A review of current constitutive formulations for Network materials is presented in this chapter. Network materials are composed from discrete elements and are not continua. Their behavior is somewhat similar to that of mechanisms. Furthermore, deformation is generally nonaffine due to the stochastic network structure. These observations render difficult the adaptation of classical constitutive equations for this class of materials. These issues are discussed in detail in the opening section. Further, the chapter is divided into four sections, each presenting models of a certain type. The first category includes phenomenological models defined based on a free energy functional and examples relevant for thermal networks (elastomers and gels) are presented. The next three categories encompass mechanism-based models, which are divided based on the degree to which the respective models account for nonaffinity in affine, quasi-affine, and nonaffine models. An outline of the challenges and opportunities related to the development of mechanism-based constitutive models for Network materials is presented in closure.

Affine models have been used traditionally to describe the deformation of networks. Due to their prevalence, this chapter is dedicated to the review of such formulations. The chapter begins with a brief review of finite kinematics of continua and the definition of stress measures. Further, the affine deformation is defined and several parameters used to quantify the degree of nonaffinity are introduced. An expression is derived to quantify the evolution of preferential fiber orientation during affine deformation. Several constitutive models based on the affine deformation assumption are discussed: The affine models for molecular networks of flexible and semi-flexible filaments, and the affine model for athermal networks. The stress–optical law is reviewed, and its relation to the affine deformation models is discussed.