A New Updated Reference Lagrangian Smooth Particle Hydrodynamics Framework for Large Strain Solid Dynamics and its Extension to Dynamic Fracture

Abstract

This work presents a new updated reference Lagrangian Smooth Particle Hydro-dynamics algorithm for the analysis of large deformation by introducing a novel system of first order conservation laws. Both isothermal and thermally-coupled sce-narios are considered within the elasticity and elasto-plasticity domains. Taking as point of departure a total Lagrangian setting and considering as referential config-uration an intermediate configuration of the deformation process, the equation of conservation of linear momentum and three geometric conservation laws (for the de-formation gradient, its cofactor and its determinant) are rewritten leading to a very generic (incremental) system of first order conservation laws, which can be degen-erated into a total Lagrangian system or into a purely updated Lagrangian system. 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 rewritten in terms of incremental updated conservation variables which are evolved in time. Taking advantage of this new updated reference Lagrangian formalism, a second order (in space and time) entropy-stable upwiding stabilisation method derived by means of the use of the Rankine Hugoniot jump conditions is introduced. No ad-hoc algo-rithmic regularisation procedures are needed. To demonstrate the robustness and applicability of the methodology, a wide spectrum of challenging problems are pre-sented and compared, including benchmarks in hyperelasticity, elasto-plasticity and dynamic fracture problems. A new dynamic fracture approach is proposed in this work. The spark for fracture is based on the maximum principal stresses. Once fracture takes place, the particle is split into two new particles and post-fracture velocities and deformation gradients are computed locally, ensuring conservation of mass, linear momentum and total energy. The work explores the use of a series of novel expressions for the evaluation of kernels and the gradients of kernels, all lead-ing to equally robust results and circumventing the issues faced by classic isotropic (spherical) kernels in the presence of strong anisotropic changes in volume.