Localized growth of layered tissues

Due to their structural heterogeneity, stratified media usually loose their initial symmetry when they grow. Mismatching expansions between distinct layers create elastic stresses that lead to the emergence of wavy patterns. In the context of morphogenesis, this process of constrained growth has been used to explain patterns in biological soft-layered tissues such as the fingerprints of the skin. Discovery of morphogens has however revealed that growth is not always a homogeneous process and can be highly localized. Spatially concentrated growth processes are thought to be responsible for the formation of various structures, from placodes in the chicken embryo to nevus and skin tumours. On a different ground, the processing of complex textured surfaces requires to understand the connection between the local variation of mass and the resulting surface profile. In this paper, we investigate a model of a locally growing thin sheet bound to a soft infinite substrate. We show that, in two dimensions, bending effects select the shape of the growing sheet, therefore preventing the fabrication of an arbitrary surface. In three dimensions, the high energetic cost of stretching sets a stronger constraint on the attainable patterns and provides a simple relation between the profile of the surface and the local properties of the growth process.