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International Journal of Impact Engineering 32 (2006) 2066–2096

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Model-based simulation of the synergistic effects of blastand fragmentation on a concrete wall using the MPM

Wenqing Hua, Zhen Chenb,

aBaker Engineering and Risk Consultants, Inc., 3330 Oakwell Court, San Antonio, TX 78218-3084, USAbDepartment of Civil and Environmental Engineering, University of Missouri-Columbia,

Center for Explosion Resistant Design, Columbia, MO 65211-2200, USA

Received 5 February 2004; received in revised form 18 May 2005; accepted 22 May 2005

Available online 20 July 2005

Abstract

With the development of the material point method (MPM) that is an extension from computationalfluid dynamics (CFD) to computational structural dynamics (CSD), a model-based simulation is performedin this paper to investigate the synergistic effects of blast and fragmentation on structural failure. As can befound from the open literature, the synergistic effects of blast and fragmentation have been usuallysimulated via a combined approach through an interface between CFD codes and CSD codes. As aconsequence, numerical solutions are very sensitive to the choices of different time steps and spatial meshesfor different physical phenomena, especially for the multi-physics involved in the initiation and evolution ofstructural failure. Hence, a coupled approach within a single computational domain seems to be necessaryif objective results are needed. In this paper, a numerical procedure is proposed with the use of the MPM,so that different kinds of gradient and divergence operators could be discretized in a single computationaldomain without involving fixed mesh connectivity. To simulate the evolution of impact failure, thetransition from continuous to discontinuous failure modes is identified via the bifurcation analysis. Thepotential of the proposed model-based simulation procedure is demonstrated through 1D and 2Disothermal cases including cased bomb expansion and fragmentation, blast wave expansion through abroken case, and blast and fragment impact on a concrete wall. The preliminary results obtained in thisnumerical study provide a better understanding of the synergistic effects on impact/blast-resistant

see front matter r 2005 Elsevier Ltd. All rights reserved.

ijimpeng.2005.05.004

ding author. Tel.: +1 573 882 0311; fax: +1 573 882 4784.

ress: [email protected] (Z. Chen).

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W. Hu, Z. Chen / International Journal of Impact Engineering 32 (2006) 2066–2096 2067

structural design. An integrated experimental, analytical and computational effort is required to furtherimprove the proposed procedure for general applications.r 2005 Elsevier Ltd. All rights reserved.

Keywords: Blast; Impact; Material point method; Synergistic effects; Coupled CFD and CSD

1. Introduction

Explosive devices have been used for hundreds of years, yet little information is available on thesynergistic effects of blast and fragmentation on targets [1,2]. When the air driven by the explosiveflows at high speed, the blast wave as a steep pressure occurs, which is followed by an exponentialdecay of pressure. In addition to the blast wave, explosions produce fragments and missiles (forinstance, part of bomb case) that could cause structural damage. A metal case in contact withexplosive is usually broken into chunky fragments with the dimension in one direction being nomore than a few times larger than those in the other directions. The initial velocity for a fragmentmay be as high as 3 or 4 km/s. The blast wave interacting with and loading the fragments causesthe damage of the target synergistically [3], which involves highly non-linear transient phenomena.Although physical experiments play a vital role in the characterization of such problems, theycould be very costly, and often difficult to instrument. Numerical simulation, on the other hand,offers an alternative approach. There are two major problems to be considered in the numericalanalysis of such events. One is the need for an accurate material characterization in terms of theconstitutive models, since the structural failure due to explosion involves plasticity, damage,localization, thermal softening, phase transition and fragmentation. The other is that a robustspatial discretization method must be developed for large-scale simulation of multi-physicalphenomena involved in blast responses. Based on the previous research on the constitutivemodeling of failure evolution, the focus of this paper is on the spatial discretization of multi-physics involved, with the use of the material point method (MPM).As can be found from the open literature, the synergistic effects of blast and fragmentation on

structural failure have been usually simulated via an uncoupled approach or a combined approachwith an interface between computational fluid dynamics (CFD) codes and computationalstructural dynamics (CSD) codes. Baum [4] employed two codes, i.e. FEFLO98 (CFD) andDYNA3D (CSD), to simulate the weapon detonation/fragmentation and the resulting blast andfragments interaction with the target. With the combined CFD/CSD methodology, the structurewas used as the ‘‘master-surface’’ to define the extent of the fluid region, while the fluid was usedas the ‘‘master-surface’’ to define the loads. The transfer of loads, displacements, and velocitieswas carried out via a fast interpolation algorithm. Several types of commercially availablesoftware for dynamic failure simulation, such as AUTODYN-2D & 3D [5], have combined theirmulti-processors to include all the materials under consideration. In AUTODYN-2D & 3D codes,the Lagrange processor is typically used for modeling solid continua and structures, and the Eulerprocessor for modeling gases, fluids and the large distortion of solids. In addition, an arbitraryLagrange Euler (ALE) processor is included, which can be used to provide automatic rezoning ofdistorted grids. Although these processors are powerful in certain situations, it is necessary tocombine them to simulate the synergistic effects of blast and fragmentation on targets. Thus, a

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W. Hu, Z. Chen / International Journal of Impact Engineering 32 (2006) 2066–20962068

mapping technique is required to link the data transportation among different computationaldomains. While it is fairly easy for the remapping algorithm to generate a Cartesian Euler gridfrom a Lagrangian approach to an Euler one, it is usually very difficult to fill the custom-fittedshapes [6]. As a consequence of such a combined approach through an interface communication,numerical solutions are very sensitive to the choices of different time steps and spatial meshes fordifferent physical phenomena, especially for the multi-physics involved in the initiation andevolution of structural failure. Hence, a coupled approach without interface treatment within asingle computational domain seems to be necessary for model-based simulation.Particle, element-free, and meshless methods have been receiving considerable attention

recently. One of such methods is the MPM. The MPM has evolved from a Particle-in-Cell (PIC)method called FLIP, which was originally developed at Los Alamos National Laboratory forfluid dynamics problems involving shock waves [7,8]. Starting from the early 1990s, Sulsky et al.modified FLIP into the MPM for solid dynamics problems, such as impact and penetration, asshown by representative Refs. [9–12]. Since the MPM is an extension from the discretizationmethod for CFD to that for CSD, it is natural to design a model-based simulation tool, based onthe MPM, to investigate the synergistic effects of blast and fragmentation on structural failure.Structural failure involves multi-physical phenomena such as instantaneous responses, diffusion

and wave propagation, which is characterized by the evolution of localized failure [12]. As can befound from the open literature, two different kinds of approaches have been proposed over thelast 20 years to model and simulate the evolution of localization, namely, continuous anddiscontinuous ones. Decohesion and fracture-mechanics-based models are representative ofdiscontinuous approaches, in which strong discontinuities are introduced into a continuum bodysuch that the governing differential equations remain well posed for given boundary and/or initialdata. On the other hand, nonlocal (integral or strain gradient) models, Cosserat continuummodels and rate-dependent models are among the continuous approaches proposed to regularizethe localization problems, in which the higher order terms in space and/or time are introducedinto the stress–strain relations so that the mathematical model is well-posed in a higher ordersense for given boundary and/or initial data. Only weak discontinuities in the kinematical fieldvariables are allowed in the continuous approaches, i.e., the continuity of displacement field musthold in the continuum during the failure evolution. Since the discontinuous bifurcation identifiesthe transition from continuous to discontinuous failure modes, it appears that a combined rate-dependent local continuum damage/plasticity and decohesion approach could be sound in physicsand efficient in computation [13]. As a result, the gap between the continuous and discontinuousapproaches could be bridged to simulate a complete failure evolution process without invokinghigher order terms in space. Based on the previous research [12,13], a rate-dependent localcontinuum damage/plasticity model is combined with a rate-dependent decohesion model via thebifurcation analysis in this paper to simulate the dynamic failure of concrete with the use of theMPM. The remaining sections of the paper are organized as follows.The governing equations for solids and fluids along with the constitutive relations are described

in Section 2. Based on the basic framework of the MPM, a numerical scheme for the fluid–solidcoupling in a single computational domain is proposed in Section 3. In Section 4, several examplesare given to show the features of the proposed model-based simulation tool. In the first example,we test the algorithm for the strong shock with the Riemann problem in one spatial dimensionand the Sedov–Taylor blast wave problem in two spatial dimensions. In the second example, a

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concrete wall subjected to a high-velocity steel flyer is considered to demonstrate thecharacteristics of the rate-dependent transition from continuous to discontinuous failure modes.Then we examine the algorithm for fluid–structure interaction via a piston-container problem anda piston-gas problem. In Section 5, the proposed model-based simulation tool is employed tosimulate several interacting physical phenomena in a 2D computational domain, such as casedbomb expansion and fragmentation, blast wave expansion through the broken case, and blast andfragment impact on a concrete wall. As a result, a better understanding of the synergistic effectson impact/blast-resistant structural design could be obtained through this numerical study.Finally, conclusions and the future work are discussed based on the current research results.

2. Governing differential equations

To describe the equations in the paper, scalars are denoted by plain letters, and the first orhigher order tensors by bold-faced letters or conventional symbols, respectively, except as notedotherwise. The dot represents the inner product between tensor quantities of the first or higherorders.

2.1. Conservation equations

As illustrated in Fig. 1, a region of gas, fluid or solid material points occupies a volume O0p

initially and Otp at later times. For the current position x 2 Ot

p, at time t, let rðx; tÞ be the mass

density, vðx; tÞ be the particle velocity, rðx; tÞ be the Cauchy stress tensor, and bðx; tÞ be the specificbody force in the current configuration. The conservation equation of mass takes the form of

_r ¼ rr v, (1)

X

Y

0pΩ

tpΩ

0px

tpx

Fig. 1. Spatial discretization with the MPM.

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where the superposed dot represents the material time derivative, and r is the gradient operatorwith respect to the current configuration. The conservation of linear momentum is given by

r rþ rb ¼ r_v. (2)

Neglecting the conduction of heat, the conservation of energy can be written as

r_e ¼ r : _e (3)

with e being the specific internal energy and _e ¼ 12½rvþ ðrvÞT denoting the strain rate. Gases,

fluids and solids are distinguished by the constitutive equations. The governing differentialequations can be formulated based on the conservation equations, constitutive models and certainkinematic conditions for given initial and boundary data.

2.2. Constitutive equations for solids

Standard elastoplasticity models are used for metal flyers. A rate-dependent local continuumdamage/plasticity model is combined with a rate-dependent decohesion model via the bifurcationanalysis to simulate the evolution of dynamic failure of concrete, as described below.To estimate stress-wave-induced fracturing, a combined rate-dependent local continuum

damage/plasticity model has evolved over a number of years, which was primarily applied to thecase of rock fragmentation [14,15]. Within the loading regime of the model, an isotropic elasticitytensor governs the elastic material behavior; a scalar measure of continuum damage is activethrough the rate-dependent degradation of the elasticity tensor if the confining pressure PX0(tensile regime); and a pressure-dependent perfectly plastic model is used if Po0 (compressiveregime). The evolution equation for rate-dependent tensile damage can be described as follows:

Cd ¼5k

2

K IC

rC_max

2

mv ; v ¼ v 116

9Cd

,

D ¼16ð1 v2Þ

9ð1 2vÞCd; K ¼ ð1DÞK

in which Cd is a crack-density parameter, KIC the fracture toughness, v the mean volumetricstrain, _max the maximum volumetric strain rate experienced by the material at fracture, C ¼ffiffiffiffiffiffiffiffiffi

E=rp

the uniaxial wave speed with E being Young’s modulus, and D a single damage parameter.Also, K and v are the original bulk modulus and Poisson’s ratio, respectively, for the undamagedmaterial, and the barred quantities represent the corresponding parameters of the damagedmaterial. The model parameters k and m can be determined by using the fracture stress versusstrain rate curve.The rate form of stress–strain relation for the rate-dependent tensile damage model is given by

_r ¼ Ted : _e (4)

in which Ted is the tangent stiffness tensor. To identify the transition from continuous to

discontinuous failure modes, and the corresponding normal to the surface of discontinuity, thebifurcation analysis of the acoustic tensor must be performed based on Eq. (4). As shown in theprevious work [16], the failure angle is rate-independent although the transition level is rate-dependent for the tensile damage model.

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Based on the representative research on decohesion models [17,18], a bifurcation-based rate-dependent decohesion model is formulated here to predict decohesion or separation ofcontinuum. As shown in Fig. 2 for 2D cases, n and t denote the unit normal and tangentvectors to the cohesive surface, respectively. To determine the constitutive relation between thetraction s and decohesion (displacement jump) u

d, the following equations, which satisfy thethermodynamic restrictions, must be solved simultaneously for a given strain rate:

_r ¼ Ted : ð_e _edÞ continuum elastodamage; (5a)

_s ¼ _r n traction equilibrium; (5b)

_ud ¼ _lm evolution of decohesion; (5c)

_ed ¼_l

2Leðnmþm nÞ decohesion strain; (5d)

Fd ¼ te U0ð1 lyÞ ¼ 0 consistency condition (5e)

in which l is a dimensionless monotonically increasing variable parametrizing the evolution ofdecohesion, and Le is the effective length representing the ratio of the volume to the area of thedecohesion within a material element. For the purpose of simplicity, an associated evolutionequation is employed, namely, m ¼ u0ðAd s=ðs Ad sÞ

1=2Þ, so that the effective traction takes the

form of

se ¼ s m ¼ u0ðs Ad sÞ1=2 (6)

with the reference surface energy U0 being the product of the reference decohesion scalar u0 andcorresponding scalar traction t0. The components of the positive definite tensor of materialparameters, Ad, with respect to the n–t basis, are given by

½Ad ¼ t20

1t2np

0

0 1t2tp

264

375. (7)

t

n

m

du.

Fig. 2. A 2D material element with decohesion.

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At the initiation of decohesion, it follows from Eqs. (5e), (6) and (7) with l ¼ 0 that

t2nbt2npþ

t2tbt2tp¼ 1, (8)

where the normal and tangential tractions, tnb and ttb, are determined from the discontinuousbifurcation analysis, and depend on both the strain state and strain rate, i.e. tnb ¼ tnbðe; _eÞ andttb ¼ ttbðe; _eÞ. By letting Cm ¼ ttp=tnp, different failure modes can be simulated by using differentvalues of Cm and Eq. (8). The relation between the traction and decohesion can be adjusted viachanging the value of y, as shown in Fig. 3.With the proposed computational approach, the rate-dependent decohesion model is used to

predict the evolution of microcracking based on the bifurcation analysis of continuum damage,and a fracture-mechanics-based model can then be employed to trace the crack propagation if amacrocrack occurs. Since the location and the orientation of the cohesive surface is determinedvia the bifurcation analysis, the mesh-independent (objective) results could be obtained.

2.3. Constitutive equations for compressible fluids

The stress tensor for a viscous fluid point takes the form of

r ¼ PIþ d, (9)

where I is the second-order unit tensor, and the scalar P would be the hydrostatic pressure if thefluid is brought to rest or the hydrodynamic pressure if the fluid is in motion. Note that the scalarP in Eq. (9) for an incompressible fluid is simply the mechanical pressure, ðs11 þ s22 þ s33Þ=3,while the scalar P for a compressible fluid is identified with an equation of state. The tensor d iscalled the deviatoric stress tensor and it defines the state of stress in the fluid due solely to itsmotion. By postulating the deviatoric stress tensor to be a linear function of the strain rate tensor,the constitutive equation of a fluid is

r ¼ 2mN _eþ lN _ekkI PI, (10)

where _ekk is the trace of the strain rate tensor _e, mN is the coefficient of (dynamic) viscosity, and lN

is the second coefficient of viscosity.

λ

0U

e=

1.0

1.0

>1.0

<1.0 =1.0

e*

Fig. 3. Decohesion relation in terms of l.

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2.3.1. Equation of state for a perfect gas

The equation of state for a perfect gas, where pressure P is related to material density r andspecific internal energy e, is given by

P ¼ ðg 1Þre (11)

in which g is the ratio of specific heat. In general, g decreases with the increase of temperature andincreases with the increase of pressure. For the sake of simplicity, g is assumed as a constant in thefollowing examples. At each time step, the density and the specific internal energy are updated foreach gas material point as rtþDt

p ¼ rtp=ð1þ _ekk DtÞ, and from Eq. (3),

etþDtp ¼ et

p þ rtþDtp : _etþDt

p =rtþDtp Dt. (12)

2.3.2. Equation of state for explosivesTo design devices using high explosives, simple equations of state that can be easily calibrated

with experiments are needed. An example of such an equation of state is the Jones–Wilkins–Lee(JWL) form. The JWL equation of state models the pressure generated by chemical energy in anexplosive, and can be written as

P ¼ C1 1o

r1V

er1V þ C2 1

or2V

er2V þ

ocV

, (13)

where C1;C2; r1; r2, and o are constants, V ¼ r0=r the relative volume with r0 and r being theinitial and current density, respectively. c denotes the internal energy. Tables of the constants forthe JWL equation of state are available in [19].

2.4. Artificial viscosity

It is well known that most numerical simulations of compressible-fluid shocks would yield moreaccurate results if certain type of artificial viscosity is used at the shock front. The artificialviscosity implemented here with the MPM is similar to that first proposed in 1950 by vonNeumann and Richtmyer [20], and later modified by other researchers [21,22], i.e.,

q ¼ rlðc0l _e2kk c1a_ekkÞ if _ekko0, (14a)

q ¼ 0 if ekk40, (14b)

where l ¼ffiffiffiffiAp

in two spatial dimensions with A being the area of computational element, a ¼ffiffiffiffiffiffiffiffiffiffiffigP=r

pis the local sound speed, and c0 and c1 are dimensionless constants.

3. The MPM for fluid–structure interaction

3.1. The MPM

To accommodate the discontinuities of different degrees involved in structural failure, a robustspatial discretization method is a necessity without invoking a fixed mesh connectivity. Althoughthe MPM is still under development, sample calculations have demonstrated the robustness and

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potential of this method for the challenging problems of current interests [9–12,23]. The essentialidea of the MPM is summarized as follows for the completeness of the proposed procedure.The MPM discretizes a continuum body with the use of a finite set of Np material points in the

original configuration that are tracked throughout the deformation process, as illustrated inFig. 1. Let xp

t (p ¼ 1; 2; . . . ;Np) denote the current position of material point p at time t. Eachmaterial point at time t has an associated mass Mp, density rt

p, velocity vpt , Cauchy stress tensor rp

t ,strain ep

t , and any other internal state variables necessary for constitutive modeling. Thus, thesematerial points provide a Lagrangian description of the continuum body. Since each materialpoint contains a fixed amount of mass for all time, the conservation of mass, Eq. (1), isautomatically satisfied. At each time step, the information from the material points is mapped to abackground computational mesh (grid). This mesh covers the computational domain of interest,and is chosen for computational convenience. After the information is mapped from the materialpoints to the mesh nodes, the discrete equations of the conservation of linear momentum, Eq. (2),can be solved on the mesh nodes. The weak form of the conservation of linear momentum can befound, based on the standard procedure used in the finite-element method, to beZ

Orw adO ¼

ZOrss : rwdO

þ

Zsc

rcs wdS þ

ZOrw bdO, ð15Þ

in which w denotes the test function, a is the acceleration, ss is the specific stress (i.e. stress dividedby mass density), cs is the specific traction vector (i.e. traction divided by mass density), b is thespecific body force, O is the current configuration of the continuum, and Sc is that part of theboundary with a prescribed traction. The test function w is assumed to be zero on the boundarywith a prescribed displacement. Since the whole continuum body is described with the use of afinite set of material points (mass elements), the mass density term can be written as

rðx; tÞ ¼XNp

p¼1

Mpdðx xtpÞ, (16)

where d is the Dirac delta function with dimension of the inverse of volume. The substitution ofEq. (16) into Eq. (15) converts the integrals to the sums of quantities evaluated at the materialpoints, namely,

XNp

p¼1

Mp½wðxtp; tÞ aðx

tp; tÞ

¼XNp

p¼1

Mp½ssðxt

p; tÞ : rwjxtp

þ wðxtp; tÞ c

sðxtp; tÞh

1þ wðxt

p; tÞ bðxtp; tÞ ð17Þ

with h being the thickness of the boundary layer. As can be seen from Eq. (17), the inter-actions among different material points are reflected only through the gradient terms. Becausethere is no fixed mesh connectivity in the MPM, impact, localization and the transition from

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continuous to discontinuous failure modes can be simulated without the difficultiesassociated with remeshing and master/slave nodes. The momentum equation is solved at thegrid nodes. On the boundary between the fluid and the structure, the internal forces in mixedsolid–fluid elements determine the interface conditions. Material points within the samecomputational grid element at a given time step would have the same increment in strain, buthave different stress increments because each material point follows its own constitutive equation,as discussed next.

3.2. Fluid–structure interaction

Based on the previous work [24], a numerical scheme with the MPM algorithm forfluid–structure interaction is developed in this paper. No matter whether the materials are solidsor fluids, they are discretized to be a set of material points. The initial information, such as mass,density, velocity, stress state, pressure, internal energy and other properties, is assigned at eachmaterial point. The effect of the fluid on the solid and vice versa will be determined on the gridwhen the momentum equation is solved at each grid node. The coupling of the fluid and solid isindirect in the sense that the pressure from a fluid material point is not directly applied to theneighboring solid material points. Instead, the forces from fluid and solid material points, f i, arecalculated together at grid node, i, where the divergence of the material point stress, based onEq. (17), is summed as follows:X

f

ðr rf ÞVf ;p þX

s

ðr ssÞVs;p ) f i (18)

using the fluid and solid stresses, rf and rs, and the respective material point volumes, Vf ;p andVs;p. Eq. (18) and Fig. 4 symbolically show accumulation of the grid forces from fluid (subscript f)and solid (subscript s) material points.The net effect of the force summation is that the grid forces cause accelerations of neighboring

fluid and solid material points. There is no interpenetration between fluid and solid materialpoints, which is due to the fact that the material points move in a continuous velocity fieldobtained by solving the momentum equation. The continuity of the velocity field implies thatmaterial points in the same computational cell will never overlap with each other, and that theymove as dictated by the linear velocity field.Since the material point is history-dependent, it is convenient to carry strain and stress, as well

as history variables along with the material points. The MPM applies constitutive equations atmaterial points, which allows easy evaluation and tracking of history-dependent variables. It alsoallows computations with solids and fluids to be performed independently since each materialpoint retains its identity (material properties) throughout the computation. For solids, standardsolution schemes for elasticity, plasticity and continuum damage constitutive models are used toevaluate the stress increment and update history variables. For fluids, a variety of equations ofstate can be employed at the material points for different materials. Since two unknowns, etþDt

p

and rtþDtp , appear in Eq. (12), an iterative procedure is generally required to solve this equation

together with Eqs. (10) and (11) to make the whole system accurate up to O½ðDtÞ2 þO½ðDsÞ2 [25]with Ds being the computational mesh size. The detailed description of the proposed algorithmcan be found in the Appendix A.

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X

Y

air

concrete

steel case

explosive

fluid material point

carrying stress σf

solid material pointcarrying stress σs

Fig. 4. Fluid–structure interaction.


4. Demonstration of the proposed procedure

In this section, several numerical examples are given to demonstrate the proposed model-basedsimulation tool. First, we verify the algorithm for the strong shock through the Riemann problemin one spatial dimension and the Sedov–Taylor blast wave problem in two spatial dimensions.Next, a concrete wall subjected to a high-velocity steel flyer is considered to demonstrate thepotential of the proposed procedure in simulating impact, localization and the rate-dependenttransition from continuous to discontinuous failure modes. We then examine the algorithm forfluid–structure interaction through the piston-container problem and the piston-gas problem. Ineach simulation, an explicit time integration scheme is used with the time step satisfying thestability criteria.

4.1. Simulation of shock propagation in fluid

4.1.1. Riemann problem

The simulation of the Riemann problem, a 1D shock tube problem [26], could test theimplementation of the MPM fluid formulation of strong shocks. The initial state for the Riemannproblem consists of a region of high density and pressure on the left, separated by a diaphragmfrom a region of low density and pressure on the right as shown in Fig. 5. When the diaphragm issuddenly removed, the pressure difference forces the contact discontinuity to move to the right.

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t = 0

t > 0

u = 0.0

u = 0.0

PL= 1.00

PL= 1.00

ρR

= 0.125

PR

= 0.001ρ

L= 1.00

PR

= 0.001

Fig. 5. Riemann fluid propagation problem.


The contact discontinuity acts like a piston, creating a shock that propagates to the right throughthe less dense region and a rarefaction wave that propagates to the left through the denser region.The initial conditions for the Riemann problem are given in Fig. 5, where the subscripts L and R

refer to the left and right regions, respectively. An equation of state for perfect gas is used withg ¼ 1:40. For these initial conditions, the initial shock waves have a relatively high Mach numberM 14:43 [27].This is a 1D problem, but it is solved with the MPM in two dimensions, and the solution

variables are constant with respect to the y (vertical) direction. A square background grid is usedwith 200 1 square elements of dimension size of 0.005, and nine material points are initiallyplaced in each grid cell, for a total of 1800 material points. The governing differential equationsare nondimensionalized.Fig. 6 shows the density, velocity, pressure and energy profiles with both the quasi-analytical

solution (dotted line) and simulation results (solid line) at time t ¼ 0:143. The simulation isperformed with artificial viscosity which smoothes the oscillations at the shock front, but alsosmears the shock slightly. The parameters for the artificial viscosity in Eq. (14) are c0 ¼ 2:0 andc1 ¼ 1:0. All data are plotted at the material points. The significant unbalanced physical variablesare observed in the rarefaction wave when the material points are crossing the cell boundary,which results in the noise in the rarefaction wave. In general, the simulation matches the quasi-analytical solution quite well.

4.1.2. Sedov–Taylor blast wave problem

The classical Sedov–Taylor blast wave is created on a rectangular grid by setting the pressure atthe central zone to certain very high value. The expansion of this high-pressure region drives aspherical blast wave into the surrounding uniform medium. The radial profile of this blast waveshould match the quasi-analytical solution of a point explosion in a uniform medium given bySedov [28].Fig. 7 illustrates the initial configuration. The computational domain is 0.6m 0.6m with

square cell size of 0.005m. The initial high pressure is set as 1.0GPa at the central four cells with1.0 Pa at the surrounding medium. An equation of state for perfect gas is used with g ¼ 1:67. Inorder to save computational time as well as fulfill the accuracy requirements, we assign various

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Fig. 6. Results of Riemann problem simulation with the MPM (dashed line for analytic solution).


numbers of material points into different computational domains: 25 material points are placed ineach of central four grid cells; between the radius r ¼ 0:01 and 0:05m, 16 material points areplaced in each grid cell; between the radius r ¼ 0:05 and 0:10m, 9 material points are placed ineach grid cell. Between the radius r ¼ 0:10 and 0:20m, 4 material points are placed in each gridcell; otherwise, one material point per grid cell is used.The density and pressure distribution at 41.0ms of the MPM simulations are shown in Fig. 8.

Artificial viscosity is also employed in this simulation with c0 ¼ 2:0 and c1 ¼ 1:0 so that the shockspreads over several cells. As can be seen in Table 1, the peak shock pressure and shock traveldistance nearly match the quasi-analytical solution given by Sedov [28].

4.2. Impact on concrete

The computational domain for the model problem is illustrated in Fig. 9 in which a concretewall is subjected to a high-velocity steel flyer. The bifurcation analysis is performed to model the

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0.6 m

0.6

m

1.0 GPa

Fig. 7. Initial configuration of Sedov–Taylor blast wave problem.


transition from continuous failure mode (governed by rate-dependent tensile damage) todiscontinuous failure mode (governed by rate-dependent decohesion). Since there is a lack ofexperimental data in the open literature to calibrate all the model parameters involved, thesimulation results presented here could only qualitatively demonstrate the failure evolution in theconcrete wall under impact. The following assumptions are made for the purpose of simplicity:

–
No crack closure effect is taken into account. – No strain rate effect is considered in the compressive regime. – No bifurcation analysis is performed in the compressive regime.
The size of the entire computational domain is 1.5m 1.5m, which is discretized with a 60 60uniform grid. Initially, there are four material points in each grid cell. The plane strain state isconsidered with free boundary conditions.A rate-independent elastic–perfectly plastic von Mises model with an associated flow rule is

used to describe the steel flyer with Young’s modulus E ¼ 200:0GPa, mass density r ¼7850kg=m3 and Poisson’s ratio u ¼ 0:29. The yield strength of steel is 500MPa. For the concretewall, the model parameters are assigned the following values: Young’s modulus E ¼ 25:0GPa,mass density r ¼ 2320 kg=m3, and Poisson’s ratio u ¼ 0:15, with the fracture toughnessK IC ¼ 1:0Mpa

ffiffiffiffimp

. Based on the work [14], the model parameters m and k for concrete arechosen to be 6 and 5 1026 l/m3 in the tensile regime, respectively.An elastic–perfectly plastic Drucker–Prager model with a non-associated flow rule is used in the

compressive regime of the concrete wall. Note that the rate-dependent damage as described inSection 2 is active in the tensile regime of the concrete wall before the discontinuous bifurcation

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10

8

6

4

2

00

0.050.1

0.150.2

0.250.3 0.3

0.250.2

0.150.1

0.050

0.3

0.2

0.1

0 00.05

0.10.15

0.20.25

0.3

10

8

6

4

2

0

10

8

6

4

2

00

0.1

0.2

0.30 0.05

0.1 0.15 0.20.30.25

10

5

0

0.60.5

0.40.3

0.20.1

00

0.1

0.2

0.3

0.4

0.5

0.6

10

6

4

2

0

8

0.30.2

0.10

0

0.1

0.2

0.3

x (m)

y (m)

Pre

ssur

e (P

a)

x 106

00

0.1

0.1 0.2 0.4 0.5 0.6

0.2

0.3

0.4

0.5

0.6

(a)

(b)

Fig. 8. (a) Density distribution of Sedov–Taylor blast wave simulation with the MPM and (b) pressure distribution of

Sedov–Taylor blast wave simulation with the MPM.


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125 100

5010

0

1000

Fig. 9. The initial configuration of a plane-strain impact problem (unit: mm).

Table 1

Comparison of theoretical and numerical solutions of Sedov–Taylor blast wave problem at 41.0 ms

Peak overpressure (MPa) Shock travel distance (m)

Quasi-analytical solution 5.23 0.1677

Numerical solution 4.60 0.1750


occurs. At each time step, the bifurcation analysis of acoustic tensor must be performed until thelocalization line in the strain space of Mohr osculates the major principal circle of strain [16]. Assoon as the transition from continuous to discontinuous failure modes is identified, thedecohesion model is active with mode I failure being dominant. The failure initiation value, tnp, isfound to be the largest principal stress based on the bifurcation analysis. Cm is chosen to be 10,and t0 ¼ tnp is used for the mode I failure. The reference decohesion scalar, u0, is assigned to be6 104m. The parameter y is chosen to be 1.0, which provides a linear relation between thetraction and decohesion. The main flowchart of implementation of constitutive models for theimpact problem is illustrated in Fig. 10.The failure pattern of the concrete wall due to the impact of steel flyer at time 0.442ms is shown

in Fig. 11. Note that the normal to decohesion surface at the center point of the concrete wall isparallel to the x-coordinate based on the bifurcation analysis. Also, as can be observed fromFig. 12, the impact induces significant material failure close to the middle section of the concretewall. Coupled with the presence of multiple wave reverberations due to geometric effects, the most

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1.5

0.5

00 0.5 1 1.5

1

x (m)

y (m

)

Fig. 11. The failure pattern with impact velocity being 1500m/s.

Pressure (P)

Drucker-Prager

Model

Bifurcation

Continuum

Damage Model

Decohesion Model

Compressive regime (P < 0)

Ten

sile

Reg

ime

Before Bifurcation

Aft

er B

ifur

catio

n (P

> 0

)

Fig. 10. The main flowchart of implementation of constitutive models for the impact problem.


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intense deformations occur within 2–3 times of the characteristic dimensions of the flyer. Withinthis region, strain rates in excess of 10,000/s are common, as are large strains, and pressures well inexcess of the material strength. In the simulation, we did not make use of the initial symmetry ofthe problem. The problem as a whole was analyzed and motion of all material points was traced.It is worth noting that the initial symmetry of the system is in general preserved although localizedloss of symmetry is observed. As can be seen, the MPM does not exhibit the finite elementpathology associated with distorted meshes and instabilities, and the MPM does not exhibit theorientation effect often seen with finite elements when discontinuities are allowed to propagate atvarious angles to the mesh sizes. Hence, characteristics of impact, localization and the rate-dependent transition from continuous to discontinuous failure modes could be qualitativelypredicted by the proposed model-based simulation procedure.

4.3. Fluid–structure interaction

4.3.1. Piston-container problemTo verify the algorithm for the fluid–structure interaction, a piston-container problem is

considered first. Fig. 13 shows the piston-container and its ideal MPM representation. Attachedto the piston is a spring of constant K and length L with little mass. The piston can move withoutfriction to compress or expand a compressible, inviscid, and adiabatic fluid with density r andbulk modulus B. As can be seen, the interaction between the fluid and solid (spring) occurs only atthe single interface between the different materials. At this interface, the fluid and solid stressescontribute internal forces for the solution of the momentum equation. The objective is to comparethe theoretical frequency of vibration [29] with that in the MPM simulation.

1.2

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.5 0.55 0.65 0.75 0.85 0.950.6 0.7 0.8 0.9 1

1.1

1

X (m)

y (m

)

0.1

0.2

0.5

0.6

0.1

0.1

Fig. 12. The damage contour corresponding to Fig. 11.

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fluid

(density ,

bulk modulus B )

piston mass m, area Ap

K, L

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

“heavy” material point

interface

Fig. 13. Configuration for the piston-container MPM simulation.


The continuity, energy, and constitutive relations combine to give the relationship betweenpressure P and fluid particle displacement U as follows:

P ¼ Bðr UÞ. (19)

The analytical expression for the natural frequency, o, of vibration of the mass satisfies theequation

offiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½K þ rocAp cotðoL=cÞ=m

q¼ 0, (20)

where c is the wave speedffiffiffiffiffiffiffiffiffiB=r

p, m is the piston mass, and Ap is the cross-sectional area of the

piston.The parameters used in Eqs. (19) and (20) for the MPM simulation are assigned the following

values: K ¼ 1:75 104 N=m, L ¼ 0:508m, Ap ¼ 6:45 104 m2, r ¼ 1000:0 kg=m3, and B ¼

1:09 1010 Pa. The mass density of the piston is 100 times that of the spring. To determine thefrequency of vibration, the heavy material point representing the piston is given a small initialvelocity and the position of this material point is monitored.

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0.015

0.01

0.005

0

-0.005

-0.01

-0.0150 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015

time (sec) (m = 17.51 kg solid line)

time (sec) (m = 175.10 kg dashed line)

mas

s d

efle

ctio

n (

mm

) (m

= 1

7.51

kg

so

lid li

ne)

mas

s d

efle

ctio

n (

mm

) (m

= 1

75.1

0 kg

das

hed

lin

e)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 0.0340.04

0.03

0.02

0.01

0

-0.01

-0.02

-0.03

-0.04

Fig. 14. Piston deflection in the piston-container simulation.

Table 2

Piston-container periods of vibration

Mass (kg) Theoretical o (rad/s) Theoretical period (s) MPM simulation period (s)

17.51 886.8 0.00708 0.0071

175.1 281.2 0.02234 0.0224


Fig. 14 shows the time history of displacements for two different piston masses. The verticallines in the plot represent the theoretical periods of vibration, as given from the solution ofEq. (20). Table 2 lists the theoretical natural frequencies and periods along with the periodsobserved in the MPM simulation. As can be seen, the observed periods matches the theoreticalones quite well.

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2.5

1.5

0.5

-0.5

-1.5

-2.50.04 0.05 0.06 0.07 0.080.045 0.055 0.065 0.075

-1

-2

0

1

2

X (m)

stre

ss (

Pa)

x 109

shock in the gasshock in the piston

inte

rfac

e be

twee

n th

e ga

s

and

the

right

pis

ton

Fig. 16. A strong shock occurs in a perfect gas at time t ¼ 1:3ms.

steel piston perfect gas

0.05 m 0.05 m 0.05 m

u0 = 0 u0 = 0 u0 = 26.532 km/s steel piston

Fig. 15. Configuration of piston-gas problem.


4.3.2. Piston-gas problemTo further test the ability of the MPM simulation of fluid–structure interaction for a strong

shock, a piston-gas device is designed as shown in Fig. 15. The perfect gas, with initial velocityu0 ¼ 0:0, initial pressure P0 ¼ 1:0 106 Pa, initial density r0 ¼ 1:2kg=m3 and g ¼ 1:4, is filledbetween two steel pistons with the left piston being stationary, the right one at initial velocitybeing 26.352 km/s. The experiment is designed such that the steel piston remains elasticthroughout the whole impact process. A square background grid is used with 600 1 squareelements of dimension size of 0.25mm, and one material point is initially placed in each grid cell.According to the Hugoniot equations, a 1.0GPa shock in the perfect gas is developed, whichproceeds from right to left. As can be seen, Fig. 16 illustrates an incident shock pressure in the gaswith the magnitude being 1.0GPa is moving to the left, and a shock wave in the solid piston withthe same value along the x-direction is propagating to the right.

5. Simulation of the synergistic effects of blast and fragmentation on a concrete wall

The synergistic effects of blast and fragmentation on targets are of great interest in relation towarhead design as well as to safety analysis. In this section, we will use the proposed model-based

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simulation tool to conduct a preliminary investigation on the synergistic effects of blast andfragmentation on a concrete wall in a simplified geometry. Due to the lack of experimental data inthe open literature to calibrate all the model parameters involved, only qualitative features couldbe simulated for the interacting physical phenomena such as cased bomb expansion andfragmentation, blast wave expansion through a broken case, and blast and fragment impact onthe concrete wall.

5.1. Model problem description

As illustrated in Fig. 17, a 25-mm-thick-walled cylindrical bomb is placed in the center of thesquare box with dimension size of 1.5m 1.5m. Because it is a plane-strain problem, the length ofthe cased bomb and concrete box could be imagined as being infinite along the z-direction. Insidethe steel case, there is high-explosive material. Between the cased bomb and concrete box, theambient air is filled.Besides the assumptions made in Section 4.2, the following assumptions are made for the

purpose of simplicity:

–
The whole process is assumed under the adiabatic condition. – Turbulence flow is not taken into account.
100

100

100100

y

x

z

1500

1500

150

250

Fig. 17. Initial configuration of cased bomb in the concrete box filled with air (unit: mm).

ARTICLE IN PRESS


The size of the entire computational domain is 2.0m 2.0m, which is discretized with an80 80 uniform grid. Initially, there are nine material points per grid cell in the explosive charge
with one material point per grid cell for the concrete wall, gas and steel case, respectively. Anexplicit time integration scheme is used with the time step satisfying the stability criteria.The constitutive models and the material properties for concrete and steel case are the same as
those given in Section 4.2. The parameters in the JWL equation of state, Eq. (13), for explosiveTNT used in the present study are listed in Table 3, in which c0 is the initial Chapman–Jouguet(C–J) energy per volume as the total chemical energy of the explosive, and VOD is C–J detonationvelocity of the explosive.For the ambient air, the model parameters are assigned the following values: the ratio of

specific heat g ¼ 1:4, the material density r ¼ 1:29 kg=m3 and the initial specific internal energye ¼ 2:0 106 J=kg.

5.2. Evolution of the explosion

In order to demonstrate capabilities of the proposed numerical procedure on simulation of thesynergistic effects of blast and fragmentation on the concrete wall, the evolution of the explosionwith and without steel case is shown in Figs. 18a–f and 19a–f for different time steps. Note thatthe material points for ambient air are not plotted in the figures for the sake of simplicity ofillustration. With steel case, the high pressure of the explosive expands the steel case into severalchunky fragments. Initially the cylinder is placed in greater and greater hoop stress when it isexplosively expanded. A fracture eventually occurs at some point. The fracture presents a freesurface, and a relief wave can travel away from it. Fracture can no longer occur in the relievedregion, but tensile stress and plastic flow are still growing in the unrelieved region where a newfracture is free to form. Once the steel cylinder breaks up, the explosive material squeezes out thesteel case and moves faster than the steel fragments. The concrete wall is first hit by air pressure.Then it is impacted by steel fragments with high kinetic energy. The dramatically localizeddeformation can be observed. On the other hand, without steel case, the explosive drives the airfaster by comparing Fig. 19b with Fig. 18b. The shock overpressure arrives the concrete wallearlier than that with steel case, but the damage pattern is significantly different from that withsteel case by comparing Fig. 19f with Fig. 18f. It is found that the whole concrete wall expands atthe same time without case. Also, the above observations can be inferred from the damagecontours in one quarter of the square concrete box, which are illustrated in Figs. 20–21. With steelcase the damage develops slowly, and localized damage occurs close to the middle section of theconcrete wall. On the other hand, without steel case, the damage evolves uniformly along theconcrete wall. The considerable discrepancy with and without case is due to the inertia effect ofthe steel fragments, which changes the energy distribution of the whole system. At the initial time,

Table 3

JWL parameters used for modeling explosive TNT in the present study

C1 (GPa) C2 (GPa) r1 r2 o c0 (MJ/m3) VOD (m/s) r0 (kg/m3)

373.8 3.747 4.15 0.9 0.35 6000 6930 1630

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00 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 21

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 21

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 21 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 21

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 21

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 21

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1

STEP 1

STEP 5

STEP 9 STEP 11

STEP 7

STEP 3

(a) (b)

(c) (d)

(e) (f)

Fig. 18. (a) Explosive profile at time step I with case; (b) explosive profile at time step III with case; (c) explosive profile

at time step V with case; (d) explosive profile at time step VII with case; (e) explosive profile at time step IX with case;

and (f) explosive profile at time step XI with case.


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00

0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 21

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 21

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 21 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 21

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 21

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 21

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1

0

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

1

STEP 1 STEP 3

STEP 5 STEP 7

STEP 9 STEP 11

(a) (b)

(c) (d)

(e) (f)

Fig. 19. (a) Explosive profile at time step I without case; (b) explosive profile at time step III without case; (c) explosive

profile at time step V without case; (d) explosive profile at time step VII without case; (e) explosive profile at time step

IX without case; and (f) explosive profile at time step XI without case.


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Step 7 Step 131.8

1.55

1.05

0.80.8 1.05 1.551.3 1.8 0.8 1.05 1.551.3 1.8

1.3

1.8

1.55

1.05

0.8

1.3

(a) (b)

Fig. 20. (a) Damage contour at time step VII with case and (b) damage contour at time step XIII with case.

Step 7 Step 131.8

1.55

1.3

1.05

0.8

1.8

1.55

1.3

1.05

0.80.8 1.05 1.551.3 1.8 0.8 1.05 1.551.3 1.8

(a) (b)

Fig. 21. (a) Damage contour at time step VII without case and (b) damage contour at time step XIII without case.


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the case fragments absorb the explosive energy as the kinetic energy. Then they release it whenthey impact the concrete wall. It is worth noting that no interpenetration occurs and the impact/contact is automatic with no additional computational cost. The interface conditions forfluid–structure interaction are satisfied naturally. Based on the simulation results with andwithout case, a better understanding of the synergistic effects of blast and fragmentation on theconcrete wall in explosion could be obtained through this numerical study. The proposed MPMprocedure could be a potential numerical tool to explore the synergistic effects on impact/blast-resistant structural design.

6. Conclusion

A moving discretization method in a single computational domain, which can accommodatedifferent discontinuities and material failures, has been developed based on the MPM. The model-based simulation tool proposed for the evaluation of explosive effects could couple CFD andCSD within a single computational domain. A rate-dependent local continuum damage/plasticitymodel is combined with a rate-dependent decohesion model via the bifurcation analysis so thatthe governing differential equations remain well posed for given boundary and/or initial data.Since no higher order terms in space are introduced into the stress–strain relations, the proposedprocedure is sound in physics and efficient in computation.To demonstrate the potential of the proposed procedure, we have performed a series of 1D and

2D simulations such as a concrete-steel impact problem in solid mechanics, the Riemann problemand the Sedov–Taylor blast wave problem for gas dynamics, and the piston-container problemand the piston-gas problem for fluid–structure interaction. As shown from the preliminary results,the proposed model-based simulation procedure could be used to study the synergistic effects onimpact/blast-resistant structural design in a single computational domain. An integratedexperimental, analytical and computational effort is required to verify and validate the proposedprocedure for general applications.

Acknowledgements

This research is partially sponsored by the US-NSF. The authors would like to thank Prof. SamA. Kiger at the University of Missouri-Columbia for valuable joint discussion on explosiveengineering. The authors are also grateful to reviewers for discerning comments on this paper.

Appendix A

In order to further clarify the main idea and originality of the proposed procedure, a detaileddescription of the proposed MPM for the fluid–structure interaction is given here at given time,with the left superscript k denoting an object.(I) For each material point, perform the mapping operation from the material point to the

nodes of the cell containing the material point.

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For any object k, the mapping operator for the particle mass is given by

kmti ¼

XkNp

p¼1

kMpNiðkxt

pÞ, (A.1)

where kmti is the mass of object k at node i at time t, kMp the particle mass of object k, kxt

p thelocation of the material point of object k at time t, and kNp the number of the material points ofobject k.The mapping operator for the particle momentum takes the form of

kðmvÞti ¼XkNp

p¼1

kðMvÞtpNiðkxt

pÞ, (A.2)

where kðmvÞti denotes the nodal momentum of object k at node i at time t, kðMvÞtp the particlemomentum of object k at time t.Find the internal force vector at the cell nodes,

kðftiÞint¼

XkNp

p¼1

Giðkxt

pÞ krt

p

kMp

krtp

(A.3)

in which Giðkxt

pÞ is the gradient of the shape function associated with node i evaluated at kxtp,

krtp

the particle stress tensor of object k at time t, krtp the particle mass density of object k at time t.

(II) Apply the essential and natural boundary conditions to the cell nodes, and compute thenodal force vector,

kfti ¼ ð

kftiÞintþ ðkft

iÞext, (A.4)

where ðkftiÞext represents the external nodal force vector at node i.

(III) Update the momenta at the cell nodes,

kðmvÞtþDti ¼ kðmvÞti þ

kftiDt. (A.5)

(IV) For each material point, perform the mapping operation from the nodes of the cellcontaining the material point to the material point.Map the nodal accelerations back to the material point,

katp ¼

XNn

i¼1

kfti

kmti

Niðkxt

pÞ (A.6)

with Nn being the number of mesh nodes. Map the current nodal velocities back to the materialpoint,

kvtþDtp ¼

XNn

i¼1

kðmvÞtþDti

kmti

Niðkxt

pÞ. (A.7)

Compute the current particle velocity to evaluate the strain increments,

kvtþDtp ¼ kvt

p þkat

pDt. (A.8)

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Compute the current particle position,

kxtþDtp ¼ kxt

p þkvtþDt

p Dt. (A.9)

Eq. (A.9) represents a backward integration. Compute the particle displacement vector,

kutþDtp ¼ kxtþDt

p kx0p, (A.10)

where kx0p is the initial location of the material point of object k.(V) Map the updated particle momenta back to the nodes of the cell containing these material

points,

kðmvÞtþDti ¼

XkNp

p¼1

kðMvÞtþDtp Nið

kxtpÞ. (A.11)

(VI) Find the updated nodal velocities,

kvtþDti ¼

kðmvÞtþDti

kmti

. (A.12)

(VII) Apply the essential and natural boundary conditions to the nodes of the cells containingthe boundary material points.(VIII) Find the current gradient of particle velocity,

kLtþDtp ¼

XkNp

i¼1

kvtþDti Gið

kxtpÞ (A.13)

and the particle strain increment,

kDetþDtp ¼ ðsymkLtþDt

p ÞDt, (A.14)

k _etþDtp ¼ kDetþDt

p =Dt, (A.15)

krtþDtp ¼ krt

p=ð1þ traceðkDetþDtp ÞÞ. (A.16)

(IX) Update the stress of the material point based on the specific constitutive models relating tothe fluids and solids.(IX(a)) For the solid material point, obtain the stress increments by

kDrtþDtp ¼ T : kDetþDt

p , (A.17)

where T is the tangential stiffness tensor, then update the particle stress tensor,

krtþDtp ¼ krt

p þkDrtþDt

p . (A.18)

(IX(b)). For the fluid material point, obtain the initial specific internal energy of the materialpoints by

ketþDtp ¼ ket

p þkrt

p :k _etþDt

p =krtþDtp Dt (A.19)

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and update the pressure of the fluid material point through the equation of state as

kPtþDtp ¼ ðg 1ÞðkrtþDt

p ÞketþDtp . (A.20)

Next compute the particle stress tensor for the fluid material point

krtþDtp ¼ 2mN

k _etþDtp þ lNtraceð

k _etþDtp ÞI kPtþDt

p I. (A.21)

Then update the internal energy by

ketþDtp ¼ ket

p þ

krtp þ

krtþDtp

2: k _etþDt

p =krtþDtp Dt. (A.22)

Repeat Eqs. (A.20)–(A.22) until the specific internal energy solution is converged.(X) Identify which cell each material point belongs to, and update the natural coordinates of the

material point. This is the convective phase for the next time increment. The computational cycleis complete for this time increment.

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[10] Sulsky D, Zhou SJ, Schreyer HL. Application of a particle-in-cell method to solid mechanics. Comput Phys

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