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Analytic solution for an enhanced theory of bending of plates. (English) Zbl 1101.74041
Constanda, C. (ed.) et al., Integral methods in science and engineering. Analytic and numerical techniques. Papers from the 7th international conference (IMSE 2002), University of Saint Étienne, Saint Étienne, France, August 7–10, 2002. Boston, MA: Birkhäuser (ISBN 0-8176-3228-X/hbk). 151-156 (2004).
Let \(S\) be a domain in \(\mathbb R^2\) bounded by a simple closed \(C^2\)-curve \(\partial S\), and let \(h_0=\text{const}\) be such that \(0< h_0\ll\text{diam\,}S\). An elastic body that occupies the region \(\overline S\times[-h_0/2, h_0/2]\) is called a thin plate and \(h_0\) the plate thickness. \(x= (x_1, x_2)\) is a generic point in \(\mathbb R^2\), and \(z= x_1+ ix_2\in \mathbb{C}\), \(\partial_\alpha= \partial/\partial x_\alpha\), \(\alpha= 1,2\), \(\partial_z= \partial/\partial z\). Let \(S^+\) and \(S^-\) be the domains interior and exterior to \(\partial S\), respectively.
The authors consider the problem of bending of an elastic plate with transverse shear deformation and transverse normal strain, accounted for through displacements of the form
\[ U_\alpha= x_3 u_\alpha(x_1, x_2),\quad U_3= u_{31}(x_1, x_2)+ x^2_3 u_{32}(x_1, x_2),\quad \alpha= 1,2. \] The general form of a rigid displacement is \(u= Fk\), where \(F(x)\) is the \((4\times 3)\)-matrix of columns \((1,0, -x_1,0)^T\), \((0,1,-x_2, 0)^T\), \(k\) is an arbitrary constant \((3\times 1)\)-vector, and the superscript \(T\) indicates matrix transposition.
In [P. Schiavone (ed.) et al., Integral methods in science and engineering. Proceedings of the 6th international conference, IMSE 2000, Banff, Canada, June 12–15, 2000. Boston, MA: Birkhäuser. 191–196 (2002; Zbl 1104.74318)], the authors considered the system of equilibrium equations in terms of displacements, with Dirichlet and Neumann boundary conditions. They reduced these boundary value problems to singular integral equations on the contour of the domain and solved them in spaces of smooth functions. But the authors were not able to elucidate the physical meaning of certain analytic constraints imposed on the asymptotic behavior of the solutions. This question is answered here, where the complete integral of the system is constructed through complex variable treatment.
For the entire collection see [Zbl 1055.35005].
74K20 Plates
74G05 Explicit solutions of equilibrium problems in solid mechanics