Ogden-type constitutive equations in finite elasticity of elastomers

An explicit expression is derived for the strain energy of a chain under three-dimensional deformation with finite strains. For a Gaussian chain, this relation implies the Mooney-Rivlin constitutive law, while for non-Gaussian chains it results in novel constitutive equations. Based on a three-chain approximation, a formula is derived for the strain energy of a chain with excluded-volume interactions between segments. It is demonstrated that for self-avoiding chains with a stretched exponential distribution function of end-to-end vectors, the strain energy density of a network is described by the Ogden law with two material constants. For the des Cloizeaux distribution function, a constitutive equation is derived that involves three adjustable parameters. The governing equations are verified by fitting observations at uniaxial tension-compression and biaxial tension of elastomers. Good agreement is demonstrated between the experimental data and the results of numerical analysis.