Many aquatic animals propel themselves efficiently through the water by oscillating flexible fins. These fins are, however, not homogeneously flexible, but instead their flexural stiffness varies along their chord and span. Here, we develop a simple model of these functionally graded materials where the chordwise flexibility of the foil is modeled by one or two torsional springs along the chord line. The torsional spring structural model is then strongly coupled to a boundary element fluid model to simulate the fluid–structure interactions. We show that the effective flexibility of the combined fluid–structure system scales with the ratio of the added mass forces acting on the passive portion of the foil and the elastic forces defined by the torsional spring hinge. Importantly, by considering this new scaling of the effective flexibility, the propulsive performance is then detailed for a foil with a flexible hinge that is actively pitching about its leading edge. The scaling allows for the resonance frequency of the fluid–structure system and the bending pattern of the propulsor to be independently varied by altering the effective flexibility and the location of a single torsional spring along the chord, respectively. It is shown that increasing the flexion ratio, by moving the spring away from the leading edge, leads to enhanced propulsive efficiency, but compromises the thrust production. Proper combination of two flexible hinges, however, can result in a gain in both the thrust production and propulsive efficiency.

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