Moment-based, univariate n-point quadrature rules in application to the full network model of rubber elasticity

Abstract

In this contribution we provide numerical methods to implement full network models with particular application to affine isotropic networks as they are frequently applied in theories of rubber elasticity. Unlike the common approaches, the average of the single chains’ responses is not obtained by spherical integration but by solving a univariate integral expressed in terms of the squared stretch of a fibre's or chain's end-to-end vector. In addition to the free energy function of these individual elements the methods are informed by the statistical moments of the distribution of stretch in the network, which throughout the work is assumed to be determined by affine kinematics. We exemplify the proposed procedure for two quadrature methods, which distinguish in terms of the positions of the n integration points and the corresponding weights. While the first method uses constant equal weights of 1/n and hence only requires the computation of n integration points, the second, Gauss-type method also requires the determination of the corresponding weights and builds on a recent development, previously implemented for up to 3 points (Britt & Ehret, Comput. Methods Appl. Mech. Engrg. 415, 2023). However, the structure of the solution strategy applies to a wider range of univariate quadrature rules. Both methods exemplified here can be made exact for polynomial chain free energy functions of arbitrary order, and are illustrated in application to the affine full network model of rubber elasticity with non-Gaussian chains. The results indicate high accuracy of the new methods and therefore identify them as useful and efficient alternatives to the existing approaches for computing the full network response.