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Eigenmode asymptotics in thin elastic plates. (English) Zbl 0966.74027

Summary: We investigate the behavior of the vibration modes (eigenvalues) of an isotropic homogeneous plate as its thickness tends to zero. As lateral boundary conditions, we consider clamped or free edges. We establish distinct asymptotics for bending and membrane modes: the smallest bending eigenvalues behave as the square of the thickness, whereas the membrane eigenvalues tend to nonzero limits. Moreover, we prove that all these eigenvalues have an expansion in power series with respect to the thickness regardless of their multiplicities or of multiplicities of the limit in-plane problems.

MSC:

74H10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics
74K20 Plates
74H45 Vibrations in dynamical problems in solid mechanics
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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