A theory of rods 2 or, equivalently, a one-dimensional theory of solids is a characterization of the behavior of slender three-dimensional solid bodies by a set of equations having the parameter of a certain curve and the time as the only independent variables. The formidable mathematical obstacles presented by three-dimensional continuum theories dictate the need for tractable, accurate, and illuminating one-dimensional models.

The Theory of Rods

https://people.maths.ox.ac.uk/ball/Papers/Ball%20J%20Math%20Anal%20Appl%201973.pdf

Initial-boundary value problems for an extensible beam

On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations

http://link.springer.com/content/pdf/10.1007/978-3-642-69569-8_2.pdf

R.W Dickey

Papers

Free vibrations and dynamic buckling of the extensible beam

The suspension bridge deflection equations

Asymptotic behavior of solutions of hyperbolic, semilinear partial differential equations☆

On the transformation of the Föppl membrane equations to ordinary differential equations