13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2. Governing Equations The coupled formulation of fully saturated porous media used in this work is so-called p u formulation, in which the displacement of solid u and pore fluid pressure p are treated as two primary unknown variables [5]. The governing equations are derived from the equation of momentum equilibrium of solid-fluid mixture and equation of mass conservation of fluid as follows: , ( ) 0 ij ij j i i u p b (1) , , ( ) 0 ij j f j ii i f b k np p K (2) where ij is the effective stress tensor, which is related to the total stress ij and pore fluid pressure p via ij ij ij p . The effective stress in tension and pressure in compression are assigned with positive sign. and fare the mass densities of the mixture and fluid, respectively. ib is unit body force, ij k is permeability tensor, which has a dimension of 3 [length][time]/[mass]and is related to the conventional permeability ' k (dimension: [length] / [time]) by / f k k g ; ij is the strain tensor; n is the porosity; f K is the bulk modulus of the fluid. It should be noted that the governing equations presented here are simplified from [5] with the assumption that solid grains are incompressible. The effective stress ij can be calculated from the constitutive relationship in rate form as follows: ij ijkl kl C (3) where ijkl C is the tangent modulus of the solid and ij is strain tensor. The solid-fluid behaviors are coupled in that changing pore pressure influences the mechanical equilibrium state of the mixture, and the volumetric strain rate of the mixture affects the mass balance of the fluid. While the coupling problem can be approximated and solved in a decoupled manner in some special cases, the coupling effect is an important for a strongly coupled hydro-mechanical system and it cannot be neglected. 3. Spatial Discretization with RKPM Denote uN and pN as the shape functions for displacement u and pore water pressure p , respectively. The semi-discrete form of the governing equations (1) and (2) can be obtained by multiplying the governing equations (1) and (2) by u T N and p T N respectively, followed by integration by part: 0 u Mu Ku Qp f (4) 0 T p Q u Hp Sp f (5) where
RkJQdWJsaXNoZXIy MjM0NDE=