Time-Dependent Bending of a Plate with Mixed Boundary Conditions

Abstract

Consider a homogeneous and isotropic elastic plate of thickness h _o = const > 0, which occupies a region $\bar Sx[ - h_0 /2,h_0 /2]$ in ℝ³, where S is a domain in ℝ² with a simple, closed boundary ∂S. In the transverse shear deformation model proposed in [1] it is assumed that the displacement vector at (x ₁, x ₃), x = (x ₁, x ₂) ∈ ℝ², at time t ≥ 0, is of the form

$$ (x_3 u_1 (x,t),x_3 u_2 (x,t),u_3 (x,t))^T , $$

where the superscript T denotes matrix transposition. Then the vector u = (u ₁, u ₂, u ₃)^T satisfies the equation of motion

$$ B(\partial _t^2 u)(x,t) + (Au)(x,t) = q(x,t),(x,t) \in G = Sx(0,\infty ); $$

here B = diag{ρh ², ρh ², ρ}, h2 = h ² ₀/12, ρ is the plate density, ∂ _{t =} ∂/∂t,

$$ A = \left( \begin{gathered} - h^2 \mu \Delta - h^2 (\lambda + \mu )\partial _1^2 + \mu \hfill \\ - h^2 (\lambda + \mu )\partial _1 \partial _2 \hfill \\ - u\partial _1 \hfill \\ \end{gathered} \right.\left. \begin{gathered} - h^2 (\lambda + \mu )\partial _1 \partial _2 \mu \partial _1 \hfill \\ - h^2 \mu \Delta - h^2 (\lambda + \mu )\partial _2^2 + \mu \mu \partial _2 \hfill \\ - u\partial _2 - \mu \Delta \hfill \\ \end{gathered} \right), $$

∂_α = ∂/∂x _α, α = 1,2, λ and µ are the Lamé constants satisfying λ + µ > 0 and ε > 0, and q is a combination of the forces and moments acting on the plate and its faces x ₃ = ±h ₀/2.