Stochastic modeling of the Ogden class of stored energy functions for hyperelastic materials: the compressible case

This paper is devoted to the modeling of compressible hyperelastic materials whose response functions exhibit uncertainties at some scale of interest. The construction of parametric probabilistic representations for the Ogden class of stored energy functions is specifically considered and formulated within the framework of Information Theory. The overall methodology relies on the principle of maximum entropy, which is invoked under constraints arising from existence theorems and consistency with linearized elasticity. As for the incompressible case discussed elsewhere, the derivation essentially involves the conditioning of some variables on the stochastic bulk and shear moduli, which are shown to be statistically dependent random variables in the present case. The explicit construction of the probability measures is first addressed in the most general setting. Subsequently, particular results for classical Neo-Hookean and Mooney-Rivlin materials are provided. Salient features of the probabilistic representations are finally highlighted through forward Monte-Carlo simulations. In particular, it is seen that the models allow for the reproduction of typical experimental trends, such as a variance increase at large stretches. A stochastic multiscale analysis, where uncertainties on the constitutive law of the matrix phase are taken into account through the proposed approach, is also presented.

[1]  Julien Yvonnet,et al.  The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains , 2007, J. Comput. Phys..

[2]  Christian Soize,et al.  Stochastic Model and Generator for Random Fields with Symmetry Properties: Application to the Mesoscopic Modeling of Elastic Random Media , 2013, Multiscale Model. Simul..

[3]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[4]  L. Walpole,et al.  Fourth-rank tensors of the thirty-two crystal classes: multiplication tables , 1984, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[5]  T. Zohdi Propagation of Microscale Material Uncertainty in a Class of Hyperelastic Finite Deformation Stored Energy Functions , 2001 .

[6]  Brian Staber,et al.  Stochastic modeling of a class of stored energy functions for incompressible hyperelastic materials with uncertainties , 2015 .

[7]  Jean-Philippe Ponthot,et al.  AN OVERVIEW OF NONINTRUSIVE CHARACTERIZATION, PROPAGATION, AND SENSITIVITY ANALYSIS OF UNCERTAINTIES IN COMPUTATIONAL MECHANICS , 2014 .

[8]  Jiří Souček,et al.  Cartesian currents and variational problems for mappings into spheres , 1989 .

[9]  Stefan Müller,et al.  On a new class of elastic deformations not allowing for cavitation , 1994 .

[10]  J. Ball Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[11]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[12]  Oscar Lopez-Pamies,et al.  Second-Order Estimates for the Macroscopic Response and Loss of Ellipticity in Porous Rubbers at Large Deformations , 2004 .

[13]  Alan Edelman,et al.  The efficient evaluation of the hypergeometric function of a matrix argument , 2006, Math. Comput..

[14]  R. Ogden Large deformation isotropic elasticity – on the correlation of theory and experiment for incompressible rubberlike solids , 1972, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[15]  Christian Soize,et al.  Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials , 2013 .

[16]  Christian Soize,et al.  Computational nonlinear stochastic homogenization using a nonconcurrent multiscale approach for hyperelastic heterogeneous microstructures analysis , 2012, International Journal for Numerical Methods in Engineering.

[17]  Daya K. Nagar,et al.  Matrix variate Kummer-Dirichlet distributions , 2001 .

[18]  Christian Soize,et al.  Itô SDE-based Generator for a Class of Non-Gaussian Vector-valued Random Fields in Uncertainty Quantification , 2014, SIAM J. Sci. Comput..

[19]  Christian Soize,et al.  On the Statistical Dependence for the Components of Random Elasticity Tensors Exhibiting Material Symmetry Properties , 2012, Journal of Elasticity.

[20]  Monica L. Skoge,et al.  Packing hyperspheres in high-dimensional Euclidean spaces. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Christian Miehe,et al.  Computational homogenization analysis in finite elasticity: material and structural instabilities on the micro- and macro-scales of periodic composites and their interaction , 2002 .

[22]  G. Chagnon,et al.  Hyperelastic Energy Densities for Soft Biological Tissues: A Review , 2015 .

[23]  Omar M. Knio,et al.  Spectral Methods for Uncertainty Quantification , 2010 .

[24]  Ivonne Sgura,et al.  Fitting hyperelastic models to experimental data , 2004 .

[25]  E. Jaynes Information Theory and Statistical Mechanics , 1957 .