Applications of the Finite Points Method in the Strain-Gradient Plasticity

Abstract

Abstract. In this paper, the meshless Finite Points Method (FPM) is used for numerical simulation of one and two-dimensional elastoplastic problems, which develop a softening response after to exceed the elastic range. In such problems, a pathological behaviour of the solutions is induced by the ill-conditioning of the partial differential equations after reaching the yield limit, producing the localization of deformations.Specifically, this work presents the implementation of a regularization technique through the enrichment of the constitutive equations using strain gradients, with a characteristic length based on a non-local plastic strain formulation in order to maintain the ellipticity of the differential equations.The strain localization phenomenon is replicated objectively in the computational simulation considering an isotropic Von Mises (J2) softening model. This localization phenomenon is induced weakening a region of the material and also developing problems with geometrical discontinuities.The FPM performs the domain's discretization in a finite number of points in each of which the punctual collocation of the equations is performed. For these reasons the FPM applies the strong formulation, allowing the use of high-order differentiability shape functions (C2 class or higher). This features can improve the computational cost because of the same shape functions are used in order to approximate non-local strain and the displacement nodal fields. Both fields are obtained by using the iterative Newton-Raphson method. To ensure convergence of the iterative method, the algorithmic tangent operator is obtained by Perturbation Method.The theory is developed for one-dimensional geometries, being extended to two and three-dimensional problems.In order to validate the proposal, a benchmarking is developed with typical problems extracted from literature.