPM can findJ integral by eitherapproach and both have computational efficiency. Thearea integral can be found by using the particles enclosedin the contour as integration points. Finally, some con-vergence tests suggest thatJ-integral calculations are ac-curate even with fairly coarse meshes. The introductionoutlined three key problems to numerical analysis of dy-namic fracture. The results in Nairn (2003) and the re-sults in this paper show that MPM can solve the first twoproblems. Future work should be aimed at the third prob-lem or full MPM analysis of fracture problems includingcrack propagation.

Acknowledgement: This work was supported bya grant from the Department of Energy DE-FG03-02ER45914 and by the University of Utah Centerfor the Simulation of Accidental Fires and Explo-sions (C-SAFE), funded by the Department of Energy,Lawrence Livermore National Laboratory, under Sub-contract B341493.

References

Atluri, S. N.; Shen, S. (2002): The Meshless Lo-cal Petrov-Galerkin (MLPG) method: A Simple & Less-Costly Alternative to the Finite Element and BoundaryElement Methods.Computer Modeling in Engineering& Sciences, vol. 3, pp. 11–52.

Bardenhagen, S. G.; Brackbill, J. U.; Sulsky, D.(2000): The Material Point Method for Granular Ma-terials. Computer Methods in Applied Mechanics andEngineering, vol. 187, pp. 529–541.

Bardenhagen, S. G.; Guilkey, J. E.; Roessig, K. M.;Brackbill, J. U.; Witzel, W. M.; Foster, J. C. (2001):An Improved Contact Algorithm for the Material PointMethod and Application to Stress Propagation in Gran-ular Material. Computer Modeling in Engineering &Sciences, vol. 2, pp. 509–522.

Bardenhagen, S. G.; Kober, E. M.(2004): The Gen-eralized Interpolation Material Point Method.ComputerModeling in Engineering & Sciences, vol. 5, pp. 477–496.

Batra, R. C.; Ching, H.-K. (2002): Analysis ofElastodynamic Deformations Near a Crack/Notch Tip bythe Meshless Local Petrov- Galerkin (MLPG) Method.Computer Modeling in Engineering & Sciences, vol. 3,pp. 717–730.

Belytschko, T.; Lu, Y. Y.; Gu, L. (1994): Element-FreeGalerkin Methods. Int. J. Num. Meth. Engrg., vol. 37,pp. 229–256.

Carlsson, L. A.; Gillespire, J. W.; Pipes, R. B.(1986):On The Analysis and Design of The End Notched Flex-ure (ENF) Specimen for Mode II Testing. J. Comp.Mater., vol. 20, pp. 594–604.

Chen, Y. M.; Wilkens, M. L. (1977): NumericalAnalysis of Dynamic Crack Problems, pp. 317–325. InElastodynamic Crack Problems, Noordhoff InternationalPublishing, The Netherlands, 1977.

Cherepanov, G. P.(1967): The Propagation of Cracksin A Continuous Media. J. Appl. Math. Mech., vol. 31,pp. 503–512.

Cherepanov, G. P.(1979): Mechanics of Brittle Frac-ture. McGraw-Hill, New York.

Cook, R. D.; Malkus, D. S.; Plesha, M. E.(1989):Concepts and Applications of Finite Element Analysis.John Wiley & Sons, New York.

Ferretti, E. (2003): Crack Propagation Modeling byRemeshing Using the Cell Method (CM).ComputerModeling in Engineering & Sciences, vol. 4, pp. 51–72.

Guo, Y.; Nairn, J. A. (2004): Three-Dimensional Dy-namic Fracture Analysis in the Material Point Method. inpreparation, 2004.

Guo, Y. J.; Tang, J. M. (1993): Finite Element Anal-ysis of Strain-Energy-Release Rate for The End NotchedFlexure (ENF) Specimen. InProc. 38th Int. SAMPESymp. & Exhibition, pp. 857–867.

Han, Z. D.; Atluri, S. N. (2003): SGBEM (for CrackedLocal Subdomain) FEM(for uncracked global Structure)Alternating Method for Analyzing 3D Surface Cracks


and Their Fatigue-Growth.Computer Modeling in En-gineering & Sciences, vol. 3, pp. 699–716.

Han, Z. D.; Atluri, S. N. (2003): Truly Meshless LocalPetrov-Galerkin (MLPG) Solutions of Traction & Dis-placement BIEs.Computer Modeling in Engineering &Sciences, vol. 4, pp. 665–678.

Kalthoff, J. F.; Winkler, S. (1987): Failure Mode Tran-sition at High Rates of Shear Loading. InProc. Int. Conf.on Impact Loading and Dynamic Behaviour of Materials,pp. 185–195.

Kanninen, M. F. (1973): An Augmented Double Can-tilever Beam Model for Studying Crack Propagation andArrest. Int. J. Fract., vol. 9, pp. 83–92.

Nairn, J. A. (2003): Material point method calculationswith explicit cracks. Computer Methods in Applied Me-chanics and Engineering, vol. 4, pp. 649–664.

Nishioka, T. (1983): A Numerical Study of the Useof Path Independent Integrals in Elastic-Dynamic CrackPropagation.Engr. Fract. Mech., vol. 18, pp. 23–33.

Nishioka, T. (1995): Recent Developments in Com-putational Dynamic Fracture Mechanics, pp. 1–58. InDynamic Fracture Mechanics, Computational Mechan-ics Publications, Southampton, UK, 1995.

Nishioka, T. (1997): Computational Dynamic FractureMechanics. Int. J. Fract., vol. 86, pp. 127–159.

Nishioka, T.; Atluri, S. N. (1983): Path-IndependentIntegrals, Energy Release Rates, and General Solutionsof Near-Tip Fields in Mixed-Mode Dynamic FractureMechanics.Engr. Fract. Mech., vol. 18, pp. 1–22.

Nishioka, T.; Murakami, R.; Takemoto, Y. (1990):The Use of The Dynamic J Jntegral (J) in Finite-ElementSimulation of Mode I and Mixed-Mode Dynamic CrackPropagation. International Journal of Pressure Vesselsand Piping, vol. 44, pp. 329–352.

Nishioka, T.; Stan, F. (2003): A HybridExperimental-Numerical Study on the Mechanism ofThree-Dimensional Dynamic Fracture.Computer Mod-eling in Engineering & Sciences, vol. 4, pp. 119–140.

Nishioka, T.; Tokudome, H.; Kinoshita, M. (2001):Dynamic Fracture-Path Prediction in Impact FracturePhenomena Using Moving Finite Element method based

on Delaunay Automatic Mesh Generation.Int. J. SolidsStruct., vol. 38, pp. 5273–5301.

Organ, D. J.; Fleming, M.; Belytschko, T. (1996):Continuous Meshless Approximations for NonconvexBodies by Diffraction and Transparency.ComputationalMechanics, vol. 18, pp. 225–235.

Rice, J. R. (1968): A Path Independent Integral andthe Approximate Analysis of Strain Concentration byNotches and Cracks.J. Applied Mech., vol. June, pp.379–386.

Rybicki, E. F.; Kanninen, M. F. (1977): A Finite El-ement Calculation of Stress Intensity Factors By a Mod-ified Crack Closure Integral.Eng. Fract. Mech., vol. 9,pp. 931–938.

Sulsky, D.; Chen, Z.; Schreyer, H. L.(1994): A Par-ticle Method for History-Dependent Materials.Comput.Methods Appl. Mech. Engrg., vol. 118, pp. 179–186.

Sulsky, D.; Schreyer, H. K. (1996): AxisymnmetricForm of the Material Point Method with Applications toUpsetting and Taylor Impact Problems.Comput. Meth-ods. Appl. Mech. Engrg, vol. 139, pp. 409–429.

Sulsky, D.; Zhou, S.-J.; Schreyer, H. L.(1995): Appli-cation of a Particle-in-Cell Method to Solid Mechanics.Comput. Phys. Commun., vol. 87, pp. 236–252.

Tan, H.; Nairn, J. A. (2002): Hierarchical AdaptiveMaterial Point Method in Dynamic Energy Release RateCalculations. Comput. Meths. Appl. Mech. Engrg., vol.191, pp. 2095–2109.

Wen, P. H.; Aliabadi, M. H.; Rooke, D. P. (1998):Cracks in Three Dimensions: A Dynamic Dual Bound-ary Element Analysis. Comput. Methods Appl. Mech.Engrg, vol. 167, pp. 139–151.

calculation of j-integral and stress intensity factors ... · pdf filecalculation of...

Documents