, Volume 53, Issue 9, pp 2363–2378 | Cite as

Probabilistic and stochastic aspects of rubber hyperelasticity

  • Marcin Kamiński
  • Bernd Lauke


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.


Rubber elasticity Monte-Carlo simulation Stochastic perturbation technique 



The first Author would like to acknowledge the research fellowship from Leibniz-Institut für Polymerforschung Dresden e.V. in Germany where this work has been partially completed. The Authors would like to declare that they both actively participated in preparation of any part of this work. It has been submitted nowhere else and any part of this work is entirely intellectual property of both Authors. The Authors would like to acknowledge a discussion of some ideas contained in this work with prof. G. Heinrich from IPF Dresden.


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Authors and Affiliations

  1. 1.Chair of Structural Reliability, Department of Structural Mechanics, Faculty of Civil Engineering, Architecture and Environmental EngineeringTechnical University of ŁódźŁódźPoland
  2. 2.Leibniz-Institut für Polymerforschung Dresden e.VDresdenGermany

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