Hyperbolic equations of Maxwell's nonlinear model of elastoplastic heat-conducting media

Romenskii, E. I.

doi:10.1007/bf00971761

Cited by 19 publications

(19 citation statements)

References 3 publications

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“…Eventually, we mention a few examples in which equations (38) were used, see also more details in Section 6. Thus, equations (38a), (38b), were used in [38,85,41,4,23,3,14] for modeling of elastic and elastoplastic deformations in metals, while the same equations constitute the unified formulation of continuum mechanics proposed in [77,17,78,18] and can be used for modeling of fluid flows. Equations (38a)-(38c) were used in [86,18] to describe the electrodynamics of moving medium, equations (38a), (38e) and (38f) can be used to model multiphase flows, heat conduction, superfluid helium flows, see [86,83,82,81].…”

Section: Energy Formulationmentioning

confidence: 99%

“…The reversible part, the compressible Euler equations, is of course implied by the SHTC and GENERIC formalisms. Taking for example Poisson bracket (79), the implied evolution equations are equations (85), which represent the Euler equations.…”

Section: Navier-stokes Dynamicsmentioning

confidence: 99%

“…Thus, evolution of the distortion matrix A gives, in fact, exact evolution of the Burgers tensor, defining dynamics of dislocations [39,42,41]. The evolution equation of the distortion matrix coupled with mass, momentum and energy conservation, thus represents an Eulerian framework covering elastic and elasto-plastic solids as well as fluids, see also [77,17,78,85,4,14,23]. In practice, the transition between the elastic and plastic response of solids is governed by the dependence of the strain relaxation time on the state parameters, τ = τ (A, B, D, ρ, s), see details in [78,85,4,14,23].…”

Section: Elastic and Elastoplastic Solidsmentioning

confidence: 99%

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Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

Peshkov

Pavelka

Romenski

et al. 2018

Continuum Mech. Thermodyn.

205

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SpoilerContinuum 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).

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Section: Energy Formulationmentioning

confidence: 99%

Section: Navier-stokes Dynamicsmentioning

confidence: 99%

Section: Elastic and Elastoplastic Solidsmentioning

confidence: 99%

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Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

Peshkov

Pavelka

Romenski

et al. 2018

Continuum Mech. Thermodyn.

205

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“…The equation governing J -and its contribution to the system's total energy -are derived from Romenski's model of hyperbolic heat transfer, (see [40,57]). These concepts were later implemented in [56,53].…”

Section: The Model Of Godunov Peshkov and Romenskimentioning

confidence: 99%

A unified Eulerian framework for multimaterial continuum mechanics

Jackson

Nikiforakis

2020

Journal of Computational Physics

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A framework for simulating the interactions between multiple different continua is presented. Each constituent material is governed by the same set of equations, differing only in terms of their equations of state and strain dissipation functions. The interfaces between any combination of fluids, solids, and vacuum are handled by a new Riemann Ghost Fluid Method, which is agnostic to the type of material on either side (depending only on the desired boundary conditions).The Godunov-Peshkov-Romenski (GPR) model is used for modeling the continua (having recently been used to solve a range of problems involving Newtonian and non-Newtonian fluids, and elastic and elastoplastic solids), and this study represents a novel approach for handling multimaterial problems under this model.The resulting framework is simple, yet capable of accurately reproducing a wide range of different physical scenarios. It is demonstrated here to accurately reproduce analytical results for known Riemann problems, and to produce expected results in other cases, including some featuring heat conduction across interfaces, and impact-induced deformation and detonation of combustible materials. The framework thus has the potential to streamline development of simulation software for scenarios involving multiple materials and phases of matter, by reducing the number of different systems of equations that require solvers, and cutting down on the amount of theoretical work required to deal with the interfaces between materials.

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“…The evolution equation for J and its contribution to the energy of the system are derived from Romenski's model of hyperbolic heat transfer, originally proposed in Malyshev and Romenskii [19], Romenski [26], and implemented in Romenski et al [25,24]. In this model, J is effectively defined as the variable conjugate to the entropy flux, in the sense that the latter is the derivative of the specific internal energy with respect to J. Romenski remarks that it is more convenient to evolve J and E than the heat flux or the entropy flux, and thus the equations take the form given here.…”

Section: The Gpr Modelmentioning

confidence: 99%

A fast numerical scheme for the Godunov–Peshkov–Romenski model of continuum mechanics

Jackson

2017

Journal of Computational Physics

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A new second-order numerical scheme based on an operator splitting is proposed for the Godunov-Peshkov-Romenski model of continuum mechanics. The homogeneous part of the system is solved with a finite volume method based on a WENO reconstruction, and the temporal ODEs are solved using some analytic results presented here. Whilst it is not possible to attain arbitrary-order accuracy with this scheme (as with ADER-WENO schemes used previously), the attainable order of accuracy is often sufficient, and solutions are computationally cheap when compared with other available schemes. The new scheme is compared with an ADER-WENO scheme for various test cases, and a convergence study is undertaken to demonstrate its order of accuracy.

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Hyperbolic equations of Maxwell's nonlinear model of elastoplastic heat-conducting media

Cited by 19 publications

References 3 publications

Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

A unified Eulerian framework for multimaterial continuum mechanics

A fast numerical scheme for the Godunov–Peshkov–Romenski model of continuum mechanics

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