Euler's elastica in nonlocal theory of elasticity

TL;DR

This paper extends Euler's elastica problem to a nonlocal elasticity context, deriving nonlinear solutions using elliptic integrals and setting the stage for boundary condition applications to compare with classical models.

Contribution

It introduces a nonlocal differential model for elastica, providing explicit parametric solutions involving elliptic integrals, generalizing classical local elasticity results.

Findings

Solutions expressed via elliptic integrals for nonlocal elastica

Nonlinear governing equations solved in parametric form

Framework for applying boundary conditions to compare with classical models

Abstract

A generalization of the Euler's elastic problem, i.e., finding a stationary configuration (planar elastica) of the Bernoulli's thin ideal elastic rod with boundary conditions defined through fixed endpoints and/or tangents at the endpoints, for the chosen nonlocal differential constitutive stress-strain relation (i.e., nonlocal theory of elasticity) is considered. In the classical (local) Euler-Bernoulli's beam model, the general solutions of the governing equations (that are inhomogeneous but linear) for bending moments and shear forces in the case of large deformations can be obtained using the Jacobi elliptic functions and incomplete elliptic integrals. For the discussed nonlocal toy differential model, the general solutions of the governing equations (that are this time nonlinear) can also be expressed in the parametric form through the linear combinations of all three incomplete elliptic integrals. As further research, we plan to apply some boundary conditions (clamped, simply supported, etc.) for the obtained nonlocal general solutions in order to compare them to the local solutions for the corresponding boundary value problems.