Quantitative stochastic homogenization: periodic representative volume element approximations in non-linear elasticity
Final Report Abstract
Polycrystalline materials, reinforced rubber, foams, and biological tissues are examples for the huge class of random heterogeneous materials (RM). Those materials feature microstructural uncertainties: e.g. the distribution, geometry, and constitutive parameters of the individual phases of a composite might be known on a statistical level only. In many cases, if the RM is statistically homogeneous, it displays on large length-scales an (almost) deterministic physical behavior. This allows for a tremendous reduction of complexity with regard to modeling and simulation: Instead of resolving microscopic and uncertain properties, one can consider a macroscopic and deterministic model — the so-called homogenized model — describing a homogeneous material. The derivation of the homogenized model is subject of homogenization theory for partial differential equations and in the Calculus of Variations. In the context of nonlinear elasticity, it leads to a variational model of the homogenized material that takes the form of a non-convex integral functional with a homogenized stored energy function that encodes the effective material properties. The homogenized stored energy function is de ned via a homogenization formula that invokes a non-convex minimization problem on an in nite dimensional space that models the RM. The homogenization formula is of no immediate practical use, since it requires approximation. A prominent method for appoximation is based on a representative volume element (RVE) of the random microstructure. Although RVEs are widely used, only little is known about convergence rates (e.g., for linear models), and many questions are controversially discussed (e.g. size of the RVE, boundary conditions). In the present project we obtained the rst optimal, analytic results regarding the convergence rates for periodic RVEs for nonlinear random materials. More presciely, with J. Fischer we obtained results (optia mal w.r.t. the scaling in the size of the RVE) for uniformly elliptic, montone systems, and with M. Sch¨ffner and M. Varga we obtained optimal convergence rates in the non-convex case for nonlinearly elastic, random laminates. The results further develop the state of the art in the theory of quantitative stochastic homogenization. Furthermore, they generalize recent results obtained by the PI in the deterministic periodic setting to the case of random laminates. Next to the stored energy function itself, we also studied properties of its derivatives, in particular, its gradient (the stress tensor) and the Hessian (the tangent moduli), i.e., quantities of particular interest in mechanics. For these quantities, we obtained optimal converges rates for the periodic RVE approximation, and, in a work in preparation, we prove that uctuations are Gaussian. Furthermore, in joint work with S. Haberland, O. Sander, P. Jaap, M. Varga, we investigate the convergence rate for RVEs for elasto-plastic materials. Mainly on the level of numerical studies, we observe different rates that depend on the amount of plastic deformation present in the system. To understand this analytically is an interesting, open problem.
Publications
- Lipschitz estimates and existence of correctors for nonlinearly elastic, periodic composites subject to small strains, Calculus of Variations and Partial Differential Equations 58.2, 1-51, 2019
S. Neukamm and M. Schäffner
(See online at https://doi.org/10.1007/s00526-019-1495-2) - Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems, Archive for Rational Mechanics and Analysis 242 (1), 343-452, 2021
J. Fischer and S. Neukamm
(See online at https://doi.org/10.1007/s00205-021-01686-9) - Quantitative stochastic homogenization of nonlinearly elastic, random laminates
S. Neukamm, M. Schäffner, and M. Varga
(See online at https://doi.org/10.48550/arXiv.2106.08585) - Two-scale homogenization of abstract linear time-dependent PDEs, Asymptotic Analysis 125.3-4, 247-287, 2021
S. Neukamm, M. Varga, and M. Waurick
(See online at https://doi.org/10.3233/ASY-201654) - Stochastic two-scale convergence and Young measures. Networks and Heterogeneous Media
M. Heida, S. Neukamm, and M. Varga
(See online at https://doi.org/10.3934/nhm.2022004)
