A new Updated Reference Lagrangian Smooth Particle Hydrodynamics algorithm for isothermal elasticity and elasto-plasticity

This paper presents a new Updated Reference Lagrangian Smooth Particle Hydrodynamics (SPH) algorithm for the analysis of large deformation isothermal elasticity and elasto-plasticity. Taking as point of departure a Total Lagrangian setting and considering as referential configuration an intermediate configuration of the deformation process, the equation of conservation of linear momentum and three geometric conservation laws (for the deformation gradient, its cofactor and its determinant) are re-written leading to a very generic system of first order conservation laws. The key feature of the formulation is a suitable multiplicative decomposition of the conservation variables, leading to a very simple final set of equations with striking similarities to the conventional Total Lagrangian system albeit re-written in terms of alternative Referential Updated conservation variables which are evolved in time. Taking advantage of this new Updated Reference Lagrangian formalism, a second order entropy-stable SPH upwinding stabilisation method will be introduced. With respect to previous publications by the group, a new three-stage Runge–Kutta time integration method is implemented in order to increase the CFL stability restriction. Finally, and to demonstrate the robustness and applicability of the methodology, a wide spectrum of challenging problems will be presented and compared, including some benchmark three-dimensional large deformation elasto-plasticity problems. To facilitate its ease of implementation, the paper explores the use of a series of novel expressions for the evaluation of kernels and the gradients of kernels to provide the SPH user the freedom to choose amongst various options, all leading to equally convincingly robust results