Time-domain Green’s functions for unsaturated soils. Part I: Two-dimensional solution

Abstract

In this article, primarily a brief discussion about the formulation of unsaturated soils including the equilibrium, air and moisture transfer equations is presented. Then the closed form two-dimensional GreenÕs functions of the governing differential equations for an unsaturated deformable porous medium with linear elastic behavior for a symmetric polar domain in both Laplace transform and time domains have been introduced, for the first time. Using the linear form of the governing differential equations and considering the effects of non-linearity of the governing equations, the GreenÕs functions have been derived exactly and verified in both Laplace transform and time domains.

Seismic wave propagation in laterally inhomogeneous poroelastic media via BIEM

“…The fundamental solution and the BIEM formulation for three‐dimensional dynamic poroelasticity have been given by Cheng et al 17 and by Dominguez 18 based on the analogy with thermoelasticity. Regarding two‐dimensional poroelasticity, Fukui et al 19 found the fundamental solution in a direct manner and applied it to solve wave scattering problems by BIEM, while Gatmiri and Jabbari 20, 21 derived in closed form Green's functions for two‐and three‐dimensional deformable unsaturated soils in both frequency and time domains. Also recently, in a series of papers Schanz and co‐workers 22–24 obtained some very interesting and useful results in the field of dynamic poroelasticity.…”

Analytical 3D transient elastodynamic fundamental solution of unsaturated soils

“…However, further works on the transient fundamental solution of the saturated poroelastic media were concentrated on Zienkiewicz and Shiomi [30], so-called u − p reformulation of Biot's equations in time domain for medium speed phenomena [31], as well as incorporating incompressibility of the solid grains and liquid compared to the skeleton's compressibility in the solutions [32][33][34].In spite of saturated poroelasticity, the literature for the fundamental solution of unsaturated poroelasticity is recent. Gatmiri and Jabbari [35,36] derived fundamental solutions for the static and quasi-static unsaturated soil for two-and three-dimensional problems. The fundamental solutions were presented in terms of soil skeleton displacement, water and air pressures in transformed and time domains.…”

“…In spite of saturated poroelasticity, the literature for the fundamental solution of unsaturated poroelasticity is recent. Gatmiri and Jabbari [35,36] derived fundamental solutions for the static and quasi-static unsaturated soil for two-and three-dimensional problems. The fundamental solutions were presented in terms of soil skeleton displacement, water and air pressures in transformed and time domains.…”

Three‐dimensional transient thermo‐hydro‐mechanical fundamental solutions of unsaturated soils

“…Indeed, attempting to solve numerically the boundary value problems for unsaturated soils using BEM leads one to search for the associated fundamental solutions.The comprehensive state-of-the-art review by Gatmiri and Kamalian [4], Gatmiri and Nguyen [5], Gatmiri and Jabbari [6, 7], Seyrafian et al [8] provides clearly presented information on the fundamental solution applied to the soil and the porous media. For unsaturated soils, Gatmiri and Jabbari [6,7,9,10] have derived the first fundamental solutions for the nonlinear governing differential equations for static and quasi-static poroelastic media for both two and three dimensional problems. The corresponding thermo-poro-mechanic fundamental solutions for static and quasi-static problems are, respectively, derived by Jabbari and Gatmiri (for both two and threedimensional problems) [11] and Gatmiri et al (for two-dimensional problems) [12].According to the authors' knowledge, the three-dimensional fundamental solutions of the governing partial differential equations for unsaturated porous media by considering the thermohydro-mechanical behaviour have not been developed so far, hence the development of a BEM model for unsaturated phenomena is not yet possible.In this paper first, the set of fully coupled governing differential equations of thermo-hydromechanical behaviour of unsaturated porous media subjected to quasi-static loadings is presented based on the suction-based mathematical model presented by Gatmiri [13] and Gatmiri et al [14].…”

“…The comprehensive state-of-the-art review by Gatmiri and Kamalian [4], Gatmiri and Nguyen [5], Gatmiri and Jabbari [6, 7], Seyrafian et al [8] provides clearly presented information on the fundamental solution applied to the soil and the porous media. For unsaturated soils, Gatmiri and Jabbari [6,7,9,10] have derived the first fundamental solutions for the nonlinear governing differential equations for static and quasi-static poroelastic media for both two and three dimensional problems. The corresponding thermo-poro-mechanic fundamental solutions for static and quasi-static problems are, respectively, derived by Jabbari and Gatmiri (for both two and threedimensional problems) [11] and Gatmiri et al (for two-dimensional problems) [12].…”