Some new classes of inverse coefficient problems in nonlinear mechanics

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.