In this paper, a method for the construction of a cracked beam finite element is presented. The additional flexibility due to the cracks is identified from three-dimensional finite element calculations taking into account the unilateral contact conditions between the crack lips. Based on this flexibility, which is distributed over the entire length of the element, a cracked beam finite element stiffness matrix is deduced. Considerable gain in computing efforts is reached compared to the nodal representation of the cracked section when dealing with the numerical integration of differential equations in structural dynamics. The stability analysis of a cracked shaft is carried out using the Floquet theory.

I'm in Washington DC attending a small workshop entitled Understanding and Exploiting Nonlinearity.  Yesterday several talks described recent developments of nonlinear dynamics, the kind of phenomena that can be described by a set of nonlinear ordinary differential equations.  Years ago, when the theory of chaos was in vogue, I looked at several textbooks on nonlinear dynamics, and tried to apply a few elementary ideas to evolving structures in materials.  This time I learned that many of the esoteric ideas of nonlinear dynamics have found applications in modeling natural phenomena and creating new devices.  Perhaps it is a good time to relearn nonlinear dynamics.