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)