Classes of constitutive models for landslide modelling. Failure loci In many occasions, the behaviour of the material can be considered elastic up to failure. In that case, failure coincides with the plastic yield limit and there is no post-yield behaviour, that is to say no hardening or softening of the material. In addition to the elastic component, a simple yield criterion completes in that case the idealised description of a geomaterial as elastic-perfectly plastic. In addition to Xxxxxxx, who in 1773 proposed the simple yet effective criterion which is still used today, we have the failure loci proposed by Xxxxxx (1864) and Xxx Xxxxx and Xxxxx (1913), which are pressure independent. In the family of pressure dependent yield surfaces, the Xxxxxxx-Xxxxxx (1952) surface is also very common. It can be viewed as an extension of the Xxxx-Coulomb pyramid. The surface in the space of principal stresses is a cone. Pressure independent loci can be applied only to purely cohesive soils in conditions of fast undrained loading. In general, it is much more reliable to perform a coupled analysis with a Xxxx Coulomb or Xxxxxxx-Xxxxxx criterion. One word of caution is worth addressing to the modeller dealing with the Xxxx-Coulomb and Xxxxxxx-Xxxxxx failure criterion. As shown in Figure 3 in an isotropic plane, the Xxxx Coulomb failure criterion represents a hexagon while that of Xxxxxxx-Xxxxxx is a cricle. If the analyst is given the friction angle, he will have to choose a particular value of the Xxxxxxx Xxxxxx parameter controlling the cone angle in such a way that it coincides with the Xxxx Coulomb criterion under specific conditions. It is possible to choose a cone coinciding with the outer vertexes of the hexagon, (outer cone), or with the inner (inner cone) or an intermediate position for plane strain problems. However, the choice of the outer cone results in higher friction angle in extension which in some cases may even be physically unreasonable. Classical plasticity models We will define by Classical Plasticity models those using yield surfaces which coincide with the failure loci. The size of the yield surface is controlled by the cohesion and the angle of friction in the case of Xxxx Coulomb and Xxxxxxx-Xxxxxx criteria. The important point here is that softening can be introduced just by using suitable laws describing how these strength parameters decrease with the plastic deviatoric strain, for instance. In this way, localized failure can be reproduced. Another important aspect influencing failure is the dilat...
