A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

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 resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling, and approximate the model by its viscous regularization. In fact, the analysis of the latter PDE system presents remarkable difficulties, due to its highly nonlinear character. We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with careful regularity estimates tailored to this specific system, relying on a q-Laplacian type gradient regularization of the damage variable. Hence for the viscous problem we conclude the existence of weak solutions, satisfying a suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis. The latter leads to the notion of (weak) parameterized solution to our rate-independent system, which encompasses the influence of viscosity in the description of the jump regime

Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

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 context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated

A plural indefinite quantifier on the Romance-Slavic border

This study investigates the plural form uni/une deriving from the numeral ‘one’ in the Istriot dialect of Sissano. Sissano is located in the Istrian peninsula, an area characterized by high intensity of linguistic contact. We argue that the rise of such a peculiar form is indeed induced by contact with Croatian and that uni/une is unique in the Italo-Romance domain since, generally, the plural indefinite forms derived from the Latin numeral ‘one’ are pronouns and never occur in attributive position. The use of uni/une is not attested in the few grammars of Istriot varieties because it is recent and still undergoing a process of grammaticalization. Therefore, we conducted interviews to verify how and to what extent contact with Croatian affects the meaning and the use of uni/une in Sissano. We found that this form is mostly used as a quantifier, bearing mainly the meaning ‘a pair of’, ‘one group of’, in the context of pluralia tantum and plural dominant nouns. We further observe that this quantifier has achieved a more advanced stage of grammaticalization in the younger generation of speakers than in the older ones. We discuss the role played by pluralia tantum as well as by the growing prestige of Croatian in triggering this borrowing and in fostering the grammaticalization process of uni/une on its way to become a marker of indefiniteness

Cohesive zone-type delamination in visco-elasticity : to the occasion of the 60th anniversary of Tomaš Roubícek

We study a model for the rate-independent evolution of cohesive zone delamination in a viscoelastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [OP99], 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 [Rou09] and refined in [RT15a]

A framework for probabilistic seismic risk assessment of NG distribution networks

Lifelines are essential infrastructures for human activities and the economic developm ent of a region. Lifelines vulnerability reduction is an actual question, particularly with reference to NaTech events, like earthquakes. In this regard, worldwide past seismic experiences revealed heavy damages to NG distribution networks. It is therefore essential to perform seismic risk assessment of NG buried pipelines systems with the aim to identify potential criticalities and avoid significant consequences. For such reasons, this work illustrates the proposal of a probabilistic framework for seismic risk assessment of NG lifelines. The proposed procedure is subsequently applied to a specific case study in Italy to highlight its feasibility