  We derive a model for the optimization of the bending and torsional
rigidities of non-homogeneous elastic rods. This is achieved by studying a
sharp interface shape optimization problem with perimeter penalization, that
treats both rigidities as objectives. We then formulate a phase field
approximation of the optimization problem and show the convergence to the
aforementioned sharp interface model via $\Gamma$-convergence. In the final
part of this work we numerically approximate minimizers of the phase field
problem by using a steepest descent approach and relate the resulting optimal
shapes to the development of the morphology of plant stems.
