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On Various Equations of the Analytical Mechanics of Thick-Walled Heterogeneous Shells and Some of Their Applications in Wave Dispersion Problems

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Abstract

A new variational formulation for the linear hierarchical theory of thick heterogeneous orthotropic shells resulting in the equations resolved for first-order covariant derivatives along one vector field is proposed. The shell theory of Nth order is constructed on the background of the Lagrangian formalism of the analytical mechanics of continua and the dimensional reduction approach combined with the biorthogonal expansion technique. Primo, the three-dimensional model of an elastic shell is referred to a curvilinear frame normally attached to a two-dimensional smooth oriented manifold corresponding to some base surface and represented as Lagrangian system with one field variable, the quasi-translation vector defined on the cotangent fibration of two-dimensional manifold, and the volumetric density of Lagrange’s functional. Secundo, one of the coordinates on the manifold is selected, and the quasi-stress vector on surfaces normal to the base vector corresponding to this coordinate is defined by differentiation of the Lagrangian density on covariant derivative of the quasi-translation. Application of the Legendre transform to the Lagrangian results in the volumeric density of the new mixed functional depending on the quasi-translation, quasi-stress, and time and remaining spatial derivatives of the quasi-translation. Use of the biorthogonal expansions for both quasi-translation and quasi-stress leads hence to the system of field variables of the first kind and generalized forces and consequently to the surface and contour density of the new functional interpreted as generalized Routh functional of a two-dimensional continuum system. Finally, application of the Hamilton principle results in the statement of the initial-boundary value problem for the system of equations resolved for the first-order covariant derivatives of field variables and generalized forces and interpreted as generalized Routh equations for a shell model as continuum-discrete system. The use of the obtained shell theory formulation in problems of normal wave dispersion in heterogeneous elastic waveguides for the analysis of imaginary and complex branches corresponding to evanescent wave modes is shown.

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Funding

This investigation was supported by the Russian Scientific Foundation under the grant no. 22-21-00800.

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Authors and Affiliations

  1. Institute of Applied Mechanics of the Russian Academy of Sciences, 125040, Moscow, Russia

    S. I. Zhavoronok & A. S. Kurbatov

  2. Moscow Aviation Institute, 125993, Moscow, Russia

    S. I. Zhavoronok, A. S. Kurbatov & O. V. Egorova

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  1. S. I. Zhavoronok
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  2. A. S. Kurbatov
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  3. O. V. Egorova
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Correspondence to S. I. Zhavoronok, A. S. Kurbatov or O. V. Egorova.

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(Submitted by A. M. Elizarov)

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Zhavoronok, S.I., Kurbatov, A.S. & Egorova, O.V. On Various Equations of the Analytical Mechanics of Thick-Walled Heterogeneous Shells and Some of Their Applications in Wave Dispersion Problems. Lobachevskii J Math 44, 2501–2517 (2023). https://doi.org/10.1134/S1995080223060458

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