2 MATERIAL MODELS

2.1 Yield criterion for isotropic and layered media

The most versatile yield criterion implemented in Z_Soil.PC is a generic three-parameter plasticity criterion proposed in Menétrey & Willam (1995), from which several classical models can be derived by specialization; it is adopted herein as yield criterion and for matrix failure. The yield surface is formulated as follows:

(1)

A, B, C, and m are parameters, function of f_t, the uniaxial tensile strength, f_c, the uniaxial compressive strength and e, the exentricity of the criterion. ξ and ρ are stress invariants, θ is Lode’s angle. The criterion can be specialized to several of the classical criteria, as illustrated in table 1.

Alternative definitions of the parameters are possible: in terms of C, the soil cohesion and φ, the friction angle, or in terms of Drucker-Prager material parameters. A size adjustment, dependent on the stress state, is usually necessary to obtain ultimate loads, which match the ones obtained with a Mohr-Coulomb criterion (Z_Soil 2002).

2.2 Multilaminate model

Layered media require even more advanced models, like the one described in Truty & al. (1997), Commend & al. (1998), and Z_Soil (2002), and adapted

from Zienkiewicz & Pande (1977) and Sharma & Pande (1988).

Table 1: Menétrey-Willam (M-W) parameters specializations

Up to three weakness plane orientations, which remain fixed in space, can be introduced in the proposed model. Each is characterized by a cohesion C_i, a friction angle φ_i and a dilatancy (non-associated) angle ψ_i, like any Coulomb material. A tensile cut-off can also be specified, with f_ti the maximum tensile stress. This leads to a multisurface plasticity problem which requires plasticity conditions to be simultaneously fulfilled by any stress state in the material. The flow rule is governed by a flow potential, with usually a non-associative flow rule. Plastic strains occur due to the violation of any of the plastic conditions by the elastic trial stress. The total plastic strain is the sum of each plane’s contribution. Perfectly elastic-plastic behavior (no hardening) is assumed. This leads to the following constitutive equations for the multilaminate model:

(2)

(3)

where σ is the stress, ε the strain, an upper dot indicates an incremental value, index “p” indicates a plastic component; the plastic flow is governed by equation (3), it is coaxial with the normal to the plastic potential and proportional to the plastic multiplier. The yield “F” and (un)loading conditions for each plane i = 1, ..., 3 are:

(4)

Computation of stresses requires a generalized stress-point algorithm (Truty & Zimmermann 1997, Szarlinski & Truty 1996).

2.3 Validation test: single 3D element with two sets of joints.(Commend & al. 1998)

Figure 1 shows the geometry of this 3D test with the position of the two sets of joints.

Figure 1: One element 3D test, geometry and loading

Material characteristics are as follows: rock characteristics: C_r = 10 kN/m², φ_r = 30º (Drucker-Prager specialization of MW criterion, with size adjustment through external edges); joint 1 characteristics: C_j1 = 5 kN/m², φ_j1 = 20º, ψ_j1 = 13.33º; joint 2 characteristics: C_j2 = 12 kN/m², φ_j2 = 5º, ψ_j2 = 3.33º.

A first analysis is performed without the presence of joints, i.e. in homogeneous, isotropic rock . This gives an ultimate load of σ_1rf = 50 kN/m² , which corresponds to the value of σ₁ for which the Mohr circle is tangent to the Mohr-Coulomb law defined by C_r = 10 kN/m² and φ_r = 30º (Figure 2).

When the first set of lamina is added to the model, with an inclination angle varying between 0º £ β£ 90º, failure will occur in the lamina for β between limit angles β_Min and β_Max , which can be evaluated from the Mohr circle (Parry 1995). Outside this interval, failure occurs in the rock matrix and is governed by the MW criterion (σ_1rf = 50 kN/m²).

Figure 2: Mohr circle for the test with two sets of joints

As illustrated in figure 2 we find β_Max,1 = 80.6º, β_Min,1 = 29.4º, β_Max,2 = 72.7º and β_Min,2 = 22.3º .

Four cases are analyzed next: A. Activation of the first set of joints only (0º £ β₁£ 90º), B. Activation of the second set of joints only (0º £ β₂£ 90º), C. Activation of both sets of joints (β₁= 32.5º; 0º £ β₂£ 90º), D. Activation of both sets of joints (β₂= 40º; 0º £ β₁£ 90º). Figure 3 shows the results for these four cases; σ_1f corresponding to vertical stress at failure plotted against the inclination angle of lamina. It can easily be seen that numerical simulation reproduces what can be predicted from the inspection of Mohr circles.

Figure 3. σ_1f = f(β) , failure stress function of orientation