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International Journal of Impact Engineering 36 (2009) 1269–1275

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International Journal of Impact Engineering

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

Numerical analysis of blast-induced wave propagation using FSI and ALEmulti-material formulations

Mehdi Sotudeh Chafi, Ghodrat Karami*, Mariusz ZiejewskiDepartment of Mechanical Engineering and Applied Mechanics, North Dakota State University, Dolve 106, Fargo, ND 58108-6050, USA

a r t i c l e i n f o

Article history:Received 6 June 2008Received in revised form22 February 2009Accepted 17 March 2009Available online 5 April 2009

Keywords:BlastArbitrary Lagrangian EulerianMulti-materialFluid–structure interactionWave propagation

* Corresponding author. Tel.: þ1 701 231 5859; faxE-mail address: g.karami@ndsu.edu (G. Karami).

0734-743X/$ – see front matter � 2009 Elsevier Ltd.doi:10.1016/j.ijimpeng.2009.03.007

a b s t r a c t

As explosive blasts continue to cause casualties in both civil and military environments, there is a need toidentify the dynamic interaction of blast loading with structures, to know the shock mitigating mech-anisms and, most importantly, to identify the mechanisms of blast trauma. This paper examines the air-blast simulation using Arbitrary Lagrangian Eulerian (ALE) multi-material formulation. It will explainhow the fluid–structure interaction (FSI) can be simulated using a coupling algorithm for the treatmentof the fluid as a moving media by a moving mesh using ALE formulation and how the structure is treatedon a deformable mesh using a Lagrangian formulation. To validate the numerical approach, as well as toprove its ability to simulate complicated scenarios, comparison of three distinct blast scenarios, i.e., blastfrom C-4 and TNT in open space and blast on a circular steel plate, with the experimental data wasperformed. The predicted numerical results match very well with those of experiments. This compu-tational approach is able to accurately predict the relevant aspects of the blast–structure interactionproblem, including the blast wave propagation in the medium and the response of the structure to blastloading.

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1. Introduction

An explosion is a complex phenomenon which requires efficientmodeling techniques. Studying the numerical and experimentalresponses of structures subjected to blast loading conditions havebeen a topic of investigations which have been widely reported inthe literature over the past five decades. It is well known that theblast wave produced by the detonation of an explosive is charac-terized by extremely high peak pressure and short duration [1].When an explosive charge is detonated in the air, the rapidlyexpanding gaseous reaction products compress the surrounding airand move it outwards with a velocity that is initially close to thedetonation velocity of the explosive (7–10 km/s). The radialexpansion rate of the detonation products exceeds the sound speedof air and creates a shock wave with discontinuities in pressure,density, temperature and velocity [2]. The finite element (FE)method is the most powerful general-purpose technique forcomputing approximate solutions to complex problems. In defor-mation analysis, classical Lagrangian FE methods are dominantnumerical schemes. The main advantage of a purely Lagrangianformulation is that the interface between the penetrator and targetis precisely defined and tracked, but it cannot resolve large

: þ1 701 231 8913.

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deformations accurately. ALE is an alternative approach to resolvethe mesh distortions. ALE allows the mesh to follow the movingboundaries, and preserve the element shape. The main advantageof an ALE–FE method is that the mesh geometry can be controlledindependent from the material geometry. ALE formulations havebeen presented in finite difference and finite element literatures byNoh et al. [3] and Hughes et al. [4]. The formulations have also beendiscussed in the FSI concepts. For example, Belytschko, et al. [5]described the use of ALE methods in nuclear safety calculations andAquelet et al. [6] used this method to solve the fuel slosh problem infuel tanks. The ALE formulation has also been implemented in thesimulation of blast loading analysis [7].

In this paper the multi-material ALE formulation is used to modelair-blast simulation. The formulation is applicable to the effects froman explosion because an element may contain two, or more,different materials, such as air and gases generated from an explo-sive detonation. The capability of LS-DYNA [8], as an explicit finiteelement (FE) code is employed to simulate the multi-material ALEformulation and the fluid–structure interaction behavior. FSI issimulated using a coupling algorithm; the fluid is treated on a fixed,or moving, mesh using an ALE formulation; and the structure istreated on a deformable mesh using a Lagrangian formulation. TheFSI thus couples the pressure of the unsteady fluid flow and thedeformation of the structure. Two coupling techniques in use inLS-DYNA are constraint-based formulation and penalty-based

Table 1JWL and material parameters for TNT and C-4.

Parameters C-4 [7] TNT [13]

A (GPa) 598.155 371.20B (GPa) 13.750 3.231R1 4.5 4.15R2 1.5 0.95E (GPa) 8.7 7.0D (m/s) 8040 6930u 0.32 0.3r (kg/m3) 1601 1590

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formulation. Although these are not the only coupling formulations,they are well established and accepted within the suggestedapplications. The constraint-based formulation is an algorithm thatalters the velocities of the nodes of the solid and shell elementsimplicitly and forces them to follow each other. The methodattempts to conserve momentum, but not energy, and is regarded asquite stable. The penalty-based formulation, however, applies nodalforces explicitly by tracking the relative motion of a given point. Themethod conserves energy, but is not as stable as constraint-basedformulation [12]. The coupling algorithm allows fluid material toflow around and along the structure, but not through the structure.Flow through the structure is prevented in an approximate way byapplying penalty forces to the fluid and structure. As soon as a fluidparticle penetrates through a Lagrangian structure, a force of recall isapplied to both the fluid particle and the structure node to preventpenetration from occurring [6]. The penalty method appliesa resisting force to the slave node, proportional to the penetration,through the master segment. This force is applied to both the slavenode and the nodes of the master segment in opposite directions tosatisfy equilibrium [6].

The multi-material ALE formulation allows the FE mesh to moveindependently of the material flow and where each element in themesh can contain a mixture of two, or more, different materialssuch as air and water [11]. There are two ways to implement theALE equations, and they correspond to the two approaches taken inimplementing the Eulerian viewpoint in fluid mechanics [7]. Thefirst method solves the fully coupled equations for computationalfluid mechanics. The alternative approach is the so-called ‘‘operatorsplit’’ (as described by Benson [9]) in which the ALE solver involvesa Lagrangian step, where the mesh is allowed to move and a secondstep that advects (or moves) the element state variables back ontoa reference mesh. The advection terms must be treated in a specialway within the numerical algorithm to ensure stability andaccuracy of the computational scheme. The mass transport,momentum and energy across the element boundaries should becalculated; the details of the numerical method used to solve suchequations are described by Benson [9], where first-order Donor Cellmethod and second-order van Leer algorithm [10] are used.

Employing the multi-material ALE formulation in this paper,good agreements between the test results and the predicted onesare achieved. So, a conclusion can be made, that this computationalapproach is able to correctly predict the relevant aspects of theblast–structure interaction problem, including: the propagation ofthe blast wave in the medium and the structure response subjectedto the blast loading.

2. Numerical formulations

2.1. Material models for air and explosive

The blast load imposed by an external explosion on a structure isgenerally described by time-varying pressure profiles at selectedlocations over the whole structure. The profiles usually have twophases: (i) positive phase, i.e., overpressure period – a sudden risein pressure (called a peak overpressure) above the pressure (calledthe reference or ambient pressure) before the explosion, thena quick decrease to the reference pressure, and (ii) negative phase,i.e., under-pressure period – a continuing slow decrease below thereference pressure, then an increase over the reference pressure.

In this paper, the air-blast simulations have been conductedusing LS-DYNA, an explicit FE code, a hydrocode using EulerianMulti-material and ALE formulations for the Navier–Stokesequations with the Jones–Wilkins–Lee (JWL) equation of state forgaseous products of detonation. Several explosion case studies havebeen reviewed. Air is modeled using 8 nodded brick elements using

the hydrodynamic material model. The modeling requires anequation of state, density, and a pressure cut-off and viscositycoefficient to be defined. The viscosity and pressure cut-off are setto zero because pressure cannot be negative and the viscosity forcesare negligible. The ideal gas law is used as the equationof state for air as: p ¼ ðg� 1Þ r

r0E with g ¼ 1:4; E ¼ 0:25 MPa

and r0 ¼ 1:293� 10�3g=cm3: In this expression g is thepolytropic ratio of specific heats.JWL equation of state models mosthigh explosives well and can be expressed in the form,

p ¼ A�

1� u

R1V

�eð�R1VÞ þ B

�1� u

R2V

�eð�R2VÞ þ u

VE

where p is the pressure, V is the relative volume. A, B, R1, R2 and u

are constants. These parameters for trinitrotoluene (TNT) and C-4(explosives), according to Dobratz [13] and Alia and Souli [7] aregiven in Table 1.

In the next section, three examples are presented. The first dealswith the propagation of the air-blast wave in open space, with C-4as the explosive material. The second is concerned with the air-blast wave in open space, but with TNT as the explosive material. Inthe third case a Rolled Homogeneous Armor (RHA) steel plate isembedded into the air-blast developed for the first two examples.In all cases, a spherical high explosive is surrounded by air and theignition point is placed at the center of the explosive.

2.2. Blast from C-4 in open space

This example concerns with the propagation of the air-blastwave in an open infinite domain. In order to alleviate calculations,the air and explosive are modeled by l/8th of sphere with threesymmetric planes (see Fig. 1), which consists of 56,916 elements.A shock is a narrow discontinuity in the pressure wave. To capturea reasonably accurate shock peak pressure, therefore, a fine meshresolution is required [7]. The spherical air mesh is fine enough tomatch accurately the shock pressure that originates from theexplosive. The ignition point is placed at the center of the sphere.For C-4, the radius is 4.07 cm and its weight is 1 lb. Sixteenelements are needed to span the radius of the sphere to adequatelybuild up the detonation pressure during the explosive burn.

The spherical charge is surrounded with the air mesh so there isa one-to-one node match at the boundary between the explosivemodel and the air models (see Fig. 1). In Fig. 1, the explosive-airmodel mesh and the RHA plate arrangement are shown. Theexplosive and air are meshed by ALE while the RHP is modeled byLagrangian mesh.

Two typical pressure wave propagations at different times areshown in Figs. 2 and 3. In Fig. 4, the numerical overpressure iscompared with the experimental result [14]. The arrival time andpeak magnitude shown in the graph represent a good approxima-tion for this problem. The experimental curve shows a peak pres-sure of approximately 2.96 bars at about t¼ 1.5 ms. The numericalresults converge with the experimental ones. The relative errors in

Fig. 1. Explosive-air model mesh and the RHA plate arrangement.

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arrival time and peak magnitude are negligible in an excellentapproximation for the problem.

2.3. Blast from TNT in open space

The second example is also concerned with the propagation ofthe air-blast waves in an infinite open domain but using TNT as theexplosive material. The same architecture, for air and HE, asmentioned before, is considered for this example. The sphericalcharge is surrounded with the air mesh so there is a one-to-onenode match at the boundary between the explosive model and theair models. The air and TNT are modeled by l/8th of sphere withthree symmetric planes. We examine four cases in this examplewith the radius of the highly explosive (HE) sphere being 3.3 cm,4.1 cm, 4.7 cm, 5.2 cm and the corresponding weights being 0.5 lb,1 lb, 1.5 lb, 2 lb, respectively.

In Fig. 5, the numerical pressure vs. stand-off distance iscompared with results from the Army Natick Research Develop-ment and Engineering Center [15]. The numerical results convergewith the experimental ones. The relative errors in stand-offdistance and peak magnitude are less, or equal to, 9.65%, 6.48%,5.45% and 6.27% for 0.5 lb, 1 lb, 1.5 lb and 2 lb, respectively. The

Fig. 2. A typical pressure wave propagation at the

simulated pressure characteristic for the 0.5 lb scenario hasa correlation coefficient of 0.9966 (p< 0.0001) to the experimentalone, while correlation coefficients of 0.9946 (p< 0.0001), 0.9948(p< 0.0001) and 0.9953 (p< 0.0001) are found for 1 lb, 1.5 lb and2 lb scenarios, respectively. The stand-off distances and pressuremagnitudes shown in this figure represent a good approximationfor the problem.

The relative errors in the stand-off distance and peak magnitudeare less than or equal to, 9.65%, 6.48%, 5.45% and 6.27% for 0.5 lb,1 lb, 1.5 lb and 2 lb, respectively.

Depending on the intensity of the explosion, the intensity of thepressure wave can abruptly reach several magnitudes of theatmospheric pressure. The blast-front is followed by a temporaryrelative vacuum, in the form of a negative-pressure wave, withinless than milliseconds. Fig. 6 shows four typical pressure wavepropagations in different conditions (HE weights and stand-offdistance).

2.4. Interaction of a circular RHA plate subjected to blast loading

When blast waves strike an interacting surface, an increasingoverpressure from the expanding gas builds up since there is no

air at t¼ 0.696 ms (the fringe levels in Mbar).

Fig. 3. A typical pressure wave propagation at the air at t¼ 0.933 ms (the fringe levels in Mbar).

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medium to compress and displace. The burst pressures froma surface explosion are, therefore, larger than an explosion occur-ring in the air. The size of overpressure of a surface burst mightbecome approximately twice that of open-air burst [16] dependingon the position of the plate.

Three types of loading conditions will be developed in the blast–structure interactions, as explained by Smith and Hetherington[16]. In the first type a relatively large shock wave reaches a struc-ture relatively small enough that the blast wave encloses the entirestructure. The shock wave effectively acts, simultaneously, on theentire structure. The structure is, however, massive enough to resisttranslation. The second condition also involves a relatively largeshock wave and a target much smaller than the previous case. Thesame phenomena happen during this case, but the target is suffi-ciently small enough to be moved by the dynamic pressure. In thefinal case, the shock burst is too small to surround the structure

Fig. 4. Comparison of the experimental [14] and the current algorithm numerical data.The experimental curve presents a pressure peak of 2.94 bars at about t¼ 1.5 ms. Thenumerical results converge to the experimental ones. The best relative errors in arrivaltime and peak magnitude are negligible and this represents an excellent approxima-tion for the problem.

simultaneously and the structure is too large to be shifted. Insteadof simultaneous loading, each component is affected in succession.For a typical building, the front face is loaded with a reflectedoverpressure. Neuberger et al. [17] studied the dynamic response ofclamped circular RHA plates subjected to spherical blast loadings.They used an experimental test setup so that the target plate wassupported by two thick armor steel plates with circular holes thatwere tightened together with bolts and clamps. The spherical TNTcharges were hung in the air, using fisherman’s net, and wereignited from the center of the charge.

The simulation of the detonation of a HE with an exposing RHAsteel plate subjected to the generated blast is an FSI concept. In theapproach used in this study, the pressure generated by HE deto-nation is applied to the plate directly. This approach has manyadvantages over the method based on applying a previously knownfunction of blast loading to the target. One of them can be that of

Fig. 5. Blast-front overpressure versus distance, the solid curve refers to numericaldata and the dashed curve refers to the experimental ones [15] for a TNT charges of0.5 lb, 1 lb, 1.5 lb and 2 lb.

Fig. 6. Typical pressure-time histories corresponding to different HE weights and stand-off distances. At the arrival time, following the explosion, the pressure suddenly increases toa peak pressure value. The pressure then decays to an ambient level at time and decays further to an under-pressure (creating a negative phase, or vacuum) before eventuallyreturning to ambient conditions at time.

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including direct interaction between the plate response and theblast wave, as well as avoiding additional modeling errors. Thealgorithm utilizes two major components; (a) HE detonation andpropagation in the air (ALE formulation), and (b) blast wave inter-acts with the plate (FSI) to define the procedure.

To ensure convergence with LS-DYNA, the time step iscomputed at each step as a function of the smallest mesh size, andits material properties. In LS-DYNA, by combining the ALE solverwith a Eulerian–Lagrangian penalty coupling algorithm, a struc-tural, or Lagrangian, mesh can interact with a fluid, Eulerian mesh.This technique allows the plate, constructed from a Lagrangianmesh, to interact with an ALE based air-blast impact site.

2.4.1. Material model for the RHA steel plateTo model the RHA plate, we used the Johnson–Cook (J–C)

constitutive model. Johnson and Cook [18] proposed a constitutiveequation which includes both strain rate and temperature depen-dence and requires many constants that must be determinedexperimentally. The J–C constitutive model reproduces severalimportant material responses observed in the impact and pene-tration of metals. It combines three key material responses whichare strain hardening, strain-rate effects, and thermal softening. TheJ–C constitutive model is described by the following equation:

syield ¼�

Aþ B3np

��1þ C ln

_3_30

��1� T*m�

Table 2J–C constitutive model constants for RHA steel [17].

Material A (MPa) B (MPa) C n M

RHA steel 1000 500 0.014 0.26 1

Where A, B, C, n and m are model constant and T* is given by, thenondimensional temperature, T* ¼ T�Troom

Tmelt�Troom, Where T is in Kelvin,

and the subscript ‘‘melt’’ and ‘‘room’’ represent melting and roomtemperatures, respectively. RHA density is assumed 7.8 g/cm3 andthe bulk and shear moduli for the steel are 164 and 77.5 GPa,respectively. The parameter A is the initial yield strength of thematerial at room temperature. The equivalent plastic strain rate _3 isnormalized with a reference strain rate_30. T0 is room temperature,and Tm is the melting temperature of the material, and they areassumed to be constants. While the parameter n takes into accountthe strain hardening effect, the parameter m models the thermalsoftening effect and C represents strain rate sensitivity.

Fig. 7. Normalized midpoint deflection vs. time (milliseconds), comparison of thepresent numerical solution and the one reported by Neuberger et al. [17]

Fig. 8. The vertical displacement (the fringe levels in cm) of the circular RHA plate at the time of t¼ 0.588 ms.

M.S. Chafi et al. / International Journal of Impact Engineering 36 (2009) 1269–12751274

The J–C constitutive model is a well-accepted and numericallyrobust material model and highly utilized in modeling and simu-lation studies. Various researchers have conducted split Hopkinsonpressure bar high speed compression tests to obtain the parametersA, B, C, n and m of the constitutive equation by fitting to experi-mental data. Table 2 summarizes the values of the model constantsfor the RHA plate.

2.4.2. The interaction results of the RHA plate subjected to blastThe explosive charge is placed in a distance from the clamped

circular RHA plate. The spherical high explosive TNT (radius 4.9 cm,weight 0.781 g) modeled with 940 hexahedral finite elements (seeFig. 1) is ignited at its center. The spherical charge is surroundedwith the air mesh so there is one-to-one node match at theboundary between the explosive and the air. Due to the symmetry,only 1/8 of the geometry for air and HE and 1/4 of the plate aresimulated in the FE model. Non-reflecting boundary conditions areimposed on the planes. The boundary conditions for the plate areconsidered as being perfectly clamped. The plate thickness ist¼ 0.0125 m, with a diameter of D¼ 0.5 m and is placed ata distance from the center of the charge R¼ 0.125 m. In Fig. 7 thenormalized midpoint deflection of the plate versus time is shown. Acomparison of the result of this study and the one reported byNeuberger et al. [17] are also made in this figure. From the figure, itappears that the two results are in a good agreement. Fig. 8 showsa snapshot of the vertical displacement (cm) of the circular RHAplate at the time t¼ 0.588 ms.

Based on the above validated RHA plate-blast arrangement,another three different experiments as noted by Case No.1–3 in thefollowing will be considered. The results of each case will becompared to those of experimental data for further verification ofthe algorithm.

Case No. 1: The plate thickness t¼ 0.1 m, diameter D¼ 0.5 m,the charge weight W¼ 1.094 kg TNT, and the distance from thecenter of the charge R¼ 0.065 m.Case No. 2: The plate thickness t¼ 0.1 m, diameter D¼ 0.5 m,the charge weight W¼ 1.094 kg TNT, and the distance fromcenter of the charge R¼ 0.1 m.

Table 3Experimental [17] and the computational results from the present study.

Case No. t (cm) D (cm) W (kg TNT) R (cm) d/tExperimental [17]

d/tNumerical

RelativeError %

1 10 50 1.094 6.5 7.45 6.93 6.92 10 50 1.094 10 4.85 4.33 10.73 10 50 0.468 10 2.6 2.45 5.7

Case No. 3: The plate thickness t¼ 0.1 m, diameter D¼ 0.5 m,the charge weight W¼ 0.468 kg TNT, and the distance from thecenter of the charge R¼ 0.1 m.

The radius of HE sphere for Case No. 1, 2 and 3 are, respectively4.12 cm, 5.5 cm, and 5.5 cm. Table 3 lists the experimental resultscompared to the current numerical predictions. In Case No. 1,the experimental normalized peak deflection is d/t¼ 7.45, while theequivalent numerical value is d/t¼ 6.96. In Case No. 2, the experi-mental normalized peak deflection is d/t¼ 4.85, while the equiva-lent numerical value is d/t¼ 4.33. In Case No. 3, the experimentalnormalized peak deflection is d/t¼ 2.6, while the equivalentnumerical value is d/t¼ 2.45. The relative errors for normalizedpeak deflections are, respectively 6.9%, 10.7% and 5.7%.

From Table 3, it appears that the experimental and numericalresults are in very good agreement. Fig. 9 shows the normalizedmidpoint deflection time history for the three case studies.

Fig. 9. Normalized deflection at different case studies vs. time. Case No. 1: The platethickness is t¼ 0.1 m, the plate diameter is D¼ 0.5 m, the charge weight isW¼ 1.094 kg TNT and the distance from the center of charge is R¼ 0.065 m. Case No.2: The plate thickness is t¼ 0.1 m, the plate diameter is D¼ 0.5 m, the charge weight isW¼ 1.094 kg TNT and the distance from the center of the charge is R¼ 0.1 m. Case No.3: The plate thickness is t¼ 0.1 m, the plate diameter is D¼ 0.5 m, the charge weight isW¼ 0.468 kg TNT and the distance from the center of the charge is R¼ 0.1 m.

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3. Conclusions

In this paper, the explosion processes of a highly explosivematerial were simulated based on an ALE FE formulation. For theexplosion in air, multi-material formulation was implementedwith two or several different materials (air and the generated gasby the HE material) by interface reconstruction method. Threedifferent explosion scenarios were examined, two of them forexplosive wave propagation in open-air due to the explosion ofC-4 and TNT. In the third scenario, the response of an exposingRAH steel plate to blast wave loadings was examined. Thenumerical values for parameters such as, the generated peakoverpressure, the stand-off distance-pressure variation, wavearrival time, the deflection of the exposing RHA plate wereexamined with their equivalent experimental values, and goodagreements were achieved for all cases. Apart from the accuracy ofthe numerical algorithm it can be concluded that the assumedequations of state for air, TNT, C-4 and the RAH plate constitutiverelation were reasonably chosen.

Acknowledgement

This research was supported by the Air Force Office of ScientificResearch (AFOSR).

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