Plastic Material Modelling

In the literature, there exists a great variety of constitutive models, which are able to deal with most of the observed mechanical behaviours of geomaterials. Many problems

7Solid grains are assumed to be sufficiently ductile allowing for an elasto-plastic deformation on the micro-level, whereas for brittle particles fracture might occur.

3.2 Elasto-Viscoplastic Material Behaviour 43

in geomechanics, such as slope stability, bearing capacity of retaining walls, expansive soil phenomena, and saturated soil liquefaction can be efficiently analysed using plasticity models, cf., e. g., Zienkiewicz et al. [186] for a general overview.

In the current work, the single-surface plasticity model, which is introduced by Ehlers [53, 54] within the finite elasto-plasticity framework, is discussed and applied to simulate different features of liquefaction phenomena in saturated porous media. Here, in analogy to the works by M¨ullersch¨on [134], Scholz [155] and Ehlers & Avci [60], the application of the mentioned plasticity model proceeds from an infinitesimal strain assumption.

Following this, the elements of the considered plasticity model can be recognised as follows:

(1) the yield function that encompasses the elastic domain, (2) the flow rule that describes the evolution of the plastic strain and the loading/unloading criterion in order to distin-guish between the elastic and the plastic stress steps, and (3) the hardening/softening and the failure state. In this connection, two types of hardening behaviours are reviewed, namely, the isotropic hardening that expresses the expansion/contraction of the yield sur-face, and the kinematic hardening that describes the shift of the yield surface depending on the plastic strain history through the deviatoric and the hydrostatic plastic strain directions. In the numerical examples, only the isotropic hardening is applied.

Yield Function

For the treatment of plasticity, a yield surface which enfolds the elastic domain is defined in the principle stress space. Inside this surface, stresses are assumed to be elastic whereas on the yield surface, the stress state produces plastic strains. Unlike single-phase non-porous materials (e. g. metals), granular materials exhibit volumetric as well as deviatoric plastic deformations. Therefore, the yield surface which characterises the onset of plastic deformations should be, among others, a closed surface and a function of the first stress invariant Iσ that represents the confining pressure.

Drucker & Prager [47] introduced one of the earliest pressure-dependent yield surfaces to predict the onset of plastic deformations in soil. Such an open-cone yield surface cannot capture the volumetric plastic deformations under purely hydrostatic loading. One of the solutions to tackle this issue is what has been introduced by, e. g., Vermeer [171].

Therein, a cap-type yield surface that accounts for the plastic volumetric strains and the volumetric as well as the shear hardening has been discussed. However, numerical difficulties might arise when the stress state attains the boundary between the cap and the original yield surface. This dilemma has been solved in the class of closed single-surface yield functions, which are differentiable at any point of the single-surface. A typical shape of such yield surfaces is a closed tear drop with rounded triangle cross sections in the deviatoric plane (Lade [110]). As examples, consider the well-known Cam-Clay and the modified Cam-Clay models, cf., e. g., Jeremi´c et al. [100], Abed [1] and M¨ullersch¨on [134] for a literature review.

In this work, the single-surface yield function (Figure 3.1) developed by Ehlers [53, 54] is used for the treatment of the inelastic granular material behaviour. This yield function is given in terms of the first principal stress invariant Iσ and the second and third deviatoric

-σ₁ -σ₂

-σ₃ -I_σ

Figure 3.1: Single-surface yield function in the principal stress space, cf. Ehlers [53]

stress invariants II^D_σ, III^D_σ as F(Iσ,II^D_σ,III^D_σ) =

s II^D_σ

1 +γ III^D_σ (II^D_σ)^3/2

^m

+ ¹₂αI²_σ + δ²I⁴_σ + βIσ + ǫI²_σ−κ = 0 (3.57) with the invariants

Iσ = σ^S_E ·I, II^D_σ = ¹₂σ^SD_E · σ^SD_E , III^D_σ = ¹₃ σ^SD_E · σ^SD_E σ^SD_E . (3.58) Herein, beside the stress invariants, the yield surface is also a function of two sets of ma-terial parameters, namely, the hydrostatic parameters S^h ={α, β, δ, ǫ, κ}, which control the shape of the yield surface in the hydrostatic plane, and the deviatoric parameters Sd = {γ, m}, which manage the shape of F in the deviatoric plane, cf. Ehlers [57] and Ehlers & Scholz [64] for more details.

Non-Associative Flow Rule and Loading/Unloading Conditions

Based on experimental evidences on granular materials, adoption of an associative flow rule with plastic flow direction perpendicular to the yield surface obviously leads to an overestimated dilation behaviour. Therefore, a non-associative flow rule needs to be for-mulated by introducing a plastic potential function different from the yield function. In this connection, the following potential relation as a function of the first principal stress invariant Iσ and the second deviatoric stress invariant II^D_σ is suggested (cf. Mahnkopf [121]):

G(Iσ,II^D_σ) = q

ψ1II^D_σ + ¹₂ αI²_σ + δ²I⁴_σ + ψ2βIσ + ǫI²_σ − κ (3.59) with ψ1, ψ2 as additional parameters for adjusting the dilation angle. This formulation excludes the third deviatoric invariant III^D_σ and leads to a circular shape in the deviatoric plane. Following this, a constitutive equation for the temporal evolution of

ε

Sp needs to be specified. Therefore, based on the so-called Principle of Maximum Dissipation (PMD), a dissipative optimisation problem is formulated within the potential surface leading to a

3.2 Elasto-Viscoplastic Material Behaviour 45

canonical formulation for (

ε

Sp)^′_S. In particular, theflow rule can be expressed as (

ε

Sp)^′_S = Λ ∂G

∂σ^S_E (3.60)

with Λ being the plastic multiplier. For more details about the PMD, the interested reader is referred to the works by, e. g., Lubliner [119], Simo & Hughes [157] and Miehe [131]. A necessary condition for the solution of the problem in equation (3.60) is the Karush-Kuhn-Tucker (KKT) optimality condition, which can be expressed for the case of rate-independent elasto-plasticity as

F ≤0, Λ≥0, ΛF = 0. (3.61)

In the numerical implementation of the rate-independent elasto-plasticity, instability and ill-posedness might be encountered during, e. g., shear band localisation, cf. Okaet al.[140]

and Ehlers et al. [63]. A possible method to overcome such a difficulty is to introduce a kind of material rate-dependency by use of elasto-viscoplastic models. According to the overstress concept by Perzyna [143], the following viscoplasticity ansatz can be used:

Λ = 1 ηr

DF σ0

Er

. (3.62)

Here, h·irepresent the Macauley brackets defined ash·i:= ¹₂[(·) +|(·)|] , σ0 is a reference stress, r is the viscoplastic exponent, and ηr is the viscoplastic relaxation time. As the aim of the viscosity in this case is to improve the numerical stability, choosing a small value forηr together withσ0 >0 and r= 1 allows for an elasto-viscoplastic model, which behaves very much similar to the elasto-plastic model, but with better stability characters, cf. Haupt [84] or Scholz [155] for more details.

Isotropic Hardening/Softening Evolution Laws

Experiments on dry sand show that isotropic hardening effects appear instantly after loading in triaxial compressional tests, whereas a softening behaviour might be obtained for dense sand after reaching a certain peak stress. Such isotropic hardening and softening attitudes are related to the plastic deformations and can be included in the constitutive model by introducing suitable evolution relations for a subset of the yield function param-eters pi ∈ {β, δ, ǫ, γ}. Hence, the parameters pi are chosen as functions of the dissipative plastic work, cf. de Boer [18].

According to Ehlers & Scholz [64], the evolution relations for the parameters pi are split into volumetric (p^V_i )^′_S and deviatoric (p^D_i )^′_S terms, viz.:

(pi)^′_S = (p^V_i )^′_S+ (p^D_i )^′_S = (p^⋆_i−pi) C_pi^V (ε^V_Sp)^′_S+C_pi^Dk(

ε

^D_Sp)^′_Sk

with pi(t0) =pi0. (3.63) Here, p^⋆_i and pi0 are the maximum and the initial values of pi, respectively, and C_pi^V , C_pi^D are material constants. The suggested relation (3.63) allows to distinguish between a deviatoric, positive term k(

ε

^D_Sp)^′_Sk yielding a plastic hardening behaviour, and a volu-metric part that results in positive (hardening) or negative (softening) values. Thus, the densification and the loosening attitudes in the granular structure are represented.

Kinematic Hardening

Due to the kinematic hardening effects, granular materials such as soils show an anisotropic behaviour under cyclic loading conditions. Collins [38] discussed within the critical state framework a kinematic hardening formulation, in which the plastic free energy function ψ^Sp is expressed in terms of the volumetric and the deviatoric plastic strain. Therein, the yield locus might be shifted and rotated in the principal stress space leading to an anisotropic hardening general case. Another approach is proposed in the work by Lade

& Inelb [112], which allows for rotations and intersections of the yield surfaces in order to achieve a good convergence to the experimental data.

According to, e. g., de Boer & Brauns [19], Ehlers [52] and Brauns [28], porous solid materials exhibit kinematic hardening in the sense of the Bauschinger effect. Thus, in order to simulate such behaviours, an appropriate formulation of the back-stress tensor must be specified. As shown in relation (3.25), the free energy function is decoupled for simplicity into an elasticψ^Se and a plasticψ^Sp part. In this,ψ^Spis employed to formulate the back-stress tensorY^S(see Ehlers [52]) leading to aY^S–

ε

Sprelation⁸, which represents a shift of the yield surface. Having

ε

Spsymmetric in a simple linearY^S–

ε

Sprelation, the back-stress tensor Y^S is also symmetric, and thus, the total effective solid stress tensor σ^S_E can additively be split as

σ^S_E = σ^S_E + Y^S. (3.64)

Kinematic hardening according to (3.64) can be interpreted geometrically as a transla-tion of the yield locus in the principal stress space {σ₁,σ₂,σ₃} to a shifted subspace {σ₁,σ₂,σ₃}, where both tensorsσ^S_E and σ^S_E are symmetric and have the same principal directions.

−σ₁

−σ₂

−σ₃

−σ₁

−σ₂

−σ₃

y^S_E t^S_E t^S_E

P

Figure 3.2: Geometrical interpretation of the back-stress in the principal stress space, cf.

Brauns [28]

In Figure 3.2, t^S_E represents the stress state at a point P(σ₁,σ₂,σ₃) of the principal

8In the literature, the Y^S–

ε

Sp relation could be linear or nonlinear, dependent or rate-independent, depending on the complexity of the kinematic hardening model, cf. [19].

Im Dokument Saturated porous media dynamics with application to earthquake engineering (Seite 58-63)