Theory Manual Version 3.4
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Subsection 3.2.2: Discretization Up Chapter 3: The Nonlinear FE Method Subsection 3.3.1: Linearization of Internal Virtual Work

## 3.3 Weak Formulation for Biphasic-Solute Materials

The virtual work integral for this problem is given by where is the virtual velocity of the solid, is the virtual effective fluid pressure, and is the virtual molar energy of the solute. represents the mixture domain in the spatial frame and is an elemental mixture volume in . In the last integral of , note that where is the material time derivative of a scalar function in the spatial frame, following the solid. Similarly, note that . Using the divergence theorem, the virtual work integral may be separated into internal and external contributions, , where with being evaluated on the domain's boundary surface . In the first expression represents the virtual solid rate of deformation.
To solve this nonlinear system using an iterative Newton scheme, the virtual work must be linearized at trial solutions, along increments in , and , where, for any function , represents the directional derivative of along [23]. To operate the directional derivative on the integrand of , it is first necessary to convert the integrals from the spatial to the material domain [23]: where represents the mixture domain in the material frame, is an elemental mixture volume in , and The second Piola-Kirchhoff stress tensor , and material flux vectors and , are respectively related to , and by the Piola transformations for tensors and vectors [23, 54]. Substituting (3.3-6) into (2.6.2-3) produces where and are the material representations of the permeability and diffusivity tensors, related to and via the Piola transformation, The linearization of is rather involved and a summary of the resulting lengthy expressions is provided below. In consideration of the dearth of experimental data relating and to the complete state of solid matrix strain (such as , this implementation assumes that the dependence of these functions on the strain is restricted to a dependence on the volume ratio . Furthermore, it is assumed that the free solution diffusivity is independent of the strain.
The linearization of is described in Section 3.3.2↓. Following the linearization procedure, the resulting expressions may be discretized by nodally interpolating , and over finite elements, producing a set of equations in matrix form, as described in Section 3.3.2↓.
The formulation presented in this study is implemented in FEBio by introducing an additional module dedicated to solute transport in deformable porous media. Classes are implemented to describe material functions for , , (and , and , which allow the formulation of any desired constitutive relation for these functions of and , along with corresponding derivatives of these functions with respect to and . The implementation accepts essential boundary conditions on , and , or natural boundary conditions on , and ; initial conditions may also be specified for and . Analysis results for pressure and concentration may be displayed either as and , or as and by inverting the relations of (2.6.2-1).
Subsection 3.2.2: Discretization Up Chapter 3: The Nonlinear FE Method Subsection 3.3.1: Linearization of Internal Virtual Work