The micromechanics of network deformations
Authors: Sternstein S. S. 
Published: June 2010

Application of a homogeneous stress or deformation field to a polymeric solid does not produce locally homogeneous response. Rather, the local material response frequently displays major variations in degree. For example, the phenomenon of craze and crack formation in glassy polymers and the formation of microvoids in stretched elastomers show clearly that the deformation field is highly inhomogeneous.

In elastomers and glassy high polymers one may anticipate such inhomogeneous response arising from the macromolecular structure of the system. For example, consideration of nonuniform cross‐link density in an elastomer suggests that a considerable variation in chain extensions would exist throughout the sample. The ultimate properties of such a system would be expected to depend markedly on the more highly deformed components of the system. However, the description of the macroscopic sample in terms of molecular response would entail simultaneous treatment of both linear and nonlinear (Gaussian and non‐Gaussian) chain statistics.

In order to describe the macroscopic, mechanical response of a polymeric network in terms of the behavior of its constituent elements, the macromolecules, it is necessary to develop a formalism for averaging both linear and nonlinear behavior. This paper is concerned with such a scheme. A minimum free energy approach to the network problem is presented which allows prediction of the molecular deformations without invoking the tenuous assumption that the molecular deformation field is a continuum deformation field.

The analysis involves the formulation of integral equations describing the continuum stress and deformation tensors and the macroscopic work function, in terms of appropriate averages of the deformations experienced by the elements of the network. The elastic, but not necessarily linear, response of the network. The elastic, but not necessarily linear, response of the network is characterized by a state of minimum free energy. The elemental deformations are determined such that they minimize the macroscopic work function subject to the constraints imposed on this minimum by the macroscopic network deformations. The minimizing functions, or extremals, are obtained using the calculus of variations.

This approach to the network problem provides a self‐consistent formalism for comparing various elemental behaviors and the resultant network response. The elemental deformations are functionals of the elemental potential (or strain energy) function. Thus, a different family of extremals is obtained for a particular non‐Gaussian chain than for a Gaussian chain. In this way, network stress‐strain behavior for a multitude of chain statistics can be examined using a single, self‐consistent scheme. The procedure can be applied to chains exhibiting both internal energy and entropy changes upon extension.

The original derivation of this theory has been applied to a two‐dimensional, anisotropic network of Hookean and non‐Hookean fibers and is given elsewhere [1]. The three‐dimensional treatment of the elastomer network for both Gaussian and non‐Gaussian chains is developed fully elsewhere [2] and constitutes a complete presentation of the paper given at this conference.