Thin three-dimensional plate with a crack along the clamped zone on its lateral side. (Q2773590)

The authors construct and justify some elements of the asymptotic expansion of a solution to a three-dimensional problem of the elasticity theory for a thin plate with a crack along a fixed area on its boundary surface. The three-dimensional boundary layer phenomenon is studied which appears near the ends of the crack and corresponds to special solutions in a semi-layer of the unit thickness. An existence and uniqueness theorem is proven for the problem under consideration and the asymptotic behavior of the solution obtained is studied. As an example of using of the asymptotic representations obtained, the authors calculate a potential energy differential for the strain of the plate under expansion of the crack. It is checked that the asymptotic energy formula obtained on the base of the precise plane Kirchhoff model provides a two-term asymptotic formula for the energy differential.