The role of strain gradients in continuum mechanics has been studied. Distinction is made between stabilizing and destabilizing gradients. A combined model is proposed to investigate the interaction of the two effects. Dispersion analysis and numerical simulations are used to compare the various formats of the model in a qualitative and a quantitative manner. It is shown that a destabilizing second-order gradient has a devastating effect on the numerical response, even if a stabilizing second-order gradient is present. For this class of models only the absence of destabilizing terms guarantees convergent numerical responses.