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Subsubsection 4.2.2.3: Fiber with Toe-Linear Response, Uncoupled Formulation Up Chapter 4: Materials Subsection 4.3.1: Unconstrained Continuous Fiber Distribution

## 4.3 Continuous Fiber Distribution

A continuous fiber distribution has a strain energy density that integrates the contributions from fiber bundles oriented along all directions emanating from a point in the continuum, where is the unit vector along the fiber orientation in the reference configuration, is the normal component of along (also the square of the stretch ratio along that direction), and represents the unit sphere (for 3D fiber distributions) or unit circle (for 2D fiber distributions) over which the integration is performed [91]. Thus, spans all directions from the origin to points on the unit sphere or unit circle. In the integrand, represents the strain energy density of the fiber bundle oriented along ; is the Heaviside unit step function that includes only fibers that are in tension; and is the fiber density distribution function that specifies the spatial fractional distribution of fibers. This function satisfies the constraint Fiber constitutive models may be taken from the list given in Section 4.2.1↑.
For a material with an uncoupled strain energy density the corresponding expression is where . In this case, fiber constitutive models may be taken from the list given in Section 4.2.2↑.

Subsubsection 4.2.2.3: Fiber with Toe-Linear Response, Uncoupled Formulation Up Chapter 4: Materials Subsection 4.3.1: Unconstrained Continuous Fiber Distribution