On identification of characteristics of inhomogeneous viscoelastic bodies within the framework of a fractional order model - Bogachev, Ivan Viktorovich; Vatulyan, Aleksandr Ovanesovich (Computational Continuum Mechanics, 2024)
Nowadays, the development of models of viscoelastic materials with complex inhomogeneous structure is one of the hottest problems of continuum mechanics. Along with classical models, fractional-differential models of viscoelasticity have become increasingly popular. In this paper, we present a model for describing steady-state oscillations of inhomogeneous viscoelastic bodies in terms of fractional-order differential operators. Taking into account the fractional order of the model operators, the corresponding form of the complex module describing the material properties is constructed. The module includes four characteristics: instantaneous and long-term elastic moduli (in the  case of material inhomogeneity, they are the functions of coordinates), relaxation time and fractionality parameter. The properties of the complex module are studied, and the ranges of  the model parameters, at which the rheological properties are most pronounced, are identified. A general formulation of the inverse problem (IP) on the identification of function-parameters of the model based on the acoustic sounding data is proposed. In the framework of this formulation,  the inverse problems for specific objects , namely,  an inhomogeneous rod and a round plate, are considered. In both model problems,  the influence of the fractionalization parameter on the amplitude-frequency characteristics is analyzed. It has been found that in the vicinity of viscoelastic resonances the fractionality parameter affects the parameters of the oscillatory process most significantly, which is typical for such problems. A solution to the nonlinear IPs under consideration is constructed based on the linearization method, which also serve as basis for the iterative processes supplemented with the elements of the projection approach. This  allows one to determine corrections to the required functions in the specified classes of functions using regularization. For both IPs, a series of computational experiments were carried out. The analysis of the experimental  results made it possible to formulate recommendations for the selection of optimal sensing modes.
Keywords: viscoelasticity; fractional differential models; inhomogeneous materials; acoustic method; inverse problems; identification; regularization