Theory of Plasticity - Universität Innsbruck · Theory of Plasticity ... Themathematical theory of...

Motivation and historical perspectivesThermodynamical foundations

Conventional elastoplastic models for soilsAdvanced topics: anisotropic hardening plasticity

Theory of PlasticityTheoretical fundamentals and applications to soil mechanics

Claudio Tamagnini

Dipartimento di Ingegneria Civile e AmbientaleUniversity of Perugia

Innsbruck 11.03.2011

“Theoretische Bodenmechanik”

Claudio Tamagnini Theory of Plasticity



Outline

1 Motivation and historical perspectives

2 Thermodynamical foundationsFundamental assumptionsConstitutive equations in rate formA simple 1d examplePhenomenological generalization: non–associative behavior

3 Conventional elastoplastic models for soilsPerfect plasticityIsotropic hardening plasticity

4 Advanced topics: anisotropic hardening plasticityIntroduction and definitionsExperimental evidence of anisotropic behaviorConstitutive modeling of anisotropyAnisotropic hardening plasticity for soils




Evidence of nonlinear and irreversible behavior

Granular materials exhibit verycomplex responses under appliedloads, even under relatively simpleloading conditions.

Examples of axisymmetric loading:

– isotropic compression:

σa = p σr = p

– oedometric (1d) compression:

σ1 = σa εr = 0

– drained triaxial compression:

σa > σr σr = const.




Clay sample under isotropic compression

p = (σa + 2σr)/3 = σc εv = εa + 2εr




Clay sample under drained TX compression

p = (σa + 2σr)/3 q = σa − σr σr = σc = const.




Failure locus of a coarse–grained soil

qf =Mpf




Constitutive equations and constitutive models

The main features of the observed response – nonlinearity, irreversibility,path–dependence – must be described by any constitutive equation whichaims at providing accurate predictions of the soil response under appliedloads.

Constitutive equations do not represent universal laws of nature, butrather they define “idealized materials” (costitutive models).

When available, constitutive models can be used to predict soil behaviorunder any possible circumstance, starting from the limited knowledgegathered from a few experimental observations.

The quality of predictions depends crucially on the ability of defining asuitable idealization for the real materials, for the class of loading pathsof practical interest.




Constitutive equations in rate form

To incorporate the observed non–linearity and path–dependence in aglobal stress–strain relation, non–linear and non–differentiable functionalmust be used (Owen & Williams 1969).

Therefore, rather than in global terms, constitutive equations are usuallydefined in rate form:

Oσ= F (σ, q,d) =︸︷︷︸

rate–ind.

C(σ, q,dird)d

where:

–Oσ: any objective time rate of Cauchy stress tensor;

– d := sym(∇ · v): rate of deformation tensor;

– q: additional set of state variables (internal variables), taking intoaccount the effects of the previous loading history.




The theory of plasticity: historical perspective

The mathematical theory of plasticity represent the first attempt toprovide a rational framework within which formulate the constitutiveequations in rate form, taking into account:

– the existence of a well defined limit to admissible stress states instress space;

– the flowing character of the deformation rates in ultimate failurestates, with strain rate depending on current stress rather thanstress rate.




The theory of plasticity: historical perspective

The foundations of the classical theory of plasticity can be traced back tothe fundamental works of Hill (1950) and Koiter (1960).

A thorough treatment of this subject can be found, e.g., in the treatisesby Lubliner (1990), Simo & Hughes (1997), Simo (1998), Han & Reddy(1998), Lubarda (2001), Houlsby & Puzrin (2006).

For applications to soil mechanics, good references are Desai &Siriwardane (1984), Zienkiewicz et al. (1999), Houlsby & Puzrin (2006),Nova (2010).

The application of plasticity concepts such as failure criterion and flowrule to soil mechanics are as early as the works of Coulomb (1773) andRankine (1853), referring to the problem of evaluating the earth pressureon retaining structures. These concepts still form the basis of limitequilibrium and limit analysis methods, widely used in current practice.




Fundamental assumptionsConstitutive equations in rate formA simple 1d examplePhenomenological generalization: non–associative behavior

Small–strain, rate–independent formulation

In the following, it is assumed that displacements and rotations aresufficiently small that the reference and current configurations can beconsidered almost coincident (small strain assumption).

The material behavior is assumed to be rate–independent (non viscous).

Under such hypotheses, the CE in rate form reduces to:

σ = C(σ, q,dir ε) ε

where:

– σ : Cauchy stress rate tensor;

– ε = sym(∇ · u) : infinitesimal strain rate tensor.





Assumption 1: kinematic decomposition of ε

We assume the following additive decomposition of the strain tensor intoan elastic (reversible) and a plastic (irreversible) part:

ε = εe + εp

In terms of rates, this yields:

ε = εe + εp





Assumption 2: hyperelastic behavior

We postulate the existence of a free energy density function (per unitvolume):

ψ = ψ (εe,α)

where α = strain–like internal variable.

The Clausius–Duhem inequality thus reads:

D := −ψ + σ · ε ≥ 0

⇒ D := − ∂ψ∂εe· εe − ∂ψ

∂α· α+ σ · ε ≥ 0





Assumption 2: hyperelastic behavior

Introducing the strain decomposition into the C–D inequality:

− ∂ψ∂εe·(ε− εp

)− ∂ψ

∂α· α+ σ · ε ≥ 0

and, rearranging:(σ − ∂ψ

∂εe

): ε+

∂ψ

∂εe· εp − ∂ψ

∂α· α ≥ 0

For non dissipative processes (ε = 0, α = 0) the equality must hold forany possible choice of ε. Therefore:

σ =∂ψ

∂εe





Reduced dissipation inequality

Definition: stress–like internal variable work–conjugate to α:

q = −∂ψ∂α

Taking into account the hyperelastic CE and the definition of q, theClausius–Duhem inequality yields the following reduced dissipationinequality:

D := σ · εp + q · α ≥ 0





Assumption 3: elastic domain and yield function

Let the state of the material be described by the generalized stress:

Σ = (σ, q)

Irreversibility of the material response is brought in by requiring that thestate of the material belongs to the convex set:

E ={

Σ ∈ Sym× Rnint

∣∣∣ f (Σ) ≤ 0}

For fixed q, the set Eσ represents the region of admissible states in stressspace.

The interior int{Eσ} is referred to as the elastic domain. The boundary:

∂Eσ ={σ ∈ Sym

∣∣∣ f (σ, q) = 0}

is called the yield surface. The yield surface is convex in stress space.





Assumption 3: elastic domain and yield function





Assumption 4: principle of maximum dissipation

Let the quantities:

Ep := (εp,α) Ep:= (εp, α)

be the generalized plastic strain and generalized plastic strain rate.

From the definitions of generalized stresses and strains, the RDE reducesto:

D := σ · εp + q · α = Σ · Ep

Principle of maximum dissipation. In plastic loading conditions, for agiven generalized plastic strain rate, the actual state of the material isthe one, among all admissible states, for which plastic dissipation Dattain its maximum, i.e.:

D(Σ, Ep) = Σ · Ep = max

Σ∗∈E

{D(Σ∗, E

p)}





Assumption 4: principle of maximum dissipation

According to a standard result in optimization theory, the actual state isthe stationary point of the following unconstrained Lagrangian:

L (Σ, γ) = −Σ · Ep+ γ f (Σ)

subject to the following Kuhn-Tucker optimality conditions:

γ ≥ 0 f (Σ) ≤ 0 γ f (Σ) = 0

The stationary point of L are found by setting to zero its first variationw/r to Σ:

∂L∂Σ

= −Ep+ γ

∂f

∂Σ

which yields:

Ep= γ

∂f

∂Σ





Associative flow rule and hardening law

– Associative flow rule:

εp = γ∂f

∂σ

– Associative hardening law:

α = γ∂f

∂q

– Complementarity (loading/unloading) conditions in K–T form:

γ ≥ 0 , f (σ, q) ≤ 0 , γ f (σ, q) = 0





Associative flow rule: remarks

Ep= γ

∂f

∂Σ

γ ≥ 0 , f (Σ) ≤ 0 , γ f (Σ) = 0

– A consequence of the K–T complementarity conditions is that plasticflow may take place only if the current state is on the yield surface.

– The generalized plastic flow direction Ep

is normal to the yieldsurface. Therefore the flow rule and the hardening law are said to beassociative (alternatively, the material is said to obey to thenormality rule).





Hardening law: remarks

Due to hardening, the yield surface can change in size, shape andposition in stress space.

Isotropic hardening/softening





Hardening law: remarks

Kinematic hardening

When no internal variables are present, the yield surface remains fixed instress space. The material is defined perfectly plastic.





Graphical interpretation of PMD

Max dissipation requires that:

∀ Σ∗ ∈ E ⇒ (Σ−Σ∗) · Ep ≥ 0

This is possible only if the yield surface is convex and the plastic flow isassociative.





Prager’s consistency condition

The K–T complementarity conditions are not sufficient to determinewhether plastic flow will actually occur or not for a given stress state onthe yield surface.

Prager’s consistency condition. In a plastic process, the actual value ofthe plastic multiplier is obtained by prescribing that, upon plastic loading,the stress state must remain on the yield surface.






Mathematically, the consistency condition requires that, if plastic loadingis occurring:

f =∂f

∂Σ· Σ =

∂f

∂σ· σ +

∂f

∂q· q = 0

The quantities σ and q can be obtained from the hyperelastic CEs andthe flow rule:

σ =∂ψ

∂εeσ = Ce

(ε− γ ∂f

∂σ

)Ce :=

∂2ψ

∂εe ⊗ ∂εe

q = −∂ψ∂α

q = −γH∂f∂q

H :=∂2ψ

∂α⊗ ∂α

(cont.)






Substituting:

f =∂f

∂σ· σ +

∂f

∂q· q

=∂f

∂σ· Ce

(ε− γ ∂f

∂σ

)− γ ∂f

∂q·H∂f

∂q= 0

which yields: {∂f

∂σ· Ce ∂f

∂σ+∂f

∂q·H∂f

∂q

}γ =

∂f

∂σ· Ceε

(cont.)






Solving for γ, we have:

γ =1

Kp

∂f

∂σ· Ceε

where:

Kp :=∂f

∂σ· Ce ∂f

∂σ+∂f

∂q·H∂f

∂q=∂f

∂σ· Ce ∂f

∂σ+Hp

Constitutive assumption:

Kp > 0





Plastic modulus

The quantity:

Hp :=∂f

∂q·H∂f

∂q

is called plastic modulus.

– When Hp > 0 the material is hardening (the elastic domainexpands).

– When Hp < 0 the material is softening (the elastic domain shrinks).

– For Hp = 0, the material is in a perfectly plastic condition (theelastic domain remains stationary).

Limit to allowable softening:

Kp > 0 ⇒ Hp :=∂f

∂q·H∂f

∂q> − ∂f

∂σ· Ce ∂f

∂σ





Loading/Unloading conditions

The plastic multiplier cannot be negative. Therefore, from the previousexpression we distinguish 3 possible cases:

∂f

∂σ· Ceε > 0 ⇒ plastic loading process, γ > 0

∂f

∂σ· Ceε = 0 ⇒ neutral loading process, γ = 0

∂f

∂σ· Ceε < 0 ⇒ elastic unloading, γ = 0

Hence:

γ =1

Kp

⟨∂f

∂σ· Ceε

⟩





Graphical interpretation of L/U conditions

Let:

σtr = Ceε (trial stress rate) ⇒ γ =1

Kp

⟨∂f

∂σ· σtr

⟩





Example: 1–d plasticity

Free energy function and hyperelastic behavior.

We assume an uncoupled, quadratic form:

ψ =1

2εeCeεe + 1

2αHα

Thus:

σ =∂ψ

∂εe= Ceεe q = −∂ψ

∂α= −Hα






Yield function:f(σ, q) = |σ| − (σY + q) = 0






Flow rule and hardening law:

εp = γ∂f

∂σ= γ sgn(σ) α = γ

∂f

∂q= −γ

Plastic multiplier:

Kp =∂f

∂σCe ∂f∂σ

+∂f

∂qH ∂f

∂q= Ce +H

γ =1

Kp

⟨∂f

∂σCeε⟩

=Ce

Ce +H〈sgn(σ) · ε〉






Constitutive equation in rate form.

For sgn(σ) · ε > 0 (plastic processes):

σ = Ce (ε− εp)

= Ceε− γ(Ce ∂f∂σ

)= Ceε− 1

Ce +H

(Ce ∂f∂σ

)(∂f

∂σ, Ce)ε =

(CeHCe +H

)ε






σ = Ceε− 1

Ce +H

(Ce ∂f∂σ

)(∂f

∂σ, Ce)ε =

(CeHCe +H

)ε





Non–associative flow rule

There exists a scalar function g(σ, q) (plastic potential), such that:

εp = γ∂g

∂σ





Non–associative hardening law

There exists a pseudo–vector h(σ, q) ∈ Rnint (hardening function) suchthat:

q = γ h (σ, q)

For non–associative behavior, consistency condition yields:

γ =1

Kp

⟨∂f

∂σ· Ceε

⟩

where:

Kp :=∂f

∂σ· Ce ∂g

∂σ− ∂f

∂q· h > 0

Hp := −∂f

∂q· h




Perfect plasticityIsotropic hardening plasticity

Perfect plasticity: Huber–Hencky–von Mises model

f (σ) = dev(σ) · dev(σ)− k2 = 0





Perfect plasticity: Mohr–Coulomb model

f(τ, σn) = τ − c− |σn| tanφ = 0

f(σ1, σ3) = (σ1 − σ3)− 2c cosφ− (σ1 + σ3) sinφ = 0





Perfect plasticity: remarks

Elastic–perfectly plastic models are a rather crude idealization of theactual behavior of geomaterials.

Nonetheless, they are still among the most widely used constitutivemodels in the FE analysis of practical geotechnical problems, especially insituations where site characterization is problematic.





Isotropic hardening: CS models

A radical change in perspective in the application of plasticity to soilmechanics occurred after the work of Roscoe and the Cambridge group,leading to the so–called Critical State Soil Mechanics.

In the language of plasticity, CS models are isotropic hardening modelswith associative flow rule and non–associative volumetric hardening:

– the internal variables reduce to just one scalar quantity;

– yield function and plastic potential coincide;

– its evolution depends only on volumetric plastic strain rate.

Consequence: at failure conditions, the material “flows” at constantstress and constant volume. Such conditions are called Critical States.





Modified Cam–clay model

Ref.: Roscoe & Burland (1968). Suitable for soft to medium–stifffine–grained soils.

Non–linear elastic behavior

p =p

κ∗εev

s = 2G ee

where:

s = dev(σ)

ee = dev(εe)






Yield surface and plastic potential

f(p, q, θ, ps) = p(p− ps) +{

q

M(θ)

}2

= 0

∂f

∂p= 2p− ps

∂f

∂q= 2q/M2 ∂f

∂ps= −p ≤ 0






Volumetric hardening law

Derived from exptl. observations on isotropic compression of clays.

Virgin compression line (VCL)

εv = εv0 + λ∗ logp

p0

Unloading/reloading line(URL)

εv = εv0 + κ∗ logp

p0

For the loading cycle ABA’:

εv = λ∗ psps

; εev = κ∗ psps

(cont.)








For the loading cycle ABA’:

εv = λ∗ psps

; εev = κ∗ psps

Subtracting:

εpv = (λ∗ − κ∗)psps

⇒ ps =ps

λ∗ − κ∗ εpv

= γps

λ∗ − κ∗∂f

∂p

(cont.)








Non–associative hardeninglaw:

ps = γ hs(p, q, θ, ps)

Comparing with the expressionderived for ps we obtain:

hs =ps

λ∗ − κ∗∂f

∂p






Plastic modulus

Hp = −∂f

∂pshs =

1

λ∗ − κ∗pps (2p− ps)





Example of application of CS models

A refined version of MCC: the RMP model (De Simone & T, 2005).Simulation of TX-CD tests on a pyroclastic soil.





Isotropic CS models: remarks

The application of isotropic hardening plasticity to soils – in particularthe development of Critical State soil mechanics – marks the transitionbetween classical and modern soil mechanics (a change of paradigm!).

In spite of their limitations, CS models are capable of reproducingqualitatively, and, in most cases quantitatively, the main features of theinelastic response of granular materials:

– irreversibility;

– pressure dependence;

– dilatant/contractant behavior;

– brittle/ductile transition with increasing confining stress.





Isotropic CS models: remarks

Main advantages of CS models:

– simple mathematical structure;

– limited number of material constants, easily linked to the observedresponse in standard laboratory tests;

– limited pool of internal variables (in most cases just one), of scalarnature: easy definition of initial conditions.




Introduction and definitionsExperimental evidence of anisotropic behaviorConstitutive modeling of anisotropyAnisotropic hardening plasticity for soils

Anisotropy in geomaterials

Almost all geomaterials are characterized to a certain extent by theexistence of some preferential orientations at the microstructural level.

Specimen of diatomite rock (from Boheler 1987)






These preferential orientations can be due to the geological processeswhich led to the formation of the deposit, or be the result of particlerearrangements induced by the applied engineering loads.

In some cases, the existence of particular symmetries at themicrostructural level is apparent (i.e., bedding planes).

In other situations, the detection of preferential directions requires thedefinition of suitable (statistical) indicators which quantify the geometryof grains and the directional properties of their interaction.

For example, in granular materials, such preferential orientation can beassociated to the spatial distribution of intergranular contact normals,grain shape and void shape (Oda et al. 1985).






As a consequence of the directional properties of the microstructure, thephenomenological response of geomaterials may be characterized by amore or less marked anisotropy:

The mechanical response of the material (pre–failure stiffness, shearstrength) depends on the relative orientation of the applied load w.r. tothe microstructure.





Inherent and induced anisotropy

According to Casagrande & Carillo (1944), two different kind ofanisotropic behavior can be distinguished:

a) inherent anisotropy: “[...] a physical characteristic inherent in thematerial and entirely independent of the applied strains.”

b) induced anisotropy: “[...] a physical characteristic due exclusively tothe strain associated with the applied stress”.

Some Authors tend to identify inherent anisotropy as the result of thegeological processes which led to the formation of the deposit, and of thegeometrical properties of the solid grains (e.g., Wong & Arthur 1985).

The above distinction is somewhat arbitrary, as the possibility of alteringthe microstructural properties of a granular material (or a rock) dodepend on the intensity of the applied load.





Influence of anisotropy on soil behavior

An interesting series of experimental data has been obtained in a numberof Directional Shear tests on dense sand performed by Wong & Arthur(1985).

Directional Shear Cell (DSC, Wong & Arthur, 1985)





Influence of anisotropy on soil behaviorAnisotropy due to bedding (inherent anisotropy)

DSC compression tests on dense Leighton Buzzard Sand specimens, tested inplane strain compression with different orientations of the principal directionsw.r. to the bedding plane (after Wong & Arthur 1985).

The angle δ is the inclination between the bedding plane and the max principalstress direction.





Influence of anisotropy on soil behaviorAnisotropy due to previous loading history (induced anisotropy)

DSC compression tests on dense Leighton Buzzard Sand specimens preparedwith the bedding plane coincident with the plane of deformation.

Loading program: A) plane strain compression with fixed principal directions,and unloading up to σ1 = σ3; B) plane strain compression with fixed principaldirections, oriented differently from those of stage A.





Effects of loading history on YS

Stress–path controlled TX tests on dense Aio Sand (Yasufuku et al. 1991).





Effects of loading history on YS

Stress–path controlled TX tests on Pisa clay (Callisto & Calabresi 1998).





General principles

Ref.: Boehler (1987).

Consider a general CE of the form:

σ = F (d, ξ)

in which:

– σ = Cauchy stress

– d = applied “load” (either εe or εp)

– ξ = structure tensor

This can represent an elastic CE, in which case d ≡ εe, or a flow rule, inwhich case d ≡ εp.





MFI and anisotropy

The structure tensor provides (in phenomenological terms) the directionalproperties of the microstructure which are responsible for the anisotropy.The symmetries of the structure tensor coincide with those of themicrostructure.

The principle of Material Frame Indifference requires that:

F (QdQT ,QξQT ) = QF (d, ξ)QT ∀Q ∈ O

This implies that F is an isotropic function of its tensorial arguments.However, this does not mean that F describes an isotropic mechanicalresponse!





Isotropy

A material is isotropic if no changes in the response occur for a givenload, upon an arbitrary transformation of the microstructure:

F (d,QξQT ) = F (d, ξ) ∀Q ∈ O

It can be proven that for an isotropic material:

ξ = ξ1 F (d,QξQT ) = F (d, ξ) F (QdQT ) = QFQT

that is, ξ is an isotropic tensor and F is an isotropic function w.r. to theload d.





Anisotropy: definition

A material is anisotropic if there exist a transformation of themicrostructure which affects the response to a given load:

∃Q ∈ O : σ∗ := F (d,QξQT ) 6= F (d, ξ) = σ

Let S be the group of properly orthogonal transformations that leave theresponse unchanged:

σ∗ := F (d,QξQT ) = F (d, ξ) = σ ∀Q ∈ S

S is defined symmetry group for the material, and corresponds to thegroup on invariant transformations for the structure tensor.





Symmetry group: particular cases

1 General anisotropy:S = {+1,−1}

2 Orthotropy:Sorth = {+1,−1,S1,S2,S3}

3 Transverse isotropy:

Stras = {+1,−1,S1,S2,S3,Rθ (0 ≤ θ ≤ 2π)}

where:S1, S2, S3 = reflections w.r. to the principal directions of d;Rθ (0 ≤ θ ≤ 2π) = rotation around the axis of anisotropy.





Modeling anisotropy of soils in plasticity

In the framework of the theory of plasticity, the description of anisotropyrequires the introduction of one or more structure tensors (or order ≥ 2)as material parameters (inherent) or as internal variables (induced).

The structure tensors can affect:

– the elastic behavior of the material;

– the yield function and plastic potential.

Intrinsic anisotropy is typically described by assuming that the structuretensors remains constant during plastic deformation.

Induced anisotropy is typically described by introducing in the set ofinternal variables one or more symmetric second order tensors as structuretensors, which control the position and the orientation of the yield locus.





Modeling inherent anisotropy

Inherent anisotropy can be described by means of an approach firstproposed by Hill (1950). In this approach, the structure tensors areemployed as metric tensors in the construction of the invariantsappearing in the yield function and plastic potential.

Example. Model by Nova & Sacchi (1982) for transverse isotropicmaterials, constructed starting from MCC model:

f(σ, k) := Z ·CZ − 3k2 = 0 Z :=1

µs+ (p− k)1

where C is the structure metric tensor (in principal anisotropy axis):

C :=

α 0 γ 0 0 00 α γ 0 0 0γ γ 1 0 0 00 0 0 2α 0 00 0 0 0 2β 00 0 0 0 0 2β

(α, β, γ = const.)





Modeling inherent anisotropy

Predictions of Nova & Sacchi (1985) model of experimental data ondiatomite by Allirot & Boehler (1979).





Modeling induced anisotropy

Induced anisotropy is typically described by incorporating a symmetricsecond order tensor (structure tensor) in the set of internal variables.This tensor controls the position and the orientation of the yield locus.

Most plasticity models for soils with anisotropic hardening can begrouped in two broad classes:

a) kinematic hardening models, in which the yield surface can expand,contract and translate in stress space, along the prescribed stresspath;

b) rotational hardening models, in which the yield surface can expand,contract and rotate in stress space, being oriented by the stress pathdirection.





Kinematic hardening models

Ref.: Mroz (1967); Iwan (1967).

f(σ,α, qk) = f(σ, qk) = 0 σ := σ −α





Kinematic hardening models

Several KH models have been proposed for geomaterials since the late 70.See, e.g., Prevost (1977, 1986); Mroz, Norris & Zienkiewicz (1978, 1979,1981); Mroz & Pietruszczak (1983a,b); Hashiguchi (1980, 1981, 1985,1988); Wood and coworkers (1989, 2000); Stallebrass & Taylor (1997)...

The main reason for their development stems from the need to improvethe description of cyclic behavior of soils. These model are quite effectivein modelling hysteresis and such phenomena as the so-called Bauschingereffect.

KH models have also been advocated to reproduce the observednon-linearity at small strain levels and the effects of recent stress historyon the stiffness decay with increasing strains (see Stallebrass 1990).





Rotational hardening models

Ref.: Sekiguchi & Otha (1977); Ghaboussi & Momen (1982).

f (σ,αa, p0) = f (Ia, Ja, Za, p0)

Ia := σ ·αa (Ja)2 := (1/2) sa · sa Za :=√6 tr (da)3 /[tr (da)2]3/2

sa := σ − Ia

3αa da := dev (sa) αa ·αa = 3





Rotational hardening models

RH models can be subdivided in two main classes:

i) RH models with yield surfaces open towards the positive side of thehydrostatic axis (conical YS). Best suited for sands, where plasticvolumetric strains are typically much smaller than deviatoric ones;

ii) RH models with closed YS, suitable for both coarse– andfine–grained soils.

Among the RH models for geomaterials, we mention, e.g., Ghaboussi &Momen (1982); Gajo & Wood (1999a,b); Anandarajah & Dafalias (1985,1986); Banerjee and coworkers (1985,1986); Whittle and coworkers(1994, 1999); di Prisco (1993a,b); Manzari & Dafalias (2004); Karstunenand coworkers.


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