Fracture models in SBD: Homogenization and quasistatic evolution
Final Report Abstract
The analysis of fracture models presents a variety of challenging mathematical questions, e.g. the understanding of effective mechanical behavior of brittle materials with heterogeneities. Formulations by variational methods, where solutions are determined from an energy minimization principle, provide efficient tools for modeling, analysis, and simulations. In this project we have focused on existence results and homogenization procedures for fracture models in linearized elasticity. More specifically, in the first Work Package, we have derived Γ-convergence and lower semicontinuity results for piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is constant and lies in a set without rank-one connections. In the second Work Package, we have concentrated on functionals defined on the space GSBDp (generalized special functions of bounded deformation), and we have derived homogenization results, enhancing the understanding of the effective behavior of brittle materials with fine microstructures. The problems have been tackled with advanced tools from the calculus of variations and geometric measure theory. In particular, we have profited from new fundamental results for the space GSBD, and during the project we have derived further properties of this space, e.g., fundamental estimates or integral representation results. Whereas the original research plan comprised only the analysis in dimension two, most parts of the research program could eventually be conducted in any space dimension, due to novel advancements in the literature such as a Korn-type inequality in GSBD for functions with small jump sets.
Publications
- Functionals defined on piecewise rigid functions: Integral representation and Γ-convergence. Arch. Ration. Mech. Anal. 236 (2020), 1325–1387
M. Friedrich, F. Solombrino
(See online at https://doi.org/10.1007/s00205-020-01493-8) - Integral representation for energies in linear elasticity with surface discontinuities. Adv. Calc. Var.
V. Crismale, M. Friedrich, F. Solombrino
(See online at https://doi.org/10.1515/acv-2020-0047) - Γ-convergence for free-discontinuity problems in linear elasticity: Homogenization and relaxation
M. Friedrich, M. Perugini, F. Solombrino
(See online at https://doi.org/10.48550/arXiv.2010.05461) - Lower semicontinuity for functionals defined on piecewise rigid functions and on GSBD. J. Funct. Anal. 280 (2021), 108929
M. Friedrich, M. Perugini, F. Solombrino
(See online at https://doi.org/10.1016/j.jfa.2021.108929)
