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Derivation of linear elastic from rescaled non-linear elastic quasi-static crack evolutions

Rodica Toader (University of Udine)

Crack initiation and growth: methods, applications, and challenges

Tue 10:45 - 12:15

Sayles Auditorium

In this contribution I would like to comment on the relationship between finite and linearized elasticity in fracture mechanics, specifically in the case of quasi-static crack evolutions in brittle bodies. In elasto-statics, this relationship was analysed in [DMNP] where the convergence of the rescaled nonlinear energies towards the linearized energy was shown, as well as the convergence of the corresponding minimizers. This result and its later refinements settle the static case. As fracture mechanics is concerned, the relationship between linear and nonlinear energies was studied in [NZ] under the assumption that the crack evolves along a prescribed segment. Here I would like to present a result obtained in a joint work with Matteo Negri, on the derivation of a quasi-static crack evolution in a brittle body in the framework of linear elastic fracture mechanics from finite elasticity quasi-static crack evolutions on suitably rescaled domains. We consider the case when the crack path is not a priori prescribed, but it is selected through an energy criterion among curves belonging to a suitable class. References: [DMNP] G. Dal Maso, M. Negri, and D. Percivale. Linearized elasticity as -limit of finite elasticity. Set-Valued Anal., 10, 165–183, 2002. [NZ] M. Negri and C. Zanini. From Finite to Linear Fracture Mechanics by Scaling. Preprint CVGMT, 2012.