5,267 research outputs found
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), 253â279.] is recovered. In this case, the convergence of the discrete time approximations is improved
Variational problems in fracture mechanics
We present some recent existence results for the variational model of crack
growth in brittle materials proposed by Francfort and Marigo in 1998. These
results, obtained in collaboration with Francfort and Toader, cover the case of
arbitrary space dimension with a general quasiconvex bulk energy and with
prescribed boundary deformations and applied loads.Comment: 9 page
An extension theorem in SBV and an application to the homogenization of the Mumford-Shah functional in perforated domains
The aim of this paper is to prove the existence of extension operators for SBV functions from periodically perforated domains. This result will be the fundamental tool to prove the compactness in a non coercive homogenization problem
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
Some properties of the solutions of obstacle problems with measure data
We study some properties of the obstacle reactions associated with the
solutions of unilateral obstacle problems with measure data. These results
allow us to prove that, under very weak assumptions on the obstacles, the
solutions do not depend on the components of the negative parts of the data
which are concentrated on sets of capacity zero. The proof is based on a
careful analysis of the behaviour of the potentials of two mutually singular
measures near the points where both potentials tend to infinity.Comment: 18 page
Uniqueness of solutions to Hamilton-Jacobi equations arising in the Calculus of Variations
We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi
equation associated with a Bolza problem of the Calculus of Variations,
assuming that the Lagrangian is autonomous, continuous, superlinear, and
satisfies the usual convexity hypothesis. Under the same assumptions we prove
also the uniqueness, in a class of lower semicontinuous functions, of a
slightly different notion of solution, where classical derivatives are replaced
only by subdifferentials. These results follow from a new comparison theorem
for lower semicontinuous viscosity supersolutions of the Hamilton-Jacobi
equation, that is proved in the general case of lower semicontinuous
Lagrangians.Comment: 14 page
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