  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
in January 1966
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Yu. E. Senitskii, "Dynamics of inhomogeneous nonshallow spherical shells," Mech. Solids. 37 (6), 123133 (2002) 
Year 
2002 
Volume 
37 
Number 
6 
Pages 
123133 
Title 
Dynamics of inhomogeneous nonshallow spherical shells 
Author(s) 
Yu. E. Senitskii (Samara) 
Abstract 
On the basis of a refined theory taking into account shear strains and
rotation inertia of crosssections [1], an axiallysymmetric
initialboundary value problem is formulated for an elastically
fixed inhomogeneous spherical shell. Variation of
physicalmechanical material characteristics with respect to the thickness
of the structure is caused by radiation, thermal, and chemical fields
acting on one of its surfaces and called induced inhomogeneity factors [2].
In the calculation scheme, this variation is taken into account by the
introduction of two arbitrary dimensionless positive function, f_{1}(z)
and f_{2}(z), of the thickness coordinate z.
A new exact (in the framework of the adopted model) solution is obtained
in the form of finite integral transforms [3, 4]
by the method of expansion with respect to vectorvalued eigenfunctions.
An essential point is that
the corresponding spectral problem and inversion formulas take into account
multiple eigenvalues (internal resonances) [57] which affect the
dynamic response of nonshallow shells. While solving the problem,
we also introduce internal friction forces in accordance with the
refined frequencyindependent Voigt model [8, 9].
The problem formulated in such terms has never been studied before.
Fairly complete studies of integrability in inhomogeneous elasticity problems
exist only for problems with a static loading [10]. Much less attention
has been given to dynamical problems for continuously inhomogeneous shells.
In fact, closed form solutions (in the framework of the formulated
mathematical models) of dynamical problems have been constructed only for
shallow spherical shells [11, 12]. These results were obtained without
taking into account viscoelastic resistance and internal resonances
typical of such shells. As shown in [5, 6] by the analysis of the
equations of motion, resonance interaction of flexural and tangential
shapes of vibrations in nonshallow shells is not an exceptional phenomenon
and must be taken into account when finding the frequency spectrum of such
shells. However, this question about the dynamical reaction remains open
even for homogeneous shells. 
References 
1.  E. I. Grigolyuk and I. T. Selezov,
Nonclassical Theories of Vibration of Beams, Plates, and Shells.
Advances in Science and Technology. Mechanics of Deformable Solids [in Russian],
VINITI, Moscow, 1973. 
2.  V. V. Petrov, I. G. Ovchinnikov, and Yu. M. Shikhov,
Design of Structural Elements in Corrosive Environment [in Russian], Izdvo
Saratovsk. Gos. Unta, Saratov, 1987. 
3.  Yu. E. Senitskii, "A multicomponent generalized
finite integral transform and its applications to
nonstationary problems in mechanics," Izv. Vuzov. Matematika, No. 4,
pp. 5763, 1991. 
4.  Yu. E. Senitskii, "Convergence and uniqueness of representations
defined by the formula of multicomponent generalized finite integral
transform," Izv. Vuzov. Matematika, No. 9, pp. 5356, 1991. 
5.  A. L. Goldenveiser, V. B. Lidskii, and P. E. Tovstik,
Free Vibrations of Thin Elastic Shells [in Russian], Nauka, Moscow, 1979. 
6.  E. A. Dain, S. A. Lukovenko, and N. V. Khar'kova,
On the Problem of Internal Resonances in the Theory of Vibration of Thin
Shells. Preprint No. 97 [in Russian], IPM RAN, Moscow, 1977. 
7.  A. D. Lizarev, and N. B. Rostanina,
Vibrations in Metalpolymeric and Homogeneous Spherical Shells
[in Russian], Nauka i Tekhn., Minsk, 1984. 
8.  A. I. Tseitlin, "On linear models of frequencyindependent
internal friction," Izv. AN SSSR. MTT [Mechanics of Solids], No. 3,
pp. 1828, 1978. 
9.  A. I. Tseitlin and A. A. Kusainov,
Methods for Taking into Account Internal Friction in Dynamic
Problems for Structures [in Russian], Nauka, Kaz. SSR, AmlaAta, 1987. 
10.  V. A. Lomakin, Theory of Elasticity for Inhomogeneous Bodies [in Russian],
Izdvo MGU, Moscow, 1976. 
11.  Yu. E. Senitskii,
"An axially symmetric dynamical problem for an inhomogeneous shallow
spherical shell with finite shear stiffness," Prikl. Mekhanika, Vol. 30,
No. 9, pp. 5057, 1994. 
12.  Yu. E. Senitskii,
"On the integrability of the dynamical initialboundary value problem
for an inhomogeneous shallow spherical shell,"
Vestnik Samarsk. Unta, No. 2(8), pp. 106121, 1998. 
13.  V. L. Berdichevskii,
Variational Principles in Continuum Mechanics [in Russian], Nauka, Moscow, 1983. 
14.  A. Kalnins,
"Effect of bending on vibration of spherical shell,"
J. Acoust. Soc. America, Vol. 36, No. 1, pp. 7483, 1964. 
15.  Yu. E. Senitskii,
"On some identities used for the solution of boundary value problems
by the method of finite integral transforms," Diff. Uravneniya,
Vol. 19, No. 9, pp. 16361638, 1983. 
16.  Yu. E. Senitskii and S. A. Lychev,
"On the determination of the norm of finite integral transform
kernels and their applications," Izv. Vuzov. Matematika, No. 8,
pp. 6069, 1999. 

Received 
3 May 2000 
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