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We derive a model for the optimization of the bending and torsionalrigidities of non-homogeneous elastic rods. This is achieved by studying asharp interface shape optimization problem with perimeter penalization, thattreats both rigidities as objectives. We then formulate a phase fieldapproximation of the optimization problem and show the convergence to theaforementioned sharp interface model via $\Gamma$-convergence. In the finalpart of this work we numerically approximate minimizers of the phase fieldproblem by using a steepest descent approach and relate the resulting optimalshapes to the development of the morphology of plant stems.

Phase field model for multi-material shape optimization of inextensible rods