Stress relaxation models with polyconvex entropy in Lagrangean and Eulerian coordinates

The embedding of the equations of polyconvex elastodynamics to an augmented symmetric hyperbolic system provides in conjunction with the relative entropy method a robust stability framework for approximate solutions \cite{LT06}. We devise here a model of stress relaxation motivated by the format of the enlargement process which formally approximates the equations of polyconvex elastodynamics. The model is endowed with an entropy function which is not convex but rather of polyconvex type. Using the relative entropy we prove a stability estimate and convergence of the stress relaxation model to polyconvex elastodynamics in the smooth regime. As an application, we show that models of pressure relaxation for real gases in Eulerian coordinates fit into the proposed framework.

[1]  I. Suliciu,et al.  On the thermodynamics of rate-type fluids and phase transitions. II. Phase transitions , 1998 .

[2]  R. J. Diperna.  Uniqueness of Solutions to Hyperbolic Conservation Laws. , 1978 .

[3]  Qin Tiehu,et al.  Symmetrizing Nonlinear Elastodynamic System , 1998 .

[4]  Christian Klingenberg,et al.  A multiwave approximate Riemann solver for ideal MHD based on relaxation II: numerical implementation with 3 and 5 waves , 2010, Numerische Mathematik.

[5]  B. Perthame,et al.  Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics , 1998 .

[6]  Christian Klingenberg,et al.  A multiwave approximate Riemann solver for ideal MHD based on relaxation. I: theoretical framework , 2007, Numerische Mathematik.

[7]  M. Slemrod.  Lectures on the Isometric Embedding Problem $$(M^{n},g)\rightarrow \mathrm {I\!R\!}^{m},\,m=\frac{n}{2}(n+1)$$ , 2015 .

[8]  Athanasios E. Tzavaras,et al.  Materials with Internal Variables and Relaxation to Conservation Laws , 1999 .

[9]  A. Tzavaras,et al.  Weak–Strong Uniqueness of Dissipative Measure-Valued Solutions for Polyconvex Elastodynamics , 2011, 1109.6686.

[10]  C. M. Dafermos,et al.  Hyberbolic [i.e. Hyperbolic] conservation laws in continuum physics , 2005 .

[11]  M. Mihăilescu-suliciu,et al.  The energy in one-dimensional rate-type semilinear viscoelasticity , 1987 .

[12]  Athanasios E. Tzavaras,et al.  RELATIVE ENTROPY IN HYPERBOLIC RELAXATION , 2005 .

[13]  Walter Noll,et al.  On the thermostatics of continuous media , 1959 .

[14]  Hyperbolicity of the Nonlinear Models of Maxwell’s Equations , 2004 .

[15]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[16]  Sophia Demoulini,et al.  A Variational Approximation Scheme for¶Three-Dimensional Elastodynamics¶with Polyconvex Energy , 2001 .

[17]  C. D. Levermore,et al.  Hyperbolic conservation laws with stiff relaxation terms and entropy , 1994 .

[18]  C. Dafermos.  The second law of thermodynamics and stability , 1979 .

[19]  I. Suliciu,et al.  On the thermodynamics of rate-type fluids and phase transitions. I. Rate-type fluids , 1998 .

[20]  C. Dafermos.  Quasilinear hyperbolic systems with involutions , 1986 .

[21]  Athanasios E. Tzavaras,et al.  Structural Properties of Stress Relaxation and Convergence from Viscoelasticity to Polyconvex Elastodynamics , 2006 .

[22]  Yann Brenier,et al.  Hydrodynamic Structure of the Augmented Born-Infeld Equations , 2004 .

[23]  J. Ball.  Some Open Problems in Elasticity , 2002 .

[24]  M. Gurtin,et al.  Thermodynamics with Internal State Variables , 1967 .

[25]  F. Bouchut.  Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources , 2005 .

[26]  P. Lax.  Shock Waves and Entropy , 1971 .