In this work we derive by $\Gamma$ -convergence techniques a model for brittle fracture linearly elastic plates. Precisely, we start from a brittle linearly elastic thin film with positive thickness $\rho$ and study the limit as $\rho$ tends to $0$ . The analysis is performed with no a priori restrictions on the admissible displacements and on the geometry of the fracture set. The limit model is characterized by a Kirchhoff-Love type of structure.

In this paper, we formulate and study the system of elastodynamics on domains with arbitrary growing cracks. This includes homogeneous Neumann conditions on the crack sets and mixed general Dirichlet-Neumann conditions on the boundary. The only assumptions on the crack sets are to be (n − 1)-rectifiable with finite surface measure, and increasing in the sense of set inclusions. In particular, they might be dense; hence, the weak formulation must fall outside the usual context of Sobolev spaces and Korn's inequality. We prove existence of a solution for both the damped and undamped systems, while in the damped case we are also able to prove uniqueness and an energy balance.

In this paper we provide necessary and sufficient conditions in order to guarantee the energy-dissipation balance of a Mode III crack, growing on a prescribed smooth path. Moreover, we characterize the singularity of the displacement near the crack tip, generalizing the result in [10] valid for straight fractures.

In this paper, we present a new result of continuity for the trace operator acting on functions that might jump on a prescribed (n − 1)-dimensional set Γ, with the only hypothesis of being rectifiable and of finite measure. We also show an application of our result in relation to the variational model of elasticity with cracks, when the associated minimum problems are coupled with Dirichlet and Neumann boundary conditions.

We prove a compactness result in GBD {\operatorname{GBD}} which also provides a new proof of the compactness theorem in GSBD {\operatorname{GSBD}} , due to Chambolle and Crismale. Our proof is based on a Fréchet–Kolmogorov compactness criterion and does not rely on Korn or Poincaré–Korn inequalities.