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The paper deals with the analysis of hyperbolicity of the dynamic equations for plastic solids, including one-phase solids and porous fluid-saturated solids with zero and nonzero permeability. Hyperbolicity defined as diagonalizability of the matrix of the system is necessary for the boundary value problems to be well posed. The difference between the system of equations for a plastic solid and the system for an elastic solid is that the former contains additional evolution equations for the dependent variables involved in the plasticity model. It is shown that the two systems agree with each other from the viewpoint of hyperbolicity: they are either both hyperbolic or both non-hyperbolic. Another issue addressed in the paper is the relation between hyperbolicity and the properties of the acoustic tensor (matrix). It remained unproved whether the condition for the eigenvalues of the acoustic matrix to be real and positive is not only necessary but also sufficient for hyperbolicity. It is proved in the paper that the equations are hyperbolic if and only if the eigenvalues of the acoustic matrix are real and positive with a complete set of eigenvectors. The analysis of the whole system of equations for a plastic solid can thus be reduced to the analysis of the acoustic matrix. The results are not restricted to a particular plasticity model but applicable to a wide class of models. [ABSTRACT FROM AUTHOR]

Previous studies showed that the dynamic equations for a porous fluid-saturated solid may lose hyperbolicity and thus render the boundary-value problem ill-posed while the equations for the same but dry solid remain hyperbolic. This paper presents sufficient conditions for hyperbolicity in both dry and saturated states. Fluid-saturated solids are described by two different systems of equations depending on whether the permeability is zero or nonzero (locally undrained and drained conditions, respectively). The paper also introduces a notion of wave speed consistency between the two systems as a necessary condition which must be satisfied in order for the solution in the locally drained case to tend to the undrained solution as the permeability tends to zero. It is shown that the symmetry and positive definiteness of the acoustic tensor of the skeleton guarantee both hyperbolicity and the wave speed consistency of the equations. [ABSTRACT FROM AUTHOR]

An increase in the stiffness of a solid in compression is known to lead to the steepening of the profiles of compression waves and, as a consequence, to the formation of strong discontinuities from continuous waves propagating in the solid. In this paper, the critical distance required for a continuous wave to turn into a shock wave is calculated from the evolution equation for a weak discontinuity (acceleration wave) propagating into a quiescent region. Infinite growth of the amplitude of an acceleration wave in a finite time signifies the transition to a strong discontinuity. Relations between the critical distances for plane, cylindrical and spherical waves are established. Numerical examples are presented for a particular case of the pressure-dependent stiffness typical of granular solids such as sand or soil, with emphasis placed on the influence of a small amount of free gas in the pore fluid. [ABSTRACT FROM AUTHOR]