a b s t r a c tEulerian shock-capturing schemes have advantages for modelling problems involving complex non-linear wave structures and large deformations in solid media. Various numerical methods now exist for solving hyperbolic conservation laws that have yet to be applied to non-linear elastic theory. In this paper one such class of solver is examined based upon characteristic tracing in conjunction with high-order monotonicity preserving weighted essentially non-oscillatory (MPWENO) reconstruction. Furthermore, a new iterative method for finding exact solutions of the Riemann problem in non-linear elasticity is presented. Access to exact solutions enables an assessment of the performance of the numerical techniques with focus on the resolution of the seven wave structure. The governing model represents a special case of a more general theory describing additional physics such as material plasticity. The numerical scheme therefore provides a firm basis for extension to simulate more complex physical phenomena. Comparison of exact and numerical solutions of one-dimensional initial values problems involving three-dimensional deformations is presented.
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.
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