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Theory Manual Version 3.6
 Subsection 5.2.13: Fiber with Exponential Power Law Up Section 5.2: Compressible Materials Section 5.3: Nearly-Incompressible Materials  

5.2.14 Fiber with Natural Neo-Hookean Response

This model is an adaptation of the natural neo-Hookean material presented in Section (3.3.1.1-1). Consider that the state of strain in a fiber is given by the unidirectional natural (left Hencky) strain along the fiber, where is the stretch ratio along the current fiber unit vector . For this special state of strain the invariants of the natural strain tensor reduce to , , and . In this case, a natural neo-Hookean fiber response is given by The corresponding Cauchy stress is and the elasticity tensor is where is the fiber modulus. This expression shows that the components of the elasticity tensor become negative when , or equivalently when . However, the stress always remains positive.
If we want the tensile response to engage only beyond a threshold stretch ratio , we may rewrite the strain energy density as Then and
 Subsection 5.2.13: Fiber with Exponential Power Law Up Section 5.2: Compressible Materials Section 5.3: Nearly-Incompressible Materials