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3.7.2 BFSI Linearization
Using the virtual work integral such that it may be expanded as
The linearizations of integrals are performed in the material frame of the solid domain of , allowing us to linearize along increments , , or by simply bringing the directional derivative operator inside the integrals of Eqs. (3.7.1-3)-(3.7.1-4). For notational convenience, we let and . Thus, the conversion of the internal virtual work to the material frame of the solid produces where is the second Piola-Kirchhoff stress for the solid constituent of the mixture, is the Piola transformation of , and is an elemental volume of in its material frame . In addition, is the inverse of the permeability tensor in the material frame. Note that in the material frame, the fluid acceleration is The linearization of is then performed along an increment , and the integral is reverted back to the spatial frame, yielding for the displacement equations for the fluid flux equations and for the fluid dilatation equations where , and is the fourth-order spatial elasticity tensor associated with . As done in the CFD formulation (Section 3.5.2↑) , we have introduced the fourth-order tensor representing the tangent of the viscous stress with respect to the rate of deformation, where is the fluid rate of deformation tensor (the symmetric part of ). The tensors and depend on the choice of constitutive relations for the fluid and solid constituents of , respectively. In addition, is the spatial fourth order permeability tensor with respect to right Cauchy-Green tensor . The linearized equations include the generalized- parameters because the virtual work is evaluated at the intermediate time step, while the increment itself ( in this case) is at the current time step .
Following the same procedure, the linearizations of along an increment is given by whereas the linearizations of along an increment is Here, and respectively represent the first and second derivatives of . We have also defined as the tangent of the viscous stress with respect to ,
For the external work, when , , all and are prescribed, the linearizations simplify to
As discussed in Section ↓, we define the fluid dilatation as an alternative essential variable, since initial and boundary conditions are more convenient to handle in a numerical scheme than . It follows that and . Therefore the changes to the above equations are minimal, simply requiring the substitution and .