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. 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 $L^2$-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 Francfort and Marigo and in our previous paper, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith's criterion holds at the crack tips. We prove also 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.Comment: 20 page

Precision mass measurements of very short-lived, neutron-rich Na isotopes using a radiofrequency spectrometer

Mass measurements of high precision have been performed on sodium isotopes out to $^{30}$Na using a new technique of radiofrequency excitation of ion trajectories in a homogeneous magnetic field. This method, especially suited to very short-lived nuclides, has allowed us to significantly reduce the uncertainty in mass of the most exotic Na isotopes: a relative error of 5x10$^{-7}$ was achieved for $^{28}$Na having a half-life of only 30.5 ms and 9x10$^{-7}$ for the weakly produced $^{30}$Na. Verifying and minimizing binding energy uncertainties in this region of the nuclear chart is important for clarification of a long standing problem concerning the strength of the $N$=20 magic shell closure. These results are the fruit of the commissioning of the new experimental program Mistral

Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case

We study a variational model for crack growth in elasto-plastic materials with hardening in the antiplane case. The main result is the existence of a solution to the initial value problem with prescribed time-dependent boundary conditions

A new space of generalised functions with bounded variation motivated by fracture mechanics

We introduce a new space of generalised functions with bounded variation to prove the existence of a solution to a minimum problem that arises in the variational approach to fracture mechanics in elastoplastic materials. We study the fine properties of the functions belonging to this space and prove a compactness result. In order to use the Direct Method of the Calculus of Variations we prove a lower semicontinuity result for the functional occurring in this minimum problem. Moreover, we adapt a nontrivial argument introduced by Friedrich to show that every minimizing sequence can be modified to obtain a new minimizing sequence that satisfies the hypotheses of our compactness result

Accurate mass measurements of short-lived isotopes with the MISTRAL rf spectrometer

The MISTRAL experiment has measured its first masses at ISOLDE. Installed in May 1997, this radiofrequency transmission spectrometer is to concentrate on nuclides with particularly short half-lives. MISTRAL received its first stable beam in October and first radioactive beam in November 1997. These first tests, with a plasma ion source, resulted in excellent isobaric separation and reasonable transmission. Further testing and development enabled first data taking in July 1998 on neutron-rich Na isotopes having half-lives as short as 31 ms

Visco-energetic solutions for a model of crack growth in brittle materials

Visco-energetic solutions have been recently advanced as a new solution concept for rate-independent systems, alternative to energetic solutions/quasistatic evolutions and balanced viscosity solutions. In the spirit of this novel concept, we revisit the analysis of the variational model proposed by Francfort and Marigo for the quasi-static crack growth in brittle materials, in the case of antiplane shear. In this context, visco-energetic solutions can be constructed by perturbing the time incremental scheme for quasistatic evolutions by means of a viscous correction inspired by the term introduced by Almgren, Taylor, andWang in the study of mean curvature flows. With our main result we prove the existence of a visco-energetic solution with a given initial crack. We also show that, if the cracks have a finite number of tips evolving smoothly on a given time interval, visco-energetic solutions comply with Griffith’s criterion

Existence for elastodynamic Griffith fracture with a weak maximal dissipation condition

We consider a model of elastodynamics with fracture evolution, based on energy-dissipation balance and a maximal dissipation condition. We prove an existence result in the case of planar elasticity with a free crack path, where the maximal dissipation condition is satisfied among suitably regular competitor cracks

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\u2013 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

Elastodynamic Griffith fracture on prescribed crack paths with kinks

We prove an existence result for a model of dynamic fracture based on Griffith\u2019s criterion in the case of a prescribed crack path with a kink