2.2.2. Fiber phase

The Helmholtz free energy function of the fiber phase can be expressed in terms of anisotropic pseudo-invariants of the deformation. In this study we use the following form for ΨEQF and ΨNEQF:

with

where M=V⊗V is the structure tensor defined in terms of material fiber direction vector V. In general, the fiber direction vector can be an arbitrary vector in 3D space. However, for many biological tissues the vector can reasonably be assumed to lie in a plane, especially if plane stress or strain deformations are considered, resulting in a 2D vector of the form V=[cosθsinθ0]T with θ the material fiber direction. This assumption simplifies the computations for the fiber phase since it does not contribute to the out-of-plane response. As in the matrix case, we use NODEs to approximate the derivative functions ∂ΨEQF/∂I4, ∂ΨNEQF/∂I4eF.