An alternative Biot’s displacement formulation for porous materials

Abstract

This paper proposes an alternative displacement formulation of Biot's linear model for poroelastic materials. Its advantage is a simplification of the formalism without making any additional assumptions. The main difference between the method proposed in this paper and the original one is the choice of the generalized coordinates. In the present approach, the generalized coordinates are chosen in order to simplify the expression of the strain energy, which is expressed as the sum of two decoupled terms. Hence, new equations of motion are obtained whose elastic forces are decoupled. The simplification of the formalism is extended to Biot and Willis thought experiments, and simpler expressions of the parameters of the three Biot waves are also provided. A rigorous derivation of equivalent and limp models is then proposed. It is finally shown that, for the particular case of sound-absorbing materials, additional simplifications of the formalism can be obtained.

Validity of the limp model for porous materials: A criterion based on the Biot theory

“…8 It was applied to the poroelastic formulation of Zwikker and Kosten 10 to model soft acoustical blankets in aircraft. More recently, Dazel et al 9 applied the limp approximation to an alternative displacement formulation based on Biot's poroelasticity equations. This simplified model accounts for ͑i͒ the dissipation mechanisms induced by the relative motion between the two phases and ͑ii͒ the effects of the motion of the solid phase using an inertia correction.…”

“…2, pp. 79-111͒ and ͑ii͒ the "limp" model assumes that the stiffness of the solid phase is zero but takes into account its inertial effects [7][8][9] ͑Ref. 8, Chap.…”

Weak, anisotropic symmetric formulations of Biot's equations for vibro‐acoustic modelling of porous elastic materials

“…Similarly, simulations based on transfer matrix modelling of transversely isotropic materials have recently 1521 been studied by Khurana et al [15]. Concerning recent work on isotropic material modelling, alternative formulations for solutions of Biot's equations have been studied by Dazel et al [16], Deu et al [17] and Bermudez et al [18]. Central to the current work is the paper by Atalla et al [19], an isotropic mixed displacement pressure formulation and which forms the basis for the proposed generalization in anisotropic porous material properties.…”

A new unified modeling of acoustically-coupled multilayers by (u s,u W) displacement formulation based poroelastic transfer matrix

“…Since the degrees of freedom of a poroelastic transfer matrix include those for air and elastic-solid layers, unified modeling approaches have been developed in the framework of the poroelastic transfer matrix. While the poroelastic transfer matrices in the existing unified modeling [5,[7][8][9] were derived from Biot's original ( , ) s f u u displacement formulation [12], this work employs a recent ( , ) s W u u displacement formulation [14] to build the poroelastic transfer matrix for developing a new unified modeling technique. Here, s u and f u denote the averaged solid-and fluid-phase displacement vectors in poroelastic materials, and W u is a compound displacement vector defined by a linear combination of the solidand fluid-phase displacement vectors.…”