A high strain-rate constitutive model for sand and its application in finite-element analysis of...

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. (2012)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.2153

A high strain-rate constitutive model for sand and its application infinite-element analysis of tunnels subjected to blast

William Higgins1, Tanusree Chakraborty2 and Dipanjan Basu3

1Department of Civil and Environmental Engineering, University of Connecticut, Storrs, CT, U.S.A.2Department of Civil Engineering, Indian Institute of Technology Delhi, India

3Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON, Canada

SUMMARY

The paper presents a constitutive model for simulating the high strain-rate behavior of sands. Based on theconcepts of critical-state soil mechanics, the bounding surface plasticity theory and the overstress theory ofviscoplasticity, the constitutive model simulates the high strain-rate behavior of sands under uniaxial, triaxialand multi-axial loading conditions. The model parameters are determined for Ottawa and Fontainebleau sands,and the performance of the model under extreme transient loading conditions is demonstrated throughsimulations of split Hopkinson pressure bar tests up to a strain rate of 2000/s. The constitutive model isimplemented in a finite-element analysis software Abaqus to analyze underground tunnels in sandy soilsubjected to internal blast loads. Parametric studies are conducted to examine the effect of relative densityand type of sand and of the depth of tunnel on the variation of stresses and deformations in the soil adjacentto the tunnels. Copyright © 2012 John Wiley & Sons, Ltd.

Received 28 August 2011; Revised 31 May 2012; Accepted 26 August 2012

KEY WORDS: constitutive model; sand; high strain rate; tunnel; finite-element analysis; blast

1. INTRODUCTION

The development of sustainable and resilient civil infrastructure requires that structures can not onlywithstand anticipated design loads but also encounter extreme and unanticipated loads with minimalendangerment of individuals and properties. Extreme loading can be caused by nature in the form oftornados, tsunamis or earthquakes or by human activities such as bomb blasts, collisions or industrialaccidents. A common feature of these extreme loading scenarios is that they can create very largestrains in the surrounding material in a very short period of time. Because so many structures interactwith soil, it is necessary to be able to model the effect of these extreme, high-rate loads on soil.

High strain-rate behavior of soil has been studied in the laboratory under triaxial and uniaxialconditions using various testing apparatus [1, 2]. The principal observation of the effect of strain rateon sand is that the faster the strain rate is the greater the stiffness and strength are. The increase instrength is manifested through an increase in the peak stress and initial stiffness [3]. The peak stressalso occurs at lesser values of strain as the applied strain rate increases. The effect of increased strengthand stiffness is more pronounced in samples with greater relative density and confining stress [3–6]. Inaddition to triaxial compression and uniaxial strain tests, projectile methods such as the split Hopkinsonpressure bar (SHPB) test [7–10] have also been used to understand sand behavior under very highstrain rate of the order 1000 per second.

*Correspondence to: Dipanjan Basu, Department of Civil and Environmental Engineering, University of Waterloo, 200University Avenue W, Waterloo, ON, Canada.†E-mail: [email protected], [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

W. HIGGINS, T. CHAKRABORTY AND D. BASU

A limited number of soil constitutive models have been developed to numerically simulate the highstrain-rate behavior of soil. These include the three-phase equation-of-state (EOS) models of Wanget al. [11], Laine and Sandvik [12] and Tong and Tuan [13]. The EOS soil models take into accountthe different speeds of shock wave in the solid, water and air phases of soil. In order to model thesolid phase, Tong and Tuan [13] incorporated Perzyna’s viscoplastic flow rule in the Drucker–Prager failure criterion. The model by Wang et al. [11] also features the Drucker–Prager yieldcriterion for the solid phase along with the capability of incorporating damage.

Studies on the analysis of boundary value problems related to the high strain-rate behavior of soil arerather limited in number. An et al. [14] used the constitutive of Tong and Tuan [13] for finite-element (FE)analysis of blast due to explosives embedded in soil. Nagy et al. [15] incorporated the Drucker–Pragermodel in a FE framework and simulated wave propagation through soil due to explosions on theground surface. Yang et al. [16] incorporated the soil plasticity model of Krieg [17] in a FE frameworkand simulated the propagation of blast wave in soil. Lu et al. [18] performed a coupled three-phaseanalysis using the FE method to simulate blasts propagating through soil — they used a modifiedDrucker–Prager model with a yield surface that expands with strain rate and coupled it with arheological damage model. Bessette [19] used a three-phase soil constitutive model with the FE methodto simulate the propagation of blast waves due to the explosion of buried C4. Feldgun et al. [20, 21]and Karinski et al. [22] used the variational difference method to analyze underground tunnels andcavities subjected to blast loads.

In this paper, a two-surface plasticity constitutive model is presented that simulates the mechanicalbehavior of sand subjected to strains applied with a rate of up to 2000/s. Based on the concepts ofcritical-state (CS) soil mechanics, the bounding surface plasticity theory and the overstress theory ofviscoplasticity, the model simulates the high strain-rate behavior of sand under multi-axial loadingconditions. The model is based on the rate-independent plasticity model developed by Manzari andDafalias [23] and modified by Li and Dafalias [24], Dafalias et al. [25] and Loukidis and Salgado [26].The model parameters are determined for Ottawa and Fontainebleau sands by comparing the simulationresults with the experimental data available in the literature. The constitutive model is subsequentlyused to study the response of tunnels embedded in sandy soil and subjected to internal blast loads.The FE software Abaqus (version 6.9) is used for the analyses. The explosive C4 is simulated with theJones-Wilkens-Lee (JWL) equation-of-state model. Parametric studies are performed to examine theeffect of relative density and type of sand and of the depth of tunnel on the variation of stresses anddeformations in the soil adjacent to the tunnels.

2. DEVELOPMENT OF THE CONSTITUTIVE MODEL

2.1. Critical state line

In this constitutive model, the CS line in the e–p’ space (e is the void ratio and p’ is the effective meanstress given by p’= (s’11 + s’22 + s’33)/3 where s’ij is the effective Cauchy stress tensor) is given byLi and Wang [27]

ec ¼ Γ� lp0

pa

� �z

(1)

where ec is the void ratio at the CS (Figure 1) and pa is the atmospheric pressure. The parameter Γ is theintercept of the CS line on the e axis at zero pressure, and l and zare fitting parameters. When a sandsample with a void ratio e less than its value ec at the CS (for the same mean stress p’) is sheared, thesample dilates causing an increase in e or p’ until the CS line is reached. Conversely, samples withe> ec contract with decreasing values of e or p’ until the CS line is reached. This behavior iscaptured through the use of a state parameter c(=e� ec) — the (positive or negative) sign associatedwith c governs whether the shear-induced volumetric strain is contractive or dilative [28].

Copyright © 2012 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2012)DOI: 10.1002/nag

ec =Γ− (p′ / pa)

Void Ratio, e

Critical State LineΓ

Current Stress State

Mean Stress, p‘(log scale)

01

Figure 1. Critical-state line and state parameter.

HIGH STRAIN-RATE SAND CONSTITUTIVE MODEL AND FINITE-ELEMENT ANALYSIS

2.2. Model surfaces in stress space

Figure 2 shows the constitutive model surfaces in the principal stress space s’1-s’2-s’3. The modelcontains four conical shear surfaces — the yield, bounding, dilatancy and CS surfaces — with straightedges in the meridional plane and apex at the origin. The model formulation is done in terms of stressratios, i.e. stresses normalized with respect to p’. The distance of the stress state from the yield surfaceis described by the yield function f, with the yield surface given by f = 0. The yield function in thismodel is expressed in terms of the deviatoric stress ratio tensor rij as [23]

f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirij � aij� �

rij � aij� �q

p’�ffiffiffiffiffiffiffiffi2=3

pmp’ (2)

where m is a model parameter, aij is the kinematic hardening tensor and rij = sij/p’ in whichsij (= s’ij � dijs’kk/3) is the deviatoric stress tensor (dij denotes the Kronecker’s delta). The parametersm and aij have physical meaning in the principal deviatoric stress space s1-s2-s3. The yield surface is acircle in the octahedral plane (p-plane) of the s1-s2-s3 space with the radius equal to 2m/3 and the centerlocated at the apex of the ‘vector’ aij [29]. The yield surface cannot harden isotropically (i.e. m is aconstant) but can harden kinematically through the evolution of aij given by [23]

_aij ¼ _lKP

p0

ffiffiffi23

rMb � mð Þnij � aij

!=

ffiffiffi23

rMb � mð Þ � aijnij

!(3)

Yield Surface

Hydrostatic Stress Axis

Bounding Surface

Dilatancy Surface

Critical State Surface

1′

2′

3′

Figure 2. Model surfaces in three-dimensional stress space.



where _l is the viscoplastic multiplier, KP is the viscoplastic modulus, nij ¼ sij � p0aij

� �=

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiskl � p0aklð Þ skl � p0aklð Þp � determines the direction of the projection of the current stress state on the

CS, dilatancy and bounding surfaces (i.e. nij gives the mapping rule) and Mb is the bounding surfacestress ratio in the principal deviatoric stress space given by [30]

Mb ¼ g θð ÞMcce�ckbð Þ ¼

1� 1�c1=ns1

1þc1=ns1

� �ns

1� 1�c1=ns1

1þc1=ns1

cos3θ� �ns

2664

3775Mcce

�ckbð Þ (4)

where kb is a fitting parameter,Mcc is the deviatoric stress ratio q/p’ at the CS under triaxial compression

(the deviatoric stress q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis1 � s2ð Þ2 þ s1 � s3ð Þ2 þ s2 � s3ð Þ2

q=ffiffiffi2

pwhich simplifies to q=s1� s3

for triaxial compression test), g(θ) is a function of the Lode’s angle θ and determines the shape of theCS surface in the octahedral plane of the deviatoric stress space, ns is an input parameter and controlsthe convex shape of the CS surface [30] and c1 is the ratio of the CS stress ratios in triaxial extensionand triaxial compression, given by

c1 ¼ Mce

Mcc(5)

where Mce is the deviatoric stress ratio at the CS under triaxial extension.Similar to the bounding surface, the dilatancy surface is also a function ofMcc andc, and is described by

Md ¼ g θð ÞMccekdc (6)

where kd is a fitting parameter. The CS surface is described in terms of the generic CS ratio Mc given by

Mc ¼ Mccg θð Þ (7)

2.3. Elastic moduli

The total strain tensor is divided into elastic and viscoplastic components. The incremental strains areexpressed as

_eij ¼ _eeij þ _evpij (8)

where _eij is the total strain increment, _eeij is the elastic strain increment and _evpij is the viscoplastic strainincrement. The incremental elastic stress–strain relationship is given by

_s0ij ¼ 2G _eij � _evpij

� þ K � 2

3G

� �_ekk � _evpkk� �

dij (9)

where _s0ij is the stress increment, ėij and _evpij are the total and viscoplastic deviatoric strain increments,

respectively, _ekk and _evpkk are the total and viscoplastic volumetric strain increments, respectively, and Kand G are the bulk and shear moduli, respectively. The shear modulus is given by [31]

G ¼ Cgeg � e� �21þ e

p0

� ngp1�nga (10)

where Cg, ng and eg are input parameters. The bulk modulus is related to the shear modulus through aconstant Poisson’s ratio n as



K ¼ G2þ 2n3� 6n

(11)

When the stress state is entirely within the yield surface, there is no viscoplastic strain in the soil.However, because the yield surface is very small in this model, the viscoplastic process is prevalentfor almost the entire loading duration.

2.4. Viscoplastic strain

When the stress state reaches or crosses the yield surface, the material undergoes viscoplastic strain. Inthis model, the overstress theory of Perzyna [32, 33] is used to model the viscoplastic behavior of sand.The overstress theory is based on the viscoplastic overstress function Φ defined as

Φ Fð Þh i ¼ F if F > 00 if F ≤ 0

(12)

where the parameter F quantifies the overstress, i.e. the ‘distance’ between the viscoplastic stress stateand the yield surface. In this constitutive model, F= f is assumed because, in the cutting planealgorithm used in the implementation of the model, f gives a measure of the distance of the currentstress state from the yield surface [29]. The magnitude and direction of the viscoplastic strain aredetermined by the flow rule

_evpij ¼ _lRij (13)

where the viscoplastic multiplier _l is defined as

_l ¼ Φ Fð Þh i�v

¼ f

�v(14)

with �v being the viscoplastic coefficient, and Rij is the gradient of the viscoplastic potential surface[26] given by

Rij ¼ R0ij þ 1

3Ddij (15)

where R0ij is the deviatoric component of the gradient [34] that gives the direction of the deviatoric

viscoplastic strain rate, and D is the dilatancy that controls the shear-induced volumetric viscoplasticstrain rate. In this model, the viscoplastic potential is assumed to be identical to the plastic potentialused by Dafalias and Manzari [34] and Loukidis and Salgado [26].

R0ij in equation (15) is given by

R0ij ¼ R

0 �ij=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiR0 �

klR0 �kl

q(16)

where

R0 �ij ¼ 1þ 3

21� c2c2

� �g2 θð Þ cos 3θð Þ

� �nij � 3

ffiffiffi32

r1� c2c2

� �g2 θð Þ

" #niknkj � 1

3dij

� �(17)

in which

g2 θð Þ ¼ 2c21þ c2ð Þ � 1� c2ð Þ cos 3θð Þ½ � (18)

and c2 is a user-defined parameter the value of which is set at 0.78 for all sands following Loukidis [30].The function g2(θ) determines the shape of the plastic potential surface in the deviatoric plane underdifferent loading directions.



The dilatancy D in equation (15) is given by [24]

D ¼ D0

Mcc

ffiffiffi23

rMd � mð Þ � aijnij

!(19)

where D0 is an input parameter.

2.5. Viscoplastic modulus

The viscoplastic modulusKP used in equation (3) controls the development of the viscoplastic strain and isexpressed as a function of the distance between the current stress state and the bounding surface [24, 30]:

KP ¼ h0G exp kbcð Þ

32

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirij � aij;ini� �

rij � aij;ini� �qh im

ffiffiffi23

r ffiffiffi23

rMb � mð Þ � aijnij

!(20)

where m is an input parameter and aij,ini is the initial value of the kinematic hardening tensor. The termffiffiffiffiffiffiffiffi2=3

pMb � mð Þ � aijnij ) is the distance between the current stress state and the projected stress state

on the bounding surface. The parameter h0 takes into account the effect of void ratio on KP (as loosesand develops viscoplastic strains with more ease than dense sand) and is given by

h0 ¼ e lim � e

h2

� �h1

(21)

where h1, h2 and elim are input parameters [30].

3. MODEL IMPLEMENTATION USING CUTTING PLANE ALGORITHM

The constitutive model is integrated into the FE software Abaqus/Explicit version 6.9 [35] through auser-defined material subroutine (VUMAT) using an extension of the cutting plane algorithm forviscoplasticity proposed by Ortiz and Simo [36]. The cutting plane algorithm is a semi-implicitalgorithm that uses explicit elastic predictions with an iterative viscoplastic correction loop.

Figure 3 shows a flowchart of the viscoplastic cutting plane algorithm used in this study. The inputsto the algorithm at any time t are the current values of stress (sij), strain (eij) and hardening variables xi,all denoted with a superscript t, the applied strain increment _eij and the time increment dt (dt iscontrolled from outside of the algorithm either by the user or by the FE analysis). The prime (’)associated with the stress tensor is dropped with the understanding that all the stresses calculated areeffective stresses. The calculations begin with an elastic prediction step using the current values ofthe stress state sij and the kinematic hardening variable aij. Note that aij is generically denoted by xiin Figure 3, and its evolution (equation (3)) is expressed as a function of a generic tensor hi. Duringthe elastic prediction step, the stress sij is increased based on the assumption that the strainincrement _eij is completely elastic. The stiffness tensor Dijkl used in the elastic prediction is the shearmodulus when the deviatoric stress is calculated from the deviatoric strain and is the bulk moduluswhen the mean stress is calculated from the volumetric strain. Once the stress and strain incrementshave been calculated, the tensors are updated. In addition to updating the stresses and strains,parameters such as the void ratio, stress invariants and the state parameter c are also updated (notethat Dijkl is not updated and stays at the same value as used during the elastic prediction). Using thenew values of sij and xi, the position of the stress state relative to the yield surface is checked bycalculating the overstress f (= f (i) where the superscript i within parentheses counts the iterations ofthe viscoplastic correction loop) and comparing it against the yield surface error tolerance FTOL,which is a small positive number. If the stress state is within the yield surface or sufficiently close toit such that the yield function is less than or equal to FTOL (i.e. f≤FTOL), then the increment isaccepted, and the algorithm is complete. However, if during the elastic prediction step the stress


Figure 3. Cutting plane algorithm flow chart.


state exceeds the boundary of the yield surface (i.e. f>FTOL), then the algorithm enters into aniterative viscoplastic correction loop. The value of FTOL can be determined by the user and shouldbe calibrated based on the anticipated levels of stress, the required degree of accuracy and theavailable computational resources — in this study, a value of 0.1 Pa was used.

In the viscoplastic correction loop, the incremental viscoplastic multiplier is calculated byconsidering a Taylor series expansion of the yield function as

Δl ¼ _lΔt ¼ f

�v�t (22)



where Δt(=Δt(i+ 1) = t(i+ 1)� t(i)) is the time elapsed during an iteration of the correction loop. The term�trepresents instantaneous time [36] given by

�t ¼ �v@f@sij

DijklRkl � KP

(23)

The value of Δl from equation (22) is used to quantify the change in the variables (e.g. sij and xi)between the iterations of the correction loop, and the updated values of stresses and hardeningvariables are calculated to obtain the updated yield function value f (i+1) [29]. The iterations in theviscoplastic correction loop continue until the yield function value falls within the tolerance FTOL(i.e. f (i+ 1)≤FTOL) or until the time increment dt is exhausted.

It is possible in the course of the viscoplastic correction that the position of the final, relaxed stressstate is inside the yield surface. Theoretically, this condition (i.e. f (i+ 1)≤ 0) is not possible, and it alsogives rise to numerical problems. Hence, an additional check is done to make sure that the value off (i+ 1)> 0. Therefore, if the predicted value of Δl causes the overstress to move inside the yieldsurface resulting in f (i+ 1)≤ 0, then the iteration is rejected, sij and xi are returned to the values ofthe previous iteration, and a decreased value of Δl is used to proceed further (the decreased valueof Δl is assumed to be Δl/10 in this study). It should be noted that decreasing the value of Δl doesnot affect the solution of the final stress value that is converged upon, it only affects the number ofiterations required to reach the converged value.

The actual elapsed time Δt of an iteration of the correction loop is calculated from the previous andupdated values of the yield function, f (i) and f (i+1), as [36]

Δt ¼ �t lnf ið Þ

f iþ1ð Þ

� �(24)

When the summation of the elapsed time of the iterations in the correction loopXið ÞΔt

ið Þbecomes equal

to the time increment dt of the analysis, the relaxation time expires. Thus, whenXið ÞΔt

ið Þ ¼ dt , the

program exits the viscoplastic correction loop.

If after updating the stresses it is found thatXið ÞΔt

ið Þ> dt, then too much time has elapsed and the

current stress state is invalid. If that happens (i.e. ifXið ÞΔt

ið Þ> dt), then the set of iterations is rejected

— the algorithm returns to the previous values of stresses, hardening variables and state parameters,

and starts again with a decreased value of Δl. This process is continued untilXið ÞΔt

ið Þfalls within

some tolerance of dt. This tolerance was so set that, in order for the program to exit the correction loop,

the total elapsed time has to meet the condition 1� TTOLð Þdt <Xið ÞΔt

ið Þ≤dt where TTOL=0.0001.

It is clear from the above discussion that the algorithm exits the viscoplastic correction loop if the

time increment dt is exhausted (i.e. ifXið ÞΔt

ið Þ ¼ dt) or if the viscoplastic stress state is sufficiently

close to the yield surface (i.e. f (i+ 1)≤FTOL). If the time increment gets exhausted before thecondition f (i+ 1)≤FTOL is satisfied, then the stress state remains outside the yield surface as thealgorithm moves to the next time t + dt. If, on the other hand, the condition f (i+ 1)≤FTOL issatisfied, then the algorithm is moved to the next time t + dt even before the time increment dt is



exhausted because, in the remainder of the time available for viscoplastic corrections, the change in thevalues of the stresses and hardening variables is minimal.

The implementation of the cutting plane algorithm is done in conjunction with an error controlalgorithm [29]. The error control algorithm limits the magnitude of the time increment dt by comparingthe stresses obtained by executing the cutting plane algorithm with dt as the time increment with thestresses obtained after two successive executions of the cutting plane algorithm each with a timeincrement of dt/2. If the difference between the stresses obtained from these two sets of solution islarge, then the time step dt is decreased until the difference falls within a tolerable limit.

4. MODEL CALIBRATION AND VALIDATION

The developed constitutive model was used to simulate the drained triaxial compression tests andSHPB tests performed on Ottawa and Fontainebleau sands. The parameters used in the simulationsare given in Table I. The parameters used for the Ottawa sand were mostly obtained from Loukidis[30] in which the calibrations were done based on triaxial compression tests [37, 38], triaxialextension tests [38] and bender element tests [37]. Modifications were made to the values of the CSparameters Γ, l and z so as to better capture the sand behavior at high strain rates and at highpressures (> 100 MPa) experienced in the SHPB tests. The new values of these parameters wereobtained by optimizing the CS line to capture the behavior of the SHPB tests [8] while maintaininga good agreement with the triaxial tests [29]. The model performance was checked for Ottawa sandwith the new set of parameters under both rate-independent and rate-dependent loading for triaxialand multi-axial loading conditions. The model calibration for Fontainebleau sand was done in thepresent study using the data from triaxial compression tests [39–42], triaxial extension tests [39],torsional hollow cylinder tests [43] and SHPB tests [44].

4.1. Simulation of rate-independent triaxial tests

The triaxial tests were simulated (in Abaqus/Explicit with VUMAT) using a single, axisymmetricelement. The element was fixed against vertical movement along its bottom edge. The element wasloaded with an initial hydrostatic pressure maintained as a constant load on the outer radial edge.

Table I. Parameters used in the simulations of Ottawa and Fontainebleau sand tests.

Parameters Ottawa Sand Fontainebleau Sand

n 0.15 0.3Cg 611 650eg 2.17 2.17ng 0.437 0.437Γ 0.85 2.0l 0.12 1.1z 0.275 0.1Mcc 1.31 1.157kb 1.9 3.0h1 2.2 1.2h2 0.24 0.2elim 0.81 1.0m 0.05 0.05c1 0.71 0.71ns 0.35 0.35c2 0.78 0.78D0 1.31 0.5kd 2.2 2.0m 1.2 1.2�v (kPa-s) 50 5Maximum Void Ratio emax 0.78 0.863Minimum Void Ratio emin 0.48 0.523



The analysis was driven by applying displacements at the top edge of the element. Before the finalresults were obtained, the appropriateness of using a single element to represent triaxial sampleswas checked by comparing the results of simulations obtained using a single element and usingmultiple elements. Figure 4(a) shows the simulation results of triaxial compression tests forOttawa sand samples at initial void ratio e0 = 0.7 and 0.55 subjected to a confining pressure of100 kPa. As evident, both the single- and multi-element simulations produced nearly identicalresults. Figure 4(b) shows the simulations of the Ottawa sand sample with e0 = 0.55 using a singleelement but with different time increments of 10�6, 10�5 and 10�4 s. It was found that a timeincrement of 10�5 second produced sufficiently accurate results — therefore, this time incrementwas used to produce the final results.

The triaxial compression simulations of the Ottawa sand samples were done following actuallaboratory tests performed by Carraro [37]. Figures 4(c) and (d) show the deviatoric stress versusaxial strain and the volumetric strain versus axial strain plots, respectively, for both theexperimental and simulation data. It is evident that the constitutive model differentiates betweendilative and contractive behavior of sand at different void ratios and provides a reasonably goodmatch with the experimental results.

(b)(a)

(d)(c)

Figure 4. Finite-element simulations of rate-independent drained triaxial tests on Ottawa sand with an initialconfining pressure of 100 kPa: (a) comparison between single-element and multiple-element simulations ofdeviatoric stress versus axial strain, (b) convergence study using different time increments, (c) comparisonof deviatoric stress versus axial strain results with the corresponding experimental data and (d) comparison

of volumetric strain versus axial strain results with the corresponding experimental data.


Figure 5. Finite-element simulations of rate-dependent drained triaxial tests on crushed coral sand with aninitial confining pressure of 98 kPa: comparison of deviatoric stress versus axial strain results with the

corresponding experimental data obtained from Yamamuro and Abrantes [6].

Striker Bar Incident Bar Transmitted Bar

SpecimenAxis of Symmetry

Velocity V0

Finite Element Mesh

Striker Bar Incident Bar Specimen Transmitted Bar

Enlarged Mesh

Figure 6. Schematic diagram of the SHPB test setup and the corresponding finite-element mesh.


4.2. Simulation of strain-rate-dependent triaxial tests

Strain-rate-dependent drained triaxial compression tests were simulated (in Abaqus/Explicit withVUMAT) for crushed coral sand samples following the experiments of Yamamuro and Abrantes [6].The samples were subjected to an initial confining pressure of 98 kPa, and the initial void ratio e0 ofthe samples varied over a small range of 0.92–0.94. In the simulations, the material parameters ofOttawa sand were used along with �v = 0.05 because enough experimental data on crushed coralsand are not available for model calibration. Figure 5 shows the deviatoric stress versus axial strainplots obtained from the simulations and the experiments. Although a direct comparison of the resultsof the simulations and experiments does not make much sense, the results show that the constitutivemodel is capable of simulating rate-dependent triaxial tests (i.e. tests with low to intermediate strainrates) and that the simulated results are not significantly different from the experimental results.

4.3. Simulation of SHPB tests

The SHPB tests were simulated for Ottawa and Fontainebleau sands using Abaqus/Explicit withVUMAT. Figure 6 shows a schematic diagram of the SHPB test setup and the corresponding FEmesh. Four separate axisymmetric parts were created to simulate the striker bar, incident bar, outputbar and the soil sample. First-order axisymmetric reduced integration elements in Abaqus (CAX4R)were used for meshing the geometry. The magnitude of the impulse wave was controlled byadjusting the initial velocity of the striker bar in Abaqus. In the actual experiments, the soil samplewas confined against transverse displacement with a rigid collar. In the simulations, this effect wasaccounted for by directly applying boundary conditions to the soil elements so that the transverse


Figure 8. Axial stress versus axial strain of Fontainebleau sand in split Hopkinson pressure bar tests.

20 m

2 m0.15 m

0.15 m

5.7 m

7 m

Figure 9. A typical finite-element mesh used in the analysis of tunnels (the tunnel center line is at a depth of5 m below the ground surface).

Figure 7. Axial stress versus axial strain of Ottawa sand in split Hopkinson pressure bar tests.




displacement was restrained. The contact planes between the bars and the specimen were modeledusing frictionless hard contact. A time increment of 10�9 second was used in the analysis — thevalue was selected after performing a convergence study.

The SHPB tests on dry Ottawa sand were conducted by Veyera and Ross [8]. The strain ratesachieved in these tests were between 1000/s and 2000/s. The Ottawa sand samples were compactedto a void ratio of 0.545. The samples had a diameter of 5.08 cm and lengths L0 = 1.27 cm and 0.635cm. The SHPB test set up had stainless steel bars with a diameter of 5.08 cm. The materialproperties used for simulating the bars are Young’s modulus = 207 GPa and density = 7850 kg/m3.The striker bar had a length of 0.635 m, the incident bar had a length of 3.66 m and the output barhad a length of 3.35 m. The incident and the output bars consisted of 1600 elements each while thestriker bar had 680 elements. The sample was meshed using 32 elements. By using an initial strikerbar velocity of 12 m/s in the simulations, an impulse wave comparable to the one reported byVeyera and Ross [8] was produced. Figure 7 shows the axial stress versus axial strain plots of theSHPB tests performed on Ottawa sand samples. The stress–strain plots show that a sample subjectedto a faster strain rate achieves greater stresses. There is a reasonably good match between theexperimental data and simulation results.

The simulations for Fontainebleau sand are based on the SHPB tests performed by Semblat et al.[44] on dry samples. Semblat et al. [44] ran tests with different lengths of the sand sample and withdifferent velocities of the striker bar to create different strain rates in the samples. The stress–strain

Soil stresses and strains read

along this pathd

Soil

Tunnel

Lining

BA

Figure 10. Path AB along which stresses and strains in soil are studied.

Figure 11. Pressure amplitude curve for the explosive C4 of radius 0.1 m in a tunnel with an internal radiusof 2.85 m.



plots are shown in Figure 8 for tests performed with samples of length 10 mm and diameter 40 mmwith the initial striker bar velocity V0 = 6.8 m/s, 11.6 m/s and 19.8 m/s. The samples had an initialvoid ratio of 0.667. The bars used in the SHPB test set up had a diameter of 40 mm, Young’smodulus of 70 GPa and density of 2820 Kg/m3. The striker bar had a length of 0.85 m, whilethe impulse and output bars each had a length of 2 m. The incident and the output bars consistedof 3660 elements each while the striker bar had 792 elements. The sample was meshed using78 elements. The simulated stress–strain plots match the experimental results reasonably well.

4.4. FE analysis of tunnels subjected to blast

The developed constitutive model was used to analyze underground tunnels subjected to internal blastloads. The purpose of these simulations is to demonstrate the ability of the constitutive model tosimulate real field problems and to gain insights into how soil adjacent to a tunnel behaves when ablast occurs inside the tunnel.

Two-dimensional plane strain FE analyses were performed using quadrilateral, plain strain, reducedintegration (CPE4R) elements in Abaqus, and the resulting stress waves propagating through thesurrounding soil were simulated. Two geometries were considered in this study. In one case, thecenter line of the tunnel was at 5 m below the ground surface and, in the other case, the tunnel

(a)

DR

DR

(b)

Figure 12. Temporal variation of (a) mean stress and (b) deviatoric stress at three points in soil adjacent to10 m deep tunnels in Ottawa sand subjected to explosions of C4.



center line was at a depth of 10 m below the ground surface. For both the cases, the tunnel had aninternal radius of 2.85 m with a 0.15 m thick concrete lining.

A typical FE mesh is shown in Figure 9. In order to save on the computation time, only one half ofthe actual domain was analyzed by imposing a symmetry boundary condition along the left verticalboundary of the mesh. The top horizontal boundary was free to displace, while the bottomhorizontal boundary was restrained against both vertical and horizontal displacements. Verticaldisplacements were allowed along the left and right vertical boundaries but not horizontaldisplacements. The bottom horizontal boundary and the right vertical boundary were located atsufficient distances so that they had no impact on the results of the analysis — the results wereobtained at a time when the stress wave from the blast was far from these boundaries. The mesh forthe 5 m deep tunnel consists of 1624 elements and 1718 nodes, and the mesh for the 10 m deeptunnel consists of 2306 elements and 2414 nodes.

It was assumed in the FE analyses that the soil surrounding the tunnels has properties similar toOttawa and Fontainebleau sands. Two different relative density (DR) values, 50% (whichcorresponds to an initial void ratio e0 = 0.63 for Ottawa sand and to e0 = 0.69 for Fontainebleausand) and 80% (which corresponds to e0 = 0.54 for Ottawa sand and to e0 = 0.59 for Fontainebleausand), were considered. The concrete lining of the tunnels was simulated using the concretedamaged plasticity model built into Abaqus. The material properties used for concrete are Young’s

(a)

(b)

q

Figure 13. Temporal variation of (a) mean stress and (b) deviatoric stress at three points in soil adjacent to10 m deep tunnels in Ottawa and Fontainebleau sands subjected to explosions of C4.



modulus = 31 GPa, Poisson’s ratio = 0.15, compressive yield strength = 13 MPa and tensile yieldstrength = 2.9 MPa. The stresses generated in the soil due to the explosions inside the tunnels wereinvestigated along a horizontal path AB shown in Figure 10. Geostatic (gravity) stresses wereapplied to the soil before the blast loads were applied.

Blasts due to the explosive C4 were simulated using the Jones–Wilkens–Lee (JWL) equation-of-statemodel [45] with the assumption that the explosive material is located at the center line of the tunnel.The radius of the explosive material assumed before detonation is 0.1 m — this corresponds to a massof 50.3 kg/m. Air elements were used to mesh the interior of the tunnel and the dynamic pressureacting on the inner tunnel wall due to the explosions were generated as a function of time as shown inFigure 11 [29]. The blast pressure reached the peak at 0.0014 s and then reduced close to theatmospheric pressure at 0.005 s. The analyses were run for 0.01 s.

Figures 12 (a) and (b) show the variations of the mean and deviatoric stresses with time at threedifferent points in the ground at a distance d = 0.5 m, 1.5 m and 2.5 m from the interface of thetunnel and ground along the horizontal path AB (Figure 10). For these simulations, the center lineof the tunnel is located at a depth of 10 m below the ground surface. The soil is assumed to havethe same properties as that of Ottawa sand with DR = 50% and 80%. As the stress wave propagates,the stresses at different horizontal distances increase, reach a maximum and then decrease. The

p'

(b)

(a)

q

DR

DR

Figure 14. Spatial variation of (a) maximum mean stress and (b) maximum deviatoric stress in soil adjacentto 10 m deep tunnels subjected to C4 explosions.



maximum values of the stresses experienced by a point in soil decreases as the distance of the pointfrom the tunnel increases. In the denser sand (i.e. for DR = 80%), the wave propagates faster and thestresses reach higher peaks.

Figures 13 (a) and (b) compare the temporal variations of the mean and deviatoric stresses at threedifferent points along the path AB for Ottawa and Fontainebleau sands with DR= 80%. It is evidentthat the wave speed and the peak mean stress are greater in Fontainebleau sand than in Ottawa sand.This happens because the elastic moduli and CS friction angle of Fontainebleau sand is greater thanthose of Ottawa sand. These results were obtained for the 10 m deep tunnel.

Figures 14 (a) and (b) show the spatial variations of the maximum mean and deviatoric stressesalong the horizontal path AB for the 10 m deep tunnel. In order to obtain the plots, the mean anddeviatoric stress versus time data were recorded for all the elements along the path AB, and themaximum stresses experienced over time in each element are plotted as a function of the distance ofthe element from the outer edge of the tunnel lining. The rate of decrease of the maximum meanstress with distance from the tunnel is comparable for both the sands, while the decrease withdistance of the maximum deviatoric stress is faster in Ottawa sand.

Figures 15 (a) and (b) show the p’-q and e-p’ relationships of a soil element on the path ABimmediately adjacent to the 10 m deep tunnel. The normalized void ratio e/e0 (e0 is the initial voidratio) is plotted in Figure 15(b). It is interesting to note that the changes in the stresses due to blastare quite large, but the changes in the void ratio are rather insignificant.

(a)

p'

,q

DR

DR

(b)

Figure 15. (a) Mean stress versus deviatoric stress and (b) normalized void ratio versus mean stress for thesoil element horizontally adjacent to 10 m deep tunnels exploded with C4.


(a)

(b)

q

Figure 16. (a) Maximum mean stress and (b) maximum deviatoric stress versus horizontal distance from 5 mand 10 m deep tunnels in Ottawa sand subjected to C4 explosions.

Figure 17. Spatial variation of accumulated plastic strain at 0.01 second for 10 m deep tunnel in Ottawa sandsubjected to C4 explosions.




Figures 16 (a) and (b) show how the depth of tunnel affects the soil response. For these figures, thesimulations were performed for Ottawa sand with tunnels having center lines at the depths of 5 m and10 m from the ground surface. It is evident that the decrease of the maximum mean and deviatoricstresses (along the path AB) with increasing distance from the tunnel is faster for the shallower tunnel.

Figure 17 shows the accumulated plastic octahedral deviatoric strain gpoct ¼ 23 ep11 � ep22� �2þnh

ep11� ep33� �2þ ep22 � ep22

� �2þ6 ep122 þ ep13

2þ ep232

� �g12� and plastic volumetric strain epvol ¼ ep11þ

�ep22 þ ep33Þ

at 0.01 s along the horizontal path AB (Figure 10) for the 10 m deep tunnel in soil. The plasticstrains are maximum adjacent to the tunnel and decrease with increase in distance from thetunnel.

5. CONCLUSIONS

In this paper, a constitutive model was developed which is capable of simulating the high strain-ratebehavior of sands under multi-axial loading conditions. The model is developed from the modifiedbounding surface plasticity model of Manzari and Dafalias [23] in conjunction with the overstresstheory of viscoplasticity [32, 33]. The developed model is capable of distinguishing and simulatingthe behavior of contractive and dilative sands under rate-independent, rate-dependent and high-rateloads. The parameters of the model were calibrated to simulate the mechanical behavior of Ottawaand Fontainebleau sands. The CS parameters of the model were adjusted to account for the largestresses experienced in the split Hopkinson pressure bar tests and during blast loading in soil.The model was implemented in the FE software Abaqus using the cutting plane algorithm and wasused to analyze static and transient problems. Static and rate-dependent drained triaxial tests, andhigh-rate split Hopkinson pressure bar tests on Ottawa and Fontainebleau sands were simulated forthe validation of the model.

The constitutive model was subsequently applied in two-dimensional (plane strain) FE analysis oftunnels subject to blast loads. Circular underground tunnels constructed in sandy soils were subjectedto blasts caused by the explosion of C4. The blast was simulated using the JWL equation-of-statemodel. It was found that the type and relative density of sand and the depth of tunnel influence thepropagation of the blast-induced stress waves through the soil. The wave speed was found to be greaterin Fontainebleau sand than in Ottawa sand. The rate of decrease of the maximum mean stress in soilwith increase in distance from the tunnel was comparable for both the sands while the decrease of themaximum shear stress in soil was faster for Ottawa sand. The speed of propagation of the stress wavesis faster in denser sands. The rates of decrease of the maximum mean and deviatoric stresses in soilwith increase in distance from the tunnel are greater in a 5 m deep tunnel than in a 10 m deep tunnel.

ACKNOWLEDGEMENT

The material is based upon work supported by the U.S. Department of Homeland Security underGrant Award Number 2008-ST-061-TS002. The views and conclusions contained in this document arethose of the authors and should not be interpreted as necessarily representing the official policies, eitherexpressed or implied, of the U.S. Department of Homeland Security.

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