  Spontaneous material shape changes, such as swelling, growth or thermal
expansion, can be used to trigger dramatic elastic instabilities in thin
shells. These instabilities originate in geometric incompatibility between the
preferred extrinsic and intrinsic curvature of the shell, which may be modified
by active deformations through the thickness and in plane respectively. Here,
we solve the simplest possible model of such instabilities, which assumes the
shells are shallow, thin enough to bend but not stretch, and subject to
homogeneous preferred curvatures. We consider separately the cases of zero,
positive and negative Gaussian curvature. We identify two types of
super-critical symmetry breaking instability, in which the shell's principal
curvature spontaneously breaks discrete up-down symmetry and continuous planar
isotropy respectively. These are then augmented by inversion instabilities, in
which the shell jumps sub-critically between up/down broken symmetry states,
and rotation instabilities, in which the curvatures rotate by 90 degrees
between states of broken isotropy without release of energy. Each instability
has a thickness independent threshold value for the preferred extrinsic
curvature proportional to the square-root of Gauss curvature. Finally, we show
that the threshold for the isotropy-breaking instability is the same for deep
spherical caps, in good agreement with recently published data.
