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. (2004 Phys. Rev. E 70, 015602. (doi:10.1103/PhysRevE.70.015602)) is based on a variable that describes the softening of the material. However, this model is one dimensional and it is not thermodynamically admissible. In the present article, a three-dimensional 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 constitutive 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 the continuum is thermodynamically admissible and dissipative (inelastic). In the case of small uniaxial deformations, it is shown analytically that the model reproduces qualitatively the main features of real experiments.

Dynamic compaction of granular materials.

An 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.

Mathematical and numerical model for nonlinear viscoplasticity.

A macroscopic model describing elastic-plastic solids is derived in a special case of the internal specific energy taken in separable form: it is the sum of a hydrodynamic part depending only on the density and entropy, and a shear part depending on other invariants of the Finger tensor. In particular, the relaxation terms are constructed compatible with the von Mises yield criteria. In addition, Maxwell-type material behaviour is shown up: the deviatoric part of the stress tensor decays during plastic deformations. Numerical examples show the ability of this model to deal with real physical phenomena.