On the accuracy of numerical integration over the unit sphere applied to full network models

This paper is motivated by a recent study by Verron (Mecha Mater 89:216-228, 2015) which revealed huge errors of the numerical integration over the unit sphere in application to large strain problems. For the verification of numerical integration schemes we apply here other analytical integrals over the unit sphere which demonstrate much more accurate results. Relative errors of these integrals with respect to corresponding analytical solutions are evaluated also for a full network model of rubber elasticity based on a Pad, approximation of the inverse Langevin function as the chain force. According to the results of our study, the numerical integration over the unit sphere can still be considered as a reliable and accurate tool for full network models.