Determination of Prestress in Circular Inhomogeneous Solid and Annular Plates in the Framework of the Timoshenko Hypotheses

Abstract

Determination of prestress fields in structures is of the utmost importance, since they have a significant impact on operational characteristics, and their level and distribution must be strictly controlled. In this paper, we present modeling of bending vibrations of solid and annular round inhomogeneous prestressed plates within the framework of the Timoshenko hypotheses. New inverse problems of prestress identification in plates are studied on the basis of the acoustic response subjected to some probing load. To solve direct problems on calculating oscillations and amplitude-frequency characteristics, a computational Galerkin-method-based scheme has been developed. In order to treat the inverse problems, we use a special projection approach based on the constructed weak problems statements, which makes it possible to determine the desired characteristics in the given classes of functions. The developed techniques for solving direct problems are implemented in the form of software packages realized via Maple. For both solid and annular plates, we estimate the sensitivity of the amplitude-frequency characteristics the values of which are used as the additional data in the inverse problems to a change in the prestress level; we conclude that the most favorable frequency range should be selected in the resonance vicinity. We have conducted a series of computational tests on reconstructing the plate’s prestresses of various levels and distribution patterns (decreasing, increasing, sign-changing laws). The results of computational tests revealed that the technique developed allows for the determination of the prestresses with a low error for two cases: when the cause of prestress formation and its type are known and when arbitrary prestress changing laws are considered.

Identification of Prestress in Inhomogeneous Viscoelastic Timoshenko Plates

“…According to the correspondence principle, we write the equations of vibrations of the considered prestressed plates under conditions (4), using the equations obtained in [22] for the elastic case, written in terms of dimensionless parameters and variables:…”

“…In equations ( 5), instead of the function of the cylindrical stiffness of the plate D(r) = E(r)h 3 12(1−ν 2 ) used in the elastic case [22] (where E(r) is the Young's modulus and ν is the Poisson's ratio), the complex modulus function of the form (3) is used, i.e. G(ξ, iκ) = iτ κg 2 (ξ)+g 1 (ξ) 1+iτ κ .…”

“…Thus, we can conclude that the modeling of composite materials in which there are residual stresses as well as the identification of the presence of such are very important tasks that have a focus on various practical applications. It is also worth noting that a number of inverse problems similar to those considered in this paper for identifying inhomogeneous prestress fields in plates and other bodies have also been investigated previously [20], [21], [22]. In [21], inverse problems for the identification of prestress fields occurring in bending vibrations of plates were studied.…”

“…The analysis is done within the framework of Timoshenko's hypotheses, using several techniques based on the acoustic approach. Models of circular elastic inhomogeneous prestressed Timoshenko plates were developed in [22] and the problem of identifying the prestresses was solved using the projection approach allowing to determine the desired characteristics in the given classes of functions.…”

“…This paper presents an extension of the models within Timoshenko's hypotheses for stationary bending vibrations of circular and annular inhomogeneous plates with consideration of residual stresses, as well as of the methods for solving the problem of identifying residual stresses based on acoustic sounding data proposed in [22], to the case of plates made of composite materials that possess viscoelastic (rheological) properties.…”