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The discretization of the internal work produces where We used the following interpolations: Note that the operator should be evaluated using .
The solution of the nonlinear equation is obtained by linearizing this relation as where the operator represents the directional derivative of at along an increment of , of , or of .
To linearize the virtual work, we need to express the integrals appearing in and over the material frame of the finite element solid domain. For notational convenience, we let and . Thus, where is the second Piola-Kirchhoff stress for the solid material. Similarly, Next, where .
Keep in mind that we are solving for the motions at . Therefore, In general, (and thus, ) is only a function of the solid strain, such as the right Cauchy-Green tensor or the Green-Lagrange strain . Therefore, following the standard approach in solid mechanics, the linearization of is
In general, is a function of the fluid volume ratio and the rate of deformation of the fluid, . However, since is not a degree of freedom, we need to let , where and . Thus, where