Rodica Toader

A Model for the Quasi-Static Growth of Brittle Fractures: Existence and Approximation Results

Abstract 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 Griffiths 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 cannot exclude the possibility 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.

A model for the quasi-static growth of brittle fractures based on local minimization

We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffiths criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.

A MODEL FOR CRACK PROPAGATION BASED ON VISCOUS APPROXIMATION

In the setting of antiplane linearized elasticity, we show the existence of quasistatic evolutions of cracks in brittle materials by using a vanishing viscosity approach, thus taking into account local minimization. The main feature of our model is that the path followed by the crack need not be prescribed a priori: indeed, it is found as the limit (in the sense of Hausdorff convergence) of curves obtained by an incremental procedure. The result is based on a continuity property for the energy release rate in a suitable class of admissible cracks.

Periodic solutions of radially symmetric perturbations of Newtonian systems

The classical Newton equation for the motion of a body in a gravitational central field is here modified in order to include periodic central forces. We prove that infinitely many periodic solutions still exist in this case. These solutions have periods which are large integer multiples of the period of the forcing, and rotate exactly once around the origin in their period time.

Periodic Orbits of Radially Symmetric Systems with a Singularity: the Repulsive Case

Abstract We study radially symmetric systems with a singularity of repulsive type. In the presence of a radially symmetric periodic forcing, we show the existence of three distinct families of subharmonic solutions: One oscillates radially, one rotates around the origin with small angular momentum, and the third one with both large angular momentum and large amplitude. The proofs are carried out by the use of topological degree theory.

Rate-Independent Damage in Thermo-Viscoelastic Materials with Inertia

We present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio–Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is independent of temperature.

Periodic Solutions of Pendulum-Like Hamiltonian Systems in the Plane

Abstract By the use of a generalized version of the Poincaré-Birkhoff fixed point theorem, we prove the existence of at least two periodic solutions for a class of Hamiltonian systems in the plane, having in mind the forced pendulum equation as a particular case. Our approach is closely related to the one used by Franks in [15], but the proof remains at a more elementary level.

Scaling in fracture mechanics by Bažant law: From finite to linearized elasticity

We consider crack propagation in brittle nonlinear elastic materials in the context of quasi-static evolutions of energetic type. Given a sequence of self-similar domains nΩ on which the imposed boundary conditions scale according to Bažants law, we show, in agreement with several experimental data, that the corresponding sequence of evolutions converges (for n → ∞) to the evolution of a crack in a brittle linear-elastic material.

Nonlocal Approximation of Nonisotropic Free-Discontinuity Problems

We prove that a class of free-discontinuity problems with nonisotropic bulk and surface energy densities is approximated, in the sense of

Finite element approximation

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