Solid-fluid dynamics of yield-stress fluids

On the example of two-phase continua experiencing stress induced solid-fluid phase transitions we explore the use of the Euler structure in the formulation of the governing equations. The Euler structure guarantees that solutions of the time evolution equations possessing it are compatible with mechanics and with thermodynamics. The former compatibility means that the equations are local conservation laws of the Godunov type and the latter compatibility means that the entropy does not decrease during the time evolution. In numerical illustrations, in which the one-dimensional Riemann problem is explored, we require that the Euler structure is also preserved in the discretization.Comment: 51 pages, 7 figure

Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations

Continuum mechanics with dislocations, with the Cattaneo type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov type system of the first order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov type formulation brings the mathematical rigor (the well-posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization)

Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity

The aim of this paper is to compare a hyperelastic with a hypoelastic model describing the Eulerian dynamics of solids in the context of non-linear elastoplastic deformations. Specifically, we consider the well-known hypoelastic Wilkins model, which is compared against a hyperelastic model based on the work of Godunov and Romenski. First, we discuss some general conceptual differences between the two approaches. Second, a detailed study of both models is proposed, where differences are made evident at the aid of deriving a hypoelastic-type model corresponding to the hyperelastic model and a particular equation of state used in this paper. Third, using the same high order ADER Finite Volume and Discontinuous Galerkin methods on fixed and moving unstructured meshes for both models, a wide range of numerical benchmark test problems has been solved. The numerical solutions obtained for the two different models are directly compared with each other. For small elastic deformations, the two models produce very similar solutions that are close to each other. However, if large elastic or elastoplastic deformations occur, the solutions present larger differences.Comment: 14 figure

Two-phase hyperbolic model for porous media saturated with a viscous fluid and its application to wavefields simulation

We derive and study a new hyperbolic two-phase model of a porous deformable medium saturated by a viscous fluid. The governing equations of the model are derived in the framework of Symmetric Hyperbolic Thermodynamically Compatible (SHTC) systems and by generalizing the unified hyperbolic model of continuum fluid and solid mechanics. Similarly to the unified model, the presented model takes into account the viscosity of the saturating fluid through a hyperbolic reformulation. The model accounts for such dissipative mechanisms as interfacial friction and viscous dissipation of the saturated fluid. Using the presented nonlinear finite-strain SHTC model, the governing equations for the propagation of small-amplitude waves in a porous medium saturated with a viscous fluid are derived. As in the conventional Biot theory of porous media, three types of waves can be found: fast and slow compression waves and shear waves. It turns out that the shear wave attenuates rapidly due to the viscosity of the saturating fluid, and this wave is difficult to see in typical test cases. However, some test cases are presented in which shear waves can be observed in the vicinity of interfaces between regions with different porosity

First-order hyperbolic formulation of the pure tetrad teleparallel gravity theory

Motivated by numerically solving the Einstein field equations, we derive a first-order reduction of the second-order $f(T)$-teleparallel gravity field equations in the pure-tetrad formulation (no spin connection). We then restrict our attention to the teleparallel equivalent of general relativity (TEGR) and propose a 3+1 decomposition of the governing equations that can be used in a computational code. We demonstrate that for the matter-free space-time the obtained system of first-order equations is equivalent to the tetrad reformulation of general relativity by Estabrook, Robinson, Wahlquist, and Buchman and Bardeen and therefore also admits a symmetric hyperbolic formulation. The structure of the 3+1 equations resembles a lot of similarities with the equations of relativistic electrodynamics and the recently proposed dGREM tetrad-reformulation of general relativity