Nonlinear eigenvalue problems of the elastica

Physical phenomena are usually described by nonlinear differential equations. If some of the physical parameters are unknown then adding appropriate constraints may transform a nonlinear problem to a nonlinear eigenvalue problem. This principle is illustrated by a variety of problems associated with deflections of a flexible rod.The basic problem studied is the problem formed by the natural extension of the linear buckling problem of a flexible rod when large deflections are taken into account. The unknown parameter in these problems is the load, and the constraint is the predetermined distance between the rod's ends. The problem of finding large deflections of the strongest column is also addressed. A simple numerical method for solving these problems is given.The paper gives insight into the problem of forming a non-linear eigenvalue problem. It highlights the relations between the mathematical formulation and the governing physical laws, and demonstrates the similarity between the linear and the nonlinear solutions.

• Insight on the formation of non-linear eigenvalue problems is given.
• The similarity between the solutions of the linear and the nonlinear problems is highlighted.
• Post buckling investigation of the strongest column is done.