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Subsubsection 4.1.3.27: Large Poisson's Ratio Ligament Up Chapter 4: Materials Subsection 4.2.1: Unconstrained Fiber Models

## 4.2 Fibers

Fiber materials are used to model one-dimensional structures oriented along a unit vector. The associated strain energy density is a function of the normal strain along that vector. These fiber models are constructed such that the strain energy is non-zero only when the fiber is in tension, under the idealized assumption that fibers do not sustain any load in compression. This assumption produces an inherent instability in the material response of fibers, therefore such models must be combined with elastic materials that can sustain compression and tension, thus serving as models of a ground matrix.
FEBio accommodates fiber models that can be combined with unconstrained or uncoupled models of the ground matrix. Historically, unconstrained models of a ground matrix have been combined with unconstrained fiber models, whereas uncoupled models of a ground matrix have been combined with uncoupled fiber models. The manual sections presented below follow this conventional approach.
However, it should be noted that some authors have expressed concerns about using uncoupled fiber formulations in fiber-matrix material models [84, 85, 86], due to two fundamental concerns: When highly nonlinear fibers become much stiffer than the ground matrix upon loading, it may become difficult to enforce an isochoric deformation, as would be expected in an uncoupled formulation. Furthermore, in an uncoupled formulation, the strain is split into its deviatoric (isochoric) and volumetric parts and the fiber strain is evaluated only from the deviatoric part. This deviatoric fiber strain is not the true fiber strain, yet it is used to determine whether the fiber is in tension. These factors may result in non-physical deformations, as described by Helfenstein et al. [87].
The solution advocated by these investigators has been to use an uncoupled formulation for the ground matrix only, while the fiber models should remain unconstrained [88, 89, 90]. This can be done in FEBio by using a “uncoupled solid mixture” container as described in Section 4.1.2.14↑, where the material representing the ground matrix is taken from the list of uncoupled materials in Section 4.1.2↑, whereas fiber models are taken from the list of unconstrained models in Section 4.2.1↓ or Section 4.3.1↓.

Subsubsection 4.1.3.27: Large Poisson's Ratio Ligament Up Chapter 4: Materials Subsection 4.2.1: Unconstrained Fiber Models