Modeling Uncertainties in Nonlinear Elasticity

The modeling of uncertainties in linear systems has been a subject of longstanding interest, especially within the framework of the so-called spectral approach pioneered by Ghanem and Spanos in the 90's. In this context, modeling contributions in nonlinear frameworks are still limited. Our group recently addressed the stochastic modeling of stored energy functions under constraints that are raised by:

  • physical concerns, such as phenomenological constraints and/or experimental loading curves;
  • mathematical (in)equalities, derived within the theory of nonlinear elasticity.

In addition to new models relying on a proper randomization of the Ogden class of potentials, we have assessed the relevance of the information-theoretic representations by considering:

  • the modeling of the variability exhibited by soft biomedical materias, such as brain tissues;
  • the modeling and propagation of uncertainties in nonlinear multiscale models, such as those involving hyperelastic materials at the microscale.

Examples of how our models and methods perform are provided below.

Application 1 : Modeling the Variability Exhibited by Biological Tissues

Below, the predictions of our stochastic models are compared with experimental results for two biological tissues (left panel: brain tissue; right panel: spinal cord white matter).

Variability of biological tissues (brain) Variability of biological tissues (spinal cord)

 Comparing the model predictions with experimental results for brain tissues (left panel) and spinal cord white matter (right panel)

Application 2: Uncertainty Quantification in Nonlinear Multiscale Problems

The new stochastic potentials were used to study the propagation of uncertainties in nonlinear multiscale models. More specifically, the models were used to represent uncertainties at the microscale, and the homogenized potential was represented through a polynomial chaos expansion (PCE), the coefficients of which were computed by means of a suitable quadrature rule. The comparison between our result and the reference solution obtained through Monte Carlo simulations is shown below.

Multiscale prediction using PCEs

Predicting the probability density function of the homogenized potential: PCE predictions (red circles) and reference solution (solid line)

For further details, please refer to the following (selected) publications:

  • B. Staber, J. Guilleminot, C. Soize, J. Michopoulos and A. Iliopoulos, Stochastic modeling and identification of an hyperelastic constitutive model for laminated composites, Computer Methods in Applied Mechanics and Engineering, 347, 425-444 (2019)
  • B. Staber and J. Guilleminot, A random field model for anisotropic strain energy functions and its application for uncertainty quantification in vascular mechanics, Computer Methods in Applied Mechanics and Engineering, 133, 94-113 (2018)
  • B. Staber and J. Guilleminot, Stochastic hyperelastic constitutive laws and identification procedure for soft biological tissues with intrinsic variability, Journal of the Mechanical Behavior of Biomedical Materials, 65, 743-752 (2017)
  • B. Staber and J. Guilleminot, Stochastic modelling of the Ogden class of stored energy functions for hyperelastic materials: the compressible case, ZAMM/Journal of Applied Mathematics and Mechanics, 97(3), 273-295 (2017)
  • B. Staber and J. Guilleminot, Stochastic modeling of a class of stored energy functions for incompressible hyperelastic materials with uncertainties, Comptes-Rendus Mécanique (Proceedings of the French Academy of Sciences), 343, 503-514 (2015)