Growth of an elastic body can induce mechanical instability. As an example, consider an Euler beam constrained between two simple supports placed apart at distance ℓ0. Suppose that the beam can undergo growth, in the sense that its natural length ℓ can change. The compressive force in the beam is
P = EA(ℓ0 − ℓ).
Accordingly, the beam buckles when the length reaches the critical value
$$\ell_{\rm crit}=\ell_0+\pi^2\frac{EI}{EA\ell_0^2}.$$
A more interesting instability phenomenon can take place when the evolution law that governs the length of the beam depends on the stress. This type of instability has been discussed in the paper in a broader context, inspired by experiments on actin growth.
The mechanical system studied in consists of a one-dimensional elastic bar shown clamped on one side and constrained at the other side by an elastic device, as shown in the figure.
An elastic bar clamped between a hard and a soft device, immersed in a semi-infinite channel.
Rohan Abeyaratne, Eric Puntel, and Giuseppe Tomassetti. Treadmill stability of a one-dimensional actin growth model. In preparation, 2019.