General Nonlinear-Material Elasticity in Classical One-Dimensional Solid Mechanics

We will create a class of generalized ellipses and explore their ability to define a distance on a space and generate continuous, periodic functions. Connections between these continuous, periodic functions and the generalizations of trigonometric functions known in the literature shall be established along with connections between these generalized ellipses and some spectrahedral projections onto the plane, more specifically the well-known multifocal ellipses. The superellipse, or Lam\'{e} curve, will be a special case of the generalized ellipse. Applications of these generalized ellipses shall be explored with regards to some one-dimensional systems of classical mechanics. We will adopt the Ramberg-Osgood relation for stress and strain ubiquitous in engineering mechanics and define a general internal bending moment for which this expression, and several others, are special cases. We will then apply this general bending moment to some one-dimensional Euler beam-columns along with the continuous, periodic functions we developed with regard to the generalized ellipse. This will allow us to construct new solutions for critical buckling loads of Euler columns and deflections of beam-columns under very general engineering material requirements without some of the usual assumptions associated with the Ramberg-Osgood relation.