Stochastic dynamic analysis of structures with fractional viscoelastic constitutive laws

This manuscript summarizes my main research activity in this last triennium. It adheres to a common procedure of solving a scientistic problem. That is, introduction to the problem, selection of the mathematical and physical tools to model the problem, proposed solution, and experimental/numerical validation. The main topics considered are the characterization and the modeling of the mechanical behavior of real materials, and the stochastic dynamics with special regards to the characterization of the stochastic processes. To address these two different kinds of problem an advanced mathematical tool is used. In particular, the differential calculus with non-integer operators offers several new mathematical tools that permit obtaining advanced results. Therefore, two different topics are studied: mechanical modeling of materials, and stochastic characterization of random process. Both arguments are used in order to pursue one final aim. That is, the structural dynamics of real structures and structural elements built with advanced material. This is an important goal that regards many physical and engineering problems. Indeed, in several mechanical and engineering problems it is needed to perform the dynamic analysis of such structures that are increasingly complex, and made with advanced materials. For such structures the classical models and the common tools of the structural dynamics are unable to provide an accurate description of the real problem. Obviously, to perform this kind of dynamical analysis it is necessary to define properly the global model of the structure, to describe correctly the real mechanical behavior of the building materials, and to simulate the external forced processes that load the structures during their life. The characterization of the mechanical behavior of real building materials is obtained with the aid of recent advanced models which involve fractional operators in the constitutive stress-strain relation. This kind of model is known as fractional viscoelasticity. The choice of this kind of stress-strain model is due to the fact that the fractional viscoelasticity allows to obtain the best agreement with experimental observations for a plethora of materials. After such choice, it is necessary to define the global mechanical model of the structure. In this work both continuous and discretized models of structures with fractional viscoelastic constitutive law are considered. To complete the dynamic analysis it is needed to represent the real external loads and the corresponding output in terms of displacements of the considered structures. In many cases of engineering interest the external loads are adequately modeled as random processes (earthquake excitation, wind velocity field, ocean-waves actions, impact loading, etc.). To perform this characterization, the classical tools of stochastic dynamics are developed by using fractional calculus. In particular, the fractional operators permit obtaining a new stochastic characterization of the inputs (external actions) and outputs (displacements) process.