Keywords: stress update, return mapping, consistent constitutive matrix, Mohr-Coulomb yield criterion, non-linear FEM.

The paper presents an improved method for updating the stresses in finite element calculations involving Mohr-Coulomb plasticity. The method belongs to the class of methods called "return mappings", or "backward Euler schemes", which are often used for the integration of stress increments in plastic material behaviour. The key point is a transformation from the general six-dimensional stress space into the three-dimensional principal stress space, where the mathematical manipulations simplify those based on a geometrical interpretation. The efficiency of the method is illustrated through numerical examples.

The return-map algorithm starts from an elastic stress prediction based on the current stress state and a finite strain increment. The predictor stress is then returned to the yield surface through an iteration procedure in which the first and second derivatives of the yield surface with respect to the stress components are required, see e.g. [1]. In the general case with six independent stress components the calculation is somewhat cumbersome for Mohr-Coulomb plasticity, and a transformation into the principal stress directions simplifies the calculation as the criterion is formulated directly in the principal stresses. The key observation is that the return from the predictor stress will leave the principal stress directions unchanged during the iteration. The stress state at the yield surface and the corresponding material stiffness matrix can afterwards be transformed back to the original system by standard methods.

The Mohr-Coulomb yield criterion consists in the general case of six independent yield surfaces, which gives intersection curves and an apex. The return of the predictor stress must identify the correct surface and this is very complicated in the general stress space whereas in the principal stress space it is fairly simple including the identification of intersections and apex.

In order to have a quadratic rate of convergence of the global iteration the so-called consistent constitutive matrix is applied. The calculation is in the general stress space quite complicated, but in the principal stress space the calculations can be simplified significantly. For stress points at the intersection curves it is quite difficult in the general case to include two or more independent plastic strain directions, but again in the principal direction space it can be given a straight geometric interpretation.

The method works in both associated and non-associated plasticity, and numerical examples are presented to illustrate the efficiency.

1

M.A. Crisfield, "Non-Linear Finite Element Analysis of Solids and Structures, vol 2: Advanced Topics", John Wiley & Sons, 1997.

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