and a Hoek-Brown
type yield surface, more appropriate for rocks.3. Constitutive models
Z_SOIL offers a wide choice of constitutive models for the continuum, which include elasticity and several plasticity models like Mohr-Coulomb, Von Misès and Drucker-Prager, the last one coupled with a cap-closure, essential for nonlinear consolidation. A large variety of soils, and even concrete, can be adequately represented by these models. A tension cut-off for materials with limited tensile resistance is also included in most models.
In addition, new developments include a multilaminate model for layered
media, Rankine plasticity
and a Hoek-Brown
type yield surface, more appropriate for rocks.
3.1 Plasticity
The basic plastic models, i.e Mohr-Coulomb and Drucker-Prager require
only the elastic constants E (Young's modulus), 
(Poisson's ratio), the cohesion C and friction angle 
.
Mohr-Coulomb criterion (M-C)
The Mohr-Coulomb surface is defined by two-constants C and
. The equation of the criterion in two dimensional stress space is:
(3.1)The corresponding surface in three-dimensional stress space is characterized by a cone with vertices in the deviatoric cross section. For convenience this surface is replaced by a smooth surface illustrated in figure 3.1 [MEN, 1995] and defined by equation (3.2a) to (3.2.e):
and I1 and J2 are the fundamental stress invariants.
Fig. 3.1 - Smooth Mohr-Coulomb surface [MEN, 1995]
The plastic flow direction is an essential component of the plastic model. The most appropriate flow also may depend on the type of analysis (deformation or ultimate load). Default options are provided to help the user, which for ultimate load analysis will lead to a crisp and reliable capture of failure mechanisms.
Drucker-Prager criterion (D-P)
The Drucker-Prager criterion (Eq. 3.3) is more convenient from the point
of view of numerical efficiency, it is therefore often prefered to the
Mohr-Coulomb criterion. From the comparizon of both D-P and M-C criteria
it is obvious that different size adjustments are possible which correspond
to different matching of the Mohr-Coulomb parameters C and
with the Drucker-Prager parameters a and k, and this selection obviously
affects the yield stress (Fig. 3.2):
Fig. 3.2 - Deviatoric section of Mohr-Coulomb criterion and Drucker-Prager with two different size adjustments
(3.3)I1 and J2 are stress invariants and constants a,
k can be defined from common geotechnical data : cohesion C and angle of
friction
; the adjustments proposed in [Z_SOIL] are:
The plane strain adjustment corresponds to an adjustment of failure loads. The axisymmetric adjustment results from a parametric study of the axisymmetric footing problem shown in section 6.
Cap model
A cap model coupled to the Drucker-Prager criterion is provided for the
simulation of nonlinear consolidation. An application of the model is described
in section 6. The model requires three additional
data from an oedometer test : the initial void ratio e0, the
vertical stress at yield and 
, the compression index which characterizes plastic hardening (Fig.
3.3).
Fig. 3.3 - Drucker-Prager Model with cap closure
Multilaminate model
One to three weakness planes orientations can be introduced which will
remain fixed in space. Each is characterized by a cohesion ci,
a friction angle and a dilatancy (non-associative angle 
i). A tensile cut-off can be specified with fti
the maximum tensile stress. On each plane separately, the Mohr-Coulomb plasticity
condition and the tension cut-off condition must be fulfilled.
Fig. 3.4 - Weakness plane plasticity conditions, yield function and flow potential isolines
Plasticity and flow rule conditions can be derived for each plane i=1, ..., 3 :
Additional features
Additional features include multilaminate media combined with plastic matrix behavior and contact elements.
Fig. 3.5 - Material input screen (general view)
Fig. 3.6 - Material input screen (Drucker-Prager parameters)