Nonlinearly elastic beams

In the overdamped regime (i.e. in the absence of inertia), the dimensionless equation of motion for a beam with a small deflection $y(x,t)$ is

\[\begin{aligned}\pd{y}{t}=-\pdd{}{x}\left(E\pdd{y}{x}\right),\end{aligned}\]

where $E$ represents the stiffness of beam, or how difficult it is to bend. Traditionally, this stiffness is taken to be a constant or perhaps to depend on the position $x$.

In the simulation below, we consider a beam with a stiffness that depends on the local curvature, so that

\[\begin{aligned}E = E\left(\pdd{y}{x}\right) = E^\star + \Delta_E\frac{1+\tanh{(\pdd{y}{x}/\epsilon})}{2} \end{aligned}\]

for baseline stiffness $E^\star$, stiffness change $\Delta_E$, and sensitivity $\epsilon$.

We can play with $\Delta_E$ using the slider below. The minimum value corresponds to a beam with constant stiffness, while the maximum value corresponds to a beam with a stiffness that depends strongly on the curvature. A quick exploration highlights that the dynamics of the beam depend significantly on the differential stiffness.

Play with this example in more detail in this customisable simulation.

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