Modeling collagen recruitment in hyperelastic bio-material models with statistical distribution of the fiber orientation

Gradual fiber recruitment is one of the stiffening mechanisms observed in collagen reinforced biological tissues. Given the natural statistical distribution of the fiber orientation in biological materials, in agreement with experimental findings it is reasonable to assume a stochastic nature of the fiber recruitment mechanism. In the present study, we consider the presence of a stochastic recruitment mechanism in a hyperelastic fiber reinforced material model characterized by statistical distributions of the fiber orientation. The material model is based on a second order approximation of the strain energy density, considered as a function of the fourth pseudo-invariant I4, and on the multiplicative decomposition of the deformation gradient and, consequently, of the stretch. For a planar distribution of the fiber orientation, we choose an exponential analytical expression of the strain energy density and derive the stress and stiffness tensors. The mechanical behavior of the model is assessed, through uniaxial tests, by distinguishing the mean and the variance contributions of I4 to the model is validated against experimental data.