Hyperelastic Multiphase Porous Media with Strain-Dependent Retention Laws

Abstract

This article presents poroelastic laws accounting for a retention behavior dependent also on porosity, as suggested by experimental evidence. Motivated by the numerical formulation of the corresponding boundary-value problem presented in a companion article, these constitutive equations employ displacements and fluid pressures as primary variables. The thermodynamic admissibility of the proposed rate laws for stress and fluid contents is assessed by means of symmetry and Maxwell conditions obtained from the Biot theory. In the case of strain-dependent saturation, the two elasticity tensors describing the drained response in saturated and unsaturated conditions, respectively, are proven to be in general not coincident, with their difference depending on capillary pressure and porosity. Furthermore, it is shown that besides the stress decomposition proposed by Coussy, also the stress split proposed by Lewis and Schrefler is consistent with the Biot framework. The former decomposition is obtained for retention laws depending only on capillary pressure, as expected. The Lewis-Schrefler split is proven to be consistent with retention models depending also on porosity. In these developments, the compressibility of all the phases is taken into account, in order to assess the thermodynamic consistency of an extension of the Biot's coefficient to partially saturated anisotropic porous media.

Mixed Arlequin method for multiscale poromechanics problems

“…For the linear elasticity case, B may read (cf. ), where K s is the intrinsic bulk modulus of the solid constituent, is the fourth‐order elasticity tensor, and I is the second‐order identity tensor. Note that in the isotropic case, this expression simplifies to the classical expression of Biots coefficient B = B I = (1 − K / K s ) I where K is the bulk modulus of the solid matrix (cf.…”

“…The balance of mass with compressible fluid and solid constituents reads, where M is Biot's modulus and q is the seepage velocity. For fully saturated, isotropic porous materials, M may be related to the intrinsic bulk moduli and porosity as follows : If the flow in the pore space remains laminar and the fluid movement is dominated by viscous forces, then the seepage velocity and the gradient of the pore pressure can be related by Darcy's law, where k is the effective permeability and μ is the dynamic viscosity of the pore fluid. In this work, we shall assume isotropic permeability, that is, k = k I .…”

Strong discontinuities in partially saturated poroplastic solids

“…In particular, the constitutive laws developed in [17] within a Biot thermodynamic framework [11,23] to characterize the hyperelastic response of multiphase media are generalized to account for dissipative mechanisms in the bulk and on the discontinuity.…”

“…See Remark 2 below for additional details on these considerations. The poro-elastic relations (26) and (27) are a simplified version of the more general hyperelastic laws presented in [17] based on solid skeleton displacements and fluid pressures as primary variables. In particular, Eq.…”

“…In the present work, we consider the partially saturated model presented recently in [17,18] for the continuum response of the solid. The model is formulated in a Biot thermodynamic framework [11,23] employing the same primary variables of displacements and fluid pressures (rather than e.g.…”