Theoretical and Applied Fracture Mechanics 69 (2014) 110–117

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Theoretical and Applied Fracture Mechanics

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A mass matrix formulation for cohesive surface elements

0167-8442/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.tafmec.2013.11.011

⇑ Corresponding author. Tel.: +44 (0) 114 222 5738.E-mail address: cip09jeh@sheffield.ac.uk (J. Hetherington).

Jack Hetherington ⇑, Harm AskesUniversity of Sheffield, Department of Civil and Structural Engineering, Mappin Street, Sheffield S1 3JD, UK

a r t i c l e i n f o

Article history:Available online 27 December 2013

Keywords:Interface elementCohesive surfaceExplicit dynamicsMass penaltyBipenaltyCritical time step

a b s t r a c t

A well-known method for modelling crack propagation in structural finite element analysis is the use ofinterface elements employing the theory of cohesive surfaces. However, the use of cohesive surfaces inexplicit dynamics is problematic since they have zero mass and must initially be very stiff in order toavoid the introduction of artificial compliance. These properties lead to an often drastic reduction inthe critical time step of the analysis. In this paper we use the bipenalty method to derive a mass matrixfor a 2D cohesive surface interface element that does not add net physical mass to the overall system.This allows for cohesive surfaces with very high initial stiffness that have no effect on the critical timestep of the analysis. Not only does this lead to a more robust and stable system, it also greatly simplifiesthe choice of parameters since there is no need to adjust the time step, and no need to limit the initialpenalty stiffness according to time step stability considerations.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction boundaries possessing a specially designed traction–displacement

In finite element (FE) analysis the three most common tech-niques for the modelling of fracture and crack propagation in a dy-namic setting are the element deletion method, the extendedfinite element method (XFEM), and inter-element crack methods[1]. Each of these approaches build upon standard FE formulationsto include the effects of damage and crack propagation in someway. Element deletion is the simplest of the methods and the mostwidely used in commercial codes (e.g., ANSYS [2] and LS-DYNA[3]). It requires only an alteration of the constitutive relation of a fail-ing element so that the stress in the element is reduced to zero forlarge strain, effectively removing certain elements as an analysis iscarried out. However, its reliability with regards to the predictionof crack paths has been called into question [1]. Furthermore, crackpaths and the details of crack growth are often highly mesh-depen-dent [4]. XFEM was first introduced by Belytschko and co-workers in1999 to tackle crack propagation problems in elastostatics [5,6]. Ituses shape function enrichment in order to introduce discontinu-ities within finite elements, which overcomes the high mesh depen-dence of previously existing techniques. This makes it an attractiveoption for accurately and efficiently predicting crack paths whichare not known a priori [7], but has yet to achieve widespreadadoption in commercial software.

Inter-element crack methods are a well-established group oftechniques that explicitly model cracks on the boundaries of indi-vidual finite elements. This can be achieved either by adaptiveremeshing or by the addition of interface elements at element

relationship, an approach also referred to as the cohesive zonemodel. The theory of cohesive surfaces (also known as cohesivezones) was first introduced in the 1960s [8,9] but was not appliedto dynamic crack propagation until the 1990s, with publicationsfrom Xu and Needleman [10], Camacho and Ortiz [11] and Repettoet al. [12] forming the basis for the present work. Each of these for-mulations introduces interface elements, or ‘cohesive surfaces’,into the FE continuum. A nonlinear traction–displacement rela-tionship is then chosen that approximately represents the fracturecharacteristics of the material. Cracks are thus free to coalesce andpropagate as a natural outcome of the simulation.

Using cohesive zone modelling for explicit dynamic analysis,however, leads to some unique challenges. Explicit solvers aremuch more efficient that implicit schemes per time step, but be-cause they are conditionally stable the step size must be kept be-low the so-called critical time step, Dtcrit, in order to ensurestability. For the central difference method the critical time stepis given by Dtcrit ¼ 2=xmax, where xmax is the maximum eigenfre-quency of the system. The critical time step therefore depends onmesh size, as well as material properties. Elements with high stiff-ness or low mass decrease Dtcrit, leading to extra computational ex-pense. Interface elements in a cohesive surface formulation mustinitially have very high stiffness so that they do not have any ad-verse effect on the simulation before damage has occurred; ele-ments that are not stiff enough lead to ‘artificial compliance’ inthe continuum [13,14]. In addition, they have no mass, since theyhave an initial volume of zero. These properties can lead to a dras-tic reduction in the critical time step that is required for stability.

Camacho and Ortiz [11] avoid this problem by introducingcohesive surfaces only at the onset of damage, but this requires

t

12

34

t1, 3 2, 4

Fig. 1. Line interface element in an initial (left) and deformed (right) configuration.

J. Hetherington, H. Askes / Theoretical and Applied Fracture Mechanics 69 (2014) 110–117 111

alterations to the FE discretisation (and thus to the computermemory requirements) as cracks propagate. Ortiz and Pandolfi[15] also select a cohesive law without an initial elastic region be-cause this would place ‘‘stringent restrictions’’ on the stable timestep. Espinosa and Zavattieri [13] use a large initial stiffness, butit is acknowledged by the same authors that a large penalty willhave a significant impact on the critical time step, and as a result,the time step calculation includes an additional limitation in that itmust take into account the cohesive surfaces as well as continuumelements. Because of this, a subcycling time integration routine isbuilt into the formulation, adding undesirable complexity to thesolution algorithm. This is deemed necessary because, as notedby Song et al., the original cohesive surface formulation developedby Xu and Needleman ‘‘induces artificial compliance due to theelasticity of the intrinsic cohesive law’’ [14].

Interface elements by their nature introduce large eigenvaluesinto the FE system; since the critical time step is inversely propor-tional to the maximum eigenvalue this has a detrimental effect onthe critical time step. The standard analysis states that this is dueto the high initial stiffness of the cohesive surface elements. How-ever, eigenvalues may be decreased not only by decreasing thestiffness of an element, but also by adding mass. Recently, anextension of the traditional penalty method—referred to here asthe bipenalty method—has been proposed that includes a masspenalty matrix alongside standard stiffness penalties in the formu-lation [16–20]. In the present work, we use the bipenalty methodto provide a mass matrix for a simple cohesive surface formulation.No net physical mass is added to the system; the sum of all ele-ments in the interface mass matrix is zero. The inclusion of themass matrix, however, does allow for control over the eigenvaluesintroduced by the interface elements, and therefore control overthe effect that the elements have on the critical time step. By pro-viding a mass matrix formulation alongside the traditional stiffnesspenalties, the introduced eigenvalues can be controlled even whenvery a very large initial stiffness is used, so that interface elementsand, by extension, cohesive surfaces can be used in explicit dynam-ics without having to reduce the critical time step.

2. Element formulation

We assume that initially we have a structural system, discre-tised in space by the FE method, of the form

M uþKu ¼ f ð1Þ

where M and K are the assembled mass and stiffness matrix for thecontinuum elements, u is the displacement vector, f the externalforce vector, and dot notation is used to indicate time derivatives;structural damping is neglected. A bipenalty formulation resultsin a system of equations of the form

ðMþMPÞuþðKþ KPÞu ¼ f ð2Þ

where MP and KP are mass and stiffness penalty matrices, which fora system containing cohesive surfaces are assembled from theinterface element mass and stiffness matrices, which are to bederived in this section.

The critical time step for the system is given by

Dtcrit ¼2

xmaxð3Þ

where xmax is the maximum eigenfrequency of the system. Eigen-values are related to eigenfrequencies by ki ¼ x2

i and the maximumeigenvalue is kmax. The eigenvalues can be determined by solvingthe generalised eigenvalue problem for the system. In the casewhere KP ¼ RMP (with R a scalar) it has been shown that the max-imum eigenvalue kmax of the penalised system (2) will not exceed

the maximum eigenvalue kUPmax of the unpenalised system (1) for

the case where R 6 kUPmax [20,21]. Thus, the critical time step Dtcrit

is not decreased by the addition of the interface elements forR 6 kUP

max.We will now present a standard interface element stiffness

matrix formulation, followed by the corresponding mass matrixformulation, and show that under reasonable assumptions,KP ¼ RMP (and therefore that the above analysis holds for thisbipenalty cohesive surface formulation).

2.1. Element stiffness matrix

The interface element formulation is based on the work ofSchellekens [22,23], who derives a 4-noded 2D line interfaceelement with an initial volume of zero (see Fig. 1). The stress isdefined by normal and tangential tractions across the interfaceand the stiffness of the element is controlled by user-definedparameters that describe the constitutive behaviour.

We now consider this 4-noded line interface. Each node has twodisplacement degrees of freedom (DOF), giving an element nodaldisplacement vector

d ¼ d1n;d

2n;d

3n;d

4n;d

1t ;d

2t ;d

3t ;d

4t

h iTð4Þ

where n and t denote the directions normal and tangential to theinterface, respectively, and superscripts indicate the node numbersas shown in Fig. 1. The relationship between nodal displacements dand relative displacements d ¼ ½dn; dt�T is given by

d ¼ Bd ð5Þ

where

B ¼�n n 0 00 0 �n n

� �

and n are the interpolation polynomials n ¼ ½N1;N2�. For arbitrarilyorientated elements, the matrix B should be transformed to thelocal tangential co-ordinate system of the node set.

We now introduce a matrix Ds describing the constitutive trac-tion–displacement relation, so that

t ¼ Dsd ð7Þ

where t ¼ ½tn; tt�T is the traction vector for the element (units N/m2)and Ds is a constitutive matrix of the form

Ds ¼kn 00 kt

� �

The values kn and kt (units N/m3) represent the ‘stiffness’ of theinterface in the normal and tangential directions, although a moreaccurate description is stiffness per unit area. It is these values thatfunction as the stiffness penalty parameters for the interface. In thepresent work we assume that both parameters are equal so thatkn ¼ kt ¼ as and Ds ¼ asI. We postpone until Section 2.3 a discus-sion of how these constitutive relations may change over time(due to damage).

The stiffness matrix Ke can now be obtained by minimisation ofthe total potential energy. The internal work done in the element is

112 J. Hetherington, H. Askes / Theoretical and Applied Fracture Mechanics 69 (2014) 110–117

U ¼ 12

ZS

dT t dS ð9Þ

which can be rewritten using Eqs. (5) and (7) as

U ¼ 12

dTZ

SBT DsB dS d ð10Þ

while the external work W is given by

W ¼ �dT f ð11Þ

where f is a vector containing the external forces on the element.After setting the variation of the total potential energy (U þW) tozero we find

Ked ¼ f ð12Þ

where the stiffness matrix is given by

Ke ¼Z

SBT DsBdS ð13Þ

Considering the numerical integration of such elements, we notethat the linear shape functions can be written in one isoparametricco-ordinate n as

N1 ¼12ð1� nÞ ð14Þ

N2 ¼12ð1þ nÞ ð15Þ

which means that we need integrate over only one co-ordinate. Thestiffness matrix can therefore be computed using 2-point Gaussianintegration via

Ke ¼ bZ 1

�1BT DsB

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@x@n

� �2

þ @y@n

� �2s

dn ð16Þ

where b is the width of the interface in the out-of-plane direction.

2.2. Element mass matrix

Thus far, we have formulated a stiffness matrix for the interface,based on the minimisation of total potential energy. In order to ob-tain a full bipenalty formulation, and thus obtain a suitable massmatrix, we must also consider the kinetic energy of the interface,which is related to velocity. Thus, analogous to Eq. (5) we have

_d ¼ B _d ð17Þ

Introducing a momentum vector p ¼ ½pn;pt�T we can then write a

momentum-velocity relation,

p ¼ Dm_d ð18Þ

where p represents momentum (per unit area) in the normal andtangential directions, and the matrix Dm contains mass penaltiesin the normal and tangential directions (with units kg/m2). It isassumed that this matrix is a scalar multiple of the constitutivematrix (25), so that Ds ¼ RDm, since this will simplify the imple-mentation (and in any case, there is no apparent reason for thetwo penalty types to possess different normal/tangential contribu-tions). The kinetic energy of the interface is then given by

T ¼ 12

ZS

_dT pdS ð19Þ

which, after invoking (17) and (18), becomes

T ¼ 12

_dTZ

SBT DmBdS _d ð20Þ

The equations of motion then follow from the minimisaton ofenergy

Me€d þ Ked ¼ f ð21Þ

where the mass matrix is given by

Me ¼Z

SBT DmBdS ¼ 1

RKe ð22Þ

where

R ¼ as

amð23Þ

Since the mass matrix is therefore a scalar multiple of the stiffnessmatrix, it is clear that the sum of all entries is zero, and that no netphysical mass is added to the system by the interface element massmatrices. Furthermore, the stability analysis given in Ref. [20] for anarbitrary set of multipoint constraints remains valid, which impliesthat any additional eigenvalues introduced by the addition of theinterfaces will tend to the penalty ratio R for large penalty parame-ters. Therefore, to ensure time step stability, the ratio R should bechosen such that

R 64

Dt2 ð24Þ

where Dt is the chosen time step for the analysis (i.e., a time stepsuitable for analysis of the unpenalised system, without interfaces).The magnitude of the entries in Ds and Dm can be selected by theanalyst, and may be related to the stress history in the element,as explored in Section 2.3.

It is important to note that this formulation results in a non-diagonal mass matrix. Since explicit methods are most efficientwhen a lumped, diagonal mass matrix can be used, this will inev-itably have an effect on the computational efficiency of the solu-tion algorithm. However, the system mass matrix remainsdiagonal except for those DOF which are penalised, and thereforethe effects on solution time are usually limited. For an alternativeimplementation of this methodology that avoids the inversion of(part of) the mass matrix, see the recent work of Lombardo andAskes [24].

2.3. Constitutive relations and damage law

The constitutive law for the cohesive surfaces relates traction inthe interface to the displacement jump across the surface. As sum-marised by Xu and Needleman, ‘‘the behaviour that needs to becaptured is that, as the cohesive surface separates, the magnitudeof the traction at first increases, reaches a maximum and then ap-proaches zero with increasing separation’’ [10, p. 1400]. However,this kind of cohesive law is problematic when used in explicitdynamics, since ‘‘the initial elastic slope . . .may place stringentrestrictions on the stable time step for explicit integration’’ [15].In other words, the initial penalty stiffness in the interface causesa significant decrease in the critical time step of the analysis. Thiseffect may be mitigated by decreasing the initial stiffness of theinterfaces, but this leads to an increase in artificial compliance (ageneral decrease in the stiffness of the continuum that leads tounrealistic elastic deformation), especially when cohesive surfacesare embedded throughout the finite element mesh [25,14]. Sincethe inclusion of an interface mass matrix means that such consid-erations are no longer relevant, in the present formulation there isno such restriction on the initial stiffness of the interfaces.

We begin by describing the traction–displacement relationshipto be employed, first by rewriting the constitutive matrix for thecohesive surface stiffness matrix as

Ds ¼ csI ð25Þ

where cs acts as the penalty parameter (interface stiffness), but isnow dependent on a set of damage parameters. It is initially set

J. Hetherington, H. Askes / Theoretical and Applied Fracture Mechanics 69 (2014) 110–117 113

to cs ¼ as, but may decrease over time as damage occurs and theinterface cohesion begins to lessen.

The value of cs is determined by two scalar quantities, namelythe effective opening displacement d and the effective traction t.Inspired by the fracture criteria given by Camacho and Ortiz [11],we have for the effective opening displacement

d ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2

n þ b2d2t

qif dn P 0

dn if dn < 0

(

This value gives a measure of displacement across the interface,with positive values indicating tensile strain and negative valuesindicating compression. Any non-zero d results in penalty tractionsarising in the interface (except in the case where the effective open-ing displacement has exceeded its maximum value).

The parameter b dictates to what degree tangential displace-ments are taken into account when assessing damage in the inter-face. We assume that no damage occurs in compression, and hencetangential displacements are not considered when assessing rela-tive displacement for dn < 0.

The effective traction depends on the current state of the inter-face and where it lies with regards to the cohesive law shown inFig. 2. For undamaged interfaces, the traction–displacement rela-tion is linear-elastic. By defining the history parameter dmaxas themaximum effective opening displacement reached during an anal-ysis, we can say that the interface is undamaged if dmax 6 d0, whered0is the effective displacement value corresponding to the onset ofdamage. Therefore, during this initial phase,

t ¼ asd ifdmax 6 d0 ð27Þ

where as is the initial elastic stiffness of the interface. If the effectivetraction should exceed the maximum value of tcthe interface entersa damaged state and the constitutive relation changes to reflect lin-ear softening in the material, so that

t ¼ tc 1� d� d0

dc � d0

� �if d > d0; d ¼ dmax ð28Þ

If the effective displacement reaches the critical value dc duringloading then the interface is broken irreversibly, creating a free sur-face, and

t ¼ 0 if dmax > dc ð29Þ

If the effective opening rate becomes negative ( _d < 0) at any timeafter damage has occured then the interface is said to be unloading.

Fig. 2. Cohesive law for tensile tractions, showing the loading path and a potentialunloading path.

In this state the constitutive behaviour is once again linear-elastic,but with a reduced stiffness. Then,

t ¼ tmax

dmaxd if d < dmax ð30Þ

where tmaxis the effective traction corresponding to the effectivedisplacement dmax. Together, these relations describe all four ofthe possible states for an interface: undamaged (linear-elastic),undergoing damage (softening), unloading (linear-elastic, reducedstiffness) and broken (zero traction/free surface).

The effective penalty parameter cs of (25) is given by

cs ¼as if dmax 6 d0

tmax=dmax if dmax > d0

�

Damage may only occur when the interface is in tension (d > 0). Wehere assume that the displacement gap across the interface does notclose again after damage has occurred, since a damaged interfacethat was subsequently put into compression would have a lowered(possible zero) stiffness, which could lead to excessive interpenetra-tion between elements. To model problems with the possibility forcrack re-closure the definition could be extended so that the inter-face has stiffness in compression (d < 0) even after the maximumdisplacement has been reached (dmax P dc).

At each time step, the effective relative displacement d is com-puted for each interface so that any changes in the state of theinterface may be detected. For affected elements the damage mod-el is implemented by computing the associated effective tractionsand then updating the constitutive matrix for those elements viaEq. (25).

Finally, we introduce the cohesive fracture energy Gc, a funda-mental parameter of the cohesive zone model regarded as a mate-rial constant, which represents the work of separation per unit areaof cohesive surface. It is given by the area under the traction–dis-placement curve, which, for the formulation described above, gives

Gc ¼Z dc

0t dd ¼ tcdc

2ð32Þ

This relationships allows the traction–displacement curve to befully described by tc (representing the yield strength of the mate-rial), and the fracture energy Gc, both of which can be obtained byexperimental testing of a specimen.

With the inclusion of a damage law, we must also consider howthe constitutive matrix Dm may change as damage occurs; i.e.,identify the damage parameters which determine cm in the nonlin-ear constitutive matrix

Dm ¼ cmI ð33Þ

One option is to form a new damage law specifically for the massmatrix of the element as opposed to the traditional traction–dis-placement used to govern the stiffness of the interface. However,this would effectively constitute a new set of velocity constraints,whereas our goal is to enforce the nonlinear displacement con-straints (in the form of cohesive surfaces) which have already beenderived. Consequently, we adopt the same cohesive law for bothmass and stiffness matrices, which, with the reintroduction of thepenalty ratio R gives cm ¼ cs=R. The penalty ratio may be used tocontrol the relative influence of the stiffness and mass penaltymatrices. Note by adopting this method, the assumption thatDs ¼ RDm (and therefore Ke ¼ RMe) is valid throughout theanalysis.

3. Example: elastic wave propagation

In order to test the interface element formulation, we consider arectangular region of elastic material modelled using the finite

F

2m

1m

Fig. 3. Diagram of rectangular elastic region showing fixed supports (indicated byhashed edges) and interface elements (dotted lines), for an element size ofh = 0.1 m.

114 J. Hetherington, H. Askes / Theoretical and Applied Fracture Mechanics 69 (2014) 110–117

element method. A regular grid of square elements is used to meshthe domain. Two-dimensional interface elements are then insertedbetween the FE continuum elements in the right half of the mesh toobserve what effect this has on wave propagation through themedium, as shown in Fig. 3. At this point, the interfaces are non-breaking and do not suffer damage, and the interface constitutivematrices are given simply by Ds ¼ asI and Dm ¼ amI regardless ofinterface tractions. (This corresponds to a cohesive surface formu-lation where the maximum traction tc is never exceeded.)

In the following work we define a dimensionless penalty factorthat gives a measure of penalty parameter magnitude. The param-eters as and am control the accuracy of constraint imposition, butare only effective if they are several orders of magnitude largerthan the existing entries in the system matrices K and M. Wetherefore define

ps ¼as

maxi2PðKiiÞ

ð34Þ

pm ¼am

maxi2PðMiiÞ

ð35Þ

where P is the set of all DOF numbers associated with the con-straint. Then, the stiffness and mass penalty factors ps and pm givea measure of the magnitude (and therefore effectiveness) of thepenalty parameters as and am, respectively.

Fig. 4. Von Mises stress profiles at tim

Stress wave propagation through the rectangular system isshown qualitatively in Fig. 4, for a stiffness penalised system. Thematerial has arbitrary properties Young’s modulus E ¼ 1 Pa, massdensity q = 1 kg/m3, Poisson’s ratio m ¼ 0, and plane stress is as-sumed. The point load F ¼ 10�3 N is applied from the beginningof the analysis until time t ¼ 0:1 s. The element side length ish ¼ 0:02 m for all elements (giving a total of 5000 elements).

In order to quantify errors, a reference solution (shown inFig. 4a) is first produced by omitting interface elements entirely.By introducing interface elements with low stiffness penaltyparameters, as in Fig. 4b, we can easily observe the effects of theadded interfaces. Note that the analysis concludes before any wavereflections occur at the boundaries of the region, and hence wavepropagation on the left-hand side of the material appear quiteunaffected by the interfaces. In the interface process window, how-ever, stress wave propagation is slowed considerably by the addi-tional elastic strain manifesting between the continuum elements,a phenomenon known as artificial compliance. This is clearly evi-dent in the stress error field, shown in Fig. 4c.

Fig. 4 demonstrates the need for well-enforced interface con-straints. If no damage has occurred, a continuum that includescohesive surfaces and the interface-free continuum should ideallybehave identically, with zero displacement across an interface.Using penalty methods, this is only possible in the limit as penaltyparameters tend to infinity, but cohesive surfaces can be practi-cally transparent if the initial penalty is large enough. Of course,with stiffness-type penalties this introduces concerns with regardsto time step stability.

In order to assess the performance of the stiffness, mass andbipenalty methods, we now turn our attention to the error normof the stress profiles for a number of analysis types. Fig. 5 showsthe L2 norm of the error in stress profile, kerk, between two anal-yses with and without penalty-based interface elements, for arange of penalty factors. Note that the quantity represented bythe x-axis (penalty factor, p) represents the penalty factor thathas been used as input in each case. For the stiffness penalty anal-yses this is the stiffness penalty factor ps, and for the mass andbipenalty analysis it is the mass penalty factor pm, since for thebipenalty method, the stiffness parameters are calculated basedon a suitable penalty ratio R.

Fig. 6 shows the time steps used for the analysis on a logarith-mic scale. For mass and bipenalty methods, suitable time steps areestimated using the maximum eigenvalue of all individual ele-ments, whereas for the stiffness penalty method the maximumeigenvalue of the full constrained system must be used. This data

e t = 0.8 s, including error field.

Fig. 5. Stress error norm at time t = 0.8 s for the stiffness, mass and bipenaltymethods.

Fig. 6. Time step used in each analysis (approximately 0:9Dtcrit).

70

140

45

75

7.5

ø 30

Fig. 7. Geometry of the PMMA specimen (dimensions in mm).

x [m

0.05 0 .06 0.07 0.08 0.09

Fig. 8. Experimental (red, dashed) and numerical (blue, solid) results for final crack pathreader is referred to the web version of this article.)

J. Hetherington, H. Askes / Theoretical and Applied Fracture Mechanics 69 (2014) 110–117 115

shows that achieving high accuracy using stiffness-type penaltiesquickly becomes very expensive as penalty parameters are in-creased; indeed, for stiffness penalty factors ps > 108 the tests be-came prohibitively expensive.

4. Example: crack propagation in PMMA plate

For validation of the formulation in a dynamic setting, we turnour attention to an experiment carried out by Grégoire et al. [26],who investigate crack propagation through a polymethyl methac-rylate (PMMA) plate under impact loading. The experiment usesa Hopkinson bar to apply load to the left-hand side of the PMMAspecimen shown in Fig. 7, which features a pre-existing crack ema-nating from a central hole. The specimen suffers fracture damageduring the first 500 ls after impact, and the crack tip position ismeasured during this time to obtain a detailed crack propagationhistory.

In this section we simulate the experiment numerically usingbipenalty cohesive surfaces. Only the PMMA plate is considered,since the initial contact with the Hopkinson bar on the left-handside of the specimen can be modelled with prescribed velocities,while contact at the other end is handled with absorbing boundaryconditions. Given material properties for the plate include Young’smodulus E = 4.25 GPa, mass density q = 1180 kg/m3, Poisson’s ratiom ¼ 0:42 and fracture toughness K IC ¼ 1:47 MPa

ffiffiffiffiffimp

.For the cohesive bipenalty formulation we require the damage

parameters tc and GC (from which we can calculate dc). The cohe-sive fracture energy can be calculated from the fracture toughnessvia GC ¼ K2

IC=E. The yield stress tc is not given, and so initially arange of values are tested, from 10 to 100 kPa.

The final crack path for tc ¼ 15 kPa is shown in Fig. 8. Althoughthe total length of the path is not captured by the simulation, theinitial portion of the crack is reproduced very well. Note that sincethe crack is required to move along element boundaries it cannotgenerally move in a perfectly straight line, which adds extra lengthto the simulated crack path. This in turn adds to the amount of en-ergy required to open the crack, which may help to explain whythe numerical path is somewhat shorter than the experimentalresult.

The growth of the crack over time is compared to experimentalobservations in Fig. 9. The experimental results show the crackbeginning to form at around 200 ls, stopping briefly when itreaches x � 90 mm, before continuing to its end point. While inthe numerical tests crack initiation is about 25 ls early, and crackspeed propagation is generally higher than in experiments, thestop-start behaviour of the crack propagation is captured to someextent for tc = 12–18 kPa.

]

0.1 0.11 0.12 0.13 0.14

(tc ¼ 15 kPa). (For interpretation of the references to colour in this figure legend, the

time [µs]

crac

kti

pab

scis

sa[m

m] numerical

experimental

100 200 300 400

80

90

100

110

time [µs]

crac

kti

pab

scis

sa[m

m] numerical

experimental

100 200 300 400

80

90

100

110

time [µs]

crac

kti

pab

scis

sa[m

m]

numericalexperimental

100 200 300 400

80

90

100

110

time [µs]

crac

kti

pab

scis

sa[m

m] numerical

experimental

100 200 300 400

80

90

100

110

Fig. 9. Position of crack tip (x-dir) over time: tc ¼ 12 kPa (top left), 15 kPa (top right), 18 kPa (bottom left), 21 kPa (bottom right).

116 J. Hetherington, H. Askes / Theoretical and Applied Fracture Mechanics 69 (2014) 110–117

The mostly likely reason for disparities in the numerical tests isthe accuracy of the bilinear cohesive law. By refining the traction–displacement relationship it may be possible to obtain more accu-rate results, although this would likely require access to furtherexperimental test data for the material in question. An alternativecohesive model designed for the modelling of PMMA is given byElices et al. [27].

However, in general the bipenalty cohesive surfaces are well-suited to this problem type, where high loading rates mean thatthe explicit central difference method is an obvious choice forthe solution scheme. Since stiffness-only elements lead to artificialcompliance, an artificially lowered time step, or else unexpectedtime step instabilities, their use is problematic. On the other hand,the bipenalty method allows for time steps close to the criticaltime step of the unconstrained problem, while allowing for a veryhigh initial stiffness.

5. Conclusion

The most significant advantage of the central difference methodover implicit time integration routines is that the computationscarried out at each time step are very efficient. The major disad-vantage is that for stability, the size of the time step must be rela-tively small. Traditional penalty methods, such as those used in theformulation of cohesive surfaces, have little effect on the actualcomputation, but can drastically reduce the critical time step.Alternatively, the bipenalty method has no effect on the criticaltime step, but it increases the cost of each time step computationby requiring that the mass matrix of the system be non-diagonal.

We can therefore say that the bipenalty method is an appropri-ate choice only in certain situations. If high accuracy is required(high penalty stiffness, low compliance) and the number of bipena-lised degrees of freedom is small relative to size of the problem,then the extra computational effort required to solve a small linearsystem may be offset by the fact that less time steps are needed.Furthermore, the bipenalty method may provide a more robust

solution by ensuring time step stability in all circumstances, givingthe analyst more freedom to select suitable parameters; but, ifcohesive surfaces must be introduced throughout the whole con-tinuum, a very large linear system of equations would need to besolved at each time step.

In summary, the bipenalty method is a simple way to controlthe eigenvalues introduced when adding cohesive surfaces to anFE simulation. With the introduction of a single extra parameter,the penalty ratio R, the analyst can ensure that the critical timestep of an explicit analysis is not affected by the interfaces. Butwhile the bipenalty method could theoretically be employed inany explicit dynamic problem involving cohesive surfaces, whetheror not the additional computational cost is justified depends en-tirely on the problem under consideration; specifically, the numberof constraints relative to the size of the total system should be assmall as possible.

References

[1] J.-H. Song, H. Wang, T. Belytschko, A comparative study on finite elementmethods for dynamic fracture, Comput. Mech. 42 (2007) 239–250.

[2] P. Kohnke, ANSYS, Inc. Theory Reference: ANSYS release 9.0, ANSYS, Inc.,Canonsburg, 2004.

[3] J.O. Hallquist, LS-DYNA Theory Manual, March, Livermore SoftwareTechnology Corporation, Livermore, California, 2006.

[4] R. Hambli, Comparison between Lemaitre and Gurson damage models in crackgrowth simulation during blanking process, Int. J. Mech. Sci. 43 (2001) 2769–2790.

[5] T. Belytschko, T. Black, Elastic crack growth in finite elements with minimalremeshing, Int. J. Numer. Methods Eng. 45 (1999) 601–620.

[6] N. Moes, J. Dolbow, T. Belytschko, A finite element method for crack growthwithout remeshing, Int. J. Numer. Methods Eng. 46 (1999) 131–150.

[7] A. Combescure, A. Gravouil, D. Grégoire, J. Réthoré, X-FEM a good candidate forenergy conservation in simulation of brittle dynamic crack propagation,Comput. Methods Appl. Mech. Eng. 197 (2008) 309–318.

[8] D. Dugdale, Yielding of steel sheets containing slits, J. Mech. Phys. Solids 8(1960) 100–104.

[9] G.I. Barenblatt, The mathematical theory of equilibrium cracks in brittlefracture, Adv. Appl. Mech. 7 (1962) 55–129.

[10] X.-P. Xu, A. Needleman, Numerical simulations of fast crack growth in brittlesolids, J. Mech. Phys. Solids 42 (1994) 1397–1434.

J. Hetherington, H. Askes / Theoretical and Applied Fracture Mechanics 69 (2014) 110–117 117

[11] G. Camacho, M. Ortiz, Computational modelling of impact damage in brittlematerials, Int. J. Solids Struct. 33 (1996) 2899–2938.

[12] E.A. Repetto, R. Radovitzky, M. Ortiz, Finite element simulation of dynamicfracture and fragmentation of glass rods, Comput. Methods Appl. Mech. Eng.183 (2000) 3–14.

[13] H.D. Espinosa, P.D. Zavattieri, A grain level model for the study of failureinitiation and evolution in polycrystalline brittle materials. Part I: theory andnumerical implementation, Mech. Mater. 35 (2003) 333–364.

[14] S.H. Song, G.H. Paulino, W.G. Buttlar, A bilinear cohesive zone model tailoredfor fracture of asphalt concrete considering viscoelastic bulk material, Eng.Fract. Mech. 73 (2006) 2829–2848.

[15] M. Ortiz, A. Pandolfi, Finite-deformation irreversible cohesive elements forthree-dimensional crack-propagation analysis, Int. J. Numer. Methods Eng. 44(1999) 1267–1282.

[16] J. Hetherington, H. Askes, Penalty methods for time domain computationaldynamics based on positive and negative inertia, Comput. Struct. 87 (2009)1474–1482.

[17] H. Askes, M. Caramés-Saddler, A. Rodríguez-Ferran, Bipenalty method for timedomain computational dynamics, Proc. R. Soc. A: Math. Phys. Eng. Sci. 466(2009) 1389–1408.

[18] E.A. Paraskevopoulos, C.G. Panagiotopoulos, G.D. Manolis, Imposition of time-dependent boundary conditions in FEM formulations for elastodynamics: criticalassessment of penalty-type methods, Comput. Mech. 45 (2010) 157–166.

[19] J. Hetherington, A. Rodríguez-Ferran, H. Askes, A new bipenalty formulationfor ensuring time step stability in time domain computational dynamics, Int. J.Numer. Methods Eng. 90 (2012) 269–286.

[20] J. Hetherington, A. Rodríguez-Ferran, H. Askes, The bipenalty method for arbitrarymultipoint constraints, Int. J. Numer. Methods Eng. 93 (2013) 465–482.

[21] J. Hetherington, The Bipenalty Method for Explicit Structural Dynamics, Ph.D.thesis, University of Sheffield, 2013.

[22] J.C.J. Schellekens, Computational Strategies for Composite Structures, Ph.D.thesis, Delft University of Technology, 1992.

[23] J.C.J. Schellekens, R. De Borst, On the numerical integration of interfaceelements, Int. J. Numer. Methods Eng. 36 (1993) 43–66.

[24] M. Lombardo, H. Askes, Lumped mass finite element implementation ofcontinuum theories with micro-inertia, Int. J. Numer. Methods Eng. 96 (2013)448–466.

[25] M. Tijssens, On the Cohesive Surface Methodology for Fracture of BrittleHeterogeneous Solids, Ph.D. thesis, Delft University of Technology, 2000.

[26] D. Grégoire, H. Maigre, J. Réthoré, A. Combescure, Dynamic crack propagationunder mixed-mode loading – Comparison between experiments and X-FEMsimulations, Int. J. Solids Struct. 44 (20) (2007) 6517–6534.

[27] M. Elices, G. Guinea, J. Gómez, J. Planas, The cohesive zone model: advantages,limitations and challenges, Eng. Fract. Mech. 69 (2002) 137–163.