Singular perturbations of finite dimensional gradient flows
Chiara Zanini
In this paper we give a description of the asymptotic behavior, as $\varepsilon\to 0$, of the $\varepsilon$-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 solutions of the gradient flow (fast dynamics).
keywords: gradient flow heteroclinic solutions. saddle-node bifurcation singular perturbations
Periodic solutions for a class of second order ODEs with a Nagumo cubic type nonlinearity
Chiara Zanini Fabio Zanolin
We prove the existence of multiple periodic solutions as well as the presence of complex profiles (for a certain range of the parameters) for the steady-state solutions of a class of reaction-diffusion equations with a FitzHugh-Nagumo cubic type nonlinearity. An application is given to a second order ODE related to a myelinated nerve axon model.
keywords: Grindrod and Sleeman model. subharmonics chaotic-like dynamics Nagumo type equations Reaction-diffusion equations periodic solutions stationary solutions
Towards uniformly $\Gamma$-equivalent theories for nonconvex discrete systems
Lucia Scardia Anja Schlömerkemper Chiara Zanini
In this paper 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 number of atoms, i.e., in the continuum limit.
    We show that the canonical expansion by $\Gamma$-convergence does not provide an accurate approximation of the discrete energy if the boundary conditions for the deformation are close to the threshold between elastic and fracture regimes. This suggests that a uniformly $\Gamma$-equivalent approximation of the energy should be made, as introduced by Braides and Truskinovsky, to overcome the drawback of the lack of accuracy of the standard $\Gamma$-expansion.
    In this spirit we provide a uniformly $\Gamma$-equivalent approximation of the discrete energy at first order, which arises as the $\Gamma$-limit of a suitably scaled functional.
keywords: Multi-scale modeling Gamma-convergence asymptotic expansions fracture. next-to-nearest-neighbour interactions
Quasistatic evolution of magnetoelastic plates via dimension reduction
Martin Kružík Ulisse Stefanelli Chiara Zanini
A rate-independent model for the quasistatic evolution of a magnetoelastic plate is advanced and analyzed. Starting from the three-dimensional setting, we present an evolutionary $\Gamma$-convergence argument in order to pass to the limit in one of the material dimensions. By taking into account both conservative and dissipative actions, a nonlinear evolution system of rate-independent type is obtained. The existence of so-called energetic solutions to such system is proved via approximation.
keywords: Magnetoelasticity energetic solution $\Gamma$-convergence for rate-independent processes. existence dimension reduction
Cohesive zone-type delamination in visco-elasticity
Marita Thomas Chiara Zanini

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 different responses upon loading and unloading. Due to the presence of multivalued and unbounded operators featuring non-penetration and the 'memory'-constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [41] and refined in [38].

keywords: Cohesive zone delamination weak formulation rate-independent processes semistable energetic solutions non-smooth constraint gradient systems dynamics irreversibility

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