In this work, we subject elastomers to a fixed pre-stretch in uniaxial extension, ε p , upon which a large-amplitude, ε 0 , oscillatory uniaxial extensional (LAOE) deformation is superposed. We find that if both ε p and ε 0 are large enough, the stress responds with a rich set of higher harmonics, both even and odd. We further find the Lissajous-Bowditch plots of our measured stress responses versus uniaxial strain to be without two-fold symmetry, and specifically, to be shaped like convex bananas. Our new continuum uniaxial model for this behavior combines a new nonlinear spring, in parallel with a Newtonian dashpot, and we call this the Voigt model with strain-hardening. We consider this three-parameter (Young's modulus, viscosity and strain-hardening coefficient) model to be the simplest relevant one for the observed convex bananas. We fit the parameters to our both uniaxial elongation measurements at constant extension rate, and then to our LAOE measurements. We develop analytical expressions for the Fourier components of the stress response, parts both in-phase and out-of-phase with the extensional strain, for the zeroth, first, second and third harmonics. We find that the part of the second harmonic that is out-of-phase with the strain must be negative for proper banana convexity.