Bounds of applicability of Kirchhoff theory to the problem of bending of an elastic layer by concentrated forces (Q2755261)

The author obtains an explicit solution of the equations NEWLINE\[NEWLINE2Gu=-{\partial\Phi_0\over\partial x}-y{\partial\Phi_2\over\partial x};\quad 2G\nu=(3-4\nu)\Phi_2-{\partial\Phi_0\over\partial y}-y{\partial\Phi_2\over\partial y};NEWLINE\]NEWLINE NEWLINE\[NEWLINE\sigma_{x}=-{\partial^2\Phi_0\over\partial x^2}+2\nu{\partial\Phi_2\over\partial y}-y{\partial^2\Phi_0\over\partial x^2};\quad \sigma_{y}=2(1-\nu){\partial\Phi_2\over\partial y}-{\partial^2\Phi_0\over\partial y^2}-y{\partial^2\Phi_2\over\partial y^2};NEWLINE\]NEWLINE NEWLINE\[NEWLINE\tau_{xy}=-{\partial^2\Phi_0\over\partial x\partial y}+(1-2\nu){\partial\Phi_2\over\partial x}-y{\partial^2\Phi_2\over\partial x\partial y},NEWLINE\]NEWLINE where \(G\) is the Young modulus; \(\nu\) is the Poisson ratio; \(\Phi_0,\Phi_2\) are harmonic functions \((\Phi_1\equiv 0)\). The boundary conditions have the form \(\sigma_{y}|_{y=b/2}=-P\delta(x)\); \(\sigma_{y}|_{y=-b/2}=-P[\delta(x-l)+\delta(x+l)]/2\); \(\tau_{xy}|_{y=\pm b/2}=0\), where \(\delta(\cdot)\) is the Dirac delta function. A solution to this problem is constructed by means of Fourier integral transform. Using the asymptotic methods and the numerical analysis, the author makes a comparison of the explicit solution and the approximate solution obtained on the base of Kirchhoff theory. The bounds of applicability of the Kirchhoff theory are found.