One-parameter family of equations of state for isotropic compressible solids

Applying the theorem proved by the authors in [10], we established the hyperbolicity of non-stationary equations of hyperelastic isotropic solids for a one-parameter family of equations of state containing, in particular, generalized neo-hookean solids. The hyperbolicity is equivalent to the rank-one convexity of the corresponding stored energy. The influence of the parameter on the solution properties is shown in the case of a strong shear test

A unified hyperbolic formulation for viscous fluids and elastoplastic solids

We discuss a unified flow theory which in a single system of hyperbolic partial differential equations (PDEs) can describe the two main branches of continuum mechanics, fluid dynamics, and solid dynamics. The fundamental difference from the classical continuum models, such as the Navier-Stokes for example, is that the finite length scale of the continuum particles is not ignored but kept in the model in order to semi-explicitly describe the essence of any flows, that is the process of continuum particles rearrangements. To allow the continuum particle rearrangements, we admit the deformability of particle which is described by the distortion field. The ability of media to flow is characterized by the strain dissipation time which is a characteristic time necessary for a continuum particle to rearrange with one of its neighboring particles. It is shown that the continuum particle length scale is intimately connected with the dissipation time. The governing equations are represented by a system of first order hyperbolic PDEs with source terms modeling the dissipation due to particle rearrangements. Numerical examples justifying the reliability of the proposed approach are demonstrated.Comment: 6 figure

Criterion of Hyperbolicity in Hyperelasticity in the Case of the Stored Energy in Separable Form

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An Example of a One-Parameter Family of Rank-One Convex Stored Energies for Isotropic Compressible Solids

International audienceApplying the theorem proved by the authors in Ndanou et al. (J. Elast. 115: 1-15, 2014), we established the hyperbolicity of non-stationary equations of hyperelastic isotropic solids for a one-parameter family of equations of state containing, in particular, generalized neo-Hookean solids. The hyperbolicity is equivalent to the rank-one convexity of the corresponding stored energy (Dafermos, Hyperbolic Conservation Laws in Continuum Mechanics, Springer, Berlin, 2000; Silhavy, The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin, 1996). The influence of the parameter on the solution properties is shown in the case of a strong shear test

Multi-solid and multi-fluid diffuse interface model: Applications to dynamic fracture and fragmentation

International audienceWe extend the model of diffuse solid-fluid interfaces developed earlier by authors of this paper to the case of arbitrary number of interacting hyperelastic solids. Plastic transformations of solids are taken into account through a Maxwell type model. The specific energy of each solid is given in separable form: it is the sum of a hydrodynamic part of the energy depending only on the density and the entropy, and an elastic part of the energy which is unaffected by the volume change. It allows us to naturally pass to the fluid description in the limit of vanishing shear modulus. In spite of a large number of governing equations, the model has a quite simple mathematical structure: it is a duplication of a single visco-elastic model. The model is well posed both mathematically and thermodynamically: it is hyperbolic and compatible with the second law of thermodynamics. The resulting model can be applied in the situations involving an arbitrary number of fluids and solids. In particular, we show the ability of the model to describe spallation and penetration phenomena occurring during high velocity impacts. (C) 2015 Elsevier Inc. All rights reserved