CONSTITUTIVE MODELING AND COMPUTER.pdf

7/29/2019 CONSTITUTIVE MODELING AND COMPUTER.pdf

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ACTA GEOTECHNICA SLOVENICA, 2010/1 5.

CONSTITUTIVE MODELING AND COMPUTERMETHODS IN GEOTECHNICAL ENGINEERING

CHANDRAKANT S. DESAI

About the author

Chandrakant S. DesaiTe University o Arizona,Department o Civil Engineering and Engineering Mechanicsucson, Arizona. USAE-mail: [email protected]

Abstract

Computer methods are in the oreront o the proceduresor analysis and design or geotechnical problems.Constitutive models that characterize the behavior o

geologic materials and interaces/joints play a vital rolein the solutions obtained by using computer methodsor any other solution procedure. Te literature on bothconstitutive and computer models is wide; attention in this

paper is devoted to the disturbed state concept (DSC) orconstitutive modeling and the fnite element method orcomputer solutions. Te disturbed state concept, a unifedand hierarchical approach, provides a unifed ramework

or characterization o the behavior o geologic materialsand interaces/joints. Important actors such as elastic,

plastic and creep responses, stress paths, volume change(contraction and dilation), disturbance (soening anddamage or stiening), thermal eects, partial satura-tion and liqueaction can be included in the same DSC

ramework. Because o its hierarchical nature, simplifedmodels or specifc applications can be derived rom theDSC. It has been applied successully or defning behavioro many geologic materials and interaces/joints.

Procedures or the determination o parameters or theDSC models based on laboratory tests have been devel-

oped. Various models rom the DSC have been validatedat the specimen level with respect to laboratory test data.Tey have been validated at the practical boundary value

problem level by comparing observed behavior in the feldand/or simulated problems in the laboratory with predic-tions using computer (fnite element) procedures in whichDSC has been implemented; these have been presentedin various publications by Desai and coworkers, and arelisted in the Reerences. Tree typical examples o suchvalidations at the practical problem level are included inthis paper. It is believed that the DSC can provide unifed

and powerul models or a wide range o geomechanicaland other engineering materials, and interaces/joints.

Keywords

constitutive modeling, disturbed state concept DSC,geologic materials, interaces/joints

1 INTRODUCTION

In the continuing pursuits or realistic models orgeologic materials and interaces or joints, a greatnumber o simple, and simplifed to complex constitu-tive models have been proposed. It is not the intentionhere to present a detailed review o such models, whichincludes elasticity, plasticity, creep, damage, microme-chanics, micro-cracking leading to racture and ailure,and liqueaction. Such reviews are available in manypublications, including Desai (2001b).

1.1 SCOPE

Te modeling approach based on the disturbed stateconcept (DSC) is the main objective in this paper. TeDSC provides a hierarchical and unifed ramework toinclude other models such as elasticity, conventional

Dedication

Proessor Lujo uklje was the pioneer o SlovenianSoil Mechanics, whose contributions were recognizedinternationally. He has made outstanding contributionsin rheological properties o soils including creep andanisotropy, viscoplastic models, consolidation and oun-dation engineering. His book, Rheological Aspects o Soil

Mechanics published in 1969, has le an indelible markon the literature in engineering. I dedicate, with prooundrespects, this paper to the memory o Proessor uklje.


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ACTA GEOTECHNICA SLOVENICA, 2010/16.

plasticity, continuous hardening plasticity, creep, andmicro-cracking leading to soening as special cases.Also, the number o parameters required in variousversions o the DSC model is smaller or equal to thenumber in other available models o similar capabilities.

Te DSC model has been implemented in two- andthree-dimensional problems or dry and saturatedgeologic materials under static and cyclic (dynamic)loadings. Since the DSC model results in a special ormo nonsymmetrical matrix equations, it is necessary todevelop special schemes or the implementation o theDSC models in computer (fnite element) procedures.

Te descriptions o the DSC model and the fnite elementprocedure are divided in the ollowing components:

- Basic concept and hierarchical ramework.- Parameter determination and calibration rom labo-

ratory and/or feld tests.- Brie description o a special scheme or implementa-

tion o the DSC in the fnite element procedure.- Validations with respect to tests used or calibration

and independent tests, and applications o the fniteelement method or solutions o a number o practi-cal problems in geotechnical engineering.

2 BASIC CONCEPT ANDHIERARCHICAL FRAMEWORK

Details o the DSC are given in various publications(e.g. Desai 1974, Desai 1987, Desai and Ma 1996, Desaiand oth 199, Katti and Desai 1995, Park and Desai2000), which are reerenced in Desai (2001b). A briedescription o the ramework is given below with atten-tion to the calibration or DSC parameters, numericalimplementation and validations.

2.1 EQUATIONS

Te DSC is based on the idea that the observed behavioro a material can be expressed through the behavioro reerence components in the deorming material.Te reerence components are usually called relative

intact (RI) and ully adjusted (FA). Te behavior o theRI part, which may not be aected by microstructuraldiscontinuities, can be defned by using a continuummodel, e.g. elasticity, plasticity or viscoplasticity. TeRI part is transormed continuously to the FA part,distributed at random locations, Fig. 1. Te RI and FAparts are coupled and interact to yield the observedresponse, which is expressed in terms o the RI and FA

C. DESAI: CONSTITUTIVE MODELING AND COMPUTER METHODS IN GEOTECHNICAL ENGINEERING

Figure 1. Representations o DSC.

(b) Symbolic representaion o DSC

D=0 (or D0) D>0 DDu1

Ri D=0

RI

FA

FA D=1

Dc

D

DuRc

(a) Clusters o RI and FA parts

(c) Shematic o stress-strain response

i relative intacta observedc ully adjusted


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behavior, connected through the disturbance unction,which provides or the coupling. Based on the aboveconcept, the incremental constitutive equations or DSCare derived as

( ) ( )~ ~ ~ ~ ~~ ~

1 C Ci ca i c c id D d D d dDs e e s s = - + + - (1a)

or

~ ~~

DSCa id C ds e= (1b)

where~s = stress vector,

~e = strain vector, a, i and c

denote observed, RI and FA behavior, respectively,~Ci

and~Cc = constitutive matrices or the RI and FA parts,

respectively, D = disturbance and ddenotes increment.

Te RI behavior can be expressed by using a suitablecontinuum model, e.g., linear elastic, nonlinear elastic,elastic-plastic or viscoplasticity. Here, the hierarchical

single surace (HISS) plasticity (associative) model (0),derived rom the general basis (Desai 1980), is used inwhich the yield unction, F (Desai et al. 1986), is given by

( ) ( ) 0.50n 22 1 1 rJ J 1- S 0DF J a g b-= - - + = (2)

where 2DJ = second invariant o the deviatoric stresstensor,

1 1 13 , JJ J R= + = frst invariant o the stresstensor, R = parameter related to the cohesive strength,c, Fig. 2, = parameter related to the ultimate yield

surace, = parameter related to the shape o F in

1 2 3s s s- - stress space, Sr= stress ratio = ( )27/2( )3/ 23D 2,3J /J ,J3D= third invariant o the deviatoric stresstensor, and is the growth or hardening unction. In Eq.


(2), the stress invariants are nondimensionalized withrespect topa , the atmospheric pressure constant.

In a simple orm, the growth or hardening unction isgiven by

1

1

n

aa

x

= (3a)

where a1 and 1 = hardening parameters and is thetrajectory o or accumulated plastic strains:

( )1/ 2

p pij ijd dx e e= (3b)

in which pije = plastic strain tensor, which is the sum othe deviatoric and volumetric plastic strains.

Te FA part can be defned by assuming that it has nostrength like in classical damage model (Kachanov 1986),or it has hydrostatic strength like in classical plasticity, or

it has strength corresponding to the critical state model.It can be also defned by using the behavior o other statessuch as zero suction or partially saturated soils. For soils,it is ound appropriate to use the critical state equations todefne the FA response (Roscoe, et al. 1958, Desai 2001b):

c2 1JcDJ m= (4a)

c1Jnc coa

e ep

l = -

(4b)

where c denotes the critical state, e = void ratio,coe =

initial void ratio and m, and are the critical state

parameters.

Figure 2. Plots o F in dierent stress spaces.


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2.2 DISTURBANCE FUNCTION

Te disturbance unction, D, can be defned on the basiso observed stresses, volumetric strains, pore water pres-sures or nondestructive properties such as S- and P-wavevelocities. In terms o stress, D is expressed as

i a

i cD

s s

s s

-

=

-

(5a)

Te disturbance, D, is also expressed as (Fig. 3)

( )1ZDA

uD D ex-

= - (5b)

where Du = ultimate disturbance, A and Zare distur-bance parameters, and D is the trajectory o deviatoricplastic strains.

Figure 3. Disturbance or soening, stiening behavior andcritical disturbance.

Figure 4. Hierarchical versions in DSC.



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2.3 HIERARCHICAL APPROACH

Various continuum models can be derived rom theDSC equations, Eq. (1). ID = 0, i.e., the material doesnot experience any microstructural changes leadingto discontinuities, Eq. (1) reduce to that or standard

continuum models:

~ ~~dii id Cs e= (6)

where~

iC = constitutive matrix or continuum model,e.g., elasticity, plasticity, etc. Various models such asvon Mises, Drucker-Prager, critical state, Matsuoka andNakai (1974), Lade and Kim (1988), etc. can be obtainedas special cases o the HISS model, Eq. (2). Fig. 4 showsvarious versions that can be derived rom the DSC.

2.4 INTERFACE/JOINTS

Since both the solid material (soil) and the interaces arecoupled, it is appropriate to use the same mathematicalramework or both. One o the major advantages o theDSC is that it provides the same ramework or the solidmaterial and the interaces. It has been ormulated orinterace/joint behavior with suitable yield unction orRI behavior and critical state models or FA behavior(Desai et al. 1984, Desai and Ma 1992, Desai et al. 1995,Desai 2001b, Desai et al. 2005). Tis paper essentiallyaddresses the soil behavior.

2.5 ADVANTAGES OF DSC/HISS MODELSTe DSC allows or discontinuities experienced by adeorming material, that can oen result in degradationor soening; it can also allow stiening and healingin deorming materials. It is essential to include theexistence and eect o discontinuous material in themodel; then only can it provide consistent and improvedcharacterization compared to other models based oncontinuum plasticity proposed and used to account orsuch behavior (Mroz et al. 1978), Pestana and Whittle1999, Elgamal et al 2002).

Because the DSC allows or the coupling between the RIand FA responses, it can avoid such diculties as spuri-ous mesh dependence that occurs in classical damagemodels (Kachanov 1986). Also, i such a coupling isconsidered in the classical damage model by introducingadditional enrichments (e.g. Bazant 1994, Mhlhaus1995), the resulting models may be complex.

In the racture mechanics approach, usually it isrequired to introduce cracks in advance o the loading.In contrast, the DSC does not need a priori introduction

o cracks, and initiation and growth o micro-cracking,racture and ailure can be traced at appropriate loca-tions depending upon the geometry, loading and bound-ary conditions, on the basis o critical disturbance, Dc ,obtained rom test results (Desai 2001b,Park and Desai2000, Pradhan and Desai 2006).

Te DSC allows identifcation o and initiation andgrowth o microstructural instability or liqueaction byusing the critical disturbance (Desai et al. 1998, Desai2000, Park and Desai 2000, Pradhan and Desai 2006).

Te DSC/HISS models also possess the ollowing advan-tages:

1. Te yielding in the HISS model is assumed to bedependent on total plastic strains or plastic work.Hence, in contrast to other models such as criticalstate and cap in which yielding is based on the volu-

metric strains, the HISS model includes the eect oplastic shear strains also.

2. Te yield surace allows or dierent strengths alongdierent stress paths.

3. Because o the continuous and special shape o theyield surace, it allows or dilatational strains beorethe peak stress, which can be common in manygeomaterials, Fig. 2(a), and later Fig. 7(b).

4. Te DSC allows or degradation and soening, and itcan allow also or stiening or healing.

5. Introduction o disturbance can account or (a part)the nonassociative behavior, i.e., deviation o plastic

strain increment rom normality.6. Te DSC allows intrinsically the coupling betweenRI and FA parts and the nonlocal eects. Hence, itis not necessary to add extra or special enrichments,e.g., microcrack interaction, gradient and Cosseratschemes, etc. (Mhlhaus 1995).

7. Te DSC is general and can be used or a wide rangeo materials: geologic, concrete, asphalt, ceramics,metal, alloys and silicon (Desai 2001b), i appropriatetest data is available.

8. It can also be used or repetitive loading which mayinvolve a large number o loading cycles; a procedureis described below.

2.6 REPETITIVE LOADING: ACCELER-ATED PROCEDURE

Computer analysis or 2- and 3-D idealizations ortime dependent problems can be time consuming andexpensive, especially when signifcantly greater numbero cycles o loading need to be considered. Tereore,approximate and accelerated analysis procedures havebeen developed or a wide range o problems in civil



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(pavements) (Huang 1993), mechanical engineering andelectronic packaging (Desai and Whitenack 2001). Here,the computer analysis is perormed or only a selectedinitial cycles (say, 10, 20), and then the growth o plasticstrains is estimated on the basis o empirical relationbetween plastic strains and number o cycles, obtained

rom laboratory test data. A general procedure with somenew actors has been developed and applied or analysiso pavements and chip-substrate systems in electronicpackaging (Desai 2007; Desai and Whitenack 2001).

From experimental cyclic tests on engineering materi-als, the relation between plastic strains (in the case oDSC, the deviatoric plastic strain trajectory, D), and thenumber o loading cycles, N, can be expressed as, Fig. 5:

( ) ( )b

D D r

r

NN N

Nx x

= (7)

where Nr= reerence cycle, and b is a parameter,depicted in Fig. 5. Te disturbance equation (5b) can bewritten as

( ){ }(1 exp Zu DD D A N x = - - (8)

Figure 5. Plastic strain trajectory vs. number o cycles orapproximate accelerated analysis.

Substitution o (D(N)) rom Eq. (7) in Eq. (8) leads to

( )

1/b1/

uD1 1A

Z

r

D r u

N N nN D Dx

= -

(9)

Now, Eq. (9) can be used to fnd the cycles to ailure, Nf, orchosen critical value o disturbance = i (say, 0.50, 0.75, 0.80).

Te accelerated approximate procedure or repeti-tive load is based on the assumption that during therepeated load applications, there is no inertia due todynamic eects in loading. Te inertia and timedependence can be analyzed by using the 3-D and 2-D

ln(D(N))

procedures; however, or million cycles, it can be highlytime consuming. Applications o repeated load in theapproximate procedure involve the ollowing steps:

1. Perorm ull 2-D or 3-D FE analysis or cycles up toNr, and evaluate the values oD(Nr) in all elements

(or at Gauss points).2. Using Eq. (7) compute D(N) at selected cycles in allelements.

3. Compute disturbance in all elements using Eq. (8).4. Compute cycles to ailure Nfby using Eq. (9), or the

chosen value oDc .5. Te above value o disturbance allows plot o conto-

urs oD in the fnite element mesh, and based on theadopted value oDc , it is possible to evaluate extento racture and cycles to ailure, Nf.

Loading-Unloading-Reloading: Special procedures andintegrated in the codes to allow or loading, unloadingand reloading during the repetitive loads; details aregiven in (Shao and Desai 2000, Desai 2001b).

3 PARAMETER DETERMINATION ANDCALIBRATION FROM LABORATORYAND /OR FIELD TESTS

3.1 PARAMETERS

Based on the RI (Eq. 2), FA (Eq. 4) and disturbance, D(Eq. 5), the parameters required are:

Elastic E and (or Kand G)

RI: Plasticity , , n, a1, 1FA: Critical state e0, m,

Disturbance Du ,A and Z

where E = elastic modulus, = Poissons ratio, K= bulkmodulus, and G = shear modulus. Te above parametersare averaged rom the parameters obtained rom test dataunder dierent confning pressures, and other acors.I the variation is severe, then a parameter needs to beexpressed as a unction o the actor, which increases thenumber o parameters. Te number o parameters canincrease i additional actors such as temperature andrate dependence are considered (Desai 2001b).

Te ollowing able 1 gives some o the specializedversions with related number parameters. It may benoted that the number o parameters in the DSC modelis less than or equal to those in any other available modelo comparable capabilities.



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Table 1. Specialized Versions and Parameters rom DSC/HISS Model.

SpecializationNumber

o parameters

Elastic 2

Classical Elastic-plastic:

- von Mises- Mohr Coulomb- Drucker-Prager

344

Continuous Hardening:- Critical State 5

HISS 0 Model (associative plasticity) 8*

HISS 1 Model (nonassociative plasticity) 9

HISS 0+vp Viscoplastic 11

HISS 2 Anisotropic 12

HISS 0 Disturbance 12

* I the cohesive strength is not included, the number o

parameters reduces by 1.

Note: Te number o parameters increases by about 2i pore water pressure is considered and by about 2 itemperature is included.

3.2 PARAMETER DETERMINATION

Te laboratory tests used or the calibration include consol-idation (hydrostatic), triaxial and multiaxial and/or sheartests. Te feld test may include nondestructive, e.g., S-wavemeasurements. Direct shear and simple shear (e.g., usingthe CYMDOF device, Desai and Rigby, 1997) tests are usedor DSC parameters or interaces/joints (Desai 2001b).

Te values o elastic moduli are ound rom the slopes ounloading responses. For instance, E is ound rom unload-ing slopes o 1 3s s- vs.1 plots rom triaxial tests. Similarly, can be ound rom plot o vs.1 ( = volumetric strain),G can be ound rom shear stress vs. shear strain plot, and Kcan be ound rom mean pressure (p) vs. plot. Determina-tions o various elastic parameters are depicted in Fig. 6.

Figure 6. Elastic constants rom test data; (a) stress-strain behavior, (b) volumetric-strain behavior,(c) shear stress-strain behavior, and (d) pressure -volumetric stain behavior.

a) c)

b) d)



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Te parameters and are related to the ultimateresponse, which is oen adopted rom the asymptoticvalue (about 5 to 10% higher than the peak stress) or agiven stress-strain curve, Fig. 7(a). Tey are ound romthe specialized expression or F (where = 0), Eq. (2),at the ultimate stress conditions, based on tests under

dierent confning pressures:

( )0.502

2 1 rJ 1- S 0DF J g b-

= - = (10)

By substituting dierent values o ultimate stresses, Fig.2, and are ound by using procedures such as theleast-square method.

Te phase change parameter is ound rom the point(b) in the stress-strain behavior, Fig. 7(b), where theresponse transits from compressive to dilative. One o theexpressions or n is

2

1 0

2

11

v

D

s d

nJ

J Fe

g=

= -

(11)

Figure 7. Behavior o loose and dense geo-materials: (a) stress-strain behavior, (b) volumetricstrain behavior, (c) void ratio-pressurebehavior, (d) J1 -

2DJ - critical state behavior, and (e) void ratio-critical state behavior.

c)



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Te hardening parameters are determined based on theollowing expression rom Eq. (3).

1 1an n na h x+ = (12)

Here, the stress-strain curve, Fig. 8(a) is divided intoincrements and is ound as the accumulated plasticstrain or a given increment. Ten or a given isound rom Eq. (1), i.e., F = 0. Te plot o (Fig. 8b) lnvs. lnyield the (average) values oa1 and 1.

Te critical state parameters, Eq. (4), are obtained rom

the plots o2 1DJ J- or m

and ec vs. ( )1 /

can J p or

coe

and, Figs. 7(d) and 7(e).

Figure 8. Determination o hardening or growth parameters,a1 and 1.

a)

b)

Te disturbance parameter, Du, is ound rom Eq. (5b)by substituting the value o the residual (FA) stress, Fig.7(a). Ten the values oA and Zare obtained by plotting

( )( )/u un n D D D- - vs. lnD , Fig. 9, based on theollowing expression rom Eq. (5):

( ) uDu

D DZ n nA n nD

x - + = - (13)

Figure 9. Determination o disturbance parameters,A and Z.

Te values oD can be ound rom static or cyclicresponse, Fig. 1(c) or Fig. 10.

3.3 PARAMETERS FOR TYPICAL MATE-RIALS AND INTERFACES/JOINTS

Te DSC models have been used to model a wide rangeo materials, e.g., clays, sands (dry and saturated), glacialtills, partially saturated soils, rocks, concrete, asphaltconcrete, metals, alloys (solder), silicon., and variousinteraces and joints. Application or interaces includesclay-steel, sand-steel, sand-concrete and chip-substrate.Laboratory and/or feld tests have been used to calibratethe DSC parameters by ollowing the oregoing proce-

Figure 10. Cyclic tests or disturbance and liqueaction.



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dures. Details o the parameters have been presented in anumber o publications, and also given in Desai (2001b).Parameters or the relevant geologic materials and inter-aces are given in the ollowing Examples 1 to 3.

3.4 HIERARCHICAL VERSIONSVarious plasticity models such as nonassociative (1) andanisotropic hardening (2) can be obtained by modiy-ing or adding corrections in the HISS model or theactors that inuence the specifc behavior, to the basicassociative (0) model (Desai et al. 1986, Somasundaramand Desai 1988, Wathugala and Desai 1993). Since theintroduction o disturbance can account or (some)nonassociative response, the DSC model with the basisassociative (0) version is ound to account or behavioro most geologic materials.

3.5 OTHER CAPABILITIES IN DSC

3.5.1 Creep Models

A schematic o the creep behavior is shown in Fig. 11.Various creep models, e.g., Maxwell, viscoelastic (ve),elastoviscoplastic (evp), and viscoelastic- viscoplastic(vevp), can be obtained as specialization o the Multi-component DSC (MDSC) (Desai, 2001b). For example,the evp model can be defned based on the Perzynastheory (1966), in which viscoplastic strain increment,

vpe , is expressed as

~

vp Fe fs

= G

(14)

where = uidity parameter, and < > has the meaning oa switch-on-switch-o operator as

0

0 0

o o

o

o

F Fif

F F F

F Fif F

f

f

> =

(15)

Fo = reerence value oF(e.g., yield stress orpa) and =ow unction. One o the expressions o is

N

o

F

Ff

= (16)

In above equations, and Nare the viscous parameters.Te values oand Nare ound rom a creep test,Fig. 11; details are given in (Desai et al. 1995, Desai2001b). Various creep models such as viscoelastic (ve),viscoplastic (vp), viscoelastic -viscoplastic (vevp) can beobtained rom the MDSC; they are depicted in Fig. 12.Te Overlay or mechanical sub layer models (Duwez1935, Zienkiewicz el al. 1972, Pande et al. 1977) can beconsidered to be special case o the MDSC approach.

3.5.2 Liquefaction

Liqueaction in saturated soils (Fig. 10) subjected todynamic (or static) loading can be identifed in the DSCby locating the critical value o the disturbance, Dc, Eq. (5),Fig. 3, that represents the microstructural instability. Tereis no additional parameter needed or liqueaction. Detailso the development and use o the DSC or liqueactionare given in various publications (Desai et al. 1998b, Desai2000, Park and Desai 2000, Pradhan and Desai 2006).

Figure 11. Schematic o creep behavior.



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3.5.3 Partially Saturated Soils

Te DSC model can be applied to model partially

saturated soil by including actors such as suction andsaturation (Geiser et al. 1997, Desai 2001b).

3.5.4 Stiffening and Healing

Te disturbance, D, can simulate soening or degrada-tion. It can also defne stiening or healing (Fig. 3) inthe microstructure; or example, chemical and thermal(causing particle using) eects can increase in thebonding in the microstructure (Desai et al 1998a, Desai,2001b).

3.5.5 Thermal Effects

Te thermal eects can be introduced in the DSC model.Very oen, the parameters are expressed as a unction otemperature, T, as (Desai 2001b)

c

r

r

Tp p

T

= (17)

wherep = any parameter, andpr is the value o theparameters at a reerence temperature, Tr (e.g., the roomtemperature), and c = parameter.

Figure 12. (a) MDSC creep models, (b) viscoelastic (ve), (c) viscoplastic (evp), and(d) viscoelastic-viscoplastic (vevp) models (Desai 2001b).

3.5.6 Rate Effects

Behavior o many materials is aected by the rate o

loading. For a plasticity model, the rate eect can beintroduced by modiying the static yield unction deter-mined rom tests conducted under very slow or staticrate ( Baladi and Rohani 1984, Sane et al. 2009):

Fd= (Fs - FR) / Fo (18)

where Fdrepresents the yield unction defned ordynamic rate, Fs represents the yield unction at staticor very small rate, Fo is the used to nondimensionalizethe terms, and FR is the rate dependent unction. Detailsor the determination FR and Fs are given in (Sane et al.2009).

3.6 COMMENTS ON VARIOUS ISSUES

3.6.1 Physical Meanings and Curve fitting

A constitutive model or defning behavior o a materialrequires a certain level o curve ftting or calibrationo parameters based on the test data. It is useul anddesirable that such curve ftting is reduced as much aspossible, particularly i the behavior is inuenced bymany actors. One o the ways is to develop parameters



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that have physical meanings, e.g., the parameters areassociated with physical states during the response, suchas ultimate condition, and change in volumetric strain.Most o the parameters in the DSC have such physicalmeanings; hence, the need or curve ftting is reducedand the procedures or the determination o parameters

are simplifed.

3.6.2 Simple and Simplified Models

It is oen said that a constitutive model should besimple, particularly or practical applications. Aunctional representation (e.g., parabola, hyperbola,spline, etc.) o a given (single) stress-strain response asa nonlinear elastic model, is considered to be a simplemodel. Such a unction like the hyperbola extended toinclude the eect o the confning pressure may stillbe called simple. However, since the soil behavior isaected by other practical actors such as irreversible

(plastic) deormations, stress path, volumetric strains,creep and type o loading, the resulting model maynot be simple! It is apparent that models by using onlymathematical unctions (e.g. hyperbola, spline, etc) torepresent stress-strain curves are not able to account orsuch practical actors.

Ten, it is better to develop and use simplifed models.A simplifed model should be derived on the basis obasic principles o mechanics, and is intended to charac-terize the behavior aected by signifcant actors impor-tant or practical problems. It is oen obtained by deletingless signifcant actors rom a general model developed by

considering the physical nature o the problem, loadingand other relevant conditions. In other words, a simpli-fed model could include the eect o the signifcantactors, in which the number o parameters couldincrease with the inclusion o each additional actor.

An important issue is how many parameters the modelinvolves, and what type o tests are required to deter-mine the parameters. Te other important issue is howmany signifcant actors are vital or practical problemsand must be included in the model? It is natural that thenumber o parameters would increase with the number

o actors. Hence, or a practical application, one mustadopt a model that contains the inuence o the signif-cant actors or a particular application. Ten the notiono adopting just a simple model that is not capable oaccounting or practically important actors would haveno basis!

Indeed, or practical applications, it is desirable todevelop as simplifed model as possible with smallernumber o parameters. Te DSC/HISS model hasbeen developed such that its simplifed versions can be

adopted depending upon the need o the practical appli-cation. Moreover, the DSC/HISS model is hierarchicalin nature and allows selection o a model or requiredactors by introducing additional actors and relatedparameters in the same basic model. Te hierarchicalapproach in the DSCC/HISS model can reduce the

number o parameters, and simpliy the procedure ortheir determination based on standard (e.g., triaxial andshear) tests.

3.6.3 DSC and Damage Models

Te classical damage approach (Kachanov 1986) is basedon the physical cracks (damage) in a deorming material,and the resulting eect on the observed behavior. TeDSC is dierent rom the damage approach in that itconsiders a deorming material as a mixture o two ormore reerence components (e.g., RI and FA states);then the observed behavior is expressed in terms o the

behavior o the components, which interact to yield theobserved behavior. Te disturbance represents the devia-tion o the observed behavior rom the reerence state(s).Damage model can be considered as a special case oDSC i the interaction is removed, e.g., the FA state isassumed to possess no strength. However, such a model,without interaction between RI and FA states, maynot be realistic and possesses certain (computational)diculties (Desai 2001b). Moreover, the DSC approachis general and can include both soening and stieningeects.

4 FINITE ELEMENT FORMULATIONINCLUDING SPECIAL SCHEMES

Te procedure or implementation o the DSC model incomputer (fnite element) methods is given below.

Te fnite element equations or two- or three-dimensional problems (Fig. 13) can be derived as (Desai,2001b):

~ ~ ~1 1

T an n

V

B dV Qs+ +

= (19a)

where~B = strain-displacement transormation matrix,

~ 1

a

ns

+= observed stress vector at n + 1 step and the load

vector:

1

/

~ ~~ ~~ 1

X dV d ST Tn

V S

Q N N+

= + (19b)

where~

N = matrix o interpolation unctions,~

X = body

orce vector,~

T = surace traction vector, V denotesvolume and S1 denotes a part o the surace.



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Figure 13. Finite Element Discretization: 2-D and 3-D elements.

Now with

a1 n 1

a a

n nds s s+ += + (20)

where n denotes increment, the incremental equationsare:

~~~ ~~1 1

~ ~ ~1 1

d dV dV

Q

T Ta a

n nnV V

b b

n n n

B Q B

Q Q d

s s

+ +

+ +

= -

= - =

(21)

where~

b

n

Q represents the internal balanced load at step

n, Fig. 14, and~ 1

bn

d Q+

denotes the out-o-balance orresidual load vector.

Now, by substituting Eq. (20) in Eq. (19a), the incremen-tal equations with the DSC model are:

i bn 1 n 1

~ ~~B dV dq QDSCT

nV

B C d+ + = (22)

Figure 14. Incremental iterative analysis.

a) Load-Displacement and Stress-Strain Responses b) Incremental-Iterative Scheme

where ( ) ( ) c rn n n~ ~ ~

1 D 1 C R ;DSC i Tnn n

C D C a s= - + + +

n~ ~ ~ ~~

, dD dTr c i in n n n

Rs s s e = - = (Desai, 2001b), and

( )~ ~~1 C dcc i

n nnds a e= + and is the actor or relationbetween RI and FA strains.

Now, we can write Eq. (22) as

in 1

~ ~ 1

dq QDSC bn n

k d++

= (23)

where~ ~~ ~

C VDSC DSCTn

V

k B B d = .

A number o procedures are possible or the solution o Eq.

(23) (Desai, 2001b). Since~

DSC

nk is nonsymmetrical, it may

cause computational problems. Simplifed and approximateprocedure used commonly is briey described below. Tisprocedure is based on the solution o the RI behavior inwhich the stiness matrix is symmetric:

( )

1 1~ ~ 1

dq dQ Q Qi i i b ii n n nn n

k + ++

= = - (24)



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Fig. 15 shows the schematic o the procedure where thesolution, Eq. (24) pertains to the continuously hardening(RI) behavior. Once the incremental displacements,

~ 1d qi

n+, or the RI behavior are determined, the strains

( )~ 1

d in

e

+and stresses ( )

~ 1d i

ns

+are determined. Ten the

observed stresses are ound rom the ollowing equation:

( )(j)( ) n 1 1 1~ ~1 1

1-D Di ca j i n nn n

ds s s+ + ++ +

= + (25)

wherej denotes iterations depicted in Fig. (15) (Desai,2001b). Ten the observed (applied) load is ound as:

( )

~ ~ 1~ 1

Q B dVa T a mnn

d s++

= (26)

where m denotes the last (converged) iteration.

5 VALIDATIONS

Te DSC equations (1) are integrated to predict theobserved laboratory behavior or a wide range o materi-als (Desai, 2001b). For most o the materials, the DSCmodel provides very good correlations with the test dataor the specimen leveltests. Also, the DSC model has

Figure 15. Iterative procedure in solution based on RI equations.

provided very good correlations or independent testdata, which were notused or calibration o the param-eters. Details o validations at the practical boundary

problem levelare presented in a number o publications,e.g. Desai (2001b); three typical examples o applicationsare given below.

5.1 COMPUTER CODES

A computer code have been developed or (a) thedetermination o DSC parameters based on laboratorydata. Here, the user can input points on various stress-strain-volumetric responses, and the code yields theparameters. Tis code also includes the integration oEq. (1) by using the determined parameters so as to back

predict stress-strain-volumetric responses toward valida-tions at the specimen level.

Computer codes have been developed based on theabove fnite element method or static and dynamicanalysis including liqueaction o practical boundaryvalue problems, e.g. ( i ) DSC-SS2D (Desai 1999), (ii)DSC-DYN2D (Desai 2001c), and (iii) DSC-DYN3D(Desai 2001a). Tese and other codes have been used tosolve a wide range o problems in civil, mechanical andelectronic engineering. Te predictions compare very



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well with laboratory and/or feld tests or practical prob-lems. Tree typical example problems or application othese codes are given below.

5.2 EXAMPLE 1: DYNAMIC BEHAVIOR

OF PILE TESTED IN CENTRIFUGEFig. 16 shows the pile tested in the centriuge test acility(Wilson, et al., 2000). Te pile-Nevada sand system wassubjected to base motion shown in Fig. 17. Various itemssuch as pore water pressures, accelerations, displace-ments and bending moments were recorded with timeby using instruments installed as marked in Fig. 16.

Finite element analysis was perormed using a procedurebased on generalized Biot's theory in which the DSCmodel was implemented (Desai, 2001c; Pradhan andDesai, 2006). Te DSC model parameters or the Nevada

sand were determined rom laboratory triaxial testsreported by Te Earth echnology Corporation orVELACS (Arulmoli, et al. 1992). Te test data includedthree undrained monotonic triaxial tests and threeundrained cyclic triaxial tests with continuing pressures,3

= 40, 80 and 160 kPa at relative density, Dr

= 60%.Details o the evaluation o parameters or the Nevadasand are given in Pradhan and Desai (2002); the valueso the parameters are presented in able 2.

Te interace test data or the aluminum pile-Nevadasand were not available. Hence, a neural network

approach was used or estimation o the parameter(Pradhan and Desai, 2002). Here, two sets o data ortraining o the neural network were used. Set 1 consistedo Ottawa sand parameters using triaxial tests (Park andDesai, 2000), and the Ottawa sand-steel interace usingthe cyclic multi-degree-o-reedom (CYMDOF) device

(Alanazy, 1996 and Desai and Rigby, 1997). Te secondset consisted o parameters or a marine clay rom triax-ial and multiaxial testing (Katti and Desai, 1995) andmarine clay-steel interace using the CYMDOF device(Shao and Desai, 2000). It was assumed that the steel andaluminum behave approximately similar in interaceswith the soil. Also, the Nevada sand gradation curvealls between those o the Ottawa sand and the marineclay. Finally, the parameters or Nevada sand-aluminuminterace estimated rom the neural network were oundand are given in able 2.

Te sand and interace (aluminum-sand) were modeled

using the DSC with the parameters given in able 2. Tealuminum pile itsel was assumed to be linear elasticwith elastic modulus = 70.0 GPa and Poissons ratio =0.33. Te single pile, Fig. 16, was made o aluminum pipewith diameter o 0.67 m with wall thickness o 72 mm;the total depth o sand was 20.7 m and the embedmentlength o the pile was 16.8 m. Te pile-system was ideal-ized as plane strain (Pradhan and Desai, 2006; Anandara-jah, 1992). Te fnite element mesh or the total domainin the centriuge test or the pile is shown in Fig. 18(a);details o mesh around the pile are given in Fig. 18(b).

Figure 16. Schematic o pile in centriuge test and instrumentation (rom Wilson et al. 2000).



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Te FE mesh contained 302, 4-noded elements with342 nodes. Interace elements were provided or thevicinity o pile by assuming small thickness o thesand. Boundary BC (Fig. 18a) was restrained in they-direction, and AB and CD were ree to move in bothx- and y-directions. Te base o mesh was subjected tothe acceleration-time history shown in Fig. 17 (Wilson,

et al., 2000).

Te initial (in situ) and pore-water pressures at thecenters o all elements were introduced by using:

/ / /h v o

w

; , K /(1 ) , and

p

v s oh K

h

s g s s n n

g

= = = -

=

(27)

where/vs and

/hs = eective vertical and horizontal

stresses at depth, h; s = submerged unit weight o soil;w = unit weight o water;p = (initial) pore water pres-sure, and Ko = co-ecient o earth pressure at rest.

Figure 17. Base motion acceleration (rom Wilson et al. 2000).

Table 2. Parameters or Nevada Sand and Sand-Aluminum Interace.

Material Parameter Symbol Nevada Sand Nevada Sand - Aluminium Interace

RI: ElasticE

41.0 MPa0.316

14.6 MPa0.384

RI: Plasticity

3Rna11

0.06750.000.004.10

0.12450.0725

0.2460.000.00

3.3600.6200.570

FA: Critical State

m

coe

0.220.02

0.712

0.3040.02780.791

Disturbance

Du

ZA

0.99

0.4115.02

0.990

1.1950.595

5.2 RESULTS

o identiy the inuence o interaces between structure(pile) and soil, analyses were perormed (a) withoutinteraces and (b) with interaces. Only typical results arepresented herein. Fig. 19 shows comparisons betweenmeasured pore water pressure in element 139 in the vicin-ity o the pile, and predictions with and without interace.

Te observed pore water pressure approaches the initialeective vertical pressure( )/vs , showing potentialliqueaction, afer about 9 seconds. Te analysis withoutinterace shows that the computed pore water pressureremains ar rom /vs , i.e., it does not show liqueaction.However, the analysis with the interace shows that aferabout 9 seconds the pore water pressure is almost equalto /vs , indicating liqueaction. Tus, provision o interaceincluding related relative motions is considered to beessential or realistic prediction pore water pressures,liqueaction and behavior o the pile-soil system.

ime (sec)



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Figure 18. Finite element mesh for pile test.

Interface elements - 133 to 143 and 162 to 172

Pile elements - 146 to 159

a) b)

Figure 19. Comparisons of pore water pressures for element 139 in vicinity of pile.



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able 3 shows time to liqueaction in the conventionalprocedure when the pore water pressure, Uw, reaches theinitial eective pressure /vs . Te time or reaching thecritical disturbance, Dc = 0.86 (determined rom the ploto disturbance vs. time), is considered to designate micro-structural instability. Although the values o the two times

are not too dierent, the times to reach Dc are lower thanthose to reach /vs . Tis can imply that the microstructuralinstability may occur earlier than the time to liqueac-tion obtained rom the conventional procedure. Teconventional procedure suggests that the soil does notpossess any strength when liqueaction occurs. In act, itis observed that the soil can retain a small strength whenthe liqueaction is reached; this is also shown in observa-tions during the Port Island, Kobe, Japan earthquake(Desai, 2000). Hence, it could be stated that the DSCbased on the critical disturbance, Dc , could provide moreaccurate time to liqueaction (microstructural instability).

Table 3. imes to Liqueaction rom Conventional and Distur-bance Methods.

ElementConventional

/ ( )w vU ss=Disturbance

( )cD D s=

143 1.74 1.23

130 1.83 1.29

104 2.55 1.92

78 3.36 2.67

52 3.81 2.94

26 9.03 8.22

Table 4. Parameters to Analysis o Reinorced Wall.

Material Parameter Symbol Soil Interace

RI: ElasticE or kn or ks

f1(3)a1

0.30f2(n)

b

f3(n)c

RI: Plasticity

na11

k

d

0.120.452.56

3.0E-5

0.98

0.20

2.300.002.800.031.00

0.40

DisturbanceDuZ

A

0.930.371.60

Angle o riciton and adhesion //ca = 40= 34

ca = 66 kPa

Unit weight

18.00 kN/m3

Co-ecient o earth pressure at rest K0 0.400

5.3 EXAMPLE 2: REINFORCED EARTHRETAINING WALL

Te DSC parameters or backfll (soil) and interace ora ensar reinorced wall, Fig. 20 (USDO 1989), weredefned based on comprehensive laboratory tests, triaxial

or soil, and interace shear or interaces between soiland (ensar) reinorcement (Desai and El Hoseiny,2003). Te DSC parameters or the backfll soil andinteraces are shown in able 4.

Te fnite element procedure with the DSC model wasused to analyze the earth retaining wall reinorced byusing ensar geogrid. Coarse and fne meshes were usedor the analyses; it was ound that the fne mesh analysisyields improved predictions compared to those rom thecoarse mesh. Fig. 21 shows a part o the fne mesh (with11 layers); it contained 1188 nodes, and 1370 elementsincluding 480 interace, 35 wall acings and 250 bar

(reinorcement) elements.

Te analysis included simulation o constructionsequences, in which the backfll was constructed in 11layers, as was done in the feld. Te reinorcement wasplaced on a layer aer it was compacted. Te comple-tion o the sequences o construction is reerred to asend o construction. Ten the surcharge load due totrac o 20 kPa was added uniormly on the top othe mesh, which was reerred to as aer opening totrac. Te in situ stress was introduced by using theco-ecient Ko = 0.40.

a 3 0.28362 x 10E s=

b3 0.29

n(normal stiness) 18 x 10nk s=c ks (shear stiness) =

3 0.28

n30 x 10 s d = nonassociative parameter



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Figure 20. Tensar Reinforced wall (USDOT 1989).

Figure 21. Fine Mesh (Part) for Tensar reinforced wall.



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5.4 RESULTS

Only typical results are presented herein. Figure 22(a)shows comparison between predictions and feldmeasurements or the vertical stress at elevation, 1.53 m,at the end o construction; those aer opening to trac

are shown in Fig. 22(b). Tis fgure shows also the over-burden stress, and the trapezoidal vertical stress used in

design calculations. Te FE predictions agree well withthe measured values.

Fig. 23 shows comparison between predictions andmeasurements or the horizontal stress carried bythe geogrid near the wall ace, at dierent elevations.

Te measurements are obtained rom the strain gagesinstalled on the geogrid. Te predictions or the geogridstresses compare well with the measurements.

Figure 22. Comparisons between feld measurements and predictions or vertical stress at elevation 1.53 m.

(a) End o construction

(b) Aer opening to trac



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5.5 EXAMPLE 3. MODELING FOR GLACIALTILL FOR MOTION OF GLACIERS

Constitutive modeling o glacial till and till-ice interacesplay a signifcant role in predicting the movements o icesheets or glaciers. Te glacial till occurs between the iceand the base rock, and its deormations can contributesignifcantly to the motion o glaciers.

A comprehensive laboratory triaxial testing or shearand creep behavior o two glacial tills, iskilwa and SkyPilot, were perormed and used to defne the DSC model(Desai et al., 2008, Sane, et al., 2008). Te triaxial sheartests were perormed under confning pressures, /3s = 20,50, 100, 200, 300, 500 kPa. ests were also perormed or

/3s = 150 and 400 kPa or independent validations o the

model. Creep tests with constant (vertical) stress o 20,

40, 60 and 80% o the peak stress observed in the sheartests under /3s = 100, 200 and 400 kPa; tests under

/3s = 300

kPa were perormed or independent validation. Tesetests were used to fnd the parameters or the DSC model;able 5 gives the parameters or two tills.

Note: Creep parameters or elastic-viscoplastic (evp) andviscoelastic-viscoplastic (vevp) models are given in Sane,et al. (2008).

Te conventional plasticity models such as the Mohr-Coulomb have been used or behavior o glacial andtill, and or identifcations o the motion o glaciers.However, based on laboratory tests and fnite elementcomputations, it was ound that the Mohr-Coulombcriterion that assumes ailure and motion at maximumstress reached at very small (elastic) strains may notprovide realistic behavior o till and initiation o motion.

Figure 23. Comparison o feld measurements and prediction or horizontal stress carried by geogrid near wall ace.

Table 5. Parameters or Glacial ills.

Material Parameter Symbol iskilwa ill Sky Plot ill

RI: ElasticE

57.75 MPa0.45

50.50 MPa0.45

RI: Plasticity

na11

0.01750.363.60

2.0 E-4

0.27

0.00920.526.85

1.35 E-7

0.14

FA: Critical State

m

coe

0.100.210.48

0.090.160.52

DisturbanceDuZA

1.002.68

20.20

1.001.005.50



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It was observed that the glacial tills experience thedisturbance at distributed locations in the test specimen,

and ails when the critical disturbance,D

c , occurs incritical raction o volume o the specimen. Fig. 24 showsthe deormed confgurations o till specimens which

Figure 24. Photographs o triaxial test samples (a) beore the test, and (b) aer test or Sky Pilot till,and (c) and (d) aer test,or Tiskilwa till.

do not show specifc ailure planes, as is assumed inconventionally. In act, it is considered that the ailure

in terms o the initiation o motion o the ice sheet mayoccur long aer the peak stress, and at approximately theonset o the residual condition at Dc.

Figure 25. (a) Rate o change o displacement with applied stress predicted by fnite element analysis, and (b) Rate o change owithapplied strain or triaxial test Tiskilwa till under confning stress = 100 kPa. Note: Aer the induced peak shear stress = 60 kPa, stress

reduces to about 10 kPa in the post peak region, which is indicated by the dotted line in (a) above.



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Fig. 25(b) shows the plot o plastic strain trajectory ()vs. applied axial strain or the behavior o the triaxial

specimen under typical confning stress,

/

3s

= 100 kPa.Te instability causing ailure or initiation o motionappears to occur when Dc = 0.85 is reached, which indi-cates the transition rom linear to nonlinear behavior.

Te above is also seen, Fig. 25(a), when the computeddisplacement vs. applied shear stress reaches Dc = 0.85 atthe transition rom linear to nonlinear behavior occurs.Te results in Fig. 25(a) were obtained rom the fniteelement analysis o a simulated problem involving icesheets, till layer and bed rock, Fig. 26. Te details o thefnite element analysis are given in Sane, et al., 2008 andDesai, et al., 2008.

Tus, the DSC provides a realistic model or the behav-ior o glacial till and or the prediction o movementso ice sheets or glaciers, a subject which could havesignifcant inuence on global climate change.

6 CONCLUSIONS

Accurate modeling o the behavior o geologic materi-als and interaces/joints are essential or realistic

Figure 26. Simulated section o ice sheet on till and fnite element mesh with loading. (Not to scale.)

prediction o the behavior o geotechnical problems,e.g., by using computer (fnite element) procedures.

Te disturbed state concept (DSC) provides unifedand powerul constitutive models or solution ogeotechnical and other engineering problems. Tispaper presents a description o the DSC includingtheoretical details, determination o parameters, andvalidations o the model at specimen level, and practi-cal boundary value problem level. ypical examples oapplication o fnite element procedures with the DSCmodel are presented and include comparisons betweenpredictions with observations rom feld and laboratorysimulated problems. Te DSC model is considered tobe unique and provides or successul modeling o awide range o engineering materials, and interaces and

joints.

ACKNOWLEDGMENT

Te results presented herein have been partly supportedby a number o grants rom the National ScienceFoundation, Washington, DC. Participation andcontributions o a number o students and colleagues areacknowledged; some are included in the reerences.



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