5_5. Elastic Constitutive Models

7/28/2019 5_5. Elastic Constitutive Models

http://slidepdf.com/reader/full/55-elastic-constitutive-models 1/18

5. Elastic constitutive models

5.1 Synopsis

When presenting the finite element theory in Chapter 2 the soil was assumed tobehave as an isotropic linear elastic material. This chapter and Chapters 6, 7 and8 describe alternative constitutive models for soil behaviour. This chapter beginswith a short introduction to stress invariants. The remainder of the chapter isdevoted to elastic constitutive models. For completeness, both isotropic andanisotropic linear elastic models are presented. Nonlinear elastic models in whichthe material parameters vary with stress and/or strain levels are then described.

5.2 IntroductionThe finite elem ent theory presented in Chapter2 assumed material behaviourto belinear elastic. The review of real soil behaviour presented in Chapter 4 indicatesthat such an assumption is unrealistic forsoils. For realistic predictions to be m adeof practical geotechnical problems a more complex constitutive model is required.

In this chapter and Chapters 6, 7 and 8 a variety of constitutive models thathave been, and stillare, used to represent soil behaviour are described. This chapterbegins by introducing the concept of invariant measures of stress and strain, whichare particularly useful when considering the behaviour of isotropic soils. Theremainder of the chapter is then devoted to elastic constitutive models which,although of limited use, form a useful introduction to constitutive models. Forcompleteness, both isotropic and anisotropic linear elastic constitutive matrices aregiven. Nonlinear elastic models in which the material parameters vary with stressand/or strain levels are then described.

More advanced constitutive m odels, based on elasto-plasticity, are presentedin Chapters 6, 7 and 8. The extension of the finite element theory to deal withnonlinear constitutive models forms the topic of Chapter 9. The usefulness andapplicability of the various constitutive m odels to real geotechnical problem s isdiscussed in Volume 2 of this book.

5.3 InvariantsThe magnitudes of the components of the stress vector (i.e.ax,ay,a z, rxy, xxz, ryz)depend on the chosen direction of the coordinate axes, see Figure 5.1. The



Elastic constitutive models / 11 5

50

(b)

a, = 50a, = 100

a, = 100a, = 50

principal stresses (a h a2 and <J3)however, always act on the sameplanes and have the same magnitude,no matter which direction is chosenfor the coordinate axes. They aretherefore invariant to the choice ofaxes. Consequently, the state of stresscan be fully defined by eitherspecifying the six component valuesfor a fixed direction of the coordinateaxis, or by specifying the magnitudeof the principal stresses and the

directions of the three planes onwhich these principal stresses act. Ineither case six independent pieces ofinformation are required.

For materials which are isotropic,i.e. whose properties are the same inall directions, it is often convenient toR 5 ,. £ffect Qf chapge Qf axes

consider only certa.n aspects of the Q / | / f t / { / e Qf stress compon ents

stress tensor, rather than to have to

specify all six quantities. For example, if one is only interested in the magnitudeof maximum and minimum direct stresses in the material, then only the principalvalues G] and <r3 are required. On the other hand, if one is interested in the overallmagnitude of the stress, all three principal stresses would be needed, but not thedirections of the planes on which they act. In geotechnical engineering it is oftenconvenient to work with alternative invariant quantities which are combinations ofthe principal effective stresses. A convenient choice of these invariants is:

ox=100

a, = 50a, = 100a, = 50

Mean effective stress: (5.1)

Deviatoric stress: J = + (^3 ~

Lode's angle: 6 = tan" (5.3)

This choice of invariants is not arbitrary because the above quantities havegeometric significance in principal effective stress space. The value ofp' is ameasure of the distance along the space diagonal((J\'=(72=a2f), of the currentdeviatoric plane from the origin, in principal effective stress space (a deviatoricplane is any plane perpendicular to the space diagonal). The value of J provides ameasure of the distance of the current stress state from the space diagonal in thedeviatoric plane, and the magnitude of0 defines the orientation of the stress state



1 1 6 / Finite element analysis in geotechnical engineering: Theory

within this plane. For example, consider the stress state represented by point P(cr,'p, a 2

fP , (T3'p) which has invariants/?'13, f and 9? in Figure 5.2. The distance ofthe deviatoric plane in which point P lies, from the origin,is p'?-J~3, see Figure5.2a. The distance of P from the space diagonal in the deviatoric plane is given byJ V 2 , and the orientation of P within this plane by the value of #p, see Figure 5.2b.On this figure, (<r/)pr, (o2'Y and (a3')

vr refer to the projections of the principalstress axes onto the deviatoric plane. As o1'p>G2'p>a3

/P , P is constrained to liebetween the lines marked 6 = -30° and 6 = +30°. These limiting values of6correspond to triaxial compression(ox'v > o2

lV = a3/P) and triaxial extension (ox'?

> cr3/P) respectively.

Deviatoric Plane

Deviatoric Plane

a)

Figure 5.2; Invariants in principal stress space

The principal stresses can be expressed in terms of these alternative invariantsusing the following equations:

(5.4)in 0( '

sin 0--

The above d iscussion is also applicable to the accumulated strain vector(sx, sy,sz, yxy, yxz, yyz) and its principal values(eu s2 and s3). It is also applicable to theincremental strain vector (Aex , Asy , Aez , Ayxy , Ayxz , Ayyz). However, for

geotechnical engineering only two alternative invariants are usually defined. Asuitable choice for these invariants is:

Incremental volum etric strain:

As2 (5.5)




Incremental deviatoric strain:

(5.6)

The basis for selecting the above quantities is that the incremental workAW={(r'}T{Ae}=p'Asv + JAEd. The accumulated strain invariants are then givenby sv = J Aev and Ed = \AEd. The accumulated volumetric strain,sv, can also beobtained from Equation (5.5), but with the incremental principal strains replacedby the accumulated values. At first sightit might also seem that, by substituting theaccumulated principal strains for the incremental values in Equation (5.6), thevalue of Ed can be obtained. However, in general, this is not so. The reason whysuch an approach is incorrect is shown diagrammatically in Figure 5.3a, which

shows part of a deviatoric planein principal strain space. At position 'a ' an elementof soil has been subjected to no deviatoric strain,Ed = 0. Some loading is appliedand the strain state moves to position 'b ' . The incremental deviatoric strain isAEd

and the accumulated deviatoric strain isEd = AEd. A further increment of load isapplied which causes a change in the direction of the strain path. This causes thestrain state to move to position ' c \ with an incremental deviatoric strainAEd. Theaccumulated deviatoric strainis now given byEd = AEd + AEd, and is representedby the length of the path 'abc' on Figure 5.3a. Ifan equation similar to Equation(5.6) is evaluated with the principal accum ulated strains corresponding to position

' c ' , it represents the length of the path 'ac\ This clearly does not reflect theaccumulated value of the incremental deviatoric strain as defined by Equation(5.6). The paths 'abc ' and 'a c' are identical only if the directionsofAEd and AEd

are the same in the deviatoric p lane, see Figure 5.3b. Clearly, such a situation onlyoccurs if the strain path remains straight during loading. In general, such acondition does not apply. It might, however, occur in some laboratory tests fordetermining soil properties.

Deviatoric plane Deviatoric plane

a) Change in the strain path direction b) Monotonic strain path direction

Figure 5.3: Strain path directions in the deviatoric plane



1 1 8 / Finite element analysis in geotechnical engineering: Theory

Following similar logic, it can also be shown thatJtjAJ if AJ is calculatedfrom an equation similar to Equation (5.2), but with the accumulated principalstresses replaced by the incremental values.

Alternative definitions for stress and strain invariants can be, and are, used.Some of these are discussed subsequently in Chapter 8.

5 .4 Elastic behaviourThe basic assumption of elastic behaviour, that is common to all the elastic modelsdiscussed here, is that the directions of principal increm ental stress and incrementalstrain coincide. The general constitutive matrix relates increments of total stress toincrements of strain:

(5.7)

As noted in Section 3.4, it is possible to divide the total stress constitutive

matrix [D], given above, into the sum of the effective stress matrix [/)'] and thepore fluid matrix [D f]. Consequently, the constitutive behaviour can be definedeither by [D] or [/)'].

Elastic constitutive m odels can take many forms: some assume the soil to beisotropic, others assume that it is anisotropic; some assume the soil to be linear,others that it is nonlinear, with parameters dependent on stress and/or strain level.Several of the models that have been used for geotechnical analysis are presentedbelow.

Acr x

Aay

Aoz

ATXZ

Ar yz

A ^

A,A,A ,A,A iA,

D22

A2A2A*A2

AsA 3A 3A 3A 3A 3

A4A4A4A.4A 4Aa

AsAsAsAsAsAs

A6AaA6A 6AaA6

Asx

Asy

Asz

Ay xz

Ay yz

Ay xy

5 .5 Linear isotropic elasticityAn isotropic m aterial is one that has point symmetry, i.e. every plane in the bodyis a plane of symmetry for material behaviour. In such a situation it can be shownthat only two independent elastic constantsare necessary to representthe behaviourand that the constitutive matrix becom es symm etrical. In structural engineering itis common to use Young's modulus, E', and Poisson's ratio, ju', for theseparameters. Equation (5.7) then takes the form shown in Equation (5.8).

If the material behaviour is linear thenE' and juf are constants and theconstitutive matrix expressed as a relationship between accumulated effectivestresseses {a1} and strains {e} is the same as that given in Equation(5.8). It is alsopossible to express the constitutive matrix asa relationship between total stress andstrain, either on an incremental or accumulated basis. In this case the appropriateparameters are the undrained Young's modulus,Eu, and Poisson's ratio, //„.



Elastic constitutive models / 119

Ar.,

E'

I - f t ' ft

sym

M'

f'1 - / / '

000

1-2//'

000

0

1-2//'2

000

0

o

1-2//'

A*,

(5.8)

For geotechnical purposes, it is often more convenient to characterize soil

behaviour in terms of the elastic shear modulus, G, and effective bulk modulus,K' .Equation (5.7) then becomes:

(5.9)

A<T

AO-;

A ^ z

K' + %G Kf-y3c

sym

} K'-y

} K'-y

K' + y

\G 0

G 0

\G 0

G

0000G

00000

G

where

G = E'2(1 + //')

K' = E'3(1-2 / / ' ) (5.10)

Again, it is also possible to express the constitutive matrix in terms ofundrained stress param eters. As water canno t sustain shear stresses,the undrainedand effective shear modulusare the same, hence the use of G without a prime in

Equations (5.9) and (5.10). Consequently, onlyKf

has to be replaced by K u inEquation (5.9) to obtain the total stress constitutive matrix,[/)].

These linear isotropic elastic modelsdo not simulate any of the important facetsof soil behaviour highlightedin Chapter 4. They therefore have limiteduse foranalysing geotechnical problems. Their inabilityto reproduce even basic soilbehaviour is evident from the linear deviatoric stress - strain and deviatoric strain-volum e strain curves that they predictfor an ideal drained triaxial compression test,see Figure 5.4. However, these models are often used to represent structuralelements (e.g. retaining walls, floor slabsetc.)



120 / Finite element analysis in geotechnical engineering: Theory

AG,

A a3=0.

Figure 5.4: E lastic prediction of a drained

triaxial compression test

5.6 Linear anisotropic elasticityAs shown in Chapter 4 soil behaviour is rarely truly isotropic, often exhibitinganisotropy. Mathematically, if a material is fully anisotropic, the[D ] matrix inEquation (5.7) becomes fully populated. This implies that 36 independentparameters are required to define the values ofDtj. However, thermodynam ic strain

energy considerations (Love (1927)) imply that the [D ] matrix must besymmetrical, i.e. Di} = Dyi, for i±j. The total number of independent anisotropicparameters therefore reduces to 21.

Many materials, however, show limited forms of anisotropy. For soil depositsit is often assumed that their anisotropic characteristics depend on the mode ofdeposition and stress history. Soils that are deposited normally onto a plane arelikely to exhibit an axis of symmetry in the depositional direction, i.e. theircharacteristics do not vary in the plane of deposition. In Figure 5.5 Cartesiancoordinates are allocated to a sediment such that the z-axis is in the direction ofsediment deposition,S, while x andy axes are in the plane of deposition,P.

ition

Figure 5.5: Axes orientations forconsidering transverse isotropy




The above type of anisotropy is known as 'transverse isotropy', or 'cross-anisotropy' or 'orthotropy' and reduces the number of unknown materialparameters to seven. The relationship between incremental stressand straincomponents is given by matrix [D] in Equation (5.11):

(/dpP + ju'SPjUpS)E

A(\-ju'SpM' PS )E'p

Aju' PS (\ + /u'pp)E' s

000

P Aju fSP (l + ju PP )Ep

AM' SP(l + Mpp)Ep

A(\-jUppjUpp)E' s

000

1

000

GPS

00

0000

GPS

0

00000

GPP_

(5.11)

1 - Ip'spii'ps - 2/d'spM'psMpp " Mp\

where

Es' - Yo ung 's modulus in the depositional direction;EP' - Young's modulus in the planeof deposition ;juSP' - Poisson's ratio for straining in the plane of deposition due to the stress

acting in the directionof deposition;

jUpS' - Poisson's ratio for straining in the direction of deposition due to the stressacting in the planeof deposition;

jupp' - Poisson's ratio for straining in the plane of deposition due to the stressacting in the same plane;

GPS - Shear modulus in the plane of the directionof deposition;Gpp - Shear modulus in the planeof deposition.

However, due to symm etry requirementsit can be shown that:

M SP _ MPS„ (5.13)and E's E

fP

This reduces the numberof parameters required to define transversely isotropicbehaviour from seven to five. Matrix [D] in Equation (5.11) then reducesto thesymmetrical form of Equation (5.15).

Again, it is possible to express the constitutive matrix givenby Equations(5.15) in terms of undrained stress parameters.As the material parametersare

constant, the [D] matrices given in Equations (5.11) and (5.15) also relateaccum ulated stresses and strains. While the model enables an isotropic stiffness tobe modelled, it does not satisfy any of the other important facets of soil behaviourdiscussed in Chapter 4.




AMSP(1 + M'PP)E'S °

AJU'SP

(1 + M'PP)E'S

0

4(\-JUPP^C-)E; S o

GPS

0

0

0

0GPS

0

0

0

00

E'P2(\ + 2M'PP

(5.

)_

15)

where

A=-(5.16)

u sp •

5.7 Nonlinear elasticity5.7.1 Introduction

A logical first step to improving the linear elastic models described above is tomake the material parameters depend on stress and/or strain level. By doing thisit is possible to satisfy several of the requirements discussed in Chapter 4. As thereare only two parameters required to define isotropic elastic behaviour, this isrelatively straightforward. However, it is much more difficult for anisotropicbehaviour as there are five material constants. Consequently, most of the nonlinearelastic models that are currently in use assume isotropic behaviour.

When dealing with isotropic elasticity, the two material properties can bechosen arbitrarily from E, /i, K or G. For geotechnical engineering it is often

convenient to use bulk modulus, K, and shear modulus, G. The reason for this isthat the behaviour of soil under changing mean (bulk) stress is very different to thatunder changing deviatoric (shear) stress. For instance, under increasing mean stressthe bulk stiffness of the soil will usually increase, whereas under increasingdeviatoric stress the shear stiffness will reduce. Furthermore, inspection ofEquation (5.9) indicates that, for isotropic elasticity, the two modes of deformationare decoupled, i.e. changes in mean stress, Ap', do not cause distortion (no shearstrains) and changes in deviatoric stress do not cause volume change. While thisis clearly helpful when formulating a constitutive model, it must be noted that realsoils do not usually behave in this decoupled manner. For example, the applicationof a pure deviatoric stress in a direct or simple shear test does inflict volumetricstraining.

Five particular nonlinear models are now presented. These are the bi-linear, K-G, hyperbolic and two small strain stiffness models.




5.7.2 Bi-linear modelThis model assumes that the bulk and shear stiffness are constant until the stressstate reaches the failure condition. Once this occurs, the tangential shear modulus,G, is set to a very small value. Ideally, it should be set to zero, but ifthis is donethere would not be a one to one relationship between incremental deviatoricstresses and strains. This would lead to ill conditioning of the finite elementequations and numerical instability. Therefore, in practiceG is set to a small, butfinite value. The resulting stress-strain curves for this model are shown in Figure5.6. Clearly, if on reaching failure the stress state is unloaded, the shear moduluswould revert to its initial pre-failure value.

Figure 5.6: Bi-linear model

This model requires two elastic parameters,Ke and G e (or the equivalentEe andjue), to define the pre-failure elastic behaviour. In addition, it requires parametersto define the failure surface. For example, if a Mohr-Coulomb failure surface isused, a further tw o parameters, cohesionc' and angle of shearing resistancecp', areneeded.

5.7.3 K -G modelIn many respects this model is a logical extension of the bi-linear model. Thetangential (i.e. incremental) bulk,Kt, and shear, G,, moduli are explicitly defined

in terms of stress invariants:(5.17)

j3GJ (5.18)

The model therefore needs five parameters,K(o, % , Gkn aG and/?G, to describematerial behaviour. These can be chosen to fit available soil data. As with the bi-linear model, the parameters can be selected such that the tangential (incremental)shear stiffness becomes very small when failure is approached . This is achieved by

se tti ng ^; negative. Values of cohesion, c', and angle of shearing resistance,$\ aretherefore used when determining the values of the five input parameters. Themodel can also be used for unloading in much the same wayas the bi-linear m odel.A simple way of implementing this is to setfiG to zero on unloading. The bulkmodulus, Kt, therefore remains unaffected, but the shear modulus,G,, changes



1 24 / Finite element analysis in geotechnical engineering: Theory

abruptly to a higher value. If subsequent reloading brings the stress state near tofailure again, j}G is reset to its former value. The resulting stress-strain curves forthis model are shown in Figure 5.7. This model is described in detail by Nayloret

al (1981).

Figure 5.7: K-G model

5.7 .4 Hyperbolic modelWhile the two models described above are essentially incremental in that theydirectly define the change in tangential moduli, the hyperbolic model relatesaccumulated stress to accum ulated strain. Differentiationis then required to obtainthe equivalent incremental form for use in finite element analysis.

The original model is attributed to Kondner (1963), however, it has beenextensively developed by Duncan and his co-workers and is commonly known asthe 'Duncan and Chang' model (after Duncan and Chang (1970)). In its initialform it was originally formulated to fit undrained triaxial test results andwas basedon two parameters and the implicit assumption that Po isson's ratio was 0.5. W ithuse, further refinements were added and the model was appliedto both drained andundrained boundary value problems. The number of parameters needed to definethe model also increased to nine (Seedet al. (1975)).

The original model was based on the following hyperbo lic equation:

(5.19)v ' h) a + bs

where <J, and o3 are the major and minor principal stresses,£ the axial strain andaand b are material constants. It can be shown that ifox = a 3 the reciprocal ofa is theinitial tangential Young's modulus,Ei9 see Figure 5.8a. The reciprocal ofb is thefailure value of ox-o3, approached asymptotically by the stress-strain curve, seeFigure 5.8a.

Kondner and his coworkers showed that the values ofthe material propertiesmay be determined most readily if the stress-strain data from laboratory tests areplotted on transformed axes, as shown in Figure 5.8b. Rewriting Equation (5.19)in the same form gives:

(cr.-a,)~^- = a + bs (5.20)




described in Chapter 4. Many of the constitutive models in existence could notaccount for such behaviour and this, consequently, led to further developments.One of the earliest models developed specifically to deal with the concept of small

strain behaviour was that described by Jardineet al. (1986). Because this model isbased on nonlinear isotropic elasticity, it is appropriate to briefly describe it here.

The secan t expressions that describe the variation of shear and bulk moduli are:

= A + B cos

lt™£- = R + SC0S

P'

(5.22)

(5.23)

where, G sec is the secant shear modulus,K, ec the secant bulk m od ul us ,// the meaneffective stress, and the strain invariantsEd and ev are given by:

Ed = j _

£v = £x + S2

(5.24)

(5.25)

and A, B, C, R, S, T, a, y, Sand r\ are all material constants. Due to the cyclictrigonometric nature of Equations (5.22) and (5.23) it is common to set minimum(E dmin , svm m) a n d m a x i m u m ^m a x , £ vmax ) strain limits below which and abovewhich the moduli vary withp' alone, and not with strain. Typical variations ofsecant shear and bulk moduli for London clay are shown in Figure 5.9.

0.0001 0.001 0.01 0.1Deviatoric strain,Ed (%)

Shear modulus

0.0001 0.001 0.01 0.1Volumetric strain, ev (% )

Bulk modulus

Figure 5.9: Variation of shear and bulk moduli with strainlevel

To use this model in a finite element analysis, the secant expressions given byEquations (5.22) and (5.23) are differentiatedto give the following tangent values:

— = A + B (cosa) Xr

z 2.303) X7 (5.26)




(sin<5) Yn (5.27)

where (5.28)

As noted above, this model was developed to simulate soil behaviour in the smallstrain range. It was not intended to be used to deal with behaviour at large strainswhen the soil approaches failure. Consequently, the model is usually used inconjunction with a plastic model which deals with the large strain behaviour.Elasto-plastic m odels are discussed in Chapter 6, 7 and 8. The use of this model to

analyse real engineering structures is described in Volume 2 of this book.

5.7 .6 Puzrin and Burland modelWhile the small strain stiffness model described above is capable of modelling soilbehaviour under small stress and strain perturbations, it is lacking in theoreticalrigour and fails to account for several important facets of real soil behaviour. Inparticular, it does not account for the change in stiffness observed when thedirection of the stress path changes. However, its use by Jardineet al. (1986) to

investigate the behaviour of a range of geotechnical problems highlighted theimportance of modelling the small strain behaviour of soils. As a result,considerable effort has, and still is, being directed at improving our understandingof the mechanical processes controlling stress-strain behaviour of soils at smallstrains.

Based on limited experimentalevidence, it has been postulated that,in stress space, the stress state issurrounded by a small-strain region

(SSR), where the stress-strainbehaviour is nonlinear but very stiffand fully recoverable, so that cyclesof stress form closed hysteretic loops.Inside this region the stress state issurrounded by a smaller region wherethe soil response is linear elastic (theLER). When the stresses change, boththese regions move with the stressstate as illustrated by the dotted lines

Figure 5.10: Kinematic regions ofhigh stiffness

in Figure 5.10. This concept of kinematic regions of high stiffness was put forwardby Skinner (1975 ) and developed by Jardine (1985, 1992).

Puzrin and Burland (1998) adopted the above framework and developed aconstitutive model to describe the behaviour of soil within the SSR. They assumedbehaviour within the LER to be linear elastic and that once the stress state was




outside the SSR behaviour could be explained by conventional elasto-plasticity.Their work therefore concentrated only on the behaviour within the SSR. Aspublished, the modelis restricted to conventional triaxial stress space and therefore

must be extended to general stress space before it can be used in finite elementanalyses. Addenbrookeet al. (1997) used such an extended version of this modelto investigate the influence of pre-failure soil stiffness on the numerical analysisof tunnel construction. Although they used a simple Mohr-Coulomb elasto-plasticmodel to represent soil behaviour beyond the SSR, their extension of the Puzrinand Burland model is independent of how behaviour beyond the SSR is modelled.This model is now presented.

Since the model is assumed to beisotropic, it is formulated in terms of

stress invarian ts, see Section 5.3. Thestress space surrounding a stresspoint, or local origin, is divided intothree regions, see Figure 5.11. Thefirst region is the linear elastic region(LER). In the second region, the smallstrain region (SSR), the deviatoricand volumetric stress-strain behaviouris defined by a logarithmic curve

(Puzrin and Burland (1996)). Each ofthese regions is bounded by anelliptical boundary surface. Becausethe deviatorie stress invariant,J, isalways positive, these surfaces plot ashalf ellipses in Figure5.11. In the final region, between the SSR boundary and theplastic yield surface (which is defined independent of the pre-failure model), thesoil behaviour is again elastic, with the shear, G, and bulk,K, moduli beingassociated with the SSR boundary. To be consistent with experimental data, see

Chapter 4, the stiffnesses in both the LER and SSR are mean stress dependent.The equations defining soil behaviour are now presented. Note that/?', J(the

mean and deviatoric stress invariants),ev and Ed (the volumetric and deviatoricstrain invariants) are calculated from the local origin (see Figure 5.11).

Behaviour within the LERThe elliptical boundary to the LER is given by:

Local origin

P global

Figure 5.11: The stress regionssurrounding a local origin in J-p '

space

FLER(p',J)=\+n2[jr = 0 (5.29)

where aLER is a parameter defining the size of the LER. Within this boundary, theshear and bulk moduli are defined by:

A L E

? r r e fyJVLER (p'Y /

r e fy ^ '




where K LERef and G LER

ref are values of the bulk and shear elastic moduli,respectively in the LER, at the mean effective stressp 're f.

The value of the parameteraLER can be determined from an undrained triaxialtest as a LER = n J LER U , where J LER U is the value of the deviatoric stress at the LERboundary and n = S(K LER I GLER). The parameters p and y control the manner inwhich the moduli depend on mean effective stress.

Behaviour within the SSRThe boundary to the SSR is also given by an ellipse:

= 0 (5.31)_, P ' J V p '

where aSSR is a parameter defining the size of the SSR.It can be determined froman undrained triaxial testas a SSR =nJ SSR

u , where JSSRU is the value of the deviatoric

stress at the SSR boundary, and n is defined above.Between the LER and the SSR the elastic moduli depend both on mean

effective stress and on strain and obey the following logarithmic reductions:

\-a

\-a(x D -x e)R\\n(\

- > ] *

- ) ] •

wherea = -

x - \

— v1/ • / V />

b

(5.32)

(5.33)

(5.34)

(5.35)

«SSR

(5.36)

in which xu is the normalised strain at the SSR boundary,x e the normalised strain

at the LER boundary, and b is the ratio of tangent modulus on the SSR boundaryto the value within the LER:

^ L E R ^ L E R(5.37)




and u is the incremental strain energy, which can be determined from an undrainedtrixial test, being equal to 0.5£SSRVSSR

u, where £SSRU and JSSR

U are the deviatoricstrain and deviaoric stress on the SSR boundary, respectively.

Behaviour outside the SSROutside the SSR the moduli are the bulk modulus,K = bK LER , and the shearmodulus, G=bG LER . In future developments of the model it may be possible to usea hardening/softening plastic model and associate the SSR boundary with initialyielding. The elastic parametersG and K at the SSR boundary could then be relatedto the elasto-plastic shear and bulk stiffness. Such an approach would essentiallydefine the parameter b which would then no longer be required as an inputparameter to the model. The principles behind such an approach are outlined byPuzrin and Burland (1998).

Within this model, a stress path reversal results in the relocation of the localstress origin at the reversal point, so reinvoking the small-strain high-stiffnessbehaviour and enabling the model to simulate closed hysteretic loops. In thisrespect, a stress path reversal is defined if the increment of normalised mean ordeviatoric stress is less than zero:

_ PSSR

and

SSR

PSSR

J SSR 6J-JdJ SSR

- > 0

(5.38)

y SS R

where J SSR and p'SS R are the linearprojections of the current stress stateJ a n d / ? ' on the SSR, see Figure 5.12.According to this criterion the loadingreversal can occur at any stress state

within the SSR and not only on itsboundary.

A total of eight parameters arerequired to define the model. Theseare, u/p', b, aLE R/pf, aSSR/p \ KLER/p\G LER /p',fi and y. How ever, as the bulk

Local origin

Figure 5.12: Projection of thecurrent stress state on SSR

and shear moduli depend onp\ they can become very small asp' reduces.Addenbrooke et al. (1997) therefore introduced minimum valuesG min and K min

which act as cut off values calculated from the above equations.

A major difference between this model and the small strain stiffness modeldescribed in the previous section is that in this model both the shear and bulkstiffness decay simultaneously as the stress state moves through the SSR. Incontrast, in the previous model the decay of shear and bulk stiffness depends onlyon deviatoric and volumetric strains respectively. Consequently, the small strain



Elastic constitutive Models / 131

stiffness model allows only shear stiffness reduction on deviatoric straining withno volumetric straining, and bulk stiffness reduction on volumetric straining withno deviatoric strain, while the Puzrin and Burland model forces shear and bulk

moduli reduction together, independent of straining.

5.8 Summary1. In general, six independent pieces of information are required to define the

state of stress. If the material is isotropic the magnitude of the stress can bequantified usingthree independentstress invariants. These invariants canbe thethree principal stresses or some combination of these. How ever, if the materialis anisotropic all six pieces of information are required. The above statements

also apply to strains.2. Linear isotropic elastic models, which require only two material param eters, do

not reproduce any of the important facets of real soil behaviour identified inChapter 4. Linear cross-anisotropic elastic models, which require five materialparameters, do not really improve the situation, although they can reproduceanisotropic stiffness behaviour.

3. Nonlinear elastic models, in which the material parameters vary with stressand/or strain level, are a substantial improvem ent over their linear counterparts.Due to the number of parameters involved, most nonlinear elastic models

assume isotropic behaviour. However, they still fail to model some of theimportant facets of real soil behaviour. In particular, they cannot reproduce thetendency to change volume when sheared. Also, because of the inherentassumption of coincidence of principal incremental stress and strain directions,they cannot accurately reproduce failure mechanisms.

5_5. Elastic Constitutive Models

Documents

Transcript of 5_5. Elastic Constitutive Models