A singular limit of a family of variational evolutions for a brittle elastic bi-layer. (June 2023)

• Record Type: Journal Article
• Authors: Causin, Andrea; Solci, Margherita
• Is Part Of: Nonlinear analysis. Volume 231(2023)
• Publication Date: 2023-06
• Subjects: Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112949
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider a system of one-dimensional elastic and brittle media coupled by an elastic term. Both the coupling and the fracture-energy coefficient depend on a small parameter ɛ . We describe the quasistatic motion of this system under varying boundary conditions at ɛ fixed, interpreting the fracture energy term as a dissipation as in the variational approach to fracture growth, by analysing a time-discrete approximation. The motion is characterized by the onset of successive fracture sites. We study the behaviour of these evolutions with vanishing dissipations as ɛ → 0, characterizing the possible limit of the energies by a one-parameter family of time-parameterized functions. We show that, after an initial time interval, a function in this family corresponds to a minimum of the Γ -limit of the energies of the system only for a discrete set of the values of the quantity parameterizing boundary conditions.