E.Yu. Mihajlovaa*, D.V. Tarlakovskiia,b**, G.V. Fedotenkova,b***
aMoscow Aviation Institute (National Research University), Moscow, 125993 Russia
bResearch Institute of Mechanics, Moscow State University, Moscow, 119192 Russia
E-mail: *mihe16@yandex.ru, **tdvhome@mail.ru, ***greghome@mail.ru
Received February 9, 2018
Abstract
A generalized linear model of the dynamics of a thin elastic shell of constant thickness, which takes into account the rotation and compression of the fiber sheath normal to the middle surface, has been proposed. A coordinate system has been used that includes the curvilinear coordinates of the median surface and the distance (normal coordinate) measured from the median surface in the direction of the outer normal. The connections of spatial metrics and covariant derivatives with analogous parameters of the middle surface have been found.
The field of displacements of the shell and all the characteristics have been considered in the linear approximation in the normal coordinate. It has been shown that the movements of any point of the shell are determined by the tangential and normal displacements of the middle surface, by two angles of rotation of the normal fiber and its deformation, and the deformed state of the envelope is specified by the tensors of tangential deformation and changes in curvature and deformation of the normal fiber. By means of the linearization of the equations of compatibility of deformations for a continuous medium, three analogous equations for a thin shell have been obtained. To prove their validity, the quadratic approximation of displacements has been used.
Formulas for the potential and kinetic energy have been obtained, as well as for the operation of external forces. It has been shown that taking into account the rotation of the normal fiber and compression leads to the appearance of additional internal force factors - an additional moment and normal force. In this case, distributed moments are added to standard external force factors. The physical law has been constructed for an anisotropic material that has symmetry relative to the median surface without adopting the commonly used static hypothesis of non-adherence of fibers.
The equations of motion have been constructed using the Hamiltonian principle. They consist of six tensor relations. From this principle, natural boundary conditions have been derived. It has been shown that the Kirchhoff–Love model and Tymoshenko type follow from the constructed model as special cases.
Keywords: elastic shell, rotation and compression of a normal fiber, anisotropy, equations of motion, compatibility of deformations
Acknowledgments. The study was supported by the Russian Science Foundation, project no. 14-49-00091.
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For citation: Mihajlova E.Yu., Tarlakovskii D.V., Fedotenkov G.V. A generalized linear model of dynamics of thin elastic shells. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 3, pp. 561–577. (In Russian)
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