an:01285539
Zbl 0917.35029
Nazarov, S. A.
Justification of the asymptotic theory of thin rods. Integral and pointwise estimates
RU
Probl. Mat. Anal. 17, 101-152 (1997).
00056902
1997
j
35J45 35B45 74B05 35B40 35C20
elasticity equations; integral and pointwise estimates
The author studies the following system of elliptic equations which comes from the modeling of a thin rod in the frameworks of linear elasticity:
\[
\begin{aligned} D(-\nabla_x)AD(\nabla_x)^tu &=f, \quad \text{in } \Omega_h, \\ D(n)AD(\nabla_x)^t &=g, \quad\text{on } \Gamma_h = \{x\in\partial\Omega_h: |{}z|{}<1\}, \\ u&=0,\quad\text{on } \gamma_h^{\pm}=\{x\in\partial\Omega_h:z=\pm 1\}, \end{aligned}
\]
where \(\Omega_h\) denotes a thin rod located in the strip \(\{x=(y,z):y\in\mathbb{R}^2\), \(z\in (0,1)\}\) and is given by the equality
\[
\Omega_h = \{x:|{}z|{}<1,\;\xi = (\eta,\zeta)\equiv h^{-1}(y,z)\in Q\subset \mathbb{R}^3\};
\]
here \(h=1/N\) is a small parameter, \(\xi\) is the ``fast'' variable. The rod is assumed to be nonhomogeneous and the end-walls \(\gamma_h^{\pm}\) of the rod \(\Omega_h\) are rigidly fixed, which is expressed by the last condition in the model. The aim of the article is to construct and justify the asymptotic of a solution to the above problem as \(h\to +0\). This allows us to expose an asymptotic deduction of the one-dimensional equations of the theory of thin rods and then to estimate the error which arises due to reduction of dimension. The author observes that the passage to the ``slow'' variable \(z\) is also possible.
V.Grebenev (Novosibirsk)