Probabilistic and stochastic aspects of rubber hyperelasticity

The main aim of this work is to compare various models of rubber elasticity, i.e. neo-Hookean, Mooney-Rivlin, Yeoh, Gent, Arruda-Boyce as well as the Extended Tube model in terms of their application to the probabilistic analysis. Some discussions concerning failure analysis of the rubbers according to these models is provided also. Constitutive relations following these theories are tested for the case of uniaxial tension of the incompressible material, where deformation of a rubber specimen is treated as Gaussian random variable having a priori given expectation and standard deviations varying in some interval with bounds driven by various experimentation techniques. Probabilistic analysis is provided here in two alternative ways—via traditional Monte-Carlo technique as well as using higher order stochastic perturbation method implemented both in the symbolic computer algebra software. An application of non-Gaussian distributions relevant to the considered deformation, like lognormal one for instance has been also considered. This analysis includes computational determination of the first four basic probabilistic characteristics, i.e. expectation, coefficient of variation, skewness and kurtosis, and is provided to verify the resulting probabilistic distribution of the induced stress and its entropy. Some conclusions are drawn for the generalization of this method to other stress softening materials.