Nonlinear waves in solids with slow dynamics: an internal-variable model

In heterogeneous solids such as rocks and concrete, the speed of sound diminishes with the strain amplitude of a dynamic loading (softening). This decrease known as "slow dynamics" occurs at time scales larger than the period of the forcing. Also, hysteresis is observed in the steady-state response. The phenomenological model by Vakhnenko et al. is based on a variable that describes the softening of the material [Phys. Rev. E 70-1, 2004]. However, this model is 1D and it is not thermodynamically admissible. In the present article, a 3D model is derived in the framework of the finite strain theory. An internal variable that describes the softening of the material is introduced, as well as an expression of the specific internal energy. A mechanical constitu-tive law is deduced from the Clausius-Duhem inequality. Moreover, a family of evolution equations for the internal variable is proposed. Here, an evolution equation with one relaxation time is chosen. By construction, this new model of continuum is thermodynamically admissible and dissipative (inelas-tic). In the case of small uniaxial deformations, it is shown analytically that the model reproduces qualitatively the main features of real experiments

Analytical solution to the 1D nonlinear elastodynamics with general constitutive laws

International audienceUnder the hypothesis of small deformations, the equations of 1D elastodynamics write as a 2 × 2 hyperbolic system of conservation laws. Here, we study the Riemann problem for convex and nonconvex constitutive laws. In the convex case, the solution can include shock waves or rarefaction waves. In the nonconvex case, compound waves must also be considered. In both convex and nonconvex cases, a new existence criterion for the initial velocity jump is obtained. Also, admissibility regions are determined. Lastly, analytical solutions are completely detailed for various constitutive laws (hyperbola, tanh and polynomial), and reference test cases are proposed

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

A Zener model for nonlinear viscoelastic waves

A macroscopic model describing nonlinear viscoelastic waves is derived in Eulerian formulation, through the introduction of relaxation tensors. In the limiting case of small deformations, the governing equations recover those of the linear Zener model with memory variables, which is widely used in acoustics. The structure of the relaxation terms ensures that the model is dissipative. The chosen family of specific internal energies ensures also that the model is unconditionally hyperbolic. Numerical examples are proposed to illustrate the properties of viscoelastic waves, in small and large deformations

Dynamic compaction of granular materials

International audienceAn Eulerian hyperbolic multiphase flow model for dynamic and irreversible compaction of granular materials is constructed. The reversible model is first constructed on the basis of the classical Hertz theory. The irreversible model is then derived in accordance with the following two basic principles. First, the entropy inequality is satisfied by the model. Second, the corresponding `intergranular stress' coming from elastic energy owing to contact between grains decreases in time (the granular media behave as Maxwell-type materials). The irreversible model admits an equilibrium state corresponding to von Mises-type yield limit. The yield limit depends on the volume fraction of the solid. The sound velocity at the yield surface is smaller than that in the reversible model. The last one is smaller than the sound velocity in the irreversible model. Such an embedded model structure assures a thermodynamically correct formulation of the model of granular materials. The model is validated on quasi-static experiments on loading-unloading cycles. The experimentally observed hysteresis phenomena were numerically confirmed with a good accuracy by the proposed model

A rapid numerical method for solving Serre-Green-Naghdiequations describing long free surface gravity waves

International audienceA new numerical method for solving the Serre-Green-Naghdi (SGN) equations describing dispersive waves on shallow water is proposed. From the mathematical point of view, the SGN equations are the Euler-Lagrange equations for a 'master' lagrangian submitted to a differential constraint which is the mass conservation law. One major numerical challenge in solving the SGN equations is the resolution of an elliptic problem at each time instant. It is the most time-consuming part of the numerical method. The idea is to replace the 'master' lagrangian by a one-parameter family of 'extended' lagrangians, for which the corresponding Euler-Lagrange equations are hyperbolic. In such an approach, the 'master' lagrangian is recovered by the 'extended' lagrangian in some limit (for example, when the corresponding parameter is large). The choice of such a family of extended lagrangians is proposed and discussed. The corresponding hyperbolic system is numerically solved by a Godunov type method. Numerical solutions are compared with exact solution of the SGN equations. It appears that the computational time in solving the hyperbolic system is much lower than in the case where the elliptic operator is inverted. The new method is, in particular, applied to study the 'Favre waves' which are non-stationary undular bores produced after reflection of the fluid flow with a free surface at an immobile wall

Un modèle hypoélastique bien posé dérivé à partir d'un modèle hyperélastique

Hypoelastic models are widely used in industrial and military codes for numerical simulation of high strain dynamics of solids. This class of model is often mathematically inconsistent. More exactly, the second principle is not verified on the solutions of the model, and the initial state after a reversible cycle is not recovered. In the past decades, hyperelastic models, which are mathematically consistent, have been intensively studied. For their practical use, ones needs to entirely rewrite the commercial codes. Moreover, calibration of equation of states would be needed. In this paper two hypoelastic models for isotropic solids are derived from equivalent hyperelastic models. The hyperelastic models are hyperbolic for all possible deformations. It allows us to use robust Godunov's schemes for numerical resolution of these models. Two new objective derivatives corresponding to two different equations of state and defining the evolution of the deviatoric part of the stress tensor naturally appear. These derivatives are compatible with the reversibility property of the model : it conserves the specific entropy in a continuous motion. The most used hypoelastic model (Wilkins model) is recovered in the small deformation limit

Diffuse interface model for compressible fluid – Compressible elastic–plastic solid interaction

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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