توابع گرين سه بعدي براي معادلات حاكم بر خاك غيراشباع در فضاي زمان

Three-Dimensional Time-Domain Greenʹs Functions for Unsaturated Soils

Numerical modeling has been largely developed in soil mechanics behavior by different methods. Among them, the development of the boundary element method, which is the most suitable one for domains with infinite boundary conditions like soils media, has been restricted by the necessity of deriving the Greenʹs functions of the governing differential equations. Indeed, attempting to solve the three-dimensional boundary value problems for unsaturated soils leads one to search for the associated Greenʹs functions of the governing differential equations. In this paper the governing differential equations of the phenomena are presented which consist of three main groups of equations: 1- equilibrium and constitutive equations of soilʹs solid skeleton, 2- conservation and transfer equations for air phase and 3- conservation and transfer equations for moisture phase. The associated Greenʹs functions have been manipulated with a few assumptions to first linearize the completely non-linear governing differential equations and next to make possible deriving the results, mathematically. After applying the Laplace transform to eliminate the time variable, Greenʹs functions of the governing differential equations have been derived using the straightforward Kupeadzeʹs method. Then using the inverse Laplace transforms, the completely closed-form Greenʹs functions have been derived in the time domain. Finally, for verification of the results it has been demonstrated that when the conditions approach to the poroelastostatic case, the Greenʹs functions approach to the corresponding poroelastostatic Greenʹs functions as well. As no analytical solutions are available in the literature for the mentioned Greenʹs functions, it seems to be a new experience to introduce a set of fundamental solutions for the unsaturated case for the first time.