General coupling of porous flows and hyperelastic formulations—From thermodynamics principles to energy balance and compatible time schemes

Abstract

We formulate a general poromechanics model -within the framework of a two-phase mixture theory -compatible with large strains and without any simplification in the momentum expressions, in particular concerning the fluid flows. The only specific assumptions made are fluid incompressibility and isothermal conditions. Our formulation is based on fundamental physical principles -namely, essential conservation and thermodynamics laws -and we thus obtain a Clausius-Duhem inequality which is crucial for devising compatible constitutive laws. We then propose to model the solid behavior based on a generalized hyperelastic free energy potential -with additional viscous effects -which allows to represent a wide range of mechanical behaviors. The resulting formulation takes the form of a coupled system similar to a fluid-structure interaction problem written in an Arbitrary Lagrangian-Eulerian formalism, with additional volume-distributed interaction forces. We achieve another important objective by identifying the essential energy balance prevailing in the model, and this paves the way for further works on mathematical analyses, and time and space discretizations of the formulation.

A stabilized finite element method for finite-strain three-field poroelasticity

“…For further discussion of strain energy laws for porelasticity we refer to [14] and [52]. The material parameters μ and λ in (24) can be related to the Young's modulus E and the Poisson ratio ν by μ = E/(2(1 + ν)) and λ = (Eν)/((1 + ν)(1 − 2ν)).…”

“…A comprehensive development of the macroscopic theory appears in [16]. Relationships between the two theories are explored in [14,17]. As is most common in biological applications, we use the mixture theory for poroelasticity as outlined in [8] and recently summarized in [52].…”

Two finite element approaches for Darcy and Darcy–Brinkman flow through deformable porous media—Mixed method vs. NURBS based (isogeometric) continuity

“…All equations are formulated on a macroscopic level, on which the fluid-structure-interface is not resolved, see figure 2. A detailed derivation and further theory can be found in [30,31,32,33]. In this homogenized point of view, both fluid and solid phases occupy the same domain and fraction homogenization fluid phase structure phase homogenized medium (no distinction between phases) porosity φ (volume ratio (1-φ) ) (volume ratio φ ) Figure 2.…”

Saturated porous continua in the frame of hybrid description

“…(28) by using (11), (13), (20)- (22), the transport theorem (2), and Gauss' divergence theorem, we can obtain the energy balance equation in local form:…”

“…Another type of models is based on both Lagrangian and Eulerian descriptions [12][13][14][15]. The equations for the solid are formulated by using the Lagrangian description and the equations for the fluid by using the Eulerian one.…”