This paper presents a general method for defining the macroscopic free-energy density function and its complementary forms for a porous medium saturated by two non-miscible fluids, in the case of compressible fluid and solid constituents, non-isothermal conditions and negligible interfacial surface energy. The major advantage of the proposed approach is that no limitation or simplification is posed on the choice of the free energies of the fluid constituents. As a result, a fully non-linear equation of state for the pore fluids can be incorporated within the proposed framework. The method is presented under the assumption that interfacial surface energy terms are negligible, thus recovering a Bishop parameter chi coinciding with the degree of saturation, which is expected to be applicable mostly to non-plastic soils. Moreover, small strains of the solid skeleton are assumed, but the method can be easily extended to a large strain formulation as discussed below. The paper analyzes also some particular cases concerning the incompressibility of all constituents, the geometric linearization and the incompressibility only of the solid constituent. The knowledge of the free energy density function is the starting point for the evaluation of the dissipation function, of energy and entropy balance and, in general, for the formulation of thermodynamically consistent constitutive models.

The transient response of porous media is an important aspect of dynamic research. However, existing studies seldom provide solutions to the transient response problem of layered unsaturated porous media. Based on the Biot-type unsaturated wave equations, dimensionless one-dimensional wave equations are established. An appropriate displacement function is introduced to homogenize the boundary conditions. Subsequently, the transfer matrix method is used to obtain the eigenvalues and eigenfunctions of the homogeneous governing equations. Leveraging the orthogonality of the eigenfunctions, the original problem is transformed into solving a series of initial value problems of ordinary differential equations. The temporal solution within the time domain is then obtained through an improved precise time integration method. The validity of the solution presented in this paper is verified by comparing it with existing solutions in the literature. Analysis of numerical examples shows that reflection waves of opposite phases will be generated at the hard-soft and hard-harder interface, which helps in the accurate identification of weak interlayers in practical engineering applications. With increasing saturation, there is a noticeable increase in the velocities of the P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{1}$$\end{document} and P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{3}$$\end{document} waves, whereas the velocity of the P2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{2}$$\end{document} waves tends to decrease, which can be used to assess the mechanical property of medium. The peak value of pore pressure in unsaturated can be 1.64 times higher than those in saturated condition.

Transport systems such as highways and railways are constructed on earthworks that experience fluctuating levels of saturation. This can range from dry to fully saturated, however most commonly they are in a state of partial saturation. When numerically modelling such problems, it is important to capture the response of the solid, liquid and gas phases in the material. However, multi-physics solutions are computationally demanding and as a solution this paper presents a finite element approach for the dynamic analysis of unsaturated porous media in a moving coordinate system. The first novelty of the work is the development of a principle of relative motion for a three-phase medium, where the moving load is at rest while the unsaturated porous medium moves relative to the load. This makes it particularly efficient for moving load problems such as transport. The second novelty is a parametric investigation of the three-phase response of a partially saturated medium subject to a moving load. The paper starts by presenting the time domain model in terms of its constitutive relationships and equations for mass and momentum conservation. Next the model is validated using three case studies: the consolidation of a saturated soil column, the dynamics of an unsaturated soil column and finally the response of a saturated foundation to a moving load. It is then used to study a moving 2D plane strain load problem and its performance is compared to that of a standard FEM solution which does not employ a moving coordinate system. Similar accuracy is obtained while computational efficiency is improved by a factor of ten. Finally, the model is used to investigate the effect of degree of saturation and moving load speed on the response of an unsaturated porous medium. It is found that both variables have a significant impact on the dynamic response.

In this paper, a computational framework based on the Smoothed Particle Finite Element Method is developed to study the coupled seepage-deformation process in unsaturated porous media. Governing equations are derived from the balance laws of solid and fluid phases considering partial saturation effects in porous media. Moreover, an hourglass control method is implemented to avoid the rank-deficiency issue in SPFEM and the moving least squares approximation technique (MLS) is implemented to eliminate the pore pressure oscillations when the low-order triangle element is used. The proposed coupled SPFEM formulation is validated through four elastic examples and one elasto-plastic example. Good agreement with the numerical or analytical results reported in the literature is obtained. Further, the rainfallinduced slope failure is studied, in which a suction-dependent elasto-plastic Mohr-Coulomb model is adopted to take account of the suction effect in unsaturated soil. The evolution of the suction and soil deformation during the rainfall period and the whole slope failure process are obtained. It is demonstrated that the proposed method is a promising tool in numerical investigations of both the triggering mechanisms and post-failure behavior of the rainfall-induced slope failure.

The mechanical behavior of unsaturated porous media under non-isothermal conditions plays a vital role in geo-hazards and geo-energy engineering (e.g., landslides triggered by fire and geothermal energy harvest and foundations). Temperature increase can trigger localized failure and cracking in unsaturated porous media. This article investigates the shear banding and cracking in unsaturated porous media under non-isothermal conditions through a thermo-hydro-mechanical (THM) periporomechanics (PPM) paradigm. PPM is a nonlocal formulation of classical poromechanics using integral equations, which is robust in simulating continuous and discontinuous deformation in porous media. As a new contribution, we formulate a nonlocal THM constitutive model for unsaturated porous media in the PPM paradigm in this study. The THM meshfree paradigm is implemented through an explicit Lagrangian meshfree algorithm. The return mapping algorithm is used to implement the nonlocal THM constitutive model numerically. Numerical examples are presented to assess the capability of the proposed THM mesh-free paradigm for modeling shear banding and cracking in unsaturated porous media under non-isothermal conditions. The numerical results are examined to study the effect of temperature variations on the formation of shear banding and cracking in unsaturated porous media.