A model for the quasi-static growth of brittle fractures: existence and approximation results

We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith's theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution.Comment: 27 pages, LaTe

Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach

A new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation of constant unloading response, the result contained in [G. Dal Maso, C. Zanini: Quasi-static crack growth for a cohesive zone model with prescribed crack path. Proc. Roy. Soc. Edinburgh Sect. A, 137A (2007), 253–279.] is recovered. In this case, the convergence of the discrete time approximations is improved

Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth

We prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations of N planar uncoupled systems which, e.g., model some type of asymmetric oscillators. The nonlinearities are assumed to satisfy Landesman\u2013Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is carried out by the use of a generalized version of the Poincar\ue9\u2013Birkhoff Theorem. Different situations, including Lotka\u2013Volterra systems, or systems with singularities, are also illustrated

On the Cauchy problem for the wave equation on time-dependent domains

We introduce a notion of solution to the wave equation on a suitable class of time-dependent domains and compare it with a previous definition. We prove an existence result for the solution of the Cauchy problem and present some additional conditions which imply uniqueness

A VARIATIONAL MODEL FOR THE QUASI-STATIC GROWTH OF FRACTIONAL DIMENSIONAL BRITTLE FRACTURES

Abstract. We propose a variational model for the irreversible quasi-static evolution of brittle fractures having fractional Hausdorff dimension in the setting of two-dimensional antiplane and plane elasticity. The evolution along such irregular crack paths can be obtained as Γ -limit of evolutions along one-dimensional cracks when the fracture toughness tends to zero