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.

本文分析了塑料固体动力学方程的双曲率，包括一相固体和具有零渗透率和非零渗透率的多孔流体饱和固体。定义为系统矩阵对角线化的双曲性对于很好地提出边值问题是必要的。用于塑性固体的方程组和用于弹性固体的系统之间的区别在于，前者包含可塑性模型中涉及的因变量的其他演化方程。从双曲性的观点来看，这两个系统彼此一致：它们都是双曲的，或者都是非双曲的。本文讨论的另一个问题是双曲率与声学张量（矩阵）的属性之间的关系。对于双曲性来说，不仅要使声学矩阵的特征值成为实数而且是正数的条件是否必要，还没有得到证实。在本文中证明，当且仅当声学矩阵的特征值是实函数且具有完整的特征向量集时，这些方程才是双曲型的。因此，可以将整个塑料固体方程组的分析简化为声矩阵的分析。结果不限于特定的可塑性模型，而是适用于多种模型。在本文中证明，当且仅当声学矩阵的特征值是实函数且具有完整的特征向量集时，这些方程才是双曲型的。因此，可以将整个塑料固体方程组的分析简化为声矩阵的分析。结果不限于特定的可塑性模型，而是适用于多种模型。在本文中证明，当且仅当声学矩阵的特征值是实函数且具有完整的特征向量集时，这些方程才是双曲型的。因此，可以将整个塑料固体方程组的分析简化为声矩阵的分析。结果不限于特定的可塑性模型，而是适用于多种模型。