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Since the system of equations in Eq.(3.2-2) is highly nonlinear, its solution requires an iterative scheme such as Newton's method. This requires the linearization of at some trial solution , along an increment in and an increment in , where represents the directional derivative of along . For convenience, the virtual work may be separated into its internal and external parts, where where we have substituted , and The evaluation of the directional derivatives can be performed following a standard approach . In particular, a backward difference scheme is used to evaluate , where is the value of at the previous time step. For the internal part of the virtual work, the directional derivative along yields where is the fourth-order spatial elasticity tensor for the mixture and . Based on the relation of Eq.(2.5.1-3), the spatial elasticity tensor may also be expanded as where is the spatial elasticity tensor for the solid matrix . It is related to the material elasticity tensor via where is the deformation gradient of the solid matrix, where is the Lagrangian strain tensor and is the second Piola-Kirchhoff stress tensor, related to the Cauchy stress tensor via .
Similarly, is a fourth-order tensor that represents the spatial measure of the rate of change of permeability with strain. It is related to its material frame equivalent via where and is the permeability tensor in the material frame, such that . Since and are symmetric tensors, it follows that and exhibit two minor symmetries (e.g., and . However, unlike the elasticity tensor, it is not necessary that these tensors exhibit major symmetry (e.g., in general).
The directional derivative of along is given by Note that letting and in the above equations recovers the virtual work relations for nonlinear elasticity of compressible solids. The resulting simplified equation emerging from Eq.(3.2.1-5) is symmetric to interchanges of and , producing a symmetric stiffness matrix in the finite element formulation . However, the general relations of Eqs.(3.2.1-5) and (3.2.1-9) do not exhibit symmetry to interchanges of and , implying that the finite element stiffness matrix for a solid-fluid mixture is not symmetric under general conditions.
The directional derivatives of the external virtual work depend on the type of boundary conditions being considered. For a prescribed total normal traction , where , and where are covariant basis (tangent) vectors on , such that For a prescribed normal effective traction , where and is not prescribed, then and For a prescribed normal fluid flux , and Finally, for a prescribed external body force, recognizing that and assuming that the body forces and do not depend on ,