Vibration of a Three-Dimensional, Finite Linear, Elastic Solid Containing Cracks

This paper is to determine vibrational eigensolutions $[λm2,vm(r)]m=1∞$ of a three-dimensional, finite, linear, elastic solid C containing cracks in terms of crack configuration σ_c and eigensolutions $[ωn2,un(r)n=1∞$ of a perfect elastic solid P without the cracks. Use of Betti reciprocal theorem and the Green’s function of P expands $vm(r)$ in terms of an infinite series of $un(r)$⁠. Substitution of the $vm(r)$ series representation into the Kamke quotient of C and stationarity of the quotient result in a Fredholm integral equation whose nontrivial solutions predict $λm2$⁠, and $vm(r)$ of C. Finally, natural frequencies and mode shapes of a circular shaft of finite length containing a circumferential crack under torsional vibration are predicted through a two-term Ritz approximation of the Fredholm integral equation. The results differ significantly from those predicted by the method of flexibility matrices, when the ratio of the shaft length to the shaft radius is small.