Second fundamental boundary value problem of elasticity theory for a spindle-shaped solid (Q2755269)

In the unbounded elastic medium external to a spindle-shaped solid the author considers the Lamé equation NEWLINE\[NEWLINE2{m-1\over m-2}\,grad\, div\,\vec U - rot\,rot\vec U=0NEWLINE\]NEWLINE under arbitrary value of \(\vec U\) on the boundary, where \(m\) is the Poisson ratio \((2\leq m<\infty)\); \(\vec U\) is the vector of elastic displacements. A solution of the considered problem is presented in terms of the boundary value of the dilatation function \(\theta\). The boundary value problem for the Fourier transform of \(\theta\) is obtained, and then this problem is reduced to a Fredholm equation of the second kind. As an example, the boundary value problem for the shift of the rigid spindle-shaped solid along the direction perpendicular to its symmetry axis is considered.