Homogenisation and elliptic approximation of random free-discontinuity functionals

• Applicant: Professorin Dr. Caterina Ida Zeppieri
• Subject Area: Mathematics
• Term: from 2019 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 426599264

• Final Report Year: 2023

Final Report Abstract

Motivated by primary questions arising in the variational theory of (static) fracture, the main goal of this research project was to study the large-scale behaviour of random elastic composites which can undergo fracture. From a mathematical standpoint this amounts to the development of a stochastic homogenisation theory for energy-functionals of free-discontinuity type. In this proposal we developed a comprehensive qualitative theory of stochastic homogenisation for free-discontinuity functionals. This is done by combining two complementary approaches: a ”direct” approach and an ”indirect” approximation-approach. The direct approach consist in extending the SBV-theory both to the BV-setting and to the setting of functionals with degenerate coefficients, the latter being relevant, e.g., in the study of fracture in perforated materials. The approximationapproach, instead, consist in proposing suitable elliptic phase-field approximations of random freediscontinuity functionals which provide, among other, regular-approximations of the homogenised coefficients. The problems have been tackled with advanced tools from the Calculus of Variations, Geometric Measure Theory, and Ergodic Theory.

Publications

• A global method for deterministic and stochastic homogenisation in BV. Annals of PDE, 8(1).
  Cagnetti, Filippo; Dal, Maso Gianni; Scardia, Lucia & Zeppieri, Caterina Ida
  
• Interaction Between Oscillations and Singular Perturbations in a One-Dimensional Phase-Field Model. Research in Mathematics of Materials Science, 3–31.
  Bach, Annika; Esposito, Teresa; Marziani, Roberta & Zeppieri, Caterina Ida
  
• Gradient Damage Models for Heterogeneous Materials. SIAM Journal on Mathematical Analysis, 55(4), 3567-3601.
  Bach, Annika; Esposito, Teresa; Marziani, Roberta & Zeppieri, Caterina Ida
  
• Stochastic homogenisation of free-discontinuity functionals in randomly perforated domains. Advances in Calculus of Variations, 17(3), 643-671.
  Pellet, Xavier; Scardia, Lucia & Zeppieri, Caterina Ida
  
• Γ-convergence and stochastic homogenisation of phase-transition functionals. ESAIM: Control, Optimisation and Calculus of Variations, 29(2023), 44.
  Marziani, Roberta
  
• Γ-convergence and stochastic homogenisation of singularly-perturbed elliptic functionals. Calculus of Variations and Partial Differential Equations, 62(7).
  Bach, Annika; Marziani, Roberta & Zeppieri, Caterina Ida