Higher harmonics in tapping-mode atomic force microscopy offers the potential for imaging and sensing material properties at the nanoscale. The signal level at a given harmonic of the fundamental mode can be enhanced if the cantilever is designed in such a way that the frequency of one of the higher harmonics of the fundamental mode (designated as the resonant harmonic) matches the resonant frequency of a higher-order flexural mode. Here we present an analytical approach that relates the amplitude and phase of the cantilever vibration at the frequency of the resonant harmonic to the elastic modulus of the sample. The resonant harmonic response is optimized for different samples with a proper design of the cantilever. It is found that resonant harmonics are sensitive to the stiffness of the material under investigation.