A framework for microplane models at large strain, with application to hyperelasticity

Abstract

A general framework is proposed for the formulation of microplane models at large strain. It is based on the thermodynamic approach to microplane formulation recently presented by the authors, which defines the macroscopic free energy of the material as an integral of a microplane free-energy potential over all possible orientations. By simple differentiation with respect to strain, it is possible to obtain the consistent definition of microplane stresses and integral expressions for evaluation of the macroscopic stress tensor. To apply this approach to large strains, new microplane strain measures need to be defined, including volume change, stretch of fibers, “thickening” of planes, deviatoric parts of the stretch and thickening, and distortion (shear) angles. Based on these, various microplane formulations are developed. Each formulation starts with the definition of microplane stresses and the derivation of the integral expressions which are valid for the general case of dissipative materials. Then, these expressions are particularized to specific forms of hyperelastic potentials leading to various hyperelastic models. The simplest model, with a quadratic microplane potential in terms of the fiber stretch, corresponds to the classical Gaussian statistical theory of long-chain molecules and leads to the neo-Hookean type of macroscopic free-energy potential. Many other, more complex forms of the microplane potential are investigated and their relation to existing models for rubber elasticity is analyzed. It is shown that, in the small-strain limit, they collapse into well-known small-strain microplane formulations, either with restricted or with unrestricted values of Poisson’s ratio.