Energy Minimising Configurations of Pre-strained Multilayers

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 Karman to linearised von Karman theories. While explicit formulae are available in the linearised regimes, the von Karman theory turns out to be critical and a phase transition from cylindrical (as in linearised Kirchhoff) to spherical or saddle-shaped (as in linearised von Karman) configurations is observed there. We analyse this behaviour with the help of a whole family(IvK theta)theta is an element of(0,infinity)of effective von Karman functionals which interpolates between the two linearised regimes. We rigorously show convergence to the respective explicit minimisers in the asymptotic regimes theta(R)0and theta(R)infinity Numerical experiments are performed for general theta is an element of(0,infinity)which indicate a stark transition in a critical region of parameters theta