Constitutive equations in finite elasticity of rubbers

A constitutive model is derived for the elastic behavior of rubbers at arbitrary 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 material. With reference to the concept of constrained junctions, the chain ends are assumed to be located at some distances from appropriate junctions. These distances are not fixed, but are altered under deformation. An explicit expression is developed for the distribution function of vectors between junctions (an analog of the end-to-end distribution function for a flexible chain with fixed ends). An analytical formula is obtained for the strain energy density of a polymer network, when the ratio of the mean-square distance between the ends of a chain and appropriate junctions is small compared with the mean-square end-to-end distance of chains. Stress–strain relations are derived by using the laws of thermodynamics. The governing equations involve three adjustable parameters with transparent physical meaning. These parameters are found by fitting experimental data on plain and particle-reinforced elastomers. The model ensures good agreement between the observations at uniaxial tension and the results of numerical simulation, as well as an acceptable prediction of stresses at uniaxial compression, simple shear and pure shear, when its parameters are found by matching observations at uniaxial tensile tests.