Method of Variable Material Properties for Small Elastoplastic Deformations

This work presents a mesh-free formulation that combines the method of variable material properties (MVMP) with a collocation method based on finite atomic basis functions (ABF) for the analysis of small elastoplastic deformations. Unlike existing approaches that update only the tangent modulus of material hardening, the presented method simultaneously updates the elastic modulus E and Poisson’s ratio ν (i.e., Lamé constants λ and μ) as smooth fields. This reduces the nonlinear problem to a sequence of linear elasticity problems. The algorithm is implemented using a strong formulation and the finite Fup4 basis functions from the class of algebraic ABFs. Fup4 is an infinitely differentiable function that exactly reproduces polynomials up to the fourth degree, maintains the continuity of higher derivatives, and thus ensures numerical stability and fast convergence of the collocation-based MVMP procedure. Due to the compact support of the basis functions and the choice of the positions of the collocation points, the matrix of the equations system retains its band form throughout all iterations, which improves the conditioning of the system and accelerates convergence. The accuracy of the proposed approach has been verified on two classical benchmark problems with analytical solutions: a one-dimensional bar under axial load and a thin cylindrical disc subjected to internal pressure.