The critical load for the Euler elastica under compressive thrust

The boundary-value problem governing the equilibrium of an elastic rod with bending stiffness $B$, loaded at its free end by a compressive thrust of intensity $F$ is

where $\lambda^2=F/B$. The problem admits the trivial solution $\theta=0$ for each $\lambda$. The system $\eqref{pippo}$ can be obtained by requiring that the energy functional

be stationary in the admissible set of of rotations $\theta$ which vanish at $s=0$ and whose first derivative is square integrable. The energy functional $\mathcal E$ is convex for $\lambda<\pi/2$ (see for instance Corollary 4.4 of https://arxiv.org/abs/2003.02696v1), which implies that the minimizer, and hence the solution of $\eqref{pippo}$ is unique.

We show that this bound is optimal by providing an example where uniqueness is lost as soon as $\lambda>\pi/2$.

To get a hint on how this solution may be constructed, we shall derive some necessary conditions which identify a monotone-increasing solution, and then show that these conditions are sufficient.

Thus, suppose that $\theta$ solves $\eqref{pippo}$. By quadrature we find that

for some constant $\gamma$. Since $\theta'(1)=0$, the constant $\gamma$ is given by .

Integrating we obtain

In particular,

Conversely, suppose that the equation

admits a solution $\beta>0$. Then for each $s\in[0,1]$ the equation

has a unique solution $\tilde\theta(s)$. Moreover, the function $s\mapsto\tilde\theta(s)$ solves the differential equation

along with $\tilde\theta(0)=0$ and $\tilde\theta(1)=\beta$. The latter boundary condition implies, in turn, that $\tilde\theta'(1)=0$, hence $\tilde\theta$ is a solution of $\eqref{pippo}$.

To investigate the solvability of $\eqref{eq:3}$ for $\beta$, it is expedient to express the left-hand side in terms of elliptic integrals through the change of variable [1,Eq. 1.82]

The result is

where $K(k)=\int_0^\pi \frac{d\psi}{\sqrt{1-k^2 \sin^2\psi}}$ is the complete elliptic integral of the first kind whose graph, taken from [2,Fig. 12.1], is shown below, along with that of the complete elliptic integral of the second kind:

The function $K$ is increasing with $\lim_{k\to 0-}K(k)=\pi/2$ and blows up as $k\to 1$.

We conclude that the equation $\eqref{eq:3}$ can be solved for every $\lambda>\pi/2$, and as a result the implicit formula $\eqref{eq:88}$ provides a non-trivial solution.

Remark 1.

A formal argument to show that $\pi/2$ is the limit of the integral on the right-hand side of $\eqref{eq:3}$ as $\beta\to 0$ is to carry out a Taylor expansion of the $\cos$ function at 0 up to second order, which would yield,

However, this argument should be made rigorous.

Remark 2.

We can think of $\eqref{eq:3}$ as defining the compressive thrust that is needed for the elastica to attain the end rotation $\theta(1)=\beta$. Thus, our argument shows that the minimum value of the renormalized thrust $\lambda^2$ needed to induce a non-null rotation is $\pi^2/4$.

Remark 3.

If $\beta$ does not solve $\eqref{eq:3}$, then equation $\eqref{eq:88}$ defines a function $\tilde\theta:[0,S]\to[0,\beta]$, where

This function solves

Thus, we may think of $\eqref{eq:13}$ as defining the length $S$ that the cantilever should have to attain an equilibrium configuration $\tilde\theta$ under a renormalized thrust $\lambda^2$ with end rotation $\tilde\theta(S)=\beta$.

References

[1] Davide Bigoni, Nonlinear Solid Mechanics, Cambridge University Press, 2012. DOI: https://doi.org/10.1017/CBO9781139178938

[2] Nico M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, 1996. DOI (online version): https://doi.org/10.1002/9781118032572