# Time-Dependent Bending of a Plate with Mixed Boundary Conditions

• Igor Chudinovich
• Christian Constanda

## 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.

## Keywords

Boundary Operator Boundary Integral Equation Elastic Plate Mixed Boundary Condition Trace Operator

## References

1. 1.
C. Constanda, A mathematical analysis of bending of plates with transverse shear deformation, Pitman Res. Notes Math. Ser. 215, Longman/Wiley, Harlow-New York, 1990.Google Scholar
2. 2.
I. Chudinovich and C. Constanda, Nonstationary integral equations for elastic plates, C.R. Acad. Sci. Paris Sér. I 329 (1999), 1115–1120.