Exact and approximate solutions of Riemann problems in non-linear elasticity

Abstract

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.

On Computational Modelling Of Strain-Hardening Material Dynamics

“…The origin of these artefacts is unequivocally attributable to entropy errors that manifest as a result of the numerical methods, and have been observed elsewhere [1,33] for similar testcases (also known as overheating errors due to the consequential increase in temperature). Whilst we do not set out here to modify the numerical methods to eliminate the overheating problem, we shall provide evidence to support the argument that these are directly responsible for the observed anomaly in strain-hardening.…”

“…(5.7) are evaluated from solutions of the Riemann problem arising as a result of the jump in cell averaged data at cell boundaries. A linearised Riemann solver is used [1] with increased spatial accuracy achieved through reconstruction of the piecewise constant data using a third-order weighted essentially non-oscillatory (WENO) method [18]. Near boundaries, interfacial fluxes are found through solution of the appropriate multi-component Riemann problem incorporating the relative boundary conditions (e.g.…”

“…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.…”

A conservation law formulation of nonlinear elasticity in general relativity

“…We also show that (17), together with just ∇ b T ab = 0, as used in the Newtonian formalisms [12][13][14]21], is strongly hyperbolic but not symmetric hyperbolic.…”

“…To validate our Newtonian code, and the Newtonian limit of our relativistic code, we have compared our results to two previously published studies ( [12] and [13]). These results use the Newtonian theory and the mixed framework outlined in Appendix D.…”