Some new classes of inverse coefficient problems in nonlinear mechanics

A. Kh. Khasanov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 1, pp. 20-33

Résumé

The present study deals with the following two types of inverse problems governed by nonlinear PDEs, and related to determination of unknown properties of engineering materials based on boundary/surface measured data. The first inverse problem consists of identifying the unknown coefficient $g(\xi^2)$ (plasticity function) in the nonlinear differential equation of torsional creep $-(g(|\nabla u|^2)u_{x_1})_{x_1}-(g(|\nabla u|^2)u_{x_2})_{x_2}= 2\phi$, $x\in\Omega\subset\mathbb R^2$, from the torque (or torsional rigidity) $\mathcal T(\phi)$, given experimentally. The second class of inverse problems is related to identification of the unknown coefficient $g(\xi^2)$ in the nonlinear bending equation $Au\equiv(g(\xi^2(u))(u_{x_1x_1}+u_{x_2x_2}/2))_{x_1x_1}+(g(\xi^2(u))u_{x_1x_2})_{x_1x_2}+(g(\xi^2(u))(u_{x_2x_2}+u_{x_1x_1}/2))_{x_2x_2}=F(x)$, $x\in\Omega\subset\mathbb R^2$. The boundary measured data here is assumed to be the deflections $w_i[\tau_k]:=w(\lambda_i;\tau_k)$, measured during the quasi-static bending process, given by the parameter $\tau_k$, $k=\overline{1,K}$, at some points $\lambda_i=(x_1^{(i)},x_2^{(i)})$, $i=\overline{1,M}$, of a plate. Based on obtained continuity property of the direct problem solution with respect to coefficients, and compactness of the set of admissible coefficients, an existence of quasi-solutions of the considered inverse problems are proved. Some numerical results, useful from the points of view of nonlinear mechanics and computational material science, are demonstrated. Keywords: inverse coefficient problem, material properties, quasisolution method.

Détail

Keywords: material properties, quasisolution method.
Mots-clés : inverse coefficient problem

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