1. Introduction

This elastoplastic behavior relationship, particularly dedicated to granular geo-materials, such as sands, describes the following phenomenological aspects: cohesion, expansion and internal friction. This model is designed especially to deal with monotonous loading paths, up to the ruin of the structure, knowing that the absence of work hardening causes losses in stability, cf. [5], and also in load controllability, and therefore in solutions depending on the mesh.

The original Mohr-Coulomb criterion is a pyramid characterized by the intersection of 6 planes in the space of the main constraints; the main major and minor constraints intervene but not the intermediate constraint. It is not bounded in the direction of the spherical compression stresses.

The edges and the top of the Mohr-Coulomb pyramid constitute difficulties for the resolution algorithm, in particular because this requires management of the evolution mechanisms dedicated to each facet of the pyramid and also for the summit regime, cf. [R7.01.28]. These difficulties eventually cause a lack of robustness. Here we adopt the solution for smoothing (regularizing) the criterion proposed by Abbo and Sloan in [2], and implemented in the environment MFront, cf. [4], using the algorithm developed by [3]. The MFront [4] programming has been modified to add some specificities.

At the top of the Mohr-Coulomb pyramid, the smoothing is constituted by a hyperbolic approximation.

The parameters of the behavioral relationship, in addition to the elastic parameters, which are assumed to be isotropic here, are 5 in number: cohesion \(c\), the angle of friction \({\varphi }_{\mathit{pp}}\) (provided in°), the angle of expansion \(\psi\) (provided in°), which must be less than or equal to the angle of internal friction \({\varphi }_{\mathit{pp}}\), which must be less than or equal to the angle of internal friction, identical to the original Mohr-Coulomb model, and two parameters associated with smoothing: the transition Lode angle \({\theta }_{T}\) (supplied in°), and traction truncation \({a}_{\mathit{tt}}\). The plastic flow is assumed to be unassociated and perfect. If \(\psi ={\varphi }_{\mathit{pp}}\), then the flow is associated and the coherent tangent matrix is then symmetric. We use an implicit integration method direct in time (MFront).

The internal variables are also 4 in number: norm of deviatory deformations (equivalent plastic deformation: the only true internal thermodynamic variable); volume plastic deformation; volume dissipation density; indicator of activation of plasticity (1) or not (0).

This simple, robust model is very used for initial analyses, under monotonous loading, of geotechnical structures.