Morphoelastic rods Part II: Growing birods

The general problem of determining the shape and response of two attached growing elastic Kirchhoff rods is considered. A description of the kinematics of the individual interacting rods is introduced. Each rod has a given intrinsic shape and constitutive laws, and a map associating points on the two rods is defined. The resulting filamentary structure, a growing birod, can be seen as a new filamentary structure. This kinematic description is used to derive the general equilibrium equations for the shape of the rods under loads, or equivalently, for the new birod. It is shown that, in general, the birod is not simply a Kirchhoff rod but rather, due to the internal constraints, new effects can appear. The two-dimensional restriction is then considered explicitly and the limit for small deformation is shown to be equivalent to the classic Timsohenko bi-metallic strip problem. A number of examples and applications are presented. In particular, the problem of two attached rods with intrinsic helical shape and uniform growth is computed in detail and a host of new interesting solutions and bifurcations are observed.