On three- and two-dimensional fiber distributed models of biological tissues

We describe three-dimensional and planar models of hyperelastic fiber reinforced materials characterized by statistical distribution of the fiber orientation. Our models are based on a second order approximation of the strain energy density in terms of the fourth pseudo-invariant I¯4, typically employed in the description of fiber reinforced materials. For a particular choice of the strain energy density associated to the fiber reinforcement, it is possible to derive the explicit expression of the material and spatial stress tensors and of the stress covariance tensors. The mechanical behavior of the models is assessed through uniaxial, biaxial and shear tests.