Study of the solvability of a boundary value problem for the system of nonlinear differential equations of the theory of shallow shells of the Timoshenko type (Q311190)

In a plane connected bounded domain \(\Omega\), the system of nonlinear differential equations is considered \[ T^{i\lambda}_{\alpha^{\lambda}}+R^i=0, \; i=1,2 \] \[ T^{\lambda3}_{\alpha^{\lambda}}+k_{\lambda}T^{\lambda\lambda}+(T^{\lambda \mu}w_{3\alpha^{\lambda}})_{\alpha^{\lambda}}+R^3=0, \eqno{(1)} \] \[ M^{i\lambda}_{\alpha^{\lambda}}-T^{i3}+L^i=0, \; i=1,2 \] with the conditions on the boundary \(\partial\Omega\) \[ w_3=\psi_1=0, \eqno{(2)} \] \[ T^{j1}\frac{d\alpha ^2}{ds}-T^{j2}\frac{d\alpha ^1}{ds}=P^j(s),\; j=1,2 \eqno{(3)} \] \[ M^{21}\frac{d\alpha ^2}{ds}-M^{22}\frac{d\alpha ^1}{ds}=N^2(s),\; j=1,2 \eqno{(4)} \] where \(a= (w_1,w_2,w_3, \psi_1, \psi_2),\;T^{ij}\equiv T^{ij}(a) = D_0^{ijkn}\gamma^0_{kn},\; D^{ijkn}_m = \int\limits_{-h/2}^{h/2} B^{ijkn}(\alpha^3)^m d\alpha^3\), \(B^{1111} = B^{2222} =\frac{E}{1-\mu^2}\), \(B^{1212} = \frac{\mu E}{1-\mu^2}\), \(B^{1313} = B^{2323} = \frac{ EK^2}{2(1+\mu)}\), the remaining \(B^{ijkn}\) are zero, \(\alpha^j = \alpha^j(s)\; (j = 1,2)\) are equations of the curve \(\Gamma\), \(s\) is the arc length on \(\Gamma\), and \[ \gamma^0_{jj} = w_{j\alpha^j}- k_jw_3 + w^2_{3\alpha^j}/2 \; (j = 1, 2),\; \gamma^0_{12} = w_{1\alpha^2} + w_{2\alpha^1} + w_{3\alpha^1}w_{3\alpha^2},\; \gamma^1_{jj} = \psi_{j\alpha^j} \; (j = 1, 2), \] \[ \gamma^1_{12} = \psi_{1\alpha^2} + \psi_{2\alpha^1},\; \gamma^0_{j3} = w_{3\alpha^j} + \psi_j\; (j = 1, 2), \gamma^0_{33} = \gamma^1_{k3} \equiv 0, \; k= 1, 2, 3. \] System (1) supplemented with the boundary conditions (2)--(4) describes the equilibrium of an elastic shallow isotropic homogeneous shell of the Timoshenko type with hinged edge [\textit{K. Z. Galimov}, Osnovy nelineinoi teorii tonkikh obolochek (Foundations of Nonlinear Theory of Thin Shells), Kazan (1975)] \(T^{ij}\) are the stresses, \(M^{ij}\) are the moments, \(\gamma^k_{ij}\; (i, j = 1, 2, 3,\; k = 0, 1)\) are the strain components of the midsurface \(S_0\) of the shell homeomorphic to the domain \(\Omega, w_i (j = 1, 2)\) and \(w_3\) are the tangential and normal displacements of the points of \(S_0, \psi_i\; (i = 1, 2)\) are the rotation angles of the normal cross-sections of \(S_0, a\) is the vector of generalized displacements, \(R^j (j = 1, 2, 3), L^k, P^k (k = 1, 2)\), and \(N^2\) are components of external forces acting on the shell, \(\mu = \mathrm{const}\) is the Poisson ratio, \(E = \mathrm{const}\) is the Young modulus, \(k_1, k_2 = \mathrm{const}\) are the principal curvatures, \(k_2 = \mathrm{const}\) is the shear modulus, \(h = \mathrm{const}\) is the shell thickness, and \(\alpha^1\) and \(\alpha^2\) are the Cartesian coordinates of points of the domain \(\Omega\). The used method implies the reduction of the original equations system to a single nonlinear differential equation whose solvability is proved with the use of the contraction mapping principle.