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2.5.1 Governing Equations
Consider a mixture consisting of a solid constituent and a fluid constituent. Both constituents are considered to be intrinsically incompressible, but the mixture can change volume when fluid enters or leaves the porous solid matrix [25, 57]. According to the kinematics of the continuum , each constituent of a mixture ( for the solid and for the fluid) has a separate motion which places particles of each mixture constituent, originally located at , in the current configuration according to For the purpose of finite element analyses, the motion of the solid matrix, , is of particular interest.
The governing equations that enter into the statement of virtual work are the conservation of linear momentum and the conservation of mass, for the mixture as a whole. Under quasi-static conditions, the conservation of momentum reduces to where is the Cauchy stress for the mixture, is the mixture density and is the external mixture body force per mass. Since the mixture is porous, this stress may also be written as where is the fluid pressure and is the effective or extra stress, resulting from the deformation of the solid matrix. Conservation of mass for the mixture requires that where is the solid matrix velocity and is the flux of the fluid relative to the solid matrix. Let the solid matrix displacement be denoted by , then .
To relate the relative fluid flux to the fluid pressure and solid deformation, it is necessary to employ the equation of conservation of linear momentum for the fluid, where is the solid matrix porosity, is the apparent fluid density and is the true fluid density, is the external body force per mass acting on the fluid, and is the momentum exchange between the solid and fluid constituents, typically representing the frictional interaction between these constituents. This equation neglects the viscous stress of the fluid in comparison to . The most common constitutive relation is , where the second order, symmetric tensor is the hydraulic permeability of the mixture. When combined with Eq.(2.5.1-5), it produces which is equivalent to Darcy's law. In general, may be a function of the deformation.