Theory of fluid saturated porous media with surface effects

Abstract

Fluid saturated porous media are abundant in nature and engineering. Existing constitutive theories of fluid saturated porous media have largely ignored surface effects, whereas recent experimental results demonstrate the importance of accounting for surface effects in constitutive modeling at micro- and nano-scales (such as brain tissue, cytoplasm, soil, oil shale, etc.), thus raising two critical issues: (1) is classical poromechanics (e.g., the Biot theory) valid at sufficiently small scales, and (2) how do surface effects influence the mechanical behaviors of saturated porous media? To squarely address these issues, establishing the mechanics of solid-fluid interfaces within the porous media becomes a necessity. Built upon the homogenization assumption in the mixture theory and making use of Cauchy's theorem, we first prove mathematically that stresses on solid-fluid interfaces can be expressed by a second order tensor defined at each point of a saturated porous medium with surface effects, and then establish a rigorous theoretical framework to characterize its mechanical behaviors. Restrictions on various constitutive laws of saturated porous media developed in this framework are discussed. As an application of the developed theoretical framework, we prove that, at small deformation, the classical Biot theory still holds, at least in format, for porous media with surface effects; however, relevant parameters (e.g., effective moduli) appearing in Biot's constitutive laws are dependent upon surface effects. This proof not only addresses the first issue, but also lays a solid foundation for theoretical and experimental studies of porous media with surface effects. To address the second issue, we employ the microstructures of a macromolecular network and an elastic matrix embedded with liquid inclusions as prototypes to demonstrate how their mechanical behaviors are related to the properties of three constituents (i.e., solid, fluid and solid-fluid interface) and determine explicitly the parameters appearing in their constitutive relations.