Stress-free configurations induced by a family of locally incompatible growth functions

This paper studies stress-free deformations induced by growth within a two dimensional (2D) setting. With the use of the polar decomposition of the growth tensor and properly chosen base vectors (or curvilinear coordinates) in the reference and current configurations and the virtual stress-free state, we derive a constraint condition on the growth functions in a simple form, together with the system of equations governing the current configuration and the elastic rotation. For a special case, this system is solved analytically and a family of growth functions with three parameters are obtained. Remarkably, by adjusting the initial configurations as well as these three parameters, five types of morphologies are identified, which share similarities with five biological patterns, including the growth patterns of ferns, walnut shells, human infants’ skulls, stems of certain bean plant and the leaves of water lily. It is worth mentioning that the analytical results are established purely through local kinematics, not depending on specific energy density functions and initial geometries.