# A thermodynamically compatible splitting procedure in hyperelasticity

## Abstract

A material is hyperelastic if the stress tensor is obtained by variation of the stored energy function. The corresponding 3D mathematical model of hyperelasticity written in the Eulerian coordinates represents a system of 14 conservative partial differential equations submitted to stationary differential constraints. A classical approach for numerical solving of such a 3D system is a geometrical splitting: the 3D system is split into three 1D systems along each spatial direction and solved then by using a Godunov type scheme. Each 1D system has 7 independent eigenfields corresponding to contact discontinuity, longitudinal waves and shear waves. The construction of the corresponding Riemann solvers is not an easy task even in the case of isotropic solids. Indeed, for a given specific energy it is extremely difficult, if not impossible, to check its rank-one convexity which is a necessary and sufficient condition for hyperbolicity of the governing equations. In this paper, we consider a particular case where the specific energy is a sum of two terms. The first term is the hydrodynamic energy depending only on the density and the entropy, and the second term is the shear energy which is unaffected by the volume change. In this case a very simplemore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22314882

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 270; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DENSITY; ELASTICITY; ENTROPY; LIMITING VALUES; MATHEMATICAL MODELS; NUMERICAL SOLUTION; PARTIAL DIFFERENTIAL EQUATIONS; SHEAR; SOLIDS; STORED ENERGY; STRESSES; TENSORS; VELOCITY

### Citation Formats

```
Favrie, N., E-mail: nicolas.favrie@univ-amu.fr, Gavrilyuk, S., E-mail: sergey.gavrilyuk@univ-amu.fr, and Ndanou, S., E-mail: serge.ndanou@univ-amu.fr.
```*A thermodynamically compatible splitting procedure in hyperelasticity*. United States: N. p., 2014.
Web. doi:10.1016/J.JCP.2014.03.051.

```
Favrie, N., E-mail: nicolas.favrie@univ-amu.fr, Gavrilyuk, S., E-mail: sergey.gavrilyuk@univ-amu.fr, & Ndanou, S., E-mail: serge.ndanou@univ-amu.fr.
```*A thermodynamically compatible splitting procedure in hyperelasticity*. United States. doi:10.1016/J.JCP.2014.03.051.

```
Favrie, N., E-mail: nicolas.favrie@univ-amu.fr, Gavrilyuk, S., E-mail: sergey.gavrilyuk@univ-amu.fr, and Ndanou, S., E-mail: serge.ndanou@univ-amu.fr. Fri .
"A thermodynamically compatible splitting procedure in hyperelasticity". United States.
doi:10.1016/J.JCP.2014.03.051.
```

```
@article{osti_22314882,
```

title = {A thermodynamically compatible splitting procedure in hyperelasticity},

author = {Favrie, N., E-mail: nicolas.favrie@univ-amu.fr and Gavrilyuk, S., E-mail: sergey.gavrilyuk@univ-amu.fr and Ndanou, S., E-mail: serge.ndanou@univ-amu.fr},

abstractNote = {A material is hyperelastic if the stress tensor is obtained by variation of the stored energy function. The corresponding 3D mathematical model of hyperelasticity written in the Eulerian coordinates represents a system of 14 conservative partial differential equations submitted to stationary differential constraints. A classical approach for numerical solving of such a 3D system is a geometrical splitting: the 3D system is split into three 1D systems along each spatial direction and solved then by using a Godunov type scheme. Each 1D system has 7 independent eigenfields corresponding to contact discontinuity, longitudinal waves and shear waves. The construction of the corresponding Riemann solvers is not an easy task even in the case of isotropic solids. Indeed, for a given specific energy it is extremely difficult, if not impossible, to check its rank-one convexity which is a necessary and sufficient condition for hyperbolicity of the governing equations. In this paper, we consider a particular case where the specific energy is a sum of two terms. The first term is the hydrodynamic energy depending only on the density and the entropy, and the second term is the shear energy which is unaffected by the volume change. In this case a very simple criterion of hyperbolicity can be formulated. We propose then a new splitting procedure which allows us to find a numerical solution of each 1D system by solving successively three 1D sub-systems. Each sub-system is hyperbolic, if the full system is hyperbolic. Moreover, each sub-system has only three waves instead of seven, and the velocities of these waves are given in explicit form. The last property allows us to construct reliable Riemann solvers. Numerical 1D tests confirm the advantage of the new approach. A multi-dimensional extension of the splitting procedure is also proposed.},

doi = {10.1016/J.JCP.2014.03.051},

journal = {Journal of Computational Physics},

number = ,

volume = 270,

place = {United States},

year = {Fri Aug 01 00:00:00 EDT 2014},

month = {Fri Aug 01 00:00:00 EDT 2014}

}