Buckling of a sheet

The Föppl–von Kármán (FvK) equations in 2D capture the buckling of a sheet/strip/beam. The morphology is described by the in-plane displacement u(x) and the out-of-plane displacement w(x). The governing equations are

\begin{aligned} S (u_{xx} + w_{x} w_{xx}) + f =& \ 0, \\ -B w_{xxxx} + S [ w_{x} u_{xx} + \tfrac{1}{2} w_{xx} (2 u_{x} + 3 w_{x}^2) ] + p =& \ 0. \end{aligned}

Here S = E\xi is the stretching modulus, B = EI the bending modulus, f and p external forces per unit length.

Non-dimensionalising with body length L and Young's modulus E as scales (using S \sim E\xi,\ B \sim E\xi^3):

\begin{aligned} (d_{ss} + h_{s} h_{ss}) + \mathsf{f} =& \ 0, \\ -K h_{ssss} + h_{s} d_{ss} + \tfrac{1}{2} h_{ss}(2 d_{s} + 3 h_{s}^2) + \mathsf{p} =& \ 0, \end{aligned}

where K = (\xi/L)^2 is the von Kármán number and \mathsf{f}, \mathsf{p} are non-dimensional forces. Boundary conditions: fixed ends d|_0 = h|_0^1 = 0; zero moments h_{ss}|_0^1 = 0; applied tangential strain d_s|_1 = -\alpha.

The leverage of the .auto script is that we can steer the continuation by varying different parameters. After branch-switching with ISW=-1 to the buckled branch, we can change the continuation parameter to K and then to both (K,\alpha) simultaneously using ICP and UZSTOP. The constraint NCONT = NBC + NINT - NDIM + 1 (NBC=6, NDIM=6) requires one integral constraint for two-parameter continuation. The result is isocontours in the (K,\alpha)-plane representing distinct morphologies. Code is in Tutorial2_FvK.