Static and dynamic pore-collapse relations for ductile porous materials are obtained by analysis of the collapse of a hollow sphere of incompressible elastic-plastic material, with appropriate pore radius and over-all porosity. There are three phases of the pore-collapse process: an initial phase, a transitional elastic-plastic phase, and a plastic phase. The change in porosity during the first two phases is quite small. In the plastic phase, the static pore-collapse relation is an exponential law that depends only on the yield strength of the material; the dynamic relation is a nonlinear second-order ordinary differential equation that involves the yield strength and a material constant (with the physical dimension of time) that depends on the yield strength, the density, the initial porosity, and the pore radius. Comparison of the theoretical predictions with finite-difference computer-code calculations for pore collapse of a hollow sphere of compressible material indicates that the effect of elastic compressibility on pore collapse is quite small, so that the pore-collapse relations obtained from the incompressible model should have a wide range of validity. Also, the specific internal energy at the pore boundary has a logarithmic singularity as the pore closes.

Three classes of compressible isotropic elastic solids are introduced, for each of which the strain energy, expressed as a function of the three principal invariants of the stretch tensors, is linear in two of its arguments and nonlinear in the third argument. One of these is the class of harmonic materials. Several deformation fields are examined, for which the governing equations, for general compressible isotropic elastic response, reduce to a nonlinear ordinary differential equation. For the three special classes of materials, this differential equation may be solved in closed form, giving a deformation field which is independent of the material response function, or its solution may be reduced to a single quadrature, involving the nonlinear material response function.

An effective stress law is derived analytically to describe the effect of pore fluid pressure on the linearly elastic response of saturated porous rocks which exhibit anisotropy. For general anisotropy the difference between the effective stress and the applied stress is not hydrostatic. The effective stress law involves two constants for transversely isotropic response and three constants for orthotropic response; these constants can be expressed in terms of the moduli of the porous material and of the solid material. These expressions simplify considerably when the anisotropy is structural rather than intrinsic, i.e., in the case of an isotropic solid material with an anisotropic pore structure. In this case the effective stress law involves the solid or grain bulk modulus and two or three moduli of the porous material, for transverse isotropy and orthotropy, respectively. The law reduces, in the case of isotropic response, to that suggested by Geertsma (1957) and by Skempton (1961) and derived analytically by Nur and Byerlee (1971).

A three-parameter strain energy function is developed to model the nonlinearly elastic response of rubber-like materials. The development of the model is phenomenological, based on data from the classic experiments of Treloar, Rivlin and Saunders, and Jones and Treloar on sheets of vulcanized rubber. A simple two-parameter version, similar to the Mooney-Rivlin and Gent-Thomas strain energies, provides an accurate fit with all of the data from Rivlin and Saunders and Jones and Treloar, as well as with Treloar's data for deformations for which the principal deformation invariant I 1 has values in the range 3 ≤ I 1 ≤ 20.

Motivated by the success of the spherical model in predicting the volumetric compaction behavior of both porous rocks and metals to a hydrostatic pressure, we consider the applicability of the spherical model to nonhydrostatic loading conditions. Specifically, the spherical model is used to examine the influence of the presence of a shear stress on the volumetric compression of a porous solid. We first obtain the linear, elastic solution for a hollow sphere subject to homogeneous tractions on the outer boundary. Then, assuming that the matrix material is governed by the Drucker‐Prager yield criterion, we use the elastic solution to derive an analytic expression for the onset of yield in the hollow sphere. The expression for the initial yield surface shows that the presence of a shear stress hastens the onset of yield in the sphere in comparison to a hydrostatic loading condition. This result agrees well with experimental data which shows that, for porous solids, permanent crush‐up begins at a lower mean stress under a nonhydrostatic loading than when the applied loading is a hydrostatic pressure. At this point, due to difficulty in obtaining an analytic solution, we turn to a numerical scheme (finite element method) to extend the analysis of the hollow sphere problem into the elasto‐plastic range. The spherical model results clearly exhibit the experimental finding that the presence of a shear stress tends to enhance the volumetric compaction of porous solids in comparison to a hydrostatic loading condition. For both a porous rock and metal sample, agreement between the spherical model and experimental results is excellent.