Optimizing a fin ray for stiffness

We optimize the elastic constants of the supporting structure in flexible fish fins—the fin ray—to obtain the least deflection under loading. We first solve the problem numerically for rays with shear layers of uniform thickness. The optimal distributions of bending and shear moduli are nonzero on disjoint regions. The numerical solution suggests the form of the analytical solution, which we obtain using calculus of variations on two intervals with continuity conditions at the interface. The deflection of the optimal ray is less than half that of the uniform ray. When the shear layer thickness is allowed to vary, we find a tapered thickness distribution which gives zero deflection for a given positive loading.