Rapid Evaluation of Material Properties for Materials Reinforced by Statistically Oriented Fibres

Abstract

The arrangement of the collagen fibres in biological tissues can be described by an orientation probability distribution. The overall contribution of the fibres to a given mechanical property can be estimated by an averaging integral of the constitutive function describing the mechanical property over the set of all directions in space. Except for the case of polynomials in the structure tensor (tensor product of the unit vector describing the fibre direction by itself), the averaging integral cannot be evaluated directly, as the integrand is a function of both deformation and fibre orientation. In order to avoid computationally expensive integrations at each increment of deformation, this work proposes three methods based on the direct integrability of polynomials, which are then compared against an averaging method available in the literature and against the integration made at each increment of deformation. Finally, a representation of the higher order averaged structure tensors is suggested.