Euler's elastica in nonlocal theory of elasticity

TL;DR

This work generalizes Euler's elastica to a nonlocal elasticity setting via a toy strain-driven model. Starting from the classical local elastica, it derives the standard nonlinear ODE for the tangent angle $\theta(s)$ and expresses the solution in closed form using Jacobi elliptic functions and incomplete elliptic integrals, with integration constants fixed by boundary conditions. The authors then formulate a nonlinear nonlocal elastica through $\sigma_{xx}-\mu\frac{d^2\sigma_{xx}}{ds^2}=E\varepsilon_{xx}$, yielding a modified moment-curvature relation and a nonlinear ODE for $\theta(s)$, which admits first-integral forms and multiple elliptic-function representations for the shape through $\theta(s)$ and the coordinates $(x(s),y(s))$. They further develop implicit parametrizations via a tangent-half-angle substitution, expressing $s(\theta)$ and $x(\theta)$ in terms of incomplete elliptic integrals of the first, second and third kinds, with auxiliary constants encoding the nonlocal parameter $\mu$. As a roadmap for future work, boundary-value problems (e.g., clamped or simply supported ends) will be used to compare local and nonlocal elastica shapes and to quantify nonlocal effects on planar nanobeams.

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.

Figures (2)

• Figure 1: Exemplary graph of the parametric solution $(x\left(\theta\right),y\left(\theta\right))$ defined by (\ref{['eq2.12a']}) and (\ref{['eq2.12b']}) for integration constants $x_0=y_0=0$, $C=20$ nm$^{-2}$, elliptic modulus $1/k=0.99549$, force $F=500$ nN, Young modulus $E=1000$ GPa, moment of inertia $I=0.0491$ nm$^{4}$.
• Figure 2: Exemplary graphs of the parametric solution $(x\left(\theta\right),y\left(\theta\right))$ defined by (\ref{['eq4.13a-b']}) and (\ref{['eq3.14']}) for integration constants $x_0=y_0=0$, $D=0.1$, nonlocality parameter $\mu=2.25$ nm$^2$, force $F=500$ nN, Young modulus $E=1000$ GPa, moment of inertia $I=0.0491$ nm$^{4}$, where green line: $-\pi/2+\varphi(\theta)$, red line: $\varphi(\theta)$, blue line: $\pi/2-\varphi(\theta)$.