TY - JOUR

T1 - Applicability ranges for four approaches to determination of bending stiffness of multilayer plates

AU - Morozov, Nikita F.

AU - Belyaev, Alexander K.

AU - Tovstik, Petr E.

AU - Tovstik, Tatiana P.

N1 - Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/7

Y1 - 2021/7

N2 - A linear static problem of bending of a multilayer plate with homogeneous isotropic layers is considered. The deflection is assumed to have harmonic shape in the tangential directions. For calculation of the bending stiffness we introduce two dimensionless parameters: a small thickness parameter equal to the ratio of the thickness to the deformation wavelength in the tangential directions, and a large inhomogeneity parameter equal to the ratio of the maximum and minimum Young’s moduli of the layers. The plane of these parameters is split into four regions, in which different models are available for calculating the bending stiffness. The first of these is the Kirchhoff - Love model based on the hypothesis of straight non-deformable normal. In the second region, the Timoshenko - Reissner model with a shear parameter is used, calculated by an asymptotic formula of the second order of accuracy. The third region is based on assumptions on inextensible normal fiber and large heterogeneity. Finally, in the fourth region, compression of the normal is taken into account, and an approximate formula for bending stiffness is proposed for a three-layer plate. Error estimation of the models is carried out on test examples by comparison with the exact numerical solution of the three-dimensional problem of the elasticity theory. The possibility of suitability of the bending stiffness for calculating the plate eigenfrequencies is discussed.

AB - A linear static problem of bending of a multilayer plate with homogeneous isotropic layers is considered. The deflection is assumed to have harmonic shape in the tangential directions. For calculation of the bending stiffness we introduce two dimensionless parameters: a small thickness parameter equal to the ratio of the thickness to the deformation wavelength in the tangential directions, and a large inhomogeneity parameter equal to the ratio of the maximum and minimum Young’s moduli of the layers. The plane of these parameters is split into four regions, in which different models are available for calculating the bending stiffness. The first of these is the Kirchhoff - Love model based on the hypothesis of straight non-deformable normal. In the second region, the Timoshenko - Reissner model with a shear parameter is used, calculated by an asymptotic formula of the second order of accuracy. The third region is based on assumptions on inextensible normal fiber and large heterogeneity. Finally, in the fourth region, compression of the normal is taken into account, and an approximate formula for bending stiffness is proposed for a three-layer plate. Error estimation of the models is carried out on test examples by comparison with the exact numerical solution of the three-dimensional problem of the elasticity theory. The possibility of suitability of the bending stiffness for calculating the plate eigenfrequencies is discussed.

KW - Bending stiffness

KW - Free vibration

KW - Kirchhoff - Love model

KW - Multilayer plate

KW - Timoshenko - Reissner model

KW - Love model

KW - Kirchhoff

KW - Reissner model

KW - Timoshenko

UR - http://www.scopus.com/inward/record.url?scp=85103387915&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/76dd1a91-ab19-3901-ab49-051d3e85dd74/

U2 - 10.1007/s00161-021-00996-3

DO - 10.1007/s00161-021-00996-3

M3 - Article

AN - SCOPUS:85103387915

VL - 33

SP - 1659

EP - 1673

JO - Continuum Mechanics and Thermodynamics

JF - Continuum Mechanics and Thermodynamics

SN - 0935-1175

IS - 4

ER -