A scaled boundary finite element method study on the elastoplastic contact problem based on B-differentiable Newton method

In this paper, an algorithm based on the scaled boundary finite element method (SBFEM) and the B-differentiable Newton method (BDNM) is proposed to solve the elastoplastic contact problems under small deformation and small strains. Due to the introduction of the contact flexibility matrix, the contact equations are expressed in the form of B-differentiable equations, which consist only of contact conditions in normal and tangential directions, and are independent of displacement and stress fields. To decouple the contact and material nonlinearities, we consider the governing equations as the combination of elastoplastic equations and elastic contact equations, which are solved separately by adopting the two-layer nested iteration strategy. In the external iteration, the contact forces are frozen, and the elastoplastic equations are solved by Newton method to update the displacement and stress fields, while the elastic contact equations are solved by the BDNM to derive the contact forces in the internal iteration, where the stiffness matrix remains constant. The proposed algorithm satisfies the contact conditions accurately without introducing additional variables or assumptions. Moreover, the polygons and quadtree elements are employed in mesh generation. To account for the elastoplastic properties of material, a polynomial function is adopted to locally approximate the constitutive matrix, where the stiffness matrix and the residual load vector are expressed as integrals of matrix powers to achieve the high-accurate analytical computation. Numerical examples demonstrate the accuracy of the developed algorithm for both static and dynamic problems via comparison with ANSYS, where the results involving displacements, contact forces, stress, and mesh convergence are presented.