Review of the analytical network-averaging: part I — theoretical foundation

Abstract

AbstractThe analytical network-averaging, an elegant homogenization scheme, has been proposed in Khiêm and Itskov (J. Mech. Phys. Solids 95:254–269, 2016) to capture a wide range of mechanical phenomena in soft materials. These phenomena include nonlinear elasticity observed in unfilled rubbers, anisotropic damage behaviors in mechanoresponsive elastomers, phase transition occurring in natural rubbers, cross-effect of damage in double-network hydrogels, and irreversible fiber sliding in textile reinforcements. While the effectiveness of the analytical network-averaging has been evidenced through various illustrative examples, a thorough exposition of the theory remains elusive, primarily due to the concise nature preferred in conventional scientific articles and the specific thematic emphasis of individual publications. In the first part of this review series, an exhaustive theoretical examination of the analytical network-averaging concept is provided. Such theory postulates the presence of an orientational distribution function of material directions, such as fibers or polymer chains. Utilizing this distribution function, microscopic strain measures essential for solving homogenized boundary value problems can be obtained by averaging of macroscopic strain measures. It is interesting that in all scenarios, analytical derivation of the microscopic strain measures can always be obtained. Thus, such analytical homogenization scheme offers physically based invariants that automatically guarantee realistic behaviors (such as non-affine deformation, material objectivity and material symmetry) in stress response functions. This is particularly important in the age of data science and machine learning as it allows for the choice of stress hypothesis without limitations, while ensuring a priori interpretability of results.