A conservative Eulerian formulation of the equations for elastic flow

Plohr, Bradley J.; Sharp, David H.

doi:10.1016/0196-8858(88)90025-5

Cited by 77 publications

(75 citation statements)

References 2 publications

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“…The Plohr and Sharp formulation [5] consists in viewing equation (3) as a conservation law in order to write the following homogeneous system of partial differential equations :…”

Section: The Hyperelastic Dynamic Continuum Problemmentioning

confidence: 99%

A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

Renaud¹,

Heuzé²

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. The material point method is extended in this work to the Discontinuous Galerkin approximation framework for the simulation of impacts on elastic and hyperelastic solids. The formulation is based on the weak form of conservation laws on each cell of an eulerian grid in which volume integrals are discretized on a set of material points lying in that cell, and on the computation of Godunov fluxes at cells faces. The resulting method is first derived within the small strains framework and illustrated on a one-dimensional and a two-dimensional problem of impact on an elastic media. Then a one-dimensional hyperelastic problem of a solid undergoing large strains is presented, and a comparison is performed with an analytical solution.

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“…The Plohr and Sharp formulation [5] consists in viewing equation (3) as a conservation law in order to write the following homogeneous system of partial differential equations :…”

Section: The Hyperelastic Dynamic Continuum Problemmentioning

confidence: 99%

A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

Renaud¹,

Heuzé²

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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“…One of these [14,15,21] is fully Eulerian and arises naturally as the Newtonian limit of our relativistic framework. The other [16][17][18] mixes Eulerian and Lagrangian points of view and gives rise to more complicated evolution equations. For completeness, we have proved that the two frameworks are equivalent in their weak form, and hence that the weak form of the second framework is also correct.…”

Section: Discussionmentioning

confidence: 99%

“…We now show explicitly that ∇ a j a = 0 is a linear combination of the evolution equations (18) for ψ A i , that is, the kinematic evolution equations with the hyperbolicity fix. Contracting (18) with F i A , the matrix inverse of ψ A i , and using the matrix identity δ(ln…”

Section: Matter Space Metric and Particle Number Currentmentioning

confidence: 99%

“…In Appendix C we derive the Newtonian limit of our framework. Other Newtonian papers [17,18] use the the 3 × 3 matrix inverse of ψ A i , which we shall denote by F i A , as the primary variable (denoted there by F ). We review this alternative framework in Appendix D. In the Newtonian literature, F i A is commonly called the (Lagrangian) deformation gradient, and ψ A i the inverse deformation gradient.…”

Section: Kinematicsmentioning

confidence: 99%

“…One of two existing Newtonian formalisms [14,15] is essentially the Newtonian limit of our formalism. We derive the Newtonian limit in Appendix C. In Appendix D we derive the equations of an alternative Newtonian formalism [16][17][18], and prove that the two have the same weak solutions.…”

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confidence: 99%

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A conservation law formulation of nonlinear elasticity in general relativity

Gundlach¹,

Hawke²,

Erickson³

2011

Class. Quantum Grav.

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We present a practical framework for ideal hyperelasticity in numerical relativity. For this purpose, we recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime. The resulting equations are presented as an extension of the standard Valencia formalism for a perfect fluid, with additional terms in the stress-energy tensor, plus a set of kinematic conservation laws that evolve a configuration gradient ψ A i. We prove that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the unsheared state. We discuss the Newtonian limit of our formalism and its relation to a second formalism also used in Newtonian elasticity. We validate our framework by numerically solving a set of Riemann problems in Minkowski spacetime, as well as Newtonian ones from the literature. CONTENTS

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A higher‐order Godunov method for modeling finite deformation in elastic‐plastic solids

Trangenstein

Colella

1991

Comm Pure Appl Math

117

105

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In this paper we develop a first‐order system of conservation laws for finite deformation in solids, describe its characteristic structure, and use this analysis to develop a second‐order numerical method for problems involving finite deformation and plasticity. The equations of mass, momentum, and energy conservation in Lagrangian and Eulerian frames of reference are combined with kinetic equations of state for the stress and with caloric equations of state for the internal energy, as well as with auxiliary equations representing equality of mixed partial derivatives of the deformation gradient. Particular attention is paid to the influence of a curl constraint on the deformation gradient, so that the characteristic speeds transform properly between the two frames of reference. Next, we consider models in rate‐form for isotropic elastic‐plastic materials with work‐hardening, and examine the circumstances under which these models lead to hyperbolic systems for the equations of motion. In spite of the fact that these models violate thermodynamic principles in such a way that the acoustic tensor becomes nonsymmetric, we still find that the characteristic speeds are always real for elastic behavior, and essentially always real for plastic response. These results allow us to construct a second‐order Godunov method for the computation of three‐dimensional displacement in a one‐dimensional material viewed in the Lagrangian frame of reference. We also describe a technique for the approximate solution of Riemann problems in order to determine numerical fluxes in this algorithm. Finally, we present numerical examples of the results of the algorithm.

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A conservative Eulerian formulation of the equations for elastic flow

Cited by 77 publications

References 2 publications

A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

A conservation law formulation of nonlinear elasticity in general relativity

A higher‐order Godunov method for modeling finite deformation in elastic‐plastic solids

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