Stochastic dynamic analysis of structures with fractional viscoelastic constitutive laws

The main purpose of this thesis is to provide a new way to correctly perform stochastic analysis of structures with viscoelastic constitutive law. The reason for this kind of problem relates the fact that structures with viscoelastic materials are built in many areas of mechanical, civil and aerospace engineering. To perform this kind of stochastic analysis there are two fundamental problems. That is, the mechanical description of the viscoelastic phenomenon, and the correct representation of the external loads. Both of these problems are addressed and solved by the proposed modeling that involves some advanced mathematical tools. The reason to describe materials as viscoelastic is given by the fact that the elastic model, commonly used to describe the mechanical behavior in the classic approach of the mechanics of materials is an idealization of the real mechanical behavior. Indeed, many materials are not completely elastic but have a mechanical behavior intermediate between the purely elastic behavior, that is typical of the solids, and the purely viscous behavior, which is common in the fluids. This phenomenon is known as viscoelastic behavior and is typical of many different materials used in various engineering fields. Moreover, in recent years complex materials, increasingly used in engineering applications, have been obtained with the aid of sophisticated industrial processes aiming to enhance stiffness and strength of materials. For these latter materials the elastic models do not yield an adequate representation of the real mechanical behavior and the modeling through viscoelastic laws is always greater. Besides, the viscoelastic properties cannot be neglected when it is necessary to evaluate the long-term effects in the structures. For example, concrete and some kinds of rocks, that are commonly modeled as elastic-brittle materials, for long observation times they also show viscoelastic behavior. Viscoelastic materials, commonly used in the engineering fields, are woods, pultruded elements, elastomers, fibre-reinforced-polymers, resins, bio-inspired materials, etc.. Other materials which have viscoelastic properties are some kinds of rocks, bitumens, bones, rubbers, biologic tissues, etc.. There are different ways to model the stress-strain relation of the viscoelastic material but the best results, compared to the experimental investigation are given by the use of fractional operators in the stress-strain relation. After the choice of the local stress-strain relation, the dynamic analysis of real structures is driven by introducing a proper global model which must be able to describe the behavior in terms of displacements and loads. Two kinds of global models are considered, the continuous systems and the discretized one with lumped parameters. In both cases, the choice of fractional operators in the local stress-strain relation leads to have fractional differential equations in the global displacement-load relations when the dynamic analysis is performed. This kind of differential equations are more difficult to solve respect to the integer-orders one. For this reason new methods to solve this kind of equations are developed and provided in this manuscript. As continuous system, the Euler-Bernoulli beam with fractional dampers is considered; while, as discretized case, the case of shear-type multi-degree-of freedom system with fractional viscoelastic elements is discussed. For both cases the analysis is conducted considering both stochastic and deterministic loads. Structural elements that can be modeled as continuous Euler-Bernoulli beams with viscoelastic dampers are the pultruded elements, which are composed by fibers with elastic-brittle behavior into a matrix with pronounced viscoelastic behavior. The analysis of the continuous system is performed by the proposed eigenanalysis. This problem is solved using the eigenfunctions of the classic elastic Euler-Bernoulli beam obtaining the solution of a set of uncoupled fractional differential equations in the modal space. For the fractional multi-degree-of-freedom system the motion is ruled by a set of coupled fractional differential equations. Such coupled system cannot be decoupled by the classical methods, for this reason, a novel method based on complex eigenanalysis in the expanded state variables domain. Further, to provide a characterization the response from a stochastic point of view, the fractional calculus provides some important tools that are also discussed in this thesis, with particular emphasis on the characterization of the structures with fractional viscoelastic constitutive law forced by Gaussian white noise. In particular, with the aid of the complex spectral moments, that are related to the fractional integrals of the power-spectral density function, it is possible to obtain a complete characterization of the stochastic processes. This method is used in conjunction with other results to provide a new description of the stochastic response of the fractional multi-degree-of-freedom system under Gaussian white noise.