Computers and Geotechnics 43 (2012) 72–79

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Computers and Geotechnics

journal homepage: www.elsevier .com/ locate/compgeo

Numerical simulation of surface explosions over dry, cohesionless soil

Anirban DeCivil & Environmental Engineering Department, Manhattan College, Bronx, NY, United States

a r t i c l e i n f o

Article history:Received 1 September 2011Received in revised form 16 February 2012Accepted 18 February 2012Available online 17 March 2012

Keywords:Numerical modelingCentrifuge modelingExplosionBlast loadingUnderground tunnel

0266-352X/$ - see front matter � 2012 Elsevier Ltd.doi:10.1016/j.compgeo.2012.02.007

E-mail address: anirban.de@manhattan.edu

a b s t r a c t

Numerical modeling of the effects of explosions relies on suitable material models appropriate for largedeformation problems. Available results of a wide range of static and dynamic tests on Nevada #120sand, completed as part of an earlier project (VELACS), were utilized to calibrate a numerical model forsand, suitable for modeling surface explosions. A fully-coupled Euler–Lagrange Interaction was utilizedto correctly model pressures created by the explosion simultaneously with the large deformations inthe soil. The model was used to study two cases – the first with a 2-D axisymmetric case of crater forma-tion; and the second with a 3-D case of surface explosion above an underground tunnel. The results ofnumerical analyses were found to closely match those from other analyses, field tests, and centrifugemodel tests.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

An explosion on the ground surface, such as caused by terroristactivities, can induce relatively large stresses and strains in thesubsurface soil. Therefore, it is necessary to investigate the possi-bility for underground structures, such as tunnels and pipelines,to experience significant magnitudes of stresses due to an explo-sion at the surface. While it is true that an underground explosionor an explosion inside the tunnel would be more effective incausing structural damage, potentials for such explosions can bereduced or eliminated by vigilant surveillance and controlledaccess. On the other hand, the potential for surface explosions(especially, those caused by mobile/portable charges) are moredifficult to eliminate.

Effects of explosions on a soil medium and underground struc-tures can be studied through full-scale model tests, reduced scale/enhanced acceleration physical tests (as in centrifuge tests), andnumerical modeling. Full-scale model tests involve elaborate prep-arations and are very expensive. In addition, they also require theuse of relatively large amounts of explosives, involving potentialrisks and need careful handling, which is typically not feasible incivilian research.

The effects of an explosion are influenced by the engineeringproperties of the soil medium through which the shock waves tra-vel. In this paper, a model for dry sand is proposed, which can beused to characterize most of the essential properties necessary to

All rights reserved.

predict the formation of surface craters due to an explosion anddevelopment of strains in an underground structure.

1.1. Background

Both centrifuge model tests and numerical modeling techniqueshave been employed to study the effects of blasts. Kutter et al. [1]and Davies [2–4] utilized centrifuge modeling to study the effectsof explosions on tunnels. Goodings et al. [5] conducted over 100tests between g levels of 1 g and 100 g to study formation of cratersin dry soil. They concluded that centrifuge tests provided a validmethod of modeling effects of explosion in soil. Fragaszy et al.[6] reported results of centrifuge tests on models of reinforced soilwalls subjected to blasts. Sausville [7] and Simpson et al. [8] inves-tigated the nature of damage caused to earth structures from sur-face blasts.

Numerical modeling was utilized by Wang [9], usingLS-DYNA3D (LSTC [10]) computer program to model the effectsof a buried landmine explosion on an armored personnel carrierabove ground. Results from full-scale explosions using dry sandwere used to validate numerical models, before they were usedin direct applications. Choi et al. [11] utilized Autodyn computerprogram to model the effects of explosions on undergroundstructures. They studied the influence of quantity of explosives,dimensions and nature of the underground structure, distancefrom the explosion and properties of the subsurface medium anddeveloped damage assessment charts for use in assessing vulnera-bility of tunnels. Luccioni et al. [12] used Autodyn to model cratersformed on a dry soil due to explosions using various charge sizeand correlated their results with those from full-scale field tests.

A. De / Computers and Geotechnics 43 (2012) 72–79 73

Grujicic et al. [13] developed soil models that were utilized innumerical model study of landmine detonation.

2. Computational analyses: Background and solvers

2.1. Introduction

An explosion is accompanied by a violent release of energy thatgenerates shock waves. The shock waves travel as compressionwaves and are capable of exerting large pressures on surfaces theyencounter. The passage of shock wave causes the pressure to in-crease from an ambient value to a peak incident pressure, denotedby Pso. The shock wave travels radially from the burst point withdiminishing velocity. As the shock front expands into increasinglylarger volume of a medium, the value of peak incident pressure, Pso,diminishes (USACE [14]).

Computer codes, commonly known as ‘‘hydrocode’’, are em-ployed to conduct numerical modeling of explosions. Hydrocodesemploy principles of conservation of mass, momentum, and en-ergy, using differential equations that govern unsteady, dynamicflow conditions. The numerical modeling reported in this paperwas conducted using the computer program ANSYS Autodyn 13.0(ANSYS [15]), which is a non-linear, dynamic analysis software,suitable for modeling large strain/large deformation problems,including blasts, impacts, and collisions.

2.2. Numerical solvers

2.2.1. IntroductionDifferent types of solvers are available in a hydrocode, such as

Autodyn, to model various materials and conditions, utilizing suit-able features of each solver. Common solvers available in Autodynare Euler, Lagrange, Smooth Particle Hydrodynamic (SPH), andArbitrary Lagrange Euler (ALE). In addition, fully-coupled interac-tion is allowed between Euler and Lagrange parts, as discussed ina later sub-section.

2.2.2. Euler solverIn an Euler solver, the numerical mesh remains undeformed,

while material is allowed to flow freely from one element toanother. Thus, a finite volume is assigned to each element andthe finite difference equations governing flow are solved withinthe element. This solver is suitable for modeling material in a fluidphase, such as a gas or a liquid. An Euler solver can also be used tomodel solid objects that are undergoing large deformation. In thepresent study, the air and the explosive (TNT) were modeled usingEuler solvers.

2.2.3. Lagrange solverA Lagrange solver allows the mesh, defining the object, to de-

form. There is no flow or transport of material from one elementto another. The deformation is addressed by the distortion of themesh. Very large deformations and removal of material can be han-dled either by rezoning the mesh or by introducing an erosion fea-ture that allows material to be removed from a zone when certainpre-determined criteria (such as large strain) are met. A Lagrangesolver is suitable for modeling solid objects, such as soil, tunnel,and protective barrier material, in the current study.

2.2.4. Fully-coupled Euler–Lagrange InteractionA fully-coupled Euler–Lagrange formulation is used to model

fluid–solid interaction, especially under dynamic loading, such asblast loading. The coupling of the two solvers allows for largedeformations in the fluid phase, while utilizing the solids to pro-vide boundaries which interact with the fluids. The fluids induce

stress fields on the solids, while flowing around solid boundaries.This approach has been previously reported by Choi et al. [11] inmodeling the effects of explosions on tunnels.

3. Material models

3.1. Introduction

The following materials were modeled in the study:

� Air.� Soil.� TNT (explosive).� Tunnel (only in 3-D analyses).� Barrier material (only in 3-D analyses).

Air, soil, and TNT were modeled in both the 2-D and the 3-Danalyses. Specific material models, most suitable to represent theproperties of the different materials, were selected, as discussedbelow.

3.2. Model for air

Air was modeled as an ideal gas, with a reference mass densityof 1.225 � 10�3 g/cm3 and a reference temperature of 288.2 K.These are default parameters in Autodyn [15].

3.3. Model for TNT explosive

The explosive used in the analyses reported here was TNT(trinitrotoluene). The mass density of this material is 1.658 g/cm3

(Luccioni et al. [11]). The explosive was modeled with an Eulersolver, using a material model for ideal gas. The equation of statefor this material was taken as JWL (Jones–Wilkins–Lee), which isthe most common method of modeling high explosives in dynamicanalyses (Choi et al. [11] and Luccioni et al. [12]). The internalenergy of the explosive was calibrated to the results of physicalmodel tests.

In order to model the explosion accurately, while increasingcomputational efficiency, the explosion was modeled in two-stages, as suggested by Choi et al. [16]. The explosion was modeledfirst in a 2-D analysis, using a wedge consisting of air and TNT.When this wedge was rotated in space, it created a sphere of airin three dimension, with TNT at its center. The diameter of theTNT was calculated based on the required mass and the referencedensity. A very fine mesh (typical element size 0.5 mm) was usedin this 2-D model. The analysis was continued long enough forthe shock waves formed from the explosion to reach the air thatsurrounded the TNT.

Fig. 1 shows the pressure contours on a model consisting of airand TNT, in the form of a conical wedge. The condition shown inthis figure represents a point in time when the shock wave dueto the explosion has traveled almost to the outer boundary of thewedge (note: pressure contour showing zero pressure only at theouter edge of the cone and higher pressures everywhere else). Atlarge expansions, detonation products behave like air and theJWL material model used in Autodyn to model explosives such asTNT allows transition of explosive to an ideal gas at large expan-sion (ANSYS [15]). A ‘‘Fill’’ file is created with this condition whichcontains information regarding the pressure created by the shockwave at a given instant of time. This file is used to remap the ef-fects of explosion as initial condition in a finite element file ofthe main stage of analysis in order to efficiently simulate the shockwaves created by the explosion.

The main stage of analysis consisted of models for air, soil, andother material (e.g., tunnel, etc.). The element size in this stage was

Fig. 1. Figure of air and TNT wedge used to create fill file, with 600 kg of TNT (figure from ANSYS Autodyn).

74 A. De / Computers and Geotechnics 43 (2012) 72–79

relatively large (typical element sizes were 40 mm and discussedlater). The fill file was imported into this analysis and remappedas a sphere (generated by rotating the wedge through 360�), withthe explosive located such that the ground surface was tangentto the bottom of the sphere. This two-stage analysis procedureallowed for developing the explosion in a refined model withincreased accuracy and then utilizing it in a larger model, thusmaintaining accuracy without significantly increasing the compu-tation time.

3.4. Soil material models for blast studies

The shock waves generated by a blast induce both hydrostaticstress and deviator stress as they pass through a solid medium.The total stress tensor can be expressed as the sum of the hydro-static stress tensor and the deviator stress tensor. The hydrostaticstress is responsible for influencing the density/volume of anymedium, whereas the deviator stress causes changes in the shapeof a solid medium.

The material model utilized to model the behavior of a contin-uum subjected to blast typically contains several components, suchas an equation of state (EOS), a strength model, a failure model, anderosion model. The equation of state relates the pressure, density,and energy. The strength model defines the relationship betweenthe deviator stress and consequent material deformation. This istypically done through relations established between pressure,density, yield stress, and shear modulus.

3.5. Sand models for dynamic loading due to blasts

The default sand model utilized for dynamic loading studiesusing ANSYS Autodyn was originally derived from results pub-lished by Laine and Sandvik [17]. Relevant components of materialmodel for this sand are described in the following section. Follow-ing this a calibrated model based on [17], developed for Nevada#120 sand at a relative density (Dr) of 60%, is presented.

3.5.1. Sand model by Laine and Sandvik [17]Laine and Sandvik [17] derived material properties for sand,

suitable for ground shock analyses. The material properties werebased on results of laboratory triaxial compression tests. In this

regard, two types of triaxial devices were employed. Standard soiltriaxial device was utilized up to a confining pressure of 2 MPa anda rock triaxial device was used for higher confining pressuresabove 2 MPa.

Based on the results from the consolidation stage [17], devel-oped a piecewise linear ‘‘plastic compaction curve’’, comprising10 points that define the relation between pressure and density.

In the shear stage of the triaxial tests, the axial stress wasincreased, keeping the lateral stresses on the specimen constant.In the tests on rock triaxial tests, compression wave (P-wave)velocity VP and shear wave (S-wave) velocity VS were measuredusing wave transducers. These were used in calculating the soundspeed (c).

The yield stress versus confining pressure relation was devel-oped by [17] based on results of triaxial compression tests up toa maximum confining pressure, beyond which the results wereextrapolated linearly to a maximum value corresponding to theunconfined strength of granite. The relationship between densityand shear modulus was developed through measured values ofshear wave velocities at different densities.

3.5.2. Sand model for Nevada #120 sand at Dr = 60%Nevada #120 sand at Dr = 60% was used in the VELACS (VErifi-

cation of Liquefaction Analysis by Centrifuge Studies) program,which was a multi-institutional research effort funded by theNational Science Foundation (NSF) during the early 1990s.Engineering properties of the sand used in the VELACS project atdifferent relative densities were established through an extensivelaboratory testing program, including both static and dynamictests. The results of these were presented by Arulmoli et al. [18].Subsequently, Elgamal et al. [19] developed a soil plasticity modelfor the same soil, based on the results of the VELACS tests.

The model for Nevada #120 sand at Dr = 60% (hereafter referredto as VELACS sand), was developed with a Mie–Gruneisen form ofshock equation of state, based on Rankine–Hugoniot condition, forthe hydrostatic portion and a Drucker–Prager strength model forthe shear portion.

The shock equation of state model, controlling the hydrostaticportion of loading, is based on the Rankine–Hugoniot jump condi-tion, defining a relation between material velocity behind theshock (up) and shock velocity (U) in terms of an initial velocity of

Axial strain vs. Poissons Ratio

0

0.2

0.4

0.6

0 5 10 15 20Axial Strain (%)

Pois

son'

s ra

tio

0.33

Fig. 3. Variation of Poisson’s ratio with axial strain.

Fig. 4. Density (q) versus speed of sound (c) for Nevada #120 sand at Dr = 60%.

Table 1Density (q) versus speed of sound (c) for Nevada #120sand at Dr = 60%, for use in Autodyn (data for plot inFig. 3).

Density (g/cc) Sound speed (m/s)

1.600 3011.625 5421.650 9811.750 32801.800 41601.850 50201.900 55701.950 55702.000 55702.670 5570

A. De / Computers and Geotechnics 43 (2012) 72–79 75

sound (c) and a dimensionless parameter, s through Eq. (1).Luccioni et al. [12] have previously used this model to comparecrater sizes between numerical simulation and field tests. In thepresent model, the values of the Gruneisen coefficient and dimen-sionless parameter, s were used as 0.11 and 1.5, respectively. Thevalue of initial velocity of sound was taken as 1600 m/s, which isconsistent with results from VELACS project (Arulmoli et al. [18])and similar to those used by Luccioni et al. [12].

The speed of sound was calculated from velocities of shear waveand longitudinal wave, using Eq. (2). Values of shear modulus forVELACS sand are presented by [18] from resonant column testsperformed on specimens at various confining pressures. Shearwave velocity (VS) was obtained from shear modulus (G) anddensity (q).

U ¼ c þ sup ð1Þ

c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

L �43m2

s

rð2Þ

Bulk modulus (K) can be obtained from shear modulus (G) andPoisson’s ratio (m) using Eq. (3).

K ¼ 2Gð1þ mÞ3ð1� 2mÞ ð3Þ

The possible variation of Poisson’s ratio as a function of axialstrain was investigated. Fig. 2 shows the variation of q (principalstress difference divided by two) with axial strain (DL/L) from con-solidated-drained triaxial tests conducted at three different confin-ing stresses as part of the VELACS project and reported by Arulmoliet al. [18]. The corresponding volumetric strains (DV/V) were usedto compute the Poisson’s ratio (m) using Eq. (4).

DVV¼ ð1� 2mÞDL

Lð4Þ

The variation of Poisson’s ratio with axial strain is plotted inFig. 3. At low strains, Poisson’s ratio appears to vary between alow value of approximately 0.25 and a high value just less than0.5, and tends to 0.33 beyond a strain of approximately 15%. Basedon this, a constant value of 0.33 was used in computation of bulkmodulus, K in Eq. (3), relative to the strain range of relevance toblast loading problems. This is consistent with Yang and Elgamal[20] who have reported Poisson’s ratio of VELACS sand as 0.33.

Using this value of Poisson’s ratio in Eq. (3), values of bulk mod-ulus (K) can be obtained at different densities. Longitudinal wavevelocity (VL) is calculated from bulk modulus (K) and density (q).Finally, the speed of sound (c) is calculated based on VS and VL,using Eq. (2), as a function of density. Thus, the relationship be-tween density and speed of sound, shown in Fig. 4, was developed.

0

100

200

300

420-2Strain

q (k

Pa)

Axial Strain%Volumetric Strain %

σ3 = 40 kPa (CIDC 6082)

σ3 = 80 kPa (CIDC 6075)

σ3 = 160 kPa (CIDC 6081)

Fig. 2. Variation of q with axial strain at different confining pressures.

The plot of density (q) versus speed of sound (c) is presented inFig. 4. The data are presented in Table 1.

A Drucker–Prager strength model was used to model the shearstage of loading. This soil model requires a relation between pres-sure (p) and yield stress (qmax) to be specified. The p versus qmax

data were obtained from results of triaxial compression tests underdifferent confining stresses, reported by [18]. This plot is presentedin Fig. 5. The relation between density (q) and shear modulus (G) ispresented in Fig. 6. Values of shear modulus (G) were obtainedfrom resonant column tests reported by [18]. The plot of G versusdensity was extrapolated based on the same relation as used by[17] to a maximum value equal 31.1 GPa, which is the shear mod-ulus of quartz. The maximum value of yield stress was based ontypical value for sandstone, as reported by Jizba [21] and consistentwith those used by Laine and Sandvik [17].

The soil model included a failure specified at plastic strain of15%. An erosion model is also introduced while modeling explo-sions. A maximum plastic strain of 15% was specified, beyondwhich the particular finite element cell is eroded, i.e., it is elimi-nated from further computation. The value of maximum plastic

Fig. 6. Relation between density (q) and shear modulus (G) for Nevada #120 sandat Dr = 60%.

Fig. 5. Relation between pressure (p) and yield stress (qmax) for Nevada #120 sandat Dr = 60%.

76 A. De / Computers and Geotechnics 43 (2012) 72–79

strain set for erosion was calibrated based on observation of cratersin physical model tests.

The material model for VELACS Sand developed above wasutilized in modeling the effects of surface explosions under twodifferent conditions. The first was the formation of crater on the

Fig. 7. Finite element model for 2-D axisymmetric anal

ground surface, using a 2-D, axi-symmetric analysis. The secondwas a 3-D analysis, where the effects of a surface explosion onan underground structure were analyzed.

4. 2-D axi-symmetric model for crater formation

4.1. Two dimensional model description

The formation of crater due to explosion on the ground surfacewas studied using a 2-D axi-symmetric model, consisting of soiland air, with a TNT explosive fill located within the air, directlyabove the soil surface. Fig. 7 shows a typical finite element modelused in this analysis. In this particular case, the model consists of30,000 elements, with 20,000 elements to model the air and10,000 elements to model the soil. The size of each element usedin different analyses varied from 1 mm to 40 mm.

The soil was restricted against movement along its bottom sur-face and the outer side. All external surfaces of the air were givenflow out boundary condition, which allowed transmission/flow inand out of the computational domain.

Euler solver was used to model the air and TNT explosive, whileLagrange solver was used to model the soil.

4.2. Results: Crater formation through 2-D, axi-symmetric analyses

4.2.1. Comparison with physical and numerical modelsFig. 8 shows a cross section through the crater formed in 2-D

axisymmetric numerical analyses, due to surface explosion of a400 kg TNT charge. According to Hopkinson (or cube-root) scalinglaw (Baker et al. [22]), the shock waves created at two differentscaled distances by two explosive charges, having the same geom-etry and type of explosive but different quantities of charge, wouldscale as the cube-root of the weight of explosives. The weight ofexplosives that produce similar shock waves will therefore, varyby the cube of distance.

Plots of apparent crater diameter (i.e., diameter observed afterthe explosion) versus mass of explosives are presented in Fig. 9.Also presented in the figure are results of physical explosion testsreported by other researchers. The plots include test resultsreported by, Ambrossini et al. [23]; De and Zimmie [24] Goodings

ysis to study craters (figure from ANSYS Autodyn).

Fig. 8. Cross section through the crater formed in 2-D axisymmetric analyses due to 400 kg TNT charge (figure from ANSYS Autodyn).

0

2,000

4,000

6,000

8,000

10,000

12,000

14,000

16,000

0 200 400 600 800 1000 1200Explosive mass (kg)

Cra

ter d

iam

eter

(mm

)

Numerical: De [Present work] Centrifuge 70g: De & Zimmie [24]

Centrifuge 50g: Sausville [7] Centrifuge 50g: Pena [25]

Centrifuge 1g to 100g: Goodings et al. [5] 1g: De & Zimmie [24]

1g: Pena [25] 1g: Goodings et al. [5]

1g: Ambrosini & Luccioni [23]

Fig. 9. Crater diameter, as a function of mass of explosives, comparing results ofnumerical modeling with physical model tests conducted between 1 g and 100 g.

A. De / Computers and Geotechnics 43 (2012) 72–79 77

et al. [5], Pena [25], and Sausville [7]. Some of the tests were con-ducted at 1 g, while the rest were at various g-levels, up to 100 g.

As can be seen from Fig. 9, the mean curve developed from thepresent study agrees well with the results of physical model tests,both at 1 g and at g-levels up to 100 g.

5. Three-dimensional model for tunnel

5.1. Model description

The effects of a surface explosion on an underground structurewere studied using a 3-D model, consisting of soil, air, TNT explo-sive fill and underground structure. The underground structurewas a tunnel, surrounded by a protective barrier material. Theexplosive was symmetrically located above the midspan, directlyover the centerline of the tunnel. Fig. 10 shows a typical finite ele-ment model used in this analysis. The model shown in the figurecontains 1475,898 soil elements, 315,000 air elements, 9685 ele-ments in the tunnel, and 13,824 elements in the protective barrier.

A copper tunnel was modeled in both the numerical simulationas well as the centrifuge model tests. Autodyn default properties ofcopper, using a shock equation of state and a Johnson–Cookstrength model based on Matuska [26] was used.

The model presented in Fig. 10 shows a polyurethane geofoamprotective barrier around the tunnel. In the 3-D model, the soilblock was restricted against movement along five out of its sixexternal surfaces. No restriction was specified on the top surface,which was exposed to the atmosphere. All external surfaces ofthe air were given flow out boundary condition, which allowedtransmission/flow in and out of the computational domain.

The interaction between the tunnel and surrounding soil wasmodeled by specifying a Lagrange–Lagrange interaction betweentwo solid materials. This assumes contact at the surface andenables transfer of stresses between the solid materials. Based ongeometry, the mesh was adjusted to achieve compatible elementsize in materials which were in physical contact.

The air and TNT explosive were modeled using Euler solvers.Lagrange solver was used to model the soil, as well as the tunneland barrier material.

5.2. Results: Strains on underground structures through 3-D analyses

5.2.1. Comparison with physical model tests using a geotechnicalcentrifuge

An explosion on the ground surface creates stresses and strainson an underground structure. The effects of a given explosion canbe quantitatively established using relationship between theinduced strains on the structure and the scaled distance. FollowingHopkinson’s cube-root scaling law [22], scaled distance is the dis-tance, R, from the center of mass of the explosive to the surface ofinterest, divided by the cube root of the explosive mass, W, i.e.,scaled distance equals R/W1/3.

Results of the 3-D numerical analyses were compared withthose from physical model tests. Strain measurements taken fromgeotechnical centrifuge tests conducted on a 1:70 scale model ofunderground tunnel, tested at a centrifugal acceleration of 70 gwere used for this purpose. The model tunnels were located indry sand (Nevada #120 sand at a relative density, Dr of 60%).Detailed discussions on these tests were previously presented byDe and Zimmie [24].

Fig. 10. Finite element model for 3-D analysis to study behavior of underground tunnels (figure from ANSYS Autodyn).

0

500

1000

1500

2000

2500

200 250 300 350 400 450 500 550 600Scaled Distance [R/W^(1/3)]

Mag

nitu

de o

f Pe

ak M

icro

stra

in Axial Strain at Top of Midspan (Numerical)

Axial Strain at Top of Midspan (Centrifuge)

Circumferential Strain at Midspan (Numerical)

Circumferential Strain at Midspan (Centrifuge)

Fig. 11. Plots of peak axial and circumferential strains at midspan – comparison ofnumerical (finite element) and physical (centrifuge) models.

-3000

-2000

-1000

0

1000

2000

3000

111.5 111.7 111.9 112.1 112.3 112.5 112.7 112.9Time elapsed (milliseconds)

Mic

rost

rain

Centrifuge: Test9: CS1H

Centrifuge: Test9: CS2H

Autodyn 3D

Fig. 12. Comparison of circumferential strain measurements between numericaland (finite element) and physical (centrifuge) models with 900 kg TNT explosive.

78 A. De / Computers and Geotechnics 43 (2012) 72–79

Strain measurements were obtained at various locations of theunderground structure in the centrifuge model tests during andimmediately following the explosion. The measurements were ac-quired and saved over a 15 s period at a rate of 15,000 points persecond (i.e., at 15 kHz) for each strain gage to ensure that the peakstrains during the explosion were adequately captured.

Comparison of the peak axial and circumferential strains atmidspan between the numerical model (using ANSYS Autodyn)and physical centrifuge model are shown in Fig. 11. As shown,there appears to be excellent comparison between the numericaland physical model results. The measured peak circumferentialstrain from the centrifuge test coincided with the best-fit curveof numerical results, while the peak axial strain was close to thenumerical best-fit line.

For closer comparison, the initial portion of plot, showing vari-ation of circumferential strain with time from numerical analysis is

compared with a plot from centrifuge test at 70 g (Fig. 12). In eachcase the mass of explosives was 900 kg equivalent of TNT and thetop of the tunnel was located 3.6 m below the ground surface inprototype units.

6. Conclusions

In this paper a soil model for dry sand is calibrated for use incomputational analyses of ground shocks with large strains, suchas to study the effects of explosion. The components of the sandmodel are originally derived from an existing sand model (Laineand Sandvik [17]) and modified to incorporate results from labora-tory tests conducted under the VELACS project (Arulmoli et al.[18]). The proposed constitutive model for dry Nevada #120 sand,at 60% relative density was used in numerical analyses using a

A. De / Computers and Geotechnics 43 (2012) 72–79 79

hydrocode (ANSYS Autodyn) to compare results under two con-trolled test cases.

Numerical analyses utilizing a 2-D model were conducted tostudy crater formation in dry soil. Results of these tests were com-pared with numerical and physical model tests conducted by otherresearchers and found to have good comparison. Numerical model-ing in 3-D was conducted to study the effects of surface explosionon an underground structure. Results of the numerical analyseswere compared with physical model tests using a geotechnicalcentrifuge. Strains calculated at different locations of the under-ground structure in the numerical analyses were found to be closeto those measured in the centrifuge tests.

The proposed model for dry sand appears to be capable of mod-eling effects of large strain phenomena (such as explosions) in soil.The model may be extended in the future to incorporate effects ofpore water pressure, to simulate liquefaction induced by explosion.

Acknowledgments

The work reported in this paper was funded by the Civil andMechanical Systems Program of the National Science Foundation(NSF). Major funding is provided through CMMI-0928537; fundingfor centrifuge tests reported here was provided through a SmallGrants Exploratory Research (SGER) grant and supplementaryResearch Opportunity Award (ROA) grant on CMMI-0226864,through Professor Thomas F. Zimmie of Rensselaer PolytechnicInstitute (RPI), Troy, New York. The centrifuge experiments wereconducted with the assistance of technical personnel at theGeotechnical Centrifuge Research Center at RPI.

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