Univariate Gauss quadrature for structural modelling of tissues and materials with distributed fibres

This work presents a method for computing the averaged free energy and constitutive relations in hyperelastic material models with distributed fibres, as they apply to soft fibre-reinforced materials and biological tissues. While these models are currently implemented through either spherical cubature of the fibre free energy or its Taylor series, we here propose a new method based on a univariate Gauss quadrature rule with integration points and weights informed by the statistical moments of the distribution of fibre stretch. As an intrinsic property, the new approach separates the integration of the fibre constitutive law from the integration of the orientation distribution, the latter leading to structural tensors of even order. Provided the latter 2n−1 tensors are computed accurately up to tensorial order 2(2n−1), the method integrates exactly any polynomial of order 2n−1 that agrees with the fibre law at the n integration points. After formally introducing the quadrature method for generally non-affine fibre deformations and arbitrary order, we focus on the important special case of affine fibre kinematics and discuss the rules with n≤3 integration points, for which the corresponding positions and weights are determined analytically. At a computational cost comparable to the existing approaches, the new method does not require the fibre law to be analytic and can thus robustly deal with piece-wise definitions of the fibre energy, in contrast to Taylor-series approaches, and it does not induce additional anisotropy as it can occur with spherical cubature rules. The 3-point rule is further investigated and illustrated in numerical examples relating to soft collagenous tissues based on a Fortran implementation of the method suitable for use in finite element analyses.