# What is the mathematical formulation for buckling?

Argument: Buckling is an engineering concept that can only be applied to thin columns with compressive loading.

(Is it possible to) Prove the above sentence right or wrong with mathematical formulation. Emphasis on cellular and solid materials that have no concept of "thin".

Also, why can you use buckling to describe crack growth experiments in thin sheets: out-of-plane deformation of sheets = buckling? => deformation in perpendicular direction respect to force is buckling? => 3D solids can experience buckling ? => example of this kind of material is...

Good references, rigorous treatment and mathematical approach are more than welcome.

edit: In other words, what is the mathematical definition for buckling?

edit2: So, buckling is the bifurcation of static equilibrium (see annav's comment below). And thus:

More technically, consider the continuous dynamical system described by the ODE

$\dot x=f(x,\lambda)\quad > f:\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}^n.$

A local bifurcation occurs at $(x0,λ0)$ if the Jacobian matrix $\textrm{d}f_{x_0,\lambda_0}$ has an eigenvalue with zero real part.

This also happens to coincide with ASTM E-9 standard, section 3.2.1 that says:

bucklig -- (3) a local instability, either elastic or inelastic, over a small portion of the gage length

• Buckling is an engineering concept – Georg Oct 31 '11 at 12:25
• @Georg: How about "only to thin columns" part? What about sheets? – Juha Oct 31 '11 at 12:29
• It's not just O-shaped profiles that can collapse, consider a C- or L-shaped profile. The math is of course harder since they're asymmetric. – MSalters Oct 31 '11 at 12:35
• I am amused by this question, because buckling is used as an analogue for symmetry breaking in particle physics . Look under "other examples" en.wikipedia.org/wiki/Spontaneous_symmetry_breaking "buckling". This, though – anna v Oct 31 '11 at 13:28
• Try this en.wikipedia.org/wiki/Buckling . It seems Euler devised a formula for load and buckling danger. Also some people propose improvements :books.google.gr/… – anna v Oct 31 '11 at 14:42

So, buckling is the bifurcation of static equilibrium. And thus:

More technically, consider the continuous dynamical system described by the ODE

$\dot x=f(x,\lambda)\quad > f:\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}^n.$

A local bifurcation occurs at $(x0,λ0)$ if the Jacobian matrix $\textrm{d}f_{x_0,\lambda_0}$ has an eigenvalue with zero real part.

This also happens to coincide with ASTM E-9 standard, section 3.2.1 that says:

bucklig -- (3) a local instability, either elastic or inelastic, over a small portion of the gage length

In plain english, the above mathematical formula implies: At time t, when constantly increasing a load to a column, the position of the column has more than one solution (e.g. the column bends right or left). Before time t, there is only one solution for the position.

Buckling is not limited to thin columns, it is also important, e.g., for thin shells under compression; for example, if pressure in a poorly designed tank is below atmospheric pressure, the tank can buckle under atmospheric pressure (it happens to large oil tanks, railway car tanks - you name it; you can easily find a lot of impressive images on the net).