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Exact and approximate solutions of Riemann problems in nonlinear elasticity. (English) Zbl 1172.74032
Summary: Eulerian shock-capturing schemes have advantages for modelling problems involving complex nonlinear 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 nonlinear 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, we present a new iterative method for finding exact solutions of the Riemann problem in nonlinear elasticity. Access to exact solutions enables an assessment of the performance of 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. We present a comparison of exact and numerical solutions of one-dimensional initial values problems involving three-dimensional deformations.

MSC:
74J40 Shocks and related discontinuities in solid mechanics
74B20 Nonlinear elasticity
74S10 Finite volume methods applied to problems in solid mechanics
Software:
HE-E1GODF; EISPACK
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