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 Zh. Vychisl. Mat. Mat. Fiz., 2010, Volume 50, Number 9, Pages 1640–1668 (Mi zvmmf4938)

Polyconvex potentials, invertible deformations, and a thermodynamically consistent formulation of the equations of the nonlinear theory of elasticity

V. A. Garanzha

Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333 Russia

Abstract: It is shown that the nonstationary finite-deformation thermoelasticity equations in Lagrangian and Eulerian coordinates can be written in a thermodynamically consistent Godunov canonical form satisfying the Friedrichs hyperbolicity conditions, provided that the elastic potential is a convex function of entropy and of the minors of the elastic deformation Jacobian matrix. In other words, the elastic potential is assumed to be polyconvex in the sense of Ball. It is well known that Ball’s approach to proving the existence and invertibility of stationary elastic deformations assumes that the elastic potential essentially depends on the second-order minors of the Jacobian matrix (i.e., on the cofactor matrix). However, elastic potentials constructed as approximations of rheological laws for actual materials generally do not satisfy this requirement. Instead, they may depend, for example, only on the first-order minors (i.e., the matrix elements) and on the Jacobian determinant. A method for constructing and regularizing polyconvex elastic potentials is proposed that does not require an explicit dependence on the cofactor matrix. It guarantees that the elastic deformations are quasiisometries and preserves the Lame constants of the elastic material.

Key words: elasticity equations, polyconvexity, entropy solutions, quasi-isometric mappings.

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English version:
Computational Mathematics and Mathematical Physics, 2010, 50:9, 1561–1587

Bibliographic databases:

UDC: 519.634
Revised: 27.04.2010

Citation: V. A. Garanzha, “Polyconvex potentials, invertible deformations, and a thermodynamically consistent formulation of the equations of the nonlinear theory of elasticity”, Zh. Vychisl. Mat. Mat. Fiz., 50:9 (2010), 1640–1668; Comput. Math. Math. Phys., 50:9 (2010), 1561–1587

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Garanzha V.A. Kudryavtseva L.N. Utyuzhnikov S.V., “Variational Method For Untangling and Optimization of Spatial Meshes”, J. Comput. Appl. Math., 269 (2014), 24–41
2. Dobrev V., Knup P., Kolev T., Mittal K., Tomov V., “The Target-Matrix Optimization Paradigm For High-Order Meshes”, SIAM J. Sci. Comput., 41:1 (2019), B50–B68
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