Stochastic modeling in nonlinear micromechanics – MOSAIC

As far as model uncertainties are concerned, extensive use will be made of non parametric representations constructed by invoking information theory. In addition, system-parameter uncertainties will be accounted for through functional (e.g. polynomial chaos) representations or algebraic models. These stochastic models will be subsequently calibrated by solving statistical inverse problems involving stochastic boundary value problems and/or multiscale constraints. Here, the use of L^2 and statistical methods will be favoured. The issue of random generation will be finally tackled through the construction of families of stochastic differential equations. For the modeling of random interphases, the relevance of the proposed framework will be assessed through molecular dynamics simulations.

First, and regarding the stochastic modeling of random interphases, the conditions under which the latter can be homogenized and subsequently replaced by an deterministic imperfect interface were numerically investigated. In particular, a methodology for the inverse identification of the elastic parameters associated with the equivalent interface has been successfully proposed. Further, random field models for system-parameter uncertainties were constructed and exemplified in a multiscale framework. These new models specifically allow anisotropy constraints to be accounted for and turns out to be especially relevant to the modeling of random interphases (the latter being characterized by a specific local morphology, as shown by atomistic calculations). In order to perform the calibration of these continuum models, a novel methodology for inverse identification has been proposed and relies on molecular dynamics simulations. The construction of novel random generators for the above random fields tensor values was then addressed. The proposed algorithms involve families of stochastic differential equations and do not suffer from the curse of dimensionality.

As an end-term perspective, this project will allow for the development of multiscale approaches taking into account both model and system-parameter uncertainties in the modeling of linear and non linear microstructures. The methodologies and algorithms thus obtained will therefore enhance the predictive capabilities of multiscale methods involving nanoscopic phenomena and atomistic-continuum coupling.

Publications:
[A1] J. Guilleminot, C. Soize. Stochastic model and generator for random fields with symmetry properties: Application to the mesoscopic modeling of elastic random media, SIAM Multiscale Modeling & Simulation, 11(3), pp. 840-870, 2013.
[A2] J. Guilleminot, T. T. Le, C. Soize. Stochastic framework for modeling the linear apparent behavior of complex materials: application to random porous materials with interphases, Acta Mechanica Sinica, 29(6), pp. 773-782, 2013.
[A3] J. Guilleminot, C. Soize, Generation of non-gaussian vector-valued random fields for uncertainty quantification, soumis le 10 décembre 2013.

Communications:
[C1] J. Guilleminot, C. Soize. Generation of non-Gaussian tensor-valued random fields using an ISDE-based algorithm, ICOSSAR 2013, New-York, USA, 16-20 juin 2013.
[C2] T. T. Le, J. Guilleminot, C. Soize. Modélisation probabiliste des effets de surface pour des matériaux nano-renforcés, Congrès Français de Mécanique, Bordeaux, France, 26-30 août 2013.
[C3] J. Guilleminot, C. Soize, Adaptive ISDE-based algorithm for the generation of non- Gaussian vector-valued random fields, 11th World Congress on Computational Mechanics (WCCM-XI) coupled with the 5th European Conference on Computational Mechanics (ECCM V), Barcelona, Spain, July 20-25 2014.
[C4] T. T. Le, J. Guilleminot, C. Soize, Stochastic modeling of interphase effects for nano- reinforced heterogeneous materials, 11th World Congress on Computational Mechanics (WCCM-XI) coupled with the 5th European Conference on Computational Mechanics (ECCM V), Barcelona, Spain, July 20-25 2014.

This proposal is concerned with the development of novel methodologies (including identification and validation strategies), stochastic representations and numerical methods in stochastic micromechanical modeling of nonlinear microstructures and imperfect interfaces. For the sake of feasibility, the applications will specifically focus on the modeling of hyperelastic microstructures and materials exhibiting surface effects and containing nano-inhomogeneities (such as nanoreinforcements and nanopores).
For the case of nonlinear microstructures, the project aims at developing relevant probabilistic models for quantities of interests at both the microscale and mesoscale. The consideration of the latter turns out to be especially suitable for random nonlinear microstructures (such as living tissues) for which the scale separation, which is usually assumed in nonlinear homogenization, cannot be stated. Random variable and random field models for strain-energy functions will be constructed by invoking the maximum entropy principle and propagated through stochastic nonlinear homogenization techniques. A complete methodology for identifying the proposed representations will be further introduced and validated on a simulated database.
Concerning the imperfect interface modeling, one may note that surface effects are usually taken into account by retaining an interface model (such as the widely used membrane-type model) involving several assumptions such as those related to the mechanical description of the membrane. Such arbitrary choices certainly generate model uncertainties which may be critical while propagated to coarsest scales and which may therefore penalize the predictive capabilities of the associated multiscale approaches. In this project, we propose to tackle the issue of model uncertainties in multiscale analysis of random microstructures with nano-heterogeneities by constructing nonparametric probabilistic representations for the homogenized properties.
A complementary aspect is the construction of robust random generators, able to simulate random variables taking their values in given subspaces defined by inequality constraints and non-Gaussian random fields. Whereas such random fields can typically be generated making use of point-wise polynomial chaos expansions, the preservation of the statistical dependence is hardly achievable with the currently available techniques. In this proposal, we will subsequently address the construction of new random generators relying on the definition of families of Itô stochastic differential equations. Such generators are intended to depend on a limited number of parameters (independent of the probabilistic dimension), for which tuning guidelines will be provided.
The proposed models will clearly go a step beyond what is currently done in deterministic mechanics for such materials and the expected results are in the forefront of the ongoing developments within the scopes of uncertainty quantification and material science. In addition, it worth pointing out that such theoretical derivations are absolutely required in order to support the current new developments of 3D-fields measurements and image processing at the microscale of complex materials.