An Eulerian finite‐volume scheme for large elastoplastic deformations in solids

Abstract

SUMMARYConservative formulations of the governing laws of elastoplastic solid media have distinct advantages when solved using high-order shock capturing methods for simulating processes involving large deformations and shock waves. In this paper one such model is considered where inelastic deformations are accounted for via conservation laws for elastic strain with relaxation source terms. Plastic deformations are governed by the relaxation time of tangential stresses. Compared with alternative Eulerian conservative models, the governing system consists of fewer equations overall. A numerical scheme for the inhomogeneous system is proposed based upon the temporal splitting. In this way the reduced system of non-linear elasticity is solved explicitly, with convective fluxes evaluated using high-order approximations of Riemann problems locally throughout the computational mesh. Numerical stiffness of the relaxation terms at high strain rates is avoided by utilizing certain properties of the governing model and performing an implicit update. The methods are demonstrated using test cases involving large deformations and high strain rates in one-, two-, and three-dimensions.

Mathematical and numerical model for nonlinear viscoplasticity

“…The form of equations used, for example, in Barton et al [17] is different from the present paper. In particular, they use the evolution equation for the effective deformation gradient F supplemented by the evolution equation for the vector div(rF) (here the divergence of a matrix is a vector, each component of which is the divergence of the matrix column).…”

“…An analogous conservative form can also be written for div(rF) [17]. However, a non-conservative form is preferred for numerical purposes.…”

On Computational Modelling Of Strain-Hardening Material Dynamics

“…Specifically we shall focus on a model for solid dynamics in which the thermodynamically compatible balance laws are expressible as a first order system of partial differential equations in divergent form [15,30]. For compressible problems such models are becoming increasingly popular (see for example [1][2][3]6,8,17,20,25,26,33]) over classical empirical variants based upon advection of the deviatoric stresses since numerical methods can be applied that circumvent the requirement to explicitly include artificial viscosity for resolving shocks, thus improving the accuracy of calculations. The proposed functional for the Maxwell relaxation time of tangential stress, governing the limit of allowable elastic deformations due to plastic yield, is derived based upon dislocation kinetics and calibrated for OFHC copper using available experimental data for uniaxial loading tests.…”

Frontiers in Anisotropic Shock-Wave Modeling

“…In the case of isotropy, an isentropic hyperelastic EOS in terms of the invariants of the elastic Greens tensor was considered (Miller and Colella [112]; Barton et al [113]):…”

“…The quantity P (ν) can be described by any form of isotropic EOS. Another form of the isotropic hyperelastic EOS has been put forth (Dorovskii et al [114]; Barton et al [113]; Barton and Drikakis [115]):…”