A generalized statistical approach for modeling fiber-reinforced materials

We present a generalized statistical approach for the description of fully three-dimensional fiber-reinforced materials, resulting from the composition of two independent probability distribution functions of two spherical angles. We discuss the consequences of the proposed formulation on the constitutive behavior of fibrous materials. Upon suitable assumptions, the generalized formulation recovers existing alternative models, based on averaged structure tensors both at first- and second-order approximations. We demonstrate that the generalized formulation embeds standard behaviors of fiber-reinforced materials such as planar isotropy and transverse isotropy, while any intermediate behavior is easily obtained through the calibration of two material parameters. We illustrate the performance of the model by means of uniaxial and biaxial tests. For uniaxial loading, we introduce a preliminary discussion concerning the generalized tension-compression switch procedure.