Constitutive relations of two classes are proposed for nonlinear elastic isotropic materials, which, in case of purely volumetric deformation, are reduced to the Murnaghan’s equation of state. Exact analytical solution of the Lame problem of the radially symmetric deformation of a hollow sphere is obtained for one of these material classes. Nonlinear effects are studied. The non-uniqueness of solution is obtained for the case in which the sphere radii are specified in the initial configuration. It is shown for this case that there is a limiting pressure, above which the problem has no solution. The strong ellipticity conditions are tested. The obtained results can be used in geomechanics for modeling the recrystallization of metamorphic rocks.

对于非线性弹性各向同性材料，提出了两类本构关系，在纯体积变形的情况下，将其简化为 Murnaghan 状态方程。对于这些材料类别之一，可以获得空心球径向对称变形的 Lame 问题的精确解析解。研究了非线性效应。对于在初始配置中指定球体半径的情况，获得解的非唯一性。在这种情况下表明存在一个极限压力，高于该压力问题无解。测试强椭圆度条件。获得的结果可用于地质力学中的变质岩再结晶建模。