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International Journal of Impact Engineering 74 (2014) 145e156

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International Journal of Impact Engineering

journal homepage: www.elsevier .com/locate/ i j impeng

A front tracking algorithm for hypervelocity impact problems withcrack growth, large deformations and high strain rates

Shiyu Wu a,b, Qianyi Chen a,b, Kaixin Liu a,b,*, Ning Luo a,b

a LTCS, Department of Mechanics & Aerospace Engineering, College of Engineering, Peking University, 100871 Beijing, ChinabCenter for Applied Physics and Technology, Peking University, 100871 Beijing, China

a r t i c l e i n f o

Article history:Available online 10 May 2014

Dedicated to the 70th birthday of ProfessorNK Gupta.

Keywords:Front trackingCE/SE schemeHigh-velocity impactSpall fracturePlugging failure

* Corresponding author. LTCS, Department of Meching, College of Engineering, Peking University, 1008762765844; fax: þ86 10 62751812.

E-mail addresses: wayqq@pku.edu.cn (S. Wu), kliu

http://dx.doi.org/10.1016/j.ijimpeng.2014.04.0100734-743X/� 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

In the present paper, a front tracking algorithm, which includes a front tracking part and a specified crackgrowth scheme, is proposed for tracking material interfaces and describing the formation and propa-gation of a crack. Combined with an improved spaceetime Conservation Element and Solution Element(CE/SE) scheme, the algorithm can simulate high-velocity impact problems with crack growth, largedeformations and high strain rates. The present front tracking algorithm shows high accuracy in thenumerical test of a single vortex problem. Numerical simulations are also presented for spall fractures ina plate when impacted by a spherical projectile and perforation of a cylindrical Arne tool steel projectileimpacting a plate target. The numerical results are in good agreement with the corresponding experi-mental observations. It is demonstrated that the present algorithm is feasible and reliable for analyzingfractures.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The computer codes which are used in the study of high-velocity impact problems can broadly be classified as Eulerianor Lagrangian descriptions, depending on whether the materialflows through a fixed mesh, or the mesh follows the deformationof solid. Anderson [1] summarized the advantages and disad-vantages of both Lagrangian and Eulerian descriptions.Lagrangian schemes work well at multimaterial interfaces, buthave shortcomings in problems with large deformations. Oneextreme case is that the simulation fails when a grid cell foldsover itself. In recent decades, many Lagrangian meshless/mesh-free and particle methods [2,3] have been proposed to solveproblems with large deformations. These methods also presentsome intrinsic limitations including lower computing speed thanother modern grid-based methods and difficulties in the treat-ment of boundary conditions. Eulerian codes, which allow theboundaries to flow through a fixed mesh, can solve arbitrarilylarge distortions. When combined with efficient numerical

anics & Aerospace Engineer-1 Beijing, China. Tel.: þ86 10

@pku.edu.cn (K. Liu).

methods for describing moving interfaces, Eulerian methodologyis a promising alternative.

Traditional Eulerian codes have low accuracy and cannot cap-ture strong shock waves very well. In the recent years, some re-searchers have applied high-resolution techniques which are usedin the field of modern computational fluid dynamics (CFD) toelasticeplastic flow problems. Udaykumar et al. [4] presented atechnique for the numerical simulation of high-speed multi-material impact. A high-order accurate ENO scheme was adoptedalong with the interface tracking technique to evolve sharpboundaries. After that, their methodology was updated bysubstituting the hybrid particle level set method [5] for the inter-face tracking algorithm. Sambasivan et al. [6,7] proposed a three-dimensional, Eulerian, sharp interface, Cartesian grid techniquefor simulating the response of elasto-plastic solid materials to hy-pervelocity impact, shocks and detonations. Barton et al. [8]developed an Eulerian adaptive numerical method for high-velocity impact problems in two- and three-dimensions. Wanget al. [9] have proposed an improved spaceetime ConservationElement and Solution Element (CE/SE) scheme and extended it tothe community of high-speed impact dynamics in solids. The CE/SEmethod is a novel numerical scheme for solving hyperbolic con-servation laws. It has several attractive features: (a) being mathe-matically simple; (b) a unified treatment of both space and timeand enforcement of flux conservation in both space and time; (c)very little or almost no numerical dissipation; (d) the lack of

S. Wu et al. / International Journal of Impact Engineering 74 (2014) 145e156146

directional splitting for flows in multiple spatial dimensions,resulting in a truly multidimensional scheme. The featuresmentioned above make the method substantially different fromtraditional well-established methods such as the finite differenceand the finite volume methods. Chen and Liu [10] proposed anEulerian scheme combined with the hybrid particle level setmethod for the numerical simulation of spall fracture due to high-velocity impact. These efforts have not only improved the accuracyof the numerical results for the problems of elasticeplastic flow, butalso shown a remarkable direction in the field of computationalmechanics.

Numerical methods for describing moving interfaces can begrouped into Eulerian and Lagrangian approaches. Eulerianschemes distinguish phases by characteristic functions definedon the computational grid. For example, volume-of-fluid (VOF)method uses a marker function to identify material interfaces;Level-set method represents the phase boundary implicitly asthe zero level-set of a scalar grid function. A review of the VOFmethod can be found in Scardovelli and Zaleski [11]. The level-set method is reviewed by Osher and Fedkiw [12] and bySethian [13]. Traditionally, the main difficulty in using thesemethods has been the maintenance of a sharp boundary be-tween different fluids. The so-called front-tracking methodswhich represent the interface explicitly by a set of marker par-ticles are included in the second group. Front tracking has manyadvantages, such as high accuracy and its lack of numericaldiffusion. It is found that front tracking often does not requiresuch highly refined grids, and that grid orientation does notaffect the numerical solution (no grid anisotropy). Front trackingenables a precise description of the location and geometry of theinterface. As a Lagrangian scheme, front tracking method is quitesuccessful in conserving mass since it preserves material char-acteristics for all time as opposed to regularizing them out ofexistence, which may happen with Eulerian front-capturingmethods. Front tracking also has shortcomings, such as its dif-ficulty in robustly handling interface merging and breakup.Especially, in high-velocity impact problems it cannot describethe fractures in a straightforward manner.

Many numerical methods for crack problems have beendevised in recent years. Tvergaard [14] proposed an elementvanishing technique which removes elements that meet a failurecriterion. Xu and Needleman [15] developed a mesh splittingmethod to simulate crack branching. Camacho and Ortiz [16]developed a cohesive-law model, in which adaptive meshingtechnique was employed to provide a rich enough set of possiblefracture path. More recently, cohesive finite elements have beenused to simulate dynamic fragmentation [17e19]. This techniqueis naturally more realistic than element deletion for modelingdiscrete cracks. Many Lagrangian meshless/meshfree and particlemethods [2,3,20e22] have been proposed to solve challengingdynamic crack propagation problems. Clayton [23] developed amethod that combines generalized particle algorithm (GPA) withenergy balance to simulate fragment size distribution. Wang et al.[24] presented a predictive method based on the theories ofcontinuum damage mechanics and mechanics of micro-crackdevelopment, in order to simulate fragments of masonry wallto blast loading. A few years ago, Stolarska et al. [25] introducedthe Level-set method to the field of the numerical simulation ofcrack propagation in solids. Lately, Duflot [26] reviewed in detailthe update techniques of the crack representation by level setfunctions and proposed several new techniques. In thesemethods, a crack is an open curve (an open surface in three di-

mensions) that grows from its tip (its front in three dimensions),and two level set functions are necessary to represent a crack.When it comes to the spall fracture, however, this kind of crackrepresentation becomes inappropriate. The crack appearing inthe spall fracture opens wide with the motion of the scab. If thecrack is modeled as a curve, the scab will never separate. Inaddition, it is unreasonable to put an initial crack in the targetplate before simulation. Chen et al. [27] proposed a new repre-sentation of crack by level-set method and extended the meth-odology proposed by Wang et al. [9] to simulate the spall fractureby introducing the idea of “element erosion” from the FEM. InChen’s method, the crack is represented as a two-dimensionalnarrow region which can be in any shape according to the spe-cific problem. As the element, whose damage factor reaches acritical value, is deleted in this method, it will cause mass loss. InChen’s method, the loss of mass may be severe when there aremany fractures. One of the advantages of level-set method is thatit can handle interface merging automatically. When two surfacesare close enough, they will merge and the two surfaces willdisappear. It is helpful in the simulation of fluid dynamics. But inthe simulation of hypervelocity impact problems, it is a disad-vantage as it may cause facture to disappear.

In the present paper, we propose a new front trackingmethod toconstruct a front tracking algorithm for the 2-dimensional simu-lation of hypervelocity impact problems with crack growth, largedeformations and high strain rates. The algorithm includes a fronttracking part which describes moving interfaces with high accu-racy, and a newly proposed automatic crack growth scheme whichcan describe cracks very well by marker particles. The algorithmcan be easily combined with the improved CE/SE scheme for thesimulation of hypervelocity impact problems. In the study, a singlevortex problem is used to investigate the grid dependence and theaccuracy of the front tracking algorithm. Then simulations arepresented for spall fracture in a plate when impacted by a sphericalprojectile and perforation of a cylindrical Arne tool steel projectileimpacting a plate target. To demonstrate the feasibility and reli-ability of our front tracking algorithm, the numerical results arecarefully compared with the corresponding experimentalobservations.

To the authors’ knowledge, it is the first time to apply the fronttrackingmethod in describing the fractures. Since the front trackingmethod has higher accuracy than the level-set method, the meth-odology presented here has advantages in simulating dynamicfracture such as fractures in ductile materials caused by hyperve-locity impact.

2. Governing equations

Using cylindrical coordinates (r, q, z), the Eulerian governingequations for the homogenous media without heat conducting,thermal diffusion and external forces can be written in the form ofconservation laws as

vQvt

þ vEðQ Þvr

þ vFðQ Þvz

¼ SðQ Þ; (1)

where Q is the vector of conserved variables, E and F are the con-servation flux vectors in r and z directions respectively, and S is thesource term vector. In Eulerian representation, these vectors are

where r is the density, u and v are the r and z components of thevelocity respectively, E is the total energy per unit volume, srr, szz, sqq

Q ¼

26666666666666666664

rrurvEsrrszzsqqsrz3P

TxD

37777777777777777775

; E ¼

26666666666666666664

ruru2 � srrruv� srz

uðE � srrÞ � vsrzusrruszzusqqusrzu 3P

uTuxuD

37777777777777777775

; F ¼

26666666666666666664

rvruv� srzrv2 � szz

vðE � szzÞ � usrzvsrrvszzvsqqvsrzv 3P

vTvxvD

37777777777777777775

; S ¼

26666666666666666664

SrSruSrvSESsrrSszzSsqqSsrzS 3P

STSxSD

37777777777777777775

; (2)

S. Wu et al. / International Journal of Impact Engineering 74 (2014) 145e156 147

and srz, are the components of the deviatoric stress tensor, 3P is the

equivalent plastic strain, T is the temperature, x is the void volumefraction, D is the cumulative damage. The source terms of Sr, Sru, Srv,SE, Ssrr , Sszz , Ssqq , Ssrz , S 3P and ST in Eq. (2) can be expressed as

Sr ¼ �ð1=rÞru; (3)

Sru ¼ �ð1=rÞ�ru2 þ sqq � srr

�; (4)

Srv ¼ �ð1=rÞðruv� srzÞ; (5)

SE ¼ �ð1=rÞðuðE � srrÞ � vsrzÞ; (6)

Ssrr ¼ srr

�vuvr

þ vv

vz

�þ 2Urzsrz þ 2G

�vuvr

� F

�; (7)

Sszz ¼ szz

�vuvr

þ vv

vz

�� 2Urzsrz þ 2G

�vv

vz� F

�; (8)

Ssqq ¼ sqq

�vuvr

þ vv

vz

�þ 2G

�ur� F

�; (9)

Ssrz ¼ srz

�vuvr

þ vv

vz

�þ Urzszz � Urzsrr þ G

�vuvz

þ vv

vr

�; (10)

S 3P ¼ 3P�vuvr

þ vv

vz

�þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi23_DPij_DPij

r; (11)

ST ¼ T�vuvr

þ vv

vz

�þ 0:9

_Wp

rC; (12)

where srr, szz, sqq and srz are the components of the stress tensor, Gis the shear modulus, Urz ¼ (1/2)[(vu/vz) � (vv/vr)] is the compo-nent of spin tensor, F ¼ 1/3((vu/vr) þ (u/r) þ (vv/vz)) is the volumestrain and _D

Pij is the plastic strain rate. _Wp is the rate of plastic work,

which is approximated by _Wpz_3Pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3=2Þsijsij

q, where _3P is the

equivalent plastic strain rate. C is the specific heat.A void growth (VG) model [10] which takes the Bauschinger

Effect (BE) into account is adopted in the simulation. In the VGmodel, the presence of voids is expressed in terms of the distensionratio a, which is related to the porosity x according to x ¼ 1 � 1/a.The main equations from the VG model are given in Appendix A. Inthe absence of inertial effects, the rate of change of x is given by

_x ¼ 278

ffiffiffi3

p 1hFðxÞ

�s� sg

�efx; (13)

sg ¼ 4ffiffiffi3

p

9ð1� xÞA lnð1=xÞ � Bð2=3ÞnPðxÞ� 4

3hð1� xÞGðxÞ;

(14)

PðxÞ ¼Zc2

c1

1c

���cð1þ cÞ�2=3���ndc; c1 ¼ 1

x

�x0 � x

1� x0

�;

c2 ¼ x0 � x

1� x0;

(15)

GðxÞ ¼�1� x

1� x0

�1=3"1�

�x0x

�1=3#; (16)

FðxÞ ¼�1� x01� x

�2=3"1x

�x

x0

�2=3

� 1

#: (17)

where h is the linear hardening modulus, 4 is a material parameter.Hence the source term of Sx in Eq. (2) is

Sx ¼ x

�vuvr

þ vv

vz

�þ 278

ffiffiffi3

p 1hFðxÞ

�s� sg

�efx: (18)

The yield stress and the shear modulus of ductile materials aredetermined by the JohnsoneCook (JC) model [28] in the currentwork. Themodel is based on the work by Johnson and Cook [29,30],Camacho and Ortiz [31] and Lemaitre [32]. The equivalent vonMises stress seq is given as

seq ¼ ½1� bD�½Aþ Brn�h1þ r*

ich1� T*m

i; (19)

where D is the cumulative damage in Eq. (2), and b reads 0 (nodamage coupling) or 1 (damage coupling). A, B, c, n and m arematerial constants; r is the damage accumulated plastic straingiven as _r ¼ ð1� DÞ_3P; _r* ¼ _r=_r0 is a dimensionless strain-rate,and _r0 is a reference strain-rate; T* ¼ (T � T0)/(Tm � T0) is the ho-mologous temperature, where T is the actual temperature, T0 is theroom temperature and Tm is the melting temperature respectively.The damage variableD takes values between 0 (undamaged) andDC(DC � 1, fully broken). The damage evolution rule is proposed as

Fig. 1. Interface represented by front tracking method.

S. Wu et al. / International Journal of Impact Engineering 74 (2014) 145e156148

_D ¼ DC

pf_3P (20)

where DC is the critical damage and pf is a fracture strain dependingon stress triaxiality, strain-rate and temperature. pf is given as

pf ¼hD1 þ D2 exp

�D3s

*�ih

1þ _3PiD4

h1þ D5T

*i; (21)

where D1eD5 are material constants, s* ¼ sm/seq is the stresstriaxiality ratio and sm is the mean stress. A more detaileddescription can be found in Ref. [28]. Hence the source term of SD inEq. (2) is

Fig. 2. Cracks represented by marker particles.

SD ¼ D�vuvr

þ vv

vz

�þ DC

pf_3P: (22)

The improved JohnsoneHolmquist (JH-2) model [33,34] isadopted as constitutive model for brittle materials in this paper.The JH-2 model can be expressed as

seq ¼ �1� DJH

�si þ DJHsf ; (23)

si ¼ AJH�p=PHEL þ TJH

�PHEL

�N�1þ CJH ln _3*�$sHEL; (24)

sf ¼ BJHðp=PHELÞM�1þ CJH ln _3*

�$sHEL; (25)

DJH ¼ SD 3p

DJH1�p=PHEL þ TJH

�PHEL

�DJH2; (26)

where AJH, BJH, CJH,M,N,DJH1 andDJH2 arematerial constants, sHEL isthe equivalent stress at the Hugoniot Elastic Limit (HEL), PHEL is thepressure at the HEL and TJH is the maximum tensile hydrostaticpressure thematerial canwithstand. The hydrostatic pressure p canbe expressed as

p ¼ K1$mþ K2$m2 þ K3$m3; m > 0;K1$m; m < 0;

(27)

where m ¼ r/r0 � 1, K1, K2 and K3 are material constants.The MieeGruneisen equation of state (MG EOS) is adopted for

ductile materials. The MG EOS can be expressed as

p ¼

8>>><>>>:

r0C20m

h1þ

�1� g0

2

�mi

½1� ðs� 1Þm�2þ grI m � 0;

r0C20mþ grI m < 0;

(28)

where m ¼ r/r0 � 1, r0 and C0 are the density and the bulk soundspeed at zero pressure, s is the coefficient that relate the shockspeed to the particle velocity, g is approximated by g ¼ r0g0/r andg0 is the Gruneisen parameter.

The other components in Eq. (2) are the same as those of Ref.[27]. The governing equations presented above can be used todescribe high-velocity impact problems with large deformationsand high-strain rates. We will solve the equations in a newlydeveloped high resolution scheme developed by the authors.

3. Front tracking algorithm

The front tracking method is adopted to track material interfaceand represent the crack, while the level set function is used toreconstruct the interface. The interface tracking algorithm isdivided into three steps, namely the advection step, the reseedingstep and the projection step. In the first step, the particles areadvected with the evolution equation. In the second step, theparticle distribution is readapted to the deformed interface byadding and deleting particles. In the last step, the interface infor-mation is transmitted from the Lagrangian particles to the Euleriangrid. The details are illustrated in the following sections.

3.1. The advection and reseeding step

As shown in Fig. 1, the interface is represented by a set of linkedmarker particles. The particles are advected with the characteristiccurves equation

Fig. 3. The interface simulated by front tracking method on a 200 � 200 grid.

S. Wu et al. / International Journal of Impact Engineering 74 (2014) 145e156 149

d x!p

dt¼ u!�

x!p�; (29)

where x!p is the position of the particle and u!ð x!pÞ is particle ve-locity. The particle velocities are trilinearly interpolated from thevelocities on the underlying grid. This interpolation scheme limitsthe particle evolution to second-order accuracy.

A third-order accurate RungeeKutta method is used to evolvethe particle positions forward in time. In theory, this evolution isunconditionally stable. In practice, the time step Dt is chosen underthe condition of Dt < 0.5h/vmax, where vmax is the maximummagnitude of velocity and h is the minimum value of the grid

Table 1The area loss and accuracy of different methods in the single vortex problem.

Grid cells Front tracking method Hybrid particles level setmethod

Arealoss (%)

L1 error Order Arealoss (%)

L1 error Order

50 � 50 1.5 5.0E�3 N/A 24.926 1.646E�2 N/A100 � 100 0.54 1.5E�3 1.7 2.709 2.059E�3 N/A200 � 200 0.2 6.0E�4 1.3 1.273 9.477E�4 1.1400 � 400 0.059 2.0E�4 1.5 0.636 4.670E�4 1.02

spacing. This condition guarantees that characteristic curves fromdifferent grid points do not intersect. Discrete points on the inter-face are advected, and then the interface itself is formed by con-necting these points together.

As the interface deforms, some of its parts are depleted of par-ticles, and others may become crowded with particles. To maintainan adequate resolution, the distance between two particles is keptwithin a lower boundary and an upper boundary by adding and

Fig. 4. The geometrical sketch of the hypervelocity impact problem.

Table 2Parameters of the ductile materials for the MG EOS and the JC constitutive equation.

Parameters Aluminum1100

Arne toolsteel

Weldox460Esteel

r0 (kg/m3) 2770 7850 7850C (J/kg K) 884 452 452Tm (K) 923 1800 1800G (GPa) 27.1 76.7 75.2

The MG EOS g0 1.97 1.16 1.16s 1.34 1.92 1.92C0 (m/s) 5386 4570 4570

The JC model A (MPa) 150 490B (MPa) 170 807c 0.015 0.0114m 1.03 0.94n 0.34 0.73

Table 3The material parameters of the soda-lime glass for the JH-2 model.

r0 (kg/m3) G (GPa) AJH N CJH TJH (GPa) BJH

2530 30.4 0.93 0.77 0.003 0.15 0.088

M DJH1 DJH2 K1 (GPa) K2 (GPa) K3 (GPa)

0.35 0.053 0.85 45.4 �130 290

S. Wu et al. / International Journal of Impact Engineering 74 (2014) 145e156150

deleting particles. For two neighbouring particles A and B, whenthe two particles are closer than the lower boundary, we delete oneof them. In contrast, when the distance between the two particles isbeyond the upper boundary, we add a new particle in the middle ofthe segment AB. In the present paper, the lower and upperboundary is set to 0.25d and 3d, respectively. d is the original dis-tance between two neighbouring particles. Here

d ¼ 0:25h; (30)

where h is the grid spacing. This particular choice of bounds on thedistance between the two neighbouring particles was the first setwe tried, and further experimentation might give improved results.However, as shown in the following section, these bounds alreadygive reasonable results.

Fig. 5. Effect of mesh size on the crater depth at t ¼ 15 ms.

3.2. The projection step

The level set function 4, positive in one phase and negative inthe other, is commonly defined as the signed distance of r! to thephase boundary. In this paper we use it to reconstruct the interfacein fixed grids. In fact, we do not need to solve the transport equationof the level set function. The level set function value of an arbitrarygrid point A can be given as 4A ¼ n!$BA

., where n! is the normal

vector of marker particle B which is closest to grid point A, and it isassumed that n! is perpendicular to CD. When the location of eachparticle is known, we can obtain the normal vector n! easily. BA

.is

the vector from B to A, as shown in Fig. 1.

3.3. Particle representation of a crack

As the front tracking method cannot describe the fracturing ofinterfaces in a straightforward way, we propose an automatic crackgrowth scheme which can describe the cracks by marker particles.In the simulations, wemonitor the spatio-temporal evolution of thedamage factor. Here we have two different damage models: thevoid growth model whose damage factor is x in Eq. (2) and thecumulative damage model whose damage factor is D in Eq. (2). Fordifferent problems, we can choose different models. When thedamage factor of a grid point reaches the critical value, a set ofmarker particles are placed around the grid point, as shown inFig. 2. They represent the newly-formed crack surface. Once amarker particle is added in the simulation, it will be tracked inevery time step after that. When two sets of marker particles areclose enough, they would be merged together (see Fig. 2). As moreand more grid points’ damage factor reach the critical value, thecrack becomes longer and longer. And thus the formation and thepropagation of a crack can be simulated successfully.

3.4. Grid dependence and accuracy

To test the accuracy and performance of the front tracking al-gorithm, the “vortex in a box” case is simulated. The initial interfaceis a circle of radius 0.15 centered at (0.5, 0.75) in a unit square. Onthe domain [0,1] � [0,1], we consider an incompressible velocityfield(uðx; yÞ ¼ �ðsinðpxÞÞ2 sinð2pyÞvðx; yÞ ¼ ðsinðpyÞÞ2 sinð2pxÞ: (31)

This velocity field stretches the circle into a very thin, spiralfilament, which is very difficult to capture on fixed grids. The ve-locity field is reversed at t ¼ 8 when the interface reaches themaximal deformation. Then the circle is recovered at t ¼ 16. Thisproblem is simulated by front tracking method and hybrid particlelevel set method, respectively. Fig. 3 shows the interface simulatedby the front tracking method at t ¼ 0, t ¼ 4, t ¼ 8 and t ¼ 16 on a200 � 200 grid. The thin spiral filament is well captured and theinterface at t ¼ 16 is almost the same with the initial one.

For a given sequence of grids with various grid spacing h, weevaluate both f and area loss. Table 1 shows the lost area and L1error [35] of different methods in the single vortex problem att ¼ 16. The L1 error is expressed as:

1L

Z ���H�4expected

�� H

�4computed

����dx dy; (32)

where L is the length of the interface, H(f) is the indicator functionfor f � 0; i.e., H(f) ¼ 1 if f � 0 and H(f) ¼ 0 otherwise.

From Table 1, it can be seen that the result by the front trackingalgorithm is quite accurate even on a 50 � 50 grid. It is concluded

Fig. 6. Contour plot of the equivalent plastic strain at t ¼ 15 ms: (a) by present front tracking method; (b) by level set method.

Table 5The material parameters of Weldox460E steel for the cumulative damage model.

D1 D2 D3 D4 D5

0.0705 1.732 �0.54 �0.015 0

S. Wu et al. / International Journal of Impact Engineering 74 (2014) 145e156 151

that the front tracking algorithm is a high-resolution methodwhose accuracy is better than the hybrid particles level set method.These calculations were run on an Intel Core i7-2600 CPU. The CPUtime of the simulation with front tracking method on a 200 � 200grid is 1258 s, while the level set method costs 6287 s.

4. Applications

This section presents two numerical examples to demonstratethe applicability and performance of the front tracking algorithm.The numerical results presented in this paper are obtained bysolving the hyperbolic system of Eq. (2) using a high-order CE/SEscheme. Since the CE/SE scheme implemented in this work is wellestablished and does not differ in any way from that in Ref. [27], theimplementation details are not presented here. The problemsconsidered are axisymmetric and they are solved on an rez plane inthe cylindrical coordinates.

Fig. 7. Schematic representation of a large circular plate struck by a cylindricalprojectile.

4.1. Simulation of a plate impacted by a spherical projectile

The experiment [36] that an aluminum 1100 target plateimpacted by a spherical soda-lime glass projectile at a velocity of6.0 km/s is simulated. The thickness of the target plate is 12.5 mm.The diameter of the projectile is 3.2 mm. The geometrical sketch ofa hypervelocity impact experiment [36] with spall fracture isillustrated in Fig. 4, consisting essentially of a spherical soda-limeglass projectile and a target plate.

The constitutive model and the EOS for the target plate are theJohnsoneCook model and the MG EOS, respectively. The damagemodel is the void growth (VG) model which is already provedeffective in spall fracture problems [10,27]. The porosity x accountsfor the spall fracture and its critical value is xc ¼ 0.3. The viscosity h

Table 4Comparisons of damage features between the experiment and the simulations.

a (mm) b (mm) c (mm) d (mm) e (mm) f (mm)

Experimental data 5.9 7.5 4.8 4.2 1.4 8.6Results calculated by

present method5.65 7.45 4.7 4.1 1.2 8.2

Results calculated bylevel set method

5.6 7.4 4.7 5.0 1.1 8.0Fig. 8. Effect of mesh size on the residual velocity of the projectile, for initial impactvelocity of 298 m/s.

Fig. 9. The blunt residual velocity versus its initial impact velocity.

Table 6Residual velocity of projectile (m/s).

vi 156 161 165 173 182 190 250 298

vep 52 78.7 83.7 112.0 122.6 132.3 191.7 241.4vsp 0 76 83 112 123 138 200 245

S. Wu et al. / International Journal of Impact Engineering 74 (2014) 145e156152

and parameter 4 of the material for the VG model is 1.0 Pa s and 20,respectively. The constitutive model for the soda-lime glass pro-jectile is the JH-2 model. The hydrostatic pressure of the projectileis determined by Eq. (27). Damage of the glass projectile is notconsidered in this paper. The parameters of ductile materials usedin the simulations for various kinds of models are listed in Table 2.The material parameters of the soda-lime glass are listed in Table 3.

Four different mesh sizes of 0.1 mm � 0.1 mm,0.067 mm � 0.067 mm, 0.06 mm � 0.06 mm and0.05 mm � 0.05 mm are used in studying the mesh sensitivity ofthe crater depth. Effect of mesh size on the crater depth at t ¼ 15 msis shown in Fig. 5. Nomesh dependency is observed when themeshsize is near 0.05 mm. Thus the grid size of 0.05 mm � 0.05 mm isadopted in the following simulation.

Numerical results of the equivalent plastic strain at t ¼ 15 ms areshown in Fig. 6. The plot on the left hand side is the numericalresult simulated with the newly proposed front tracking method.The automatic crack growth algorithm, which represents the cracksby marker particles, works well here and can describe the forma-tion and propagation of a crack accurately. The plot on the righthand side is the numerical result with the level set method [10]. Thegeometric parameters which describe the key deformation

Fig. 10. Particle representation of the nucleation, propagation and arrest of a crack in t

characters of the plate after impact, as shown in Fig. 6(a), arecomparedwith the experiment [36] in Table 4. The result calculatedby the newly proposed front tracking method agrees well with theexperimental data and gives errors less than 15%, while the resultcalculated by the level set method [10] gives an error of 21%. TheCPU time of the interface tracking process with front trackingmethod is 6951 s, while the level set method costs 10,003 s.

4.2. Simulation of a plate impacted by a cylindrical projectile

Ballistic impact is a complicated dynamic problem, involvingmaterial nonlinearity, geometric nonlinearity and contact nonlin-earity. In this section the perforation of a cylindrical Arne tool steelprojectile impacting a plateWeldox460E steel target is simulated todemonstrate the performance of the improved CE/SE scheme witha new front tracking method for this type of problem. The experi-ment study has been carried out by Børvik et al. [28]. There is largedeformation in the target during the perforation.

The cumulative damage model (Eq. (21)) of viscoplasticity andductile damage presented by Børvik et al. [28] is used for the targetmaterial of Weldox460E steel. The damage of the Arne tool steelprojectile is not considered in this paper. Parameters of Wel-dox460E steel for the cumulative damage model are listed inTable 5. The constitutive model for the target plate is the JohnsoneCook model. The MG EOS is adopted for the volumetric response ofthe projectile and the target. The parameters of ductile materials forthe JC model and MG EOS are listed in Table 2. The mechanicalproperties of the Arne tool steel used in the calculation are as fol-lows: the elastic modulus is 204 GPa; the Poisson ratio is 0.33; andthe yield strength is 1900 MPa. The Arne tool steel is modeled bythe linear isotropic hardening constitutive model with tangenthardening modulus Et ¼ 15000 MPa [37].

A geometrical sketch of the model is shown in Fig. 7 [38]. Thediameter, length and mass of the cylindrical projectile are 20 mm,80 mm and 0.197 kg, respectively. The thickness and diameter ofthe target plate are 8 mm and 500 mm, respectively. In theexperimental study, the initial projectile velocity varied between137.4 m/s and 298 m/s. The mesh dependency is investigated forthe case of a Weldox460E steel plate collided by an Arne tool steelprojectile at 298 m/s. Eight different mesh dimensions are tested:from 0.16 mm each side for the coarse mesh to 0.09 mm for thedense mesh. Effect of mesh size on the residual velocity of theprojectile is shown in Fig. 8. No mesh dependency is observedwhen the mesh size is below 0.1 mm. Thus the grid size of0.1 mm � 0.1 mm is adopted in the following simulation. Thematerial parameters of the target plates used in the simulations forvarious kinds of models are the same as those in Ref. [28].

The residual velocity of projectile after plugging failure is a mainmeasurable quantity of the ballistic penetration test. Results of

he simulation: (a) before crack nucleation; (b) crack propagation; (c) crack arrest.

Fig. 11. Perforation process of 8 mm thick plate impacted by a blunt projectile with a velocity of 298 m/s: (a) experimental [28] and (b) simulations.

S. Wu et al. / International Journal of Impact Engineering 74 (2014) 145e156 153

numerical simulation and experiment are compared in Fig. 9. Thecomparison is shown in Table 6, where vi, vep and vsp denote theprojectile initial velocity, the residual velocity of projectile fromexperiment and the residual velocity of the projectile from thenumerical simulation, respectively. It is observed that an excellentagreement (less than 5% error) is obtained between the numericalresults and the experimental results.

For a high-speed impact induced plugging fracture problem, thecracks will nucleate at the contact surface, and propagate until theyreach the opposite side of the specimen. The crack growth algo-rithm proposed in the present paper can automatically simulate thecrack nucleation, propagation and arrest, see Fig. 10. In this prob-lem, the two surfaces of the fracture are very close. If we use thelevel-set method to describe the fracture, the two surfaces will

Fig. 12. Illustration of the void growth model. (a) Cubic element with a random

merge together and cause a remarkable error in the simulation. Soit is difficult to simulate the plugging failure in ballistic penetrationwith the level-set method.

Fig. 11 shows the perforation processes of the 8 mm thick plateimpacted by a blunt projectile with a velocity of 298 m/s, which ismuch higher than the experimental ballistic limit. As shown inFig. 11, the simulations qualitatively agree with the high-speedcamera images from Ref. [28]. These figures clearly demonstratethat the proposed model qualitatively captures the overall physicalbehavior of the target during penetration, perforation, shearlocalization, crack propagation, and complete failure. In the simu-lation of the perforation of the 8 mm thick plate impacted by ablunt projectile with a velocity of 298 m/s, the CPU time of theinterface tracking process with front tracking method is 2416 s,

distribution of voids; (b) Spherical void surrounded by matrix material.

S. Wu et al. / International Journal of Impact Engineering 74 (2014) 145e156154

while the level set method costs 11,450 s. The front trackingmethod is more efficient than the level set method.

5. Conclusions

The present paper has proposed a front tracking algorithm tosimulate the fractures caused by high-velocity impact. The algo-rithm includes a front tracking part and a newly proposed auto-matic crack growth scheme. Through a numerical test of the singlevortex problem, it is shown that the front tracking part is of highaccuracy. The crack growth scheme can describe the formation andpropagation of cracks accurately. Combined with an improved CE/SE scheme, the front tracking algorithm can simulate hypervelocityimpact problems with crack growth, large deformations and highstrain rates very well.

The numerical simulation of spall fracture in a plate whenimpacted by a spherical projectile at a velocity of 6.0 km/s is carriedout. The results are in good agreement with the experimental data.Then the perforation of a cylindrical Arne tool steel projectileimpacting a plate Weldox460E steel target is simulated. Goodagreement between the numerical simulations and the experi-mental observations of plugging failure in ballistic penetration isalso obtained. The numerical computation presented here illus-trates that the proposed front tracking algorithm is effective andefficient.

When the two surfaces of a fracture are very close, the methodproposed here can describe it very well, while the Eulerianmethods such as level-set method will cause remarkable errors.Since the front tracking method has a higher accuracy than thelevel-set method and it can be implemented easily in 2D problems,the methodology presented here has advantages in simulatingdynamic fracture problems such as fractures in ductile materialscaused by hypervelocity impact.

Acknowledgement

The authors wish to acknowledge the financial support providedby the National Natural Science Foundation of China (grant nos.10572002, 10732010, 11332002).

Appendix A. The void growth (VG) model with Bauschingereffect [10]

The spall in ductile material is controlled by plastic deformationof the matrix around small voids which grow and coalesce to formthe spall plane. The derivation begins with the consideration of arepresentative volume element containing a random distribution ofvoids, as illustrated in Fig. 12(a). All the voids are assumed to bespherical. Imagine a uniform mean stress s acting over the surfaceof this element. Note that the cross-sectional area occupied by thevoids does not support the stress, we have

As ¼ Asss; (A1)

where A is the total area, containing the voids, of the cross section, sis the mean stress acting on A, As is the solid material part on planeA and ss is the mean stress on As. For a random distribution of voidsshapes and sizes, the following relation is acknowledged:

AAs

¼ VVs

; (A2)

ss ¼ ðV=VsÞs ¼ as; (A3)

where V is the whole volume of the element, Vs is the volume of thesolid material and a is defined as the distention ratio. If the averagemean stress ss is big enough in tension, the voids will grow byplastic deformation in the surrounding solid matrix.

Consider a single spherical void of radius a in a sphere of radius bwhere the outer boundary is subjected to the mean stress ss, asshown in Fig. 12(b). The equation of motion for the problem can bewritten as

vsrvr

þ 2rðsr � sqÞ ¼ r€r; (A4)

srða; tÞ ¼ 0; srðb; tÞ ¼ as; (A5)

where sr is the radial stress, sq is the tangential stress, r is thedensity of the solid material and €r is the radial acceleration. Notethat the inertial effect can be neglected. Integrating Eq. (A5) from ato b, we have

asþZba

2rðsr � sqÞdr ¼ 0: (A6)

The void growth is modeled as a fully plastic process. The yieldsurface of the material is usually given by

f ¼���~s� ~X

���� R� k ¼ 0; (A7)

where ~X is the back-stress for kinematic hardening, R is the increaseof yield surface size for the isotropic hardening and k is the initialyield surface size. The simplest model, linear kinematic hardening,proposed by Prager is adopted here. The evolution of the back-stress ~X is collinear with the evolution of the plastic strain

~X ¼ 23h~3P; (A8)

where h is the linear hardening modulus. Tresca yield condition isadopted here

sq � sr2

¼ ss; (A9)

where ss is the shear yield stress. For most materials ss ¼ffiffiffi3

pss,

where ss is the tensile yield stress. The tensile yield stress iscalculated by the simplified form of the JC model. From Eq. (A7), thefollowing expression is obtained

ffiffiffi3

p

2

��sq �

23h 3

Pq

���sr � 2

3h 3

Pr

��

¼ ss ¼ kþ R ¼ Aþ B�

3P�n þ h_3P;

(A10)

where A, B and n are the material parameters of the JC model, h isthe viscosity of the material, 3

P is the equivalent plastic strain, and_3P is the equivalent plastic strain rate. The equivalent plastic strainand strain rate are expressed as

3P ¼ ð2=3Þ

��� 3Pr � 3

Pq

��� ¼ ð2=3Þ����vuvr � u

r

����; _3P ¼ ð2=3Þ ddt

����vuvr � ur

����:(A11)

Then

S. Wu et al. / International Journal of Impact Engineering 74 (2014) 145e156 155

sq � sr ¼�2. ffiffiffi

3p �h

Aþ B�

3P�n þ h_3P

i� h 3

P: (A12)

Substituting Eq. (A12) into Eq. (A6), we have

as ¼�2. ffiffiffi

3p �Zb

a

2r

hAþ B

�3P�n þ h_3P

idr �

Zba

2rh 3

Pdr: (A13)

The radial displacement can be calculated by

u ¼ r � r0 ¼ r �hr3 þ BðtÞ

i1=3; (A14)

where B(t) is a function related to the rate of void growth. Itsexpression can be written as

BðtÞ ¼ a30a0 � a

a0 � 1: (A15)

Then we have

as ¼ 4ffiffiffi3

p

9

�A ln

� a

a� 1

�� B

�23

�n

PðaÞ�

� 43hjGðaÞj þ 8

ffiffiffi3

p

27h

_a

aða� 1Þ jFðaÞj;(A16)

where

PðaÞ ¼ZBðtÞ=b3

BðtÞ=a3

1c

���cð1þ cÞ�2=3���ndc; (A17)

GðaÞ ¼�a0 � 1a� 1

�1=3��a0a

�1=3;

FðaÞ ¼ ða� 1Þ�a

a0

�2=3� a

�a� 1a0 � 1

�2=3:

(A18)

When _a in Eq. (A16) approaches zero, the threshold stress forvoid growth is obtained

sg ¼ 4ffiffiffi3

p

91a

�A ln

� a

a� 1

�� B

�23

�n

PðaÞ�� 43h1ajGðaÞj:

(A19)

Then

a�s� sg

� ¼ 8ffiffiffi3

p

27h

_a

aða� 1Þ jFðaÞj: (A20)

Another variable, porosity x ¼ 1 �1/a, is also frequently used inthe published literatures. Expressing Eq. (A20) in terms of theporosity x, we have

s� sg ¼ 8ffiffiffi3

p

27hFðxÞ _x; (A21)

where

FðxÞ ¼�1� x01� x

�2=3"1x

�x

x0

�2=3

� 1

#: (A22)

To summarize, the void growth rate is expressed as

_x ¼ 278

ffiffiffi3

p 1hFðxÞ

�s� sg

�efx; (A23)

where e4x is a modification proposed by Eftis et al. [39] for the voidinteraction during the coalescence process.

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