DYNA MODELLING

International Journal of Impact Engineering 27 (2002) 709–727

DYNA-modelling of the high-velocity impact problemswith a split-element algorithm

A.D. Resnyansky*

Weapons Systems Division, DSTO Aeronautical and Maritime Research Laboratory, P.O. Box 1500, Edinburgh SA 5111,

Australia

Received 31 May 2001; received in revised form 8 January 2002

Abstract

The present work addresses the implementation of a split-element algorithm for modelling fracture interminal effects (TE) problems. The algorithm is incorporated within Vec-Dyna3D hydrocode (Technicalreport DSWA-TR-96-95. Alexandria (VA): Defense Special Weapons Agency, 1998), which is a prototypeof LS-DYNA3D (Version 950. Livermore: Livermore Software Technology Corporation, May 1999). Thisalgorithm has also been implemented in LS-DYNA2D (UCID-18756, Rev. 2. Livermore: LawrenceLivermore National Laboratory, 1984.) and it is verified numerically in the present paper. In doing so,tensile and shear modes of fracture due to high velocity impact are analysed in detail. Plate collision (thespallation problem) may be considered as a test problem for achieving the tensile mode of fracture.Encounter of a compact projectile with a plate (the plugging problem) plays similar role for the shear mode.Influence of choice of effective (equivalent) stress involved in a 3D-extension of a fracture criterion isanalysed from two points of view: (i) the mesh effects, and (ii) a role of the complex stress state. The strain-rate sensitive Maxwell-type model (J. Appl. Mech. Tech. Phys. 13(6) (1972) 868.) is employed as aconstitutive model. Workability of the algorithm in 3D case is illustrated with a numerical example for theplugging problem. The calculations being conducted show an appropriateness of the present approach forTE problems. Crown Copyright r 2002 Published by Elsevier Science Ltd. All rights reserved.

Keywords: DYNA hydrocode; High-velocity impact; Spallation; Plugging; Fracture criterion

1. Introduction

An assessment of the fragment effect due to impact is a key issue in many TEproblems. Majority of these problems could be solved with existing modern Eulerian and

*Tel.: +61-(0)8-8259-7453; fax: +61-(0)8-8259-6247.

E-mail address: [email protected] (A.D. Resnyansky).

0734-743X/02/$ - see front matter Crown Copyright r 2002 Published by Elsevier Science Ltd. All rights reserved.

PII: S 0 7 3 4 - 7 4 3 X ( 0 2 ) 0 0 0 0 8 - 8

Eulerian–Lagrangian hydrocodes. Nevertheless, Lagrangian hydrocodes are very attractive forevaluation of the fragment effect because they prevent mass diffusion through contact and freeboundaries, which allows one to estimate accurately the momentum and energy deposited to atarget. The major disadvantage of the Lagrangian approach is its intolerance to severedeformations. At the same time, many real materials do not withstand large deformations and failmuch earlier.Hydrocodes take failure into account in a number of ways. Some of the methods impart

Eulerian features into the Lagrangian hydrocodes. For example, the popular method formodelling fragmentation with the Lagrangian approach is an erosion algorithm implemented inEPIC and DYNA [1,2]. However, eroding elements due to failure results in the Eulerian-typemass diffusion through the fragment boundaries. Another radical approach, which can be usedwith any Lagrangian hydrocode, is the phenomenological representation of the crack field (see,e.g. [3]). This continuum-type approach is convenient to implement, however, the element erosionis required to calculate the brittle-type fracture followed by the fragment separation in TEproblems. The next in order of detailing the physical process of fracture is consideration of thecrack opening within the hydrocode’s numerical scheme. The advent of crack-containing finiteelements in the literature aims at incorporation of the fracture mechanics solutions into the finiteelement calculations (see, e.g. [4]). This approach seems to be more suitable for the calculationswith a few isolated cracks; numerical examples of this sort for 2D- and 3D-crack propagation canbe found in papers [5,6].The present algorithm for processing the brittle-type fracture is based on a simplified approach,

which neglects many physical peculiarities in the vicinity of the crack tip. Development of thecrack surface is accompanied by stress relaxation: within the present approach only this fact isconsidered to be essential. The split-element methodology is not aimed at determination of theexact location of cracks and shape of fragments. The primary objective of the present approach isto design an instrument for both calculation of separation of fragments from a target andlocalisation of the fracture zones. This option could enable us to assess mass and velocity offragments within the mesh accuracy and to evaluate the fragment effect on target.A similar approach, which exploits ideas of the discrete element method (DEM), has been

suggested in [7] for structural calculations, and a variety of hydrocodes with the DEM-featureshave been developed (see, e.g. [8]). The prototype of the present approach implemented into aLagrangian–Eulerian finite-volume code has been developed in [9]. The first incorporation of thisoption within LS-DYNA2D has been attempted by the author in [10]; however, the split-elementalgorithm has not been properly verified and an elastic-plastic material model with propertiesinsensitive to strain rates was employed. The strain-rate sensitive Maxwell-type model [11] is usedin the present paper. Outline of an implementation of the model into the DYNA-hydrocode isdescribed in the second part of the paper.The split-element algorithm and fracture model are briefly described in the third part of the

paper. Firstly, a fracture criterion is considered. When employing fracture criteria dependent onstress, an equivalent stress is usually employed that reduces complex stress state to the onlyvariable. This variable, which is typically an algebraic function of components of the stress tensor,is implicitly associated with a fracture mode; for example, an equivalent stress proportional to themaximum principal stress is associated with the tensile fracture mode, and so on. Fracture is acomplex process involving a number of mechanisms; therefore, treating the crack development as

A.D. Resnyansky / International Journal of Impact Engineering 27 (2002) 709–727710

one governed by a specific fracture mode is very simplistic for the case of projectile-targetinteraction. Nevertheless, the fracture experiments are usually conducted at a simple stresscondition, so, the use of single ‘effective’ or ‘equivalent’ stress is in common practice. It is yet to beunderstood how the complex stress state in the vicinity of the crack tip affects the crack opening.At present generalised phenomenological approaches are very popular: a closed or semi-closedsurface in a stress space related to quasi-static tests is considered as the failure criterion. Forexample, this type of failure surfaces for simulation of the concrete behaviour is known as a familyof cap models [12], and these surfaces are expressed as limiting dependencies between the first andsecond invariants of the stress tensor. Many materials exhibit rate sensitivity in dynamicconditions; therefore, the high strain-rate problems do not suit fully to the formulation employingquasi-static fracture criteria. In Section 3.1 we consider a possible way for taking the complexstress state into a time-dependent fracture criterion. From another angle, the effective stress in afracture criterion may be calculated either in a reference system associated with the finite-elementmesh or in an invariant form. In the subsequent Section 4 we analyse how this choice related tothe complex stress state affects numerical results.Analytical solutions of the damage fragmentation problems are very rare (e.g. see [13]); the

solutions in a finite form for the problems involving elastic and plastic flow of strain-rate sensitivematerials do not probably exist. Therefore, proper numerical testing is considered to be the onlyviable option for verification of the algorithm; again, numerical solutions exist for a number ofwell-known fragmentation problems confirmed by experiments. Examples of such problems,which might be associated with two basic tensile and shear modes of fracture, are spallation due toa plate collision and the plugging due to ballistic impact. Numerical convergence of any methodcan be checked for by calculations with refining grids. In the multi-dimensional case theconvergence formally should be verified on the meshes with arbitrary orientations, which ispractically impossible; therefore, the present consideration is restricted to meshes with a fixedorientation. Hundreds of nodes in every space direction are desirable for a good resolution ofstress waves. This requires millions of nodes and elements for a 3D-problem, which means thatnumerical verification is very difficult for the 3D-case. A compromise is considered in the paper:convergence of the method is examined for 2D-cases, and 3D-case is illustrated with an examplefollowed by comparison with 2D-calculation on similar grid.

2. Material model

According to the flowchart of the DYNA hydrocode, outlined in [14], the program blockassociated with implementation of material model involves several stages from calculation of thestress deviator from a constitutive equation further to determination of pressure from an equationof state and calculation of total stress. Usually, the primary attention is paid to the calculation ofthe stress deviator because the user-defined-material subroutine in LS-DYNA3D deals with thisstage only. The model employed in the present paper operates with a generalisation of the Mie-Gruneisen equation of state [15], so the whole material implementation block is being updated.In the current section we focus on realisation of the constitutive equation. Omitting details of

the stress rate characterisation (see, e.g. [14]), the stress rate operator in LS-DYNA is split into asum of rotational and material differential operators. Concentrating on the material stress rate,

A.D. Resnyansky / International Journal of Impact Engineering 27 (2002) 709–727 711

the Maxwell-type constitutive relation [11] can be presented as follows:

ds0ijdt

� 2G � Dij ¼ �s0ij

tðs;TÞ: ð1Þ

Here s0ij is the stress deviator, G is the shear modulus, Dij is the strain rate deviator, t is a functionresponsible for relaxation of shear stresses, T is temperature, and s is shear stress (a function ofthe second invariant of the stress tensor). When deducing Eq. (1) from the model [11], the ratechange of the shear modulus was neglected. Numerical integration of Eq. (1) is not an easyproblem because the function t varies sharply from fractions of a microsecond up to almostinfinity. The function is fitted to a set of experimental stress–strain dependencies versus strain rate,and it was designed using the dislocation theory as published in [9]. Applying the physical splittingscheme, the increment Ds0ij from a time level k up to the next one is calculated as follows:

%s0ij � s0kij

Dt¼ 2G � Dij ;

s0kþ1ij � %s0ij

Dt¼ �

s0kþ1ij

tðskþ1;TÞ; Ds0ij ¼ s

0kþ1ij � s

0kij : ð2Þ

The most important stage to achieve a satisfactory accuracy of the whole step is numericalsolution of the nonlinear stiff system of equations in Eq. (2) (the second group of the equations),which require iterations. In order to facilitate the calculation, we use the differentialrepresentation of this system. Multiplying equations of the stiff system by s0ij and then summingup the products over i and j; we have an equation for s; which takes the same form as theequations for the stress deviator. Using a finite-difference approximation, this equation acquiresthe following form, skþ1 � %s ¼ �skþ1ðDt=tðskþ1;TÞÞ; which is straightforward, calculated byiteration with the Newton’s method. Determination of the stress deviator components s0ij isfinalised with the exact formula: s

0kþ1ij ¼ %s0ijðs

kþ1= %sÞ: This relation follows from a comparison ofthe stiff system of differential equations, the finite-difference approximations of which for s0ij areshown in Eq. (2), with the corresponding equation for s: Material data for the model [11] wereobtained directly with a fitting algorithm [16] (see also Appendix). With the exception of thespecific ‘dislocation theory’ form for the function t; a similar model has been realised in [17] as‘‘Material elastic with viscosity’’ (material no. 60 in the LS-DYNA material database).Unfortunately, we could not employ the material model no. 60 in the present work because itis not available in VecDyna [18].

3. Modelling fracture

3.1. Fracture criterion

Fracture model employed in the present paper has been published in detail elsewhere [10,19].This model is based on a damage accumulation criterion [20], which is widely used for hydrocodemodelling of impact problems. The recent version of the LS-DYNA3D material model database[17] contains this criterion as a fracture model for the erosion option (*MAT ADD EROSIONoption). The first step of the split-element algorithm being realised within DYNA is generation ofinitial ‘crack’. The algorithm forms the crack when a history variable responsible foraccumulation of damage exceeds a threshold. To describe propagation of an existing crack


within the present approach a semi-empirical rule linking the crack propagation velocity and aremote stress state associated with the crack may be used [10,19]. Summarising, we can formulatethe following flowchart for processing fracture within the present model:

(1) accumulation of damage (a history variable is introduced for this purpose);(2) formation of a macrocrack when the history variable exceeds a threshold;(3) the splitting of elements of undamaged material due to either damage accumulation or crack

propagation.

It should be noted that the present algorithm of processing damage is isotropic but the fracturehaving already occurred may affect stress state so the fracture-induced anisotropy is possible.The damage criterion [20] can be rewritten in the kinetic form as follows:

dh

dt¼ F ; F ¼

ðs� s0Þn; s > s0

0; sps0;

(ð3Þ

here h is the history (damage) variable, d=dt is the particle derivative. Here h ¼ 0 at t ¼ 0 forinitially undamaged material, s0 is the minimum threshold at which the damage accumulationstarts, s is an equivalent stress, n is a material constant. We declare a macrocrack forms whenhXJ; J is a material constant. A procedure for determination of the constants has been describedin detail for a number of metals elsewhere [21].The effective or equivalent stress s is defined uniquely only at a simple stress condition. In the

case of a multi-dimensional deformation resulting in a complex stress state the effective stress canbe chosen in a number of ways as a function of the stress components. Influence of this choice onthe calculation results will be analysed in the next section. A fracture criterion at a complex stressstate is traditionally represented by a limiting curve/surface in a stress space. While employing akinetic fracture criterion, the curve is generalised to a zone of limit stresses because the strengthdepends on the time of application of load (e.g. see a schematic of the representation in [22]). Forinstance, with the effective stress linking the tensile stress sn and shear stress st linearly, this zoneis presented as a dashed area 3 in Fig. 1. The strength theories for many conventional criteria arewell developed, and, correspondingly, sets of curves/surfaces have been derived. For example,curve 1 in Fig. 1 represents a cap fracture model [12] for a concrete material. An equivalent stressemployed by a fracture criterion for ceramics [23] is represented by curve 2 in Fig. 1, and so on.

Fig. 1. Presentations of fracture criteria in a stress space.


These curves were obtained in static conditions; that is, referring to the time-dependent (kinetic)criteria, they are associated with a large time of application of load, denoted by ta; so, they arerepresented by the interior boundary of the corresponding limit stress zones (above-mentioneddashed zone in the stress space). On the contrary, at a small ta the stress causing failure isessentially larger, and it tends to the theoretical strength associated with ta approaching zero;therefore, the exterior boundary of the limit stress zone represents the corresponding criterion atta ¼ 0 (Fig. 1). Thus, a kinetic fracture criterion might schematically be presented as an infiniteseries of enclosed curves within the limit stress zone, each of them is correspondent to a specific ta:Mapping of the curves within the zone depends on a specific kinetic form of the criterion;however, shape of the curves in the stress space is determined by the functional form of theequivalent stress. For example, presentation of the equivalent stress as a linear function of thetensile and shear stresses provides us with a series of parallel straight lines bounding the dashedzone shown in Fig. 1.Summarising, the equivalent stress can be expressed as a functional relation on the basic stresses

and be used within a kinetic fracture criterion. In the present paper only the linear dependencies ofs on sn and st are considered for the damage kinetics (3):

s � Asn þ Bst: ð4Þ

It is seen, that the equivalent stress corresponding to A ¼ 0 is associated with the shear fracture(lines parallel to the line 4 in Fig. 1) and B ¼ 0—with the tensile one (lines parallel to the line 5 inFig. 1). In their turn, method of calculation of the tensile and shear stresses in Eq. (4) may dependon the finite-element mesh. First, we can choose these stresses as the normal and tangentialstresses with respect to the plane of the element interface:

sn ¼ snn; st ¼ jsntj; ðsnn > 0Þ: ð5Þ

The second choice employed in the present paper is the maximum principal stress as sn andmaximum shear stress, which relates to the shear mode of fracture:

sn ¼ maxfs1;s2;s3g; st ¼ maxfjs1 � s2j; js2 � s3j; js3 � s1jg=2; ðsn > 0Þ; ð6Þ

here s1; s2; and s3 are principal stresses. Both forms are similar, the major distinction between therepresentations (5) and (6) is that the latter has the mesh-independent formulation. The crackpropagation option is related to mesh; therefore, in order to conduct a consistent comparisonbetween various choices of the fracture criteria including the invariant version (6), the crackpropagation option has been disabled in the present study. Other choices of sn and st are alsopossible, e.g. the ‘Mises’-like one, associating the stresses with the first and second invariants ofthe stress tensor. However, for the purpose of algorithm verification we believe that one choice forevery case of the representations (mesh-dependent and invariant ones) is sufficient to illustrate therobustness of the approach.

3.2. Split-element algorithm

First, we consider a 2D-finite element mesh assuming that all elements of the mesh arequadrilateral. The finite element codes employ a standard array ix of the element interconnectivitythat provides us with the numbers of associated nodes for any element. That is, for ieth elementthe numbers ixð1; ieÞ; ixð2; ieÞ; ixð3; ieÞ; ixð4; ieÞ are the node numbers of the element ie in the


counter-clockwise direction. In order to reduce the computational cost, the present algorithmelaborates the inverse table ixc which establishes the same rule for every node with respect to thesurrounding elements. Every side between two elements of the mesh is a candidate for splitting.The failure structure of the mesh is represented by the interconnectivity arrays, which has to bemodified after every splitting event. In fact, the splitting means that the affected side is replaced bya pair of free surfaces; however, it does not mean that the nodes joining the side are automaticallysplit. The cases when it happens generate a list of the node descriptors, which are specific to thesplit-element methodology (Fig. 2). That is, for an element associated with a node from this listthe standard interconnectivity array ix and its inverse map ixc cannot provide us with actualfailure structure (an information about the split sides) around the element. Therefore, the presentalgorithm requires an auxiliary array (the array of node descriptors), with the descriptors of Fig. 2as a part of it. Similar method, employing additional information about the crack orientation, isused in the latest version of LS-DYNA3D for the material no. 17 with oriented crack [17].However, no actual separation occurs for this material in the case of fracture.Summarising, the split-element algorithm consists of the following steps:

1. Calculation of the history parameter h according to Eq. (3) for each side between two non-splitelements.

2. Check-up of the condition hXJ: If the condition takes place, then the interconnectivity andauxiliary arrays for two corresponding elements and affected nodes are updated.

3. Calculation of the crack propagation velocity for the nodes at the crack tip (the cases fromFig. 2) followed by cumulative calculation of the crack length. This means that at a nonzerocrack velocity V the current crack length related to a node is advanced by VDt and memorisedin a nodal variable L during the calculation from one time level up to the next one with the timeincrement Dt: If the length exceeds the mesh size the splitting of the affected side occurs due tothe crack propagation. In the present algorithm the crack propagation may only be calculatedfor the cracks having already formed.

Fig. 3 illustrates how the nodal configuration changes while fracture is developing. Any elemente has four nodes enumerated locally as 1,2,3,4 in the counter-clockwise direction. The nodes 1 and2 are located on a horizontal line of the local coordinate system, and the nodes 3 and 4 (side 34)are on the next horizontal line. Correspondingly, two adjacent vertical lines contain sides 14 and23. Thus, a crack initiation along side 14 for the element e means crack initiation along side 2030

for the adjacent element e0. When describing failure within the finite-element grid, we specify typesof the failure by the numbers (descriptors) from ‘1’ to ‘4’ for the central nodes in theconfigurations plotted in Fig. 2. After the element split, nodes 20 and 30 of e0 as well as nodes 1 and

Fig. 2. Descriptors for the nodes with ‘crack’ tips specific to the split-element algorithm.


4 of the element e take new descriptors (descriptors ‘1’ and ‘2’, respectively, from the list in Fig. 2),substituting the node descriptors of previously undamaged material (a ‘zero’ descriptor). For theexample in Fig. 3(a) the nodes are not split; in the next one (Fig. 3(b)) the mesh was alreadysubject to a crack formation. We have two adjacent elements with nodes 10203040 and 1234.Descriptors for the nodes 20=1 and 30=4 are ‘2’ from the list in Fig. 2 and the ‘zero’ descriptor,respectively. If either the damage variable has exceeded the threshold J or the crack lengthaccumulated in the variable L for the crack with a tip at the node 20=1 has become more thanlength of the side 14 (in the present example the latter can occur only in the Fig. 3(b) case), then:(i) new node is added into the node list (the node 20=1 split into two nodes: the node 1 takesattributes of the old one, new node 20 takes the same attributes but its node descriptor isundefined); (ii) previous descriptor ‘2’ from the list in Fig. 2 for the node 1 is replaced by thedescriptor which has free surface on the right of the node; (iii) the new descriptor for the node 20 isdefined as a similar descriptor as for (ii) but with indication of free surface on the left of the node;(iv) the ‘zero’ descriptor for the node 30=4 is modified into the descriptor ‘2’ from the list inFig. 2. In this case node splitting takes place and the node list is updated along with theinterconnectivity arrays ix and ixc for the affected elements e and e0.Three-dimensional extension of the split-element algorithm is briefly stated below. Similarly

to the 2D case we consider that the present method is applicable only to the 8-node (‘solid’)elements in DYNA3D. As above, the mesh structure is determined with the interconnectivityarray ix, which contains information for every element about the node numbers for 8 nodes ofthe element. Similarly, the failure structure is specified by a separate descriptor associatedwith every node. A node in the 3D-case is associated with 8 elements, which may surroundthe node, and 12 element sides may have this node as the common point. This nodedescriptor informs us which of the 12 associated sides associated with the node have been subjectto failure. This realisation is slightly different from that for LS-DYNA2D in the point that the 3Dnode descriptors provide us with information about associated sides, whereas the 2D descriptorsare related to the whole split structure around the node. The flowchart of the algorithm containstwo basic steps: (i) calculation of the damage variable from the damage/fracture criterion for thesides which have not yet been failed; (ii) update of the crack structure, including theinterconnectivity arrays ix and ixc, and the node descriptor, after failure on relevant sides hasoccurred.

Fig. 3. Examples of the modification of the element and node structures for initially undamaged (a) and damaged

(b) material before (left hand pictures) and after (right hand pictures) the split element processing.


4. Numerical verification

We start verification with a test spallation problem related to the tensile mode of fracture. Weconsider the case of collision between two copper plates at the velocity of 300m/s. These twocylindrical plates have the same diameter of 11.6 cm, the thicknesses of the flyer plate and targetare 3 and 5mm, respectively. A procedure for fitting constants of the function t in Eq. (1) fromexperimental dependencies of the yield stress versus strain rate for copper is described inAppendix. Three grids with ascending numbers of nodes have been used for the mesh dependenceanalysis. A successive grid in the sequence of the grids has twice as many elements in each spacedirection as the previous one. The problem simulates the plane impact and the central area of theplates is in the 1D state. Spallation is caused by reflection of the compression shock wave initiatedby the collision from the free surface of the target that may result in a strong tensile pulse beingaccompanied by the spall fracture. The criterion (3) describes the tensile mode of fracture if theequivalent stress is chosen in the form s ¼ sn; here the mesh-independent option (6) is used.Results of the hydrocode modelling at t ¼ 20 ms are shown in Fig. 4 for those three grids (the

numbers of elements of the target mesh are: (a) 100� 20; (b) 200� 40; (c) 400� 80). It is seen thatthickness of the spallation plate is reproducible for all these calculations and that the fracturezones are quite similar. A slight instability of the crack surface is observed for the coarsest grid asa result of higher numerical viscosity for this grid; however, the spall thickness is not affected bythe mesh size. Thicknesses of the calculated spall plate counted in elements and in a unit of lengthat the time t ¼ 20ms are 11 (2.82mm), 23 (2.91mm), and 45 (2.95mm) for the cases (a), (b), and(c) in Fig. 4, respectively. A magnified area of the flyer plate and target in vicinity of the symmetry

Fig. 4. Mesh dependence in the LS-DYNA2D calculation of spallation: (a) rough grid; (b) grid with moderate

refinement; (c) fine grid; (d) a magnified central area of the rough grid calculation; (e) the free surface velocities for

the rough ‘1’ and moderate ‘2’ grids, and the free surface velocity for the moderate grid case with disabled fracture

option ‘3’.


axis for the coarse grid calculation of Fig. 4(a) is shown in Fig. 4(d). This example alsodemonstrates numerical convergence of the split-element method: it is observed less differencein the calculation results between the cases of fine and moderate refinement meshes (Fig. 4(b)and (c)) than between the rough and moderate ones (Fig. 4(a) and (b)). Calculations withthe mesh-dependent choice (5) of the base stresses in the fracture criterion did not show aremarkable difference from the results in Fig. 4. The free surface velocities of target Vfs areshown in Fig. 4(e) for the coarse and moderate grids (curves 1 and 2, respectively). Curve 3 showsthe free surface velocity for the moderate grid calculation with disabled fracture criterion inorder to emphasise the onset of spallation. For the present choice of constants in the fracturecriterion the material failure starts at approximately 3.6 ms from the moment of impact for allthe grids; this time is in agreement with similar Vfs experiments. It should be noted that thetime resolution of stress waves in the present 2D calculations is rather low because even for thefine grid case the maximum number of elements in the impact direction is only 80 that is arelatively rough grid.In relation to the calculation results the question is how the brittle type spallation is agreed with

the viscous-type damage prevailing in copper; the latter may result in unclear spall plane visible asa localised zone of porous material (e.g. see [3]). It should be noted that the present algorithm doesnot detail the mechanism of fracture, and it treats fracture as a brittle-type process; therefore, theapproach provides us only with information about location of the damage zone with preservationof the failed elements. Nevertheless, this information can eventually be obtained for the both typesof fracture because both of the viscous and ductile mechanisms can be taken into account throughthe damage accumulation criterion and the brittle-type algorithm of fracture.The plugging due to ballistic impact is a suitable statement for analysis of the shear mode of

fracture at the high-velocity impact. To realise the plugging we consider the normal impact of acylindrical projectile made of hard steel to an aluminium plate. The impact velocity is 1 km/s,diameter of the projectile is 1 cm, projectile’s height is equal to its diameter, thickness of the plateis 5mm, and plate’s diameter is 8 cm. Constants of the function t in Eq. (1) for steel andaluminium are fitted in the same fashion as for copper in the previous problem. Similar set ofcalculations with refining grids has been conducted for the convergence analysis of the pluggingproblem. A grid in 40� 10 elements was allocated as the rough grid for the target in the first runand two finer grids in 80� 20 and 160� 40 elements were selected as the moderate and fine grids,respectively. Similar allocations have been made for the projectile.Result of the calculations is shown in Fig. 5. The left picture of each pair of plots in Fig. 5(a–c)

corresponds to the moment of t ¼ 20ms after the impact, the right plot is the state at t ¼ 40ms.The fracture criterion being employed in the present case involves the base stresses in the mesh-dependent form (5) and the equivalent stress is chosen to take the shear mode of fracture intoaccount: s ¼ st: It is seen from Fig. 5 that location and extension of fracture zones are very closefor all these three grids. Two crack directions are typical for the plugging problem, which havebeen analysed in details in [24]: the principal crack, aligned with the adiabatic shear zone,separates the plug from the remainder of the target, and the second (cross) system of crack arecracks in the direction orthogonal to the line of impact, which are caused by the shear bending ofthe target (Fig. 6(c)). The major attention has been traditionally paid to the first system of cracks,including extensive studies with hydrocodes (see, e.g. [5,6]). The present calculation is also in goodagreement with the availability of the cross system of cracks.


It is worth noting that the second system of cracks calculated with the present algorithm shouldbe considered as an indicator of the fracture zone only; no real cracks are calculated with thepresent approach. The ‘cracks’ with orientation along the impact axis could also be observed as aneroded zone with the erosion option in LS-DYNA; however, the second (cross) system of crackscannot be tracked with that method of calculation because the erosion option works by definitionas a separation option. On the other hand, the continuum damage approaches, which mightdescribe this damage zone, cannot simulate separation of the plug from the target. The present

Fig. 5. Mesh dependence in the LS-DYNA2D calculation of plugging: (a) rough grid; (b) grid with moderate

refinement; (c) fine grid.


method manages both the embedded cracks/damage and separation cracks. The calculations inFig. 5(a–c) show that both the systems of cracks are being developed for all the meshes.Numerical dissipation for the rough grid appears as a non-elliptical pattern of the target’s

debris. The elliptic pattern typical for the high-velocity impact is observed for the moderate gridand it conserves at further refinement that proves convergence of the algorithm. Another keyparameter is the size of fragments and dimensions of the target damage (diameter of theperforation). The diameter of the opening is mesh dependent but it is less influenced by the meshsize starting from the moderate grid that also confirms convergence of the method.Similarly to the collision problem, the choice of the base stresses sn and st in the invariant form

(6) does not make a significant impact on the plugging problem (Fig. 6(a–b)). Comparison of thelast two calculations in vicinity of the shear zone with the choices of equivalent stress in the mesh-dependent and invariant forms is shown in Fig. 7 for the rough and fine grids; here right-side plots

Fig. 6. Calculation of the plugging with the mesh-independent formulation for equivalent stress in the fracture

criterion; (a) �t ¼ 20 ms; (b) �t ¼ 37ms; (c) formation of the bending cracks near edge of the opening in an aluminium

target after ballistic impact (experiment [24]).

Fig. 7. Role of the choice of the calculation method for the base stresses in the fracture criterion; (a) rough grid, (b) fine

grid. Left hand part of each drawing is correspondent to the choice of the stresses in the invariant form (6); right hand

part—the base stresses are in the mesh-dependent form (5).


are correspondent to the results of Fig. 5(c) and left-side plots to those of Fig. 6. It is seen thatwith refining the debris patterns are getting closer regardless the method of calculation of thestresses in the fracture criterion.Up to the present the fracture calculations have been restricted to purely tensile and shear

modes. To analyse the influence of the complex stress state, calculations have been conducted forthe same problems with the equivalent stress chosen in the linear form (4) involving both tensileand shear stresses as the linear representation, the coefficients of the linear function are taken asA ¼ 0:1; B ¼ 1: The constants have been chosen on the basis of the impact test data [25] wheremeasurements of longitudinal and lateral stresses in a number of metals have been conducted. Thedata demonstrate that ratio of the normal stress to the shear one could be as high as 10; therefore,this choice of constants was believed to be realistic for the illustration purpose. The invariantchoice of the base stresses (6) was used. Results of the calculation for the plate collision problemare shown in Fig. 8. It is seen how the shear fracture is involved into the solution at the peripheralzones subject to bending.Results of calculation for the plugging problem with the same choice of the equivalent stress are

shown in Fig. 9. From observation of the plug in Fig. 9 at magnification it is seen that the tensilemode of fracture contributes mainly to the spallation near the free surface of the plug.The fracture criterion (3) has been verified in a variety of papers. In the present case the fracture

zones are also in a good agreement with experiments. For example, calculated thickness of thespall plate is close to that in experiments [3] as shown in Fig. 10. A magnified part of the target in

Fig. 8. Modelling of the plate collision problem with consideration of the complex stress state in the fracture criterion;

(a) rough grid; (b) moderate grid.

Fig. 9. Modelling of the plugging problem with consideration of the complex stress state in the fracture criterion

(moderate grid); (a) t ¼ 20ms; (b) t ¼ 35 ms.


the vicinity of the symmetry axis is shown separately (left drawing); the calculated splitting isobserved at approximately x ¼ 20:15 cm and experimental result [3] is marked by the bold line atx ¼ 20:155 cm, an undamaged area of the target from –0.8 cm down to –0.13 cm was cut out toreduce size of the plot in vertical direction. In this example two copper plates have been collidedwith the velocity of 159m/s; the plate thicknesses are 0.62 and 1.59mm. This calculationemployed the fracture criterion with the invariant choice of the equivalent stress (6) and the lineardependence (4) of the stress against the tensile and shear stresses.

5. Numerical three-dimensional example

A 3D-calculation with the split-element algorithm has been conducted for the pluggingproblem. The only distinction from the 2D-case of the previous section is the cubic shape ofprojectile. In view of symmetry, we calculate a quarter of the configuration bounded by the yz-and xz-symmetry planes. The planes are aligned with the projectile/target sides, and the impactdirection is aligned with the z-axis. To minimise the computer memory requirements, dimensionsof the target in the x- and y-directions are twice reduced against the 2D-case. Results of thecalculation for a rough grid are shown in Fig. 11((a) t ¼ 20 ms; (b) t ¼ 40ms); the grid isapproximately twice coarser than the rough grid from the 2D-calculation in the previous section.In order to compare a 3D-result with 2D-calculation the mesh was twice refined approaching

the resolution of the rough mesh for the plugging problem in Section 3. The states for the 3D- and2D-calculations being compared are shown in Fig. 12(a) and (b), respectively. For easiervisualisation of the crack system on the target surface the projectile is just outlined. The majordistinction of this crack system from that in the 2D-case is the occurrence of the longitudinal shearcracks related to the corner of the projectile.A convenient way to analyse a 3D-structure is to consider 2D-cross-sections of the structure.

Then we can assess actual shape of fragments and judge whether the separation really takes place.In the present work the cross sectioning was conducted with the new LSTC post-processor LS-POST.Fig. 13 depicts yz-cross-sections of the projectile and target at several selected x-coordinates.

The indices I and II point to the projectile and target areas, respectively. Fig. 13(a) corresponds to

Fig. 10. Comparison of computed splitting and location of experimental spallation in the plate collision problem.


xE0 (the cross-section close to the yz-plane), Fig. 13(b) is next section crossing the projectile area;Fig. 13(c–d) are cross-sections of peripheral areas of the target. It is seen that the crack system inFig. 13(a) is very close to the 2D one in Fig. 13(b). The cross-sections in Fig. 13(b–d) show thatthe plug below the projectile is actually a compact fragment completely separated from the target.

Fig. 11. Three-dimensional calculation of the plugging with VecDyna (a rough grid).

Fig. 12. Calculation of the plugging with DYNA: (a) 3D-calculation with VecDyna3D; (b) comparable 2D-calculation

with LS-DYNA2D.


In order to assess the shape of the plug, the top view is convenient to obtain by xy-cross-sections at z=const (Fig. 14). It is seen that the corner induced crack visualised in Fig. 12(a) isbeing developed into the crack separating the plug from the target, while going down. The cornerinfluence is dominant from the top area of target in z down along the whole plug zone (Fig. 14(c)).This results in a cubic shape of the plug (Fig. 14(d)).

6. Conclusions

We believe that the present approach incorporated into the DYNA hydrocode is suitable forthe TE problems because: (1) it preserves the Lagrangian approach that is essential for a clearresolution of the contact and free boundaries of fragments; and (2) it conserves mass, momentum,and energy of elements which, otherwise, could be lost during the element erosion followed by theformation of cracks and fragments. Both points are important for assessment of the fragmenteffect.The split-element method enables features of the continuum damage model and erosion method

to be combined within one approach. Thus, this allows both: (i) to assess non-fatal damage(embedded cracks), and (ii) to calculate separation of fragments from a target.It should be stressed that the present algorithm is not aimed at calculating isolated cracks

comparable with the mesh size. The only information, which is supposed to be obtained, islocalisation of the fracture zones. The 2D-analysis demonstrates that the fracture zones anddirection of fracture development are clearly localised and confirms the method convergence withthe mesh refinement.

Fig. 14. Cross-sections of the 3D-configuration at z ¼const. (a) at z ¼ z1; (b) at z ¼ z2; (c) at z ¼ z3; (d) at z ¼ z4;(z1 > z2 > z3 > z4).

Fig. 13. Cross-sections of the 3D-configuration at x ¼const. (a) at xE0; (b) at x ¼ x1; (c) at x ¼ x2; (d) at x ¼ x3;(0ox1ox2ox3).


7. Discussion

The present paper analysed a dependence of the failure zones on a choice of theequivalent stress. It was shown that the mesh-dependent and invariant choices of tensile andshear stresses involved into the equivalent stress functional dependence provide close results.On the other hand, selection of the functional dependence is not straightforward and it is notwell developed for kinetic fracture criteria. The present study exhibited that availablefunctions for the static fracture criteria, being presented as surfaces in a stress space, could beused for the choice of the equivalent stress in the kinetic fracture criterion. However, there isno yet good answer how to choose equivalent stress for a wide enough variety of impactproblems. For example, an important parameter affecting fracture at the plugging is temperaturealong with stress. There are a number of theories taking the temperature factor into accountunder the damage accumulation (e.g, relevant discussion can be found in [26]). However, thetheories can only be tested by independent experiments under controlled temperature in theshear zone that is not easy to achieve. Because the constants of the fracture criterion were fittedfrom corresponding impact tests, the results of modelling the plugging problem arequite reasonable. However, it should be stressed that with the present approach furtherinvestigations are necessary for taking the temperature factor into consideration through anequivalent stress. One problem, which will need to be looked at, is shown in Fig. 9 with theequivalent stress combining shear and normal stresses. The present criterion was fitted for theshear banding conditions where high temperature reduces flow stress; this gives us lower fracturethreshold J than that at the room temperature. As a result, this low J has resulted in unlikelyspallation within the plug zone; therefore, the temperature influence on the equivalent stress iscritical for this problem. Nevertheless, the present calculations verified convergence of thealgorithm; the shear zone is slightly larger for the coarse grid, but no essential mesh sensitivity hasbeen found.The present realisation of the split-element algorithm does not consider fragment

interaction. Allowing for the interaction would be of very high computational cost. For adominant fragment there is an opportunity in DYNA to introduce dynamically the sidesof elements of the fragment to be considered as those to interact. However, it is unrealisticat present to control automatically every element side being split, which might interact withother sides.

Acknowledgements

The author would like to express his appreciation to Livermore Software TechnologyCorporation for providing the VecDyna3D manual and a copy of the source code.

Appendix

The algorithm [16] was designed as a procedure for forcing the constants of the constitutiveEq. (1) to fit experimental data of the Split Hopkinson Pressure Bar (SHPB) test. Let us consider a


material sample in the stress–strain conditions close to those occurring in the SHPB test. Auniform deformation of the sample in x-direction is anticipated, that is, the velocity distribution inthe sample is linear: u ¼ ’e � x þ const: Therefore, the constitutive Eq. (1) can be rewritten in thefollowing form

ds1dt

� 2G � ’e ¼ �s1

tðs1;TÞ:

Here s2 ¼ 0; s3 ¼ 0 and, thus, the stress invariant s in Eq. (1) is proportional to s1; for the sake ofconvenience s1 substitutes for s in the function t throughout the section. When modelling thestress–strain response, we can take stress to be s1 and strain to be ’e � t: The dependence t isselected according to the dislocation theory of plastic deformation in the following form (see [9]for details):

t ¼ t0expðD=sÞ

N0 þ Ms=2G:

At a fixed strain rate e � ’e this function provides stress–strain response of material close to theideal elasto-plastic one. The constants N0 and M are dislocation parameters, they are usually notbeing varied when fitting; therefore, the variable fitting parameters are only D and t0: The SHPBtest data are possible to summarise as values of the yield stress Y against given strain rates e: Theyield limit Y can be associated with a sharp drop of the stress rate ds1=dt as strain increases;moreover, for the elasto-plastic response we can assume Y to be a stationary point of solution ofthe constitutive equation at which ds1=dt ¼ 0: Therefore, as an approximation, Y is taken tosatisfy the following algebraic equation:

2G’e ¼Y

tðY ;TÞ:

Ignoring temperature in the present consideration, we can calculate two constants D and t0 of thefunction tðY Þ using two experimental points (Y1; e1) and (Y2; e2). Constants of the constitutivefunction t; which have been obtained with this simple algorithm and used in the present paper, are

Steel : t0 ¼ 280 ms; D ¼ 5:3 GPa; N0 ¼ 106 1=cm2; M ¼ 1011 1=cm2;

Al : t0 ¼ 831 ms; D ¼ 1:73GPa; N0 ¼ 106 1=cm2; M ¼ 1010 1=cm2;

Cu : t0 ¼ 64:8 ms; D ¼ 2:0 GPa; N0 ¼ 106 1=cm2; M ¼ 1010 1=cm2:

References to the corresponding experimental data used for the fitting can be found in [9].

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