A system can change shape. In biology, shape changes lead to the wonderful variety of forms that surround us; in engineering, they are a tool to design soft robots and deployable structures. In this thesis, we identify and explore three fundamental paths that lead to shape change: elastic instabilities, spontaneous deformations, and the combination of these two.

Classic elastic instabilities occur when a progressive load is applied to a system which initially responds progressively, but at a critical threshold suddenly changes response and, importantly, shape. Here, we study a classic example of large strain instability — ballooning – and the process that leads to longitudinal phase separation in solids. We show that near a critical point, phase separation is described by a universal and simple energy which allows us to understand shape evolution analytically.

Typical examples of the second path towards shape change — via spontaneous defor- mations — are biological growth or muscle contraction. They sculpt the shape of flowers or power the beating of a heart. Outside of the realm of biology, the swelling of gels and the anisotropic deformations in liquid crystal elastomers (LCEs) are two excellent examples of spontaneous deformations that can be patterned, allowing scientists to program shape changes, mimic biology and design soft robots. A typical example of patterned deformation is the morphing of flat sheets into curved surfaces. Here, we show that an LCE sheet can be morphed into a variety of shapes in a reprogrammable way by patterning the activation strength of the LCE via light modulation. Moreover, we show that patterned shape changes are not limited to 2D and discuss the origin of the 3D deformation that causes twist and contraction in LCE fibres.

Finally, we recognise the last and mechanically most rich path towards shape change where elastic instabilities and spontaneous deformations combine. This new mode of defor- mation underpins, for instance, patterns in biology such as the corrugations on the mammal brain or the folding of pollen grains. In the case of soft active materials, we show that the ballooning instability and phase separation can be triggered and amplified by the spontaneous deformation of LCEs; that a coiling instability can be induced in spontaneous twisting fibres; and discuss how curvature loads govern the instabilities in thin shells.