  We investigate energetically optimal configurations of thin structures with a
pre-strain. Depending on the strength of the pre-strain we consider a whole
hierarchy of effective plate theories with a spontaneous curvature term,
ranging from linearised Kirchhoff to von K\'arm\'an to linearised von
K\'arm\'an theories. While explicit formulae are available in the linearised
regimes, the von K\'arm\'an theory turns out to be critical and a phase
transition from cylindrical (as in linearised Kirchhoff) to spherical (as in
von linearised K\'arm\'an) configurations is observed there. We analyse this
behavior with the help of a whole family $(\mathcal{I}^{\theta}_{\rm
vK})_{\theta \in (0,\infty)}$ of effective von K\'arm\'an functionals which
interpolates between the two linearised regimes. We rigorously show convergence
to the respective explicit minimisers in the asymptotic regimes $\theta \to 0$
and $\theta \to \infty$. Numerical experiments are performed for general
$\theta \in (0,\infty)$ which indicate a stark transition at a critical value
of $\theta$.
