 Thermoelastic Maxwell relation for a stretched band

Let $fdx$ be reversible work supplied to stretch a band. Its Helmholtz free energy obeys $dF=-SdT+fdx$. Equality of mixed derivatives gives the displayed relation. An entropy decrease on stretching therefore corresponds to increasing tensile force on heating at fixed extension. The sign follows the force/work convention; a restoring force exerted by the band has the opposite sign.