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Theory Manual Version 3.6
 Subsection 2.4.4: Transversely Isotropic Hyperelasticity Up Section 2.4: Hyperelasticity Section 2.5: Biphasic Material 

2.4.5 Tension-Bearing Fiber Materials

In biomechanics we often find it convenient to model fibrous or fibrillar materials using one-dimensional fibers that can only sustain tension. Typically, such fibers are represented with a strain energy density function where is the unit vector along the fiber in its reference configuration, and is the square of the stretch ratio along the fiber. The Heaviside unit step function in eq.(2.4.5-1) ensures that the fiber contributes strain energy only when it is under tension ( ); thus, the constitutive model for the tensile response of the fiber must reduce to zero when .
Using the hyperelasticity relations presented above, the Cauchy stress in this fiber material can be evaluated as and the spatial elasticity tensor is If we denote the unit vector along the fiber in the current configuration as , the above expressions may be rewritten as and . As explained in [9], these fiber models must be combined with a ground matrix in order to produce a stable material response. In FEBio this can be done by using a constrained mixture of solid constituents (for example, see Section 2.8.4↓).
In the classical fiber mechanics literature it was suggested that uncoupled fiber formulations could also be implemented, whereby and the Cauchy stress is given by eq.(2.4.3-10) where Uncoupled fiber formulations of this kind are available in FEBio. More recently however, several studies have demonstrated that using to detect whether a fiber is in tension or compression is non-physical, since and is the sole true measure of the tensile stretch in the fiber [88, 50, 49]. Therefore, uncoupled fiber formulations have fallen out of favor, even though FEBio still allows users to employ these for historical reasons. It is now recommended to use the standard (unconstrained) fiber models, also available in FEBio, with the formulation given in Eqs.(2.4.5-1)-(2.4.5-3), even when the ground matrix uses an uncoupled formulation.


 Subsection 2.4.4: Transversely Isotropic Hyperelasticity Up Section 2.4: Hyperelasticity Section 2.5: Biphasic Material