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5.2.13 Fiber with Exponential Power Law
This material model describes a constitutive model for fibers, where a single fiber family follows an exponential power law strain energy function. The Cauchy stress is given by: and the corresponding spatial elasticity tensor is where is the square of the fiber stretch, is the fiber orientation in the reference configuration, and . The function is the unit step function that enforces the tension-only contribution. Thus, the stress and elasticity tensors are non-zero only when , where is the square of the stretch at which the fiber's tensile response engages. By default we may take , though the actual value of may be set by the user. The fiber strain energy density is given by where , and . From this expression we get
Note: In the limit when , this expression produces a power law, Note: According to (5.2.13-2) and (5.2.13-5), when the fiber modulus is zero at the strain origin ( . Therefore, use when a smooth transition in the stress is desired from compression to tension.
There is an option to also add a shear modulus to account for the interaction of a fiber with the ground matrix. This additional contribution does not depend on whether the fiber is in tension. It has a strain energy density The corresponding stress is where is the left Cauchy-Green tensor. The elasticity tensor is where .