On the asymptotics of the spectrum of a thin plate problem of elasticity. (Russian, English) Zbl 1150.74367

Nazarov, S. A.

Sib. Mat. Zh. 41, No. 4, 895-912 (2000); translation in Sib. Math. J. 41, No. 4, 744-759 (2000).

Let \(\omega\) be a domain on the plane \(\mathbb R^2\) bounded by a simple closed smooth contour \(\partial\omega\) and let \(\Omega_h = \omega\times (-h/2,h/2) \subset \mathbb R^3\), where \(h\in (0,1]\) is a small parameter. The author writes down the problem on free oscillations of the plate \(\Omega_h\) with rigidly claimed lateral surface as follows: \[ \begin{aligned} &D(-\nabla_x)A(y,h^{-1}z)D(\nabla_x)^tu(x) = \Lambda\rho(y,h^{-1}z)u(z), \qquad x\in\Omega_h,\\ & D(\pm e^3)A(y,\pm 1/2)D(\nabla_x)^tu(y,\pm h/2) = 0,\qquad y\in\omega,\\ &u(x) = 0,\qquad x\in\Gamma_h. \end{aligned} \] Here \(e^j\) is the unit vector of the \(x_j\)-axis, \(\rho\) is the density of the elastic material which is positive and depends smoothly on \(y\) and \(h^{-1}z\). \(\Lambda\) is a spectral parameter (\(\Lambda^{1/2}\) is the oscillation frequency). The Neumann boundary condition means that the bases \(\omega^{\pm}\) of the plate are free of stresses.
The author studies the eigenvalues of the above-mentioned problem of elasticity for a thin three-dimensional cylinder (plate). As a result, asymptotical formulas are obtained that reveal the stratified structure of the spectrum accounting the interaction of longitudinal and transverse oscillations. The author also discusses boundary effects related to the boundary layer phenomenon.

Reviewer: V. Grebenev (Novosibirsk)

MSC:

74B20	Nonlinear elasticity
74K20	Plates
35J55	Systems of elliptic equations, boundary value problems (MSC2000)
35B34	Resonance in context of PDEs
35C20	Asymptotic expansions of solutions to PDEs

Keywords:

boundary value problem of elasticity; thin plate; asymptotics of the spectrum; stratified spectrum