Fundamental solutions for orthotropic cylindrical shell


DOI: 10.34759/trd-2022-124-11

Аuthors

Petrov I. I.*, Serdyuk D. O.**, Skopincev P. D.***

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: PetrovIlya1998@yandex.ru
**e-mail: d.serduk55@gmail.com
***e-mail: chgpashka@gmail.com

Abstract

A circular cylindrical shell of constant thickness is considered, the side surface of which is affected by non-stationary load. The shell material has symmetry with respect to the median surface, is linearly elastic, orthotropic. The motion is described in a cylindrical coordinate system associated with the axis of the cylindrical shell. The mathematical model of the object under consideration is described using the Kirchhoff — Love hypotheses. Fundamental solutions (Green’s functions, influence functions) are constructed for a cylindrical shell of great length, as well as a cylindrical shell pivotally supported at the ends. The Green function for an orthotropic shell is a solution to the problem of the effect of an instantaneous concentrated load on the shell, modeled by the Dirac delta function. To find the influence function in the case of an unlimited cylindrical shell, expansions into exponential Fourier series in angular coordinate, the integral Laplace transform in time and the integral Fourier transform in longitudinal coordinate are used. The inverse integral Laplace transform is being performed analytically, and the original integral Fourier transform is being found using numerical methods for integrating rapidly oscillating functions. In the case of a limited cylindrical shell, expansion into double trigonometric Fourier series in the angular and longitudinal coordinates is applied, as well as the integral Laplace transform in time. The inverse integral Laplace transform in this case is performed analytically. Verification of fundamental solutions has been carried out. Examples of calculations are given. The results are presented in the form of graphs.

A new numerical-analytical fundamental solution of the dynamic problem of elasticity theory for an orthotropic elastic thin unlimited cylindrical shell is obtained, as well as an analytical fundamental solution in the case of a limited Kirchhoff-Love shell. The convergence of the solution is established. To demonstrate the realism of the constructed functions, examples of calculations for one variant of the symmetry of an elastic medium are presented. The nature of the movement of non-stationary perturbations allowed us to evaluate solutions.

Fundamental solutions open up opportunities for solving new contact and inverse problems of load identification, allow performing applied research on calculating the stress and strain levels of orthotropic shells.

Keywords:

non-stationary dynamics, orthotropic material, cylindrical shell, integral transformations, generalized functions, quadrature formulas

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