Elasticity with a flow rule for geomaterials

Abstract

The modern theory of elasticity and the first law of thermodynamics are cornerstones of engineering science that share the concept of reversibility. For four decades, researchers have noted that extrapolating the more commonly accepted empirical models of geomaterial stiffness violates the first law. We propose to resolve this inconsistency by introducing a flow rule into the standard description of reversible deformation.We identify specific volume (total-volume to solid-volume ratio) as an internal state variable and divide volume change into particle compression and rearrangement components. We model the effect of particle compression on observable deformation as a scaling of mesoscopically based compression and the effect of particle rearrangement as purely macroscopic flow with no mesoscopic counterpart. We partition stress space into contraction and swelling sub-domains, derive a normality postulate for the rearrangement effect and predict degradation of shear stiffness with increasing shear strain. We identify the pressure at which particle compression surpasses particle rearrangement as pressure increases and propose the necessary and sufficient condition for free flow.Through this pressure-dependent synthesis of bottom-up and top-down contributions to observable higher-scale deformation, we offer a prototype for developing consistent continuum models of reversible deformation in the field of soft condensed matter.