# Buckling

Experience tells us that pushing hard enough on the ends of a body with a large aspect ratio will cause it to bend or dent. Pushing even further might cause it to break. This behavior is called buckling. In the first case buckling is essentially an elastic instability and can be described with methods developed for elastic systems. In the second case (breaking), buckling is a failure mode for the structure (just like rupture or mechanical overload).

The most commonly studied model for the buckling mechanism are long slender columns. As shown by Euler these structures can carry a load of

$F=\frac{\pi^2 EI}{(KL)^2}$

without buckling, where E is the modulus of elasticity, I is the area moment of inertia and L the length of the column. The constant K varies according to the boundary conditions, but is always on the order one.

In the context of soft matter, we are most often interested in the buckling of two-dimensional structures, like thin films or sheets. One example of this is the behavior of particle rafts (densely packed monolayers of particles at the interface between two fluids) under compression which behave like an ideally elastic sheet with an apparent shear modulus of

$G \approx \frac{\gamma}{d}$

where d is the particle diameter and $\gamma$ the surface tension (details for this in the paper Elasticity of interfacial particle rafts).

Another interesting example, is The Self Assembly of Flat Sheets into Closed Surfaces, which is in large parts governed by the relative energies necessary for bending and stretching, which can be compared with the Karman number

$\gamma = \frac{Y R^2}{\kappa}$