 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| V. A. Ivanov, V. N. Paimushin, and V. I. Shalashilin, "Linearized neutral equilibrium equations for nonthin sandwich shells with transversely soft filler and related problems of nonlinear elasticity," Mech. Solids. 40 (6), 81-94 (2005) |
| Year |
2005 |
Volume |
40 |
Number |
6 |
Pages |
81-94 |
| Title |
Linearized neutral equilibrium equations for nonthin sandwich shells with transversely soft filler and related problems of nonlinear elasticity |
| Author(s) |
V. A. Ivanov (Kazan)
V. N. Paimushin (Kazan)
V. I. Shalashilin (Moscow) |
| Abstract |
For sandwich shells with nonthin transversely soft filler, we construct
refined linearized stability equations, which are intended for describing
buckling modes with both zero (purely shear modes) and large (mixed bending
modes) variability of parameters of the perturbed stress-strain state in
the tangential directions. These equations comprise six neutral equilibrium
equations for the outer layers (which are constructed on the basis of the
geometrically nonlinear equations in Kirchhoff-Love theory of moderate
bending of thin shells and allow for the force interaction with the filler
on the interface) and equations describing the stress-strain state of the
transversely soft filler in the perturbed state. The latter equations are
obtained by integration, with respect to the transverse coordinate, of the
linearized elasticity equations describing the neutral equilibrium of the
transversely soft filler in the perturbed state and subject to the
kinematic matching conditions with the outer layers. If in the original
unperturbed state we accept the assumption of finiteness of the in-plane
shear strain and retain the terms nonlinear in all three displacement
components in the expression for the transverse compression strain, then
the constructed relations prove to be integro-differential equations for
two two-dimensional functions characterizing the transverse tangential
stresses in the undeformed axes and two three-dimensional filler
displacement functions (the complete model). The relations of this model
are simplified and reduced to the above-mentioned two two-dimensional
differential equations for the characteristics of transverse tangential
stresses if we introduce the assumption of small variation of the filler
deflection along the thickness in the original state (the simplified
model). The equations of the partially simplified model, intermediate
between the equations of the complete and simplified filler models, have
the same structure and complexity. We show that the equations of the
complete model, as well as those of the partially simplified model, are
inadequate, since so are the underlying well-known geometrically nonlinear
kinematic elasticity relations, which are recommended for using at small
extension strains in the scientific literature. The latter become adequate
if one omits the terms nonlinear in the displacement components in whose
direction the extension strains are to be determined. In view of the
preceding, only the simplified model is adequate for sandwich shells. All
these conclusions are based on the results obtained by solving the problem
on all possible buckling modes of a sandwich ring subjected to external (or
internal) pressure and also the elementary problem on the longitudinal
extension-compression of a rod with rectilinear axis in the exact and
approximate settings for the cases of arbitrary and small strains. |
| References |
| 1. | V. N. Paimushin, "Theory of stability for three-layer plates and
shells: stages of development, state-of-the-art, and prospects," Izv. RAN.
MTT [Mechanics of Solids], No. 2, pp. 148-162, 2001. |
| 2. | V. A. Ivanov, V. N. Paimushin, and V. I. Shalashilin, "Refined
geometrically nonlinear theory and buckling modes of sandwich shells with
transversely-soft core," Izv. RAN. MTT [Mechanics of Solids], No. 3,
pp. 167-177, 2005. |
| 3. | V. A. Ivanov and V. N. Paimushin, "Refined stability theory of
sandwich structures (the nonlinear equations of subcritical equilibrium of
shells with transversely soft filler)," Izv. Vuzov. Matem., No. 11,
pp. 29-42, 1994. |
| 4. | V. A. Ivanov, V. N. Paimushin, and T. V. Polyakova, "Refined
stability theory of sandwich structures (the linearized equations of
neutral equilibrium and elementary one-dimensional problems)," Izv. Vuzov.
Matem., No. 3, pp. 15-24, 1995. |
| 5. | V. N. Paimushin and S. N. Bobrov, "Refined geometrically nonlinear
theory of sandwich shells with transversely soft filler of medium thickness
for the analysis of mixed buckling modes," Mekh. Kompoz. Mater., Vol. 36,
No. 1, pp. 95-108, 2000. |
| 6. | V. V. Bolotin and Yu. N. Novichkov, Mechanics of Multilayered
Structures [in Russian], Mashinostroenie, Moscow, 1980. |
| 7. | V. N. Paimushin, V. A. Ivanov, S. N. Bobrov, and T. V. Polyakova,
"Stability of a sandwich circular ring under uniform external pressure,"
Mekh. Kompoz. Mater., Vol. 36, No. 3, pp. 317-328, 2000. |
| 8. | V. N. Paimushin and V. R. Khusainov, "Refined theory of sandwich
plates and shells for the analysis of dynamic deformation processes with
large variability exponents," Mekh. Kompoz. Mater. i Konstr., Vol. 7,
No. 2, pp. 215-235, 2001. |
| 9. | V. N. Paimushin, "Classical and nonclassical problems of dynamics
of sandwich shells with transversely soft filler," Mekh. Kompoz. Mater.,
Vol. 37, No. 3, pp. 289-306, 2001. |
| 10. | V. N. Paimushin, "Shear buckling mode of a sandwich circular ring
under uniform external pressure," Doklady RAN, Vol. 378, No. 1,
pp. 58-60, 2001. |
| 11. | V. V. Novozhilov, Fundamentals of Nonlinear Elasticity [in
Russian], Gostekhizdat, Moscow, 1948. |
| 12. | V. N. Paimushin and V. A. Ivanov, "Buckling modes of homogeneous
and sandwich plates under pure shear in tangential directions," Mekh.
Kompoz. Mater., Vol. 36, No. 2, pp. 215-228, 2000. |
|
| Received |
30 May 2003 |
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