Constitutive equations for the nonlinear elastic response of rubbers

A constitutive model is derived for the elastic behavior of rubbers at three-dimensional deformations with finite strains. An elastomer is thought of as an incompressible network of flexible chains bridged by permanent junctions that move affinely with the bulk medium. The constraints imposed by surrounding macromolecules on configurations of an individual chain are introduced by combining the Flory-Erman and Erman-Monnerie approaches. To describe inter-chain interactions in a tractable way, the conventional picture of a tube where a chain is confined is replaced by geometrical restrictions on the positions of its ends and center of mass. The constraints on the chain ends are formulated within the traditional Flory concept, whereas those on the position of center of mass are described following the Ronca-Allegra scenario. Stress-strain relations for a network of constrained chains are derived by using the laws of thermodynamics. The constitutive equations involve four adjustable parameters with transparent physical meaning. The material constants are found by fitting experimental data on elastomers at uniaxial and equi-biaxial tensions and pure shear. It is demonstrated that (i) the model provides an acceptable prediction of stresses in a test with one deformation mode, when its parameters are found by matching observations in an experiment with another mode, and (ii) material constants are affected by chemical composition of elastomers in a physically plausible way.