Abstract
Two numerical algorithms for solving elastoplastic problems with the finite element method are presented. The first deals with the implementation of the return mapping algorithm and is based on a fixed-point algorithm. This method rewrites the system of elastoplasticity non-linear equations in a form adapted to the fixed-point method. The second algorithm relates to the computation of the elastoplastic consistent tangent matrix using a simple finite difference scheme. A first validation is performed on a nonlinear bar problem. The results obtained show that both numerical algorithms are very efficient and yield the exact solution. The proposed algorithms are applied to a two-dimensional rockfill dam loaded in plane strain. The elastoplastic tangent matrix is calculated by using the finite difference scheme for Mohr–Coulomb’s constitutive law. The results obtained with the developed algorithms are very close to those obtained via the commercial software PLAXIS. It should be noted that the algorithm’s code, developed under the Matlab environment, offers the possibility of modeling the construction phases (i.e., building layer by layer) by activating the different layers according to the imposed loading. This algorithmic and implementation framework allows to easily integrate other laws of nonlinear behaviors, including the Hardening Soil Model.
Highlights
The use of numerical codes that account for plasticity is essential when designing geotechnical structures, as they are functional decision-making tools
The implementation of a good plasticity model requires the development of powerful algorithms to resolve the numerical difficulties arising from complex plasticity laws such as those used for soil modeling
The classical Newton–Raphson scheme is very often used to solve the nonlinearities of the plasticity models since it converges at the second order, but under the condition that the initial solution is close to the true solution
Summary
The use of numerical codes that account for plasticity is essential when designing geotechnical structures, as they are functional decision-making tools. Simo and Hughes [4] described the theoretical foundations of several formulations and algorithms for plasticity based on the Newton–Raphson procedure They show that the construction of the elastoplastic tangent matrix, which is closely related to the rate of convergence in the Finite Element Method (FEM) and guarantees the consistency of the algorithmic integration process [5], is a crucial step for the robustness of these algorithms, the difficulty of implementing certain constitutive. The resulting finite element code offers the flexibility to integrate other much more complex constitutive laws, those whose plasticity criteria evolve during loading (such as the Hardening Soil Model [6]) It can be coupled with parameter calibration algorithms. The organization of the rest of this paper is as follows: Section 2 presents the formulation of the plasticity problem, as well as the return mapping algorithm for 1D and
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