Rodica Toader

The principle of conservation of energy implies that a crack growing in a viscoelastic body must satisfy the dynamic energy-dissipation balance, an equality involving the energy dissipated by viscosity and by crack growth. Unfortunately, some models of evolution of a viscoelastic body, like the frequently used Kelvin–Voigt model, imply that the dynamic energy-dissipation balance prevents crack growth. This unrealistic result forces us to use different models to describe the evolution of a viscoelastic body in the context of dynamic fracture mechanics. In this paper, we consider an example of dynamic viscoelastic problem with memory in a two-dimensional domain with a crack growing with constant velocity along a straight line. Through a careful analysis of the singularity of the solutions around the crack tip we show that for suitable values of the material constants there exist solutions that satisfy the energy-dissipation balance. This may suggest that we should use suitable models with memory in the context of dynamic fracture mechanics of viscoelastic bodies.

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.

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.

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, and Wang 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.

We prove the existence of bounded and periodic solutions for planar systems by introducing a notion of lower and upper solutions which generalizes the classical one for scalar second order equations. The proof relies on phase plane analysis; after suitably modifying the nonlinearities, the Ważewski theory provides a solution which remains bounded in the future. For the periodic problem, the Massera Theorem applies, yielding the existence result. We then show how our result generalizes some well known theorems on the existence of bounded and of periodic solutions. Finally, we provide some corollaries on the existence of almost periodic solutions for scalar second order equations.