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

Chapter

## Abstract

Consider a homogeneous and isotropic elastic plate of thickness where the superscript T denotes matrix transposition. Then the vector here ∂

*h*_{o}= const > 0, which occupies a region \( \bar Sx[ - h_0 /2,h_0 /2] \) in ℝ^{3}, where*S*is a domain in ℝ^{2}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}) ∈ ℝ^{2}, at time*t*≥ 0, is of the form$$
(x_3 u_1 (x,t),x_3 u_2 (x,t),u_3 (x,t))^T ,
$$

*u*= (*u*_{1},*u*_{2},*u*_{3})^{T}satisfies the equation of motion$$
B(\partial _t^2 u)(x,t) + (Au)(x,t) = q(x,t),(x,t) \in G = Sx(0,\infty );
$$

*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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- 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.I. Chudinovich and C. Constanda, Nonstationary integral equations for elastic plates,
*C.R. Acad. Sci. Paris Sér. I***329**(1999), 1115–1120.MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2004