Chiara Zanini

Polytechnic University of Turin | polito · DISMA - Department of Mathematical Sciences

We study a rate-independent system with non-convex energy and in the case of a time-discontinuous loading. We prove existence of the rate-dependent viscous regularization by time-incremental problems, while the existence of the so called parameterized $BV$-solutions is obtained via vanishing viscosity in a suitable parameterized setting. In additio...

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p -power nonlinearity considered by several authors in this contex...

We study a model for the rate-independent evolution of cohesive zone delamination in a visco-elastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [32], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for...

This article is the third one in a series of papers by the authors on vanishing-viscosity solutions to rate-independent damage systems. While in the first two papers [KRZ13, KRZ15] the assumptions on the spatial domain $\Omega$ were kept as general as possible (i.e. nonsmooth domain with mixed boundary conditions), we assume here that $\partial\Ome...

In the setting of finite elasticity we study the asymptotic behaviour of a crack that propagates quasi-statically in a brittle material. With a natural scaling of size and boundary conditions we prove that for large domains the evolution with finite elasticity converges to the evolution with linearized elasticity. In the proof the crucial step is t...

This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonconvex, solutions may have jumps as a function of time, and in this situation it is known that the classical concept of energetic solutions for rate-independent systems may fail to accurately describe the behavior of the system at jumps. Therefore we res...

A quasistatic rate-independent adhesive delamination problem of laminated plates with a finite thickness is considered. By letting the thickness of the plates go to zero, a rate-independent delamination model for a laminated Kirchhoff-Love plate is obtained as limit of these quasistatic processes. The same dimension reduction procedure is eventuall...

We analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps a...

We consider a one-dimensional chain of atoms which interact with their nearest and next-to-nearest neighbours via a Lennard-Jones type potential. We prescribe the positions in the deformed configuration of the first two and the last two atoms of the chain. We are interested in a good approximation of the discrete energy of this system for a large n...

A quasistatic rate-independent brittle delamination problem and also an adhesive unilateral contact problem is considered on a prescribed normally-positioned surface in a plate with a finite thickness. By letting the thickness of the plate go to zero, two quasistatic rate-independent crack models with prescribed path for Kirchhoff-Love plates are o...

In this work we consider a one-dimensional chain of atoms which interact through nearest and next-to-nearest neighbour interactions of Lennard–Jones type. We impose Dirichlet boundary conditions and in addition prescribe the deformation of the second and last but one atoms of the chain. This corresponds to prescribing the slope at the boundary of t...

We discuss a model for crack propagation in an elastic body, where the crack path is described a priori. In particular, we develop in the framework of finite-strain elasticity a rate-independent model for crack evolution which is based on the Griffith fracture criterion. Due to the nonuniqueness of minimizing deformations, the energy-release rate i...

A two dimensional model which describes the evolution of a crack in a plate is deduced from a three dimensional linearly elastic Grif-fith's type model. The result is achieved by adopting the framework of energetic solutions for rate-independent processes, to model three dimensional fracture evolution, in conjunction with a variational di-mension r...

A modified Gilbert equation for micromagnetics is considered, obtained by augmenting the standard viscous-like dissipation with a rate-independent term. We prove existence of a weak solution both with and without viscous dissipation. By scaling time we show that, if the applied field varies very slowly, then gyromagnetic effects and viscous dissipa...

We study delamination of two elastic bodies glued together by an adhesive that can undergo a unidirectional inelastic rate-independent process. The quasistatic delamination process is thus activated by time-dependent external loadings, realized through body forces and displacements prescribed on parts of the boundary. The novelty of this work consi...

We model the evolution of a single crack as a rate–independent process based on the Griffith criterion. Three approaches are presented, namely a model based on global energy minimization, a model based on a local description involving the energy release rate and a refined local model which is the limit problem of regularized, viscous models. Finall...

We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffith's local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal o...

We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo-type equation vxx-gv+n(x)F(v)=0, previously considered by Chen and Bell in the study of the model of a nerve fiber with excitable spines. In a recent work we proved a result of nonexistence of nontrivial solutions as well as a res...

We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo type equation $v_{xx} - g v + n(x) F(v) = 0,$ previously considered by Grindrod and Sleeman and by Chen and Bell in the study of nerve fiber models. In some recent works we discussed the case of nonexistence of nontrivial solution...

In this paper we study the quasi-static crack growth for a cohesive zone model. We assume that the crack path is prescribed and we study the time evolution of the crack in the framework of the variational theory of rate-independent processes.

We introduce a new model of irreversible quasistatic crack growth in which the evolution of cracks is the limit of a suitably modified $\epsilon$-gradient flow of the energy functional, as the "viscosity" parameter $\epsilon$ tends to zero.

In this paper we give a description of the asymptotic behavior, as $\epsilon\to 0$, of the $\epsilon$-gradient flow in the finite dimensional case. Under very general assumptions we prove that it converges to an evolution obtained by connecting some smooth branches of solutions to the equilibrium equation (slow dynamics) through some heteroclinic s...

We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo type equation $v_{xx} - g v + n(x) F(v) = 0,$ previously considered by Grindrod and Sleeman and by Chen and Bell in the study of the model of a nerve fiber with excitable spines. In a recent work we proved a result of nonexistence...