The electron viscosity is popular in the dynamic deformations of metals, and it was revealed to dominate the related energy dissipation at low temperatures. The free electron model was extensively utilized to investigate the electron viscosity of the related phenomena including electron viscosity of mobile dislocations and the attenuation coefficient of elastic waves at low temperatures. However, the potential energy of the "free" electrons was neglected totally. In this work, the mechanical-electric coupling which contains both the potential energy and kinetic energy of "free" electrons was taken into account. And it was found that the attenuation coefficients of the longitudinal and shear waves of metals at cryogenic temperatures are proportional to the electrical conductivity and the square of angular frequency, which was in accord with experimental observations. The longitudinal and shear waves in a metal was found to induce the electromagnetic radiation whose frequency is the same as the stress wave. In addition, the electron viscosity was discovered to result in a temperature increase over the compression wave front. The temperature increase depends on the strain gradient, and a larger strain gradient may lead to a larger temperature rise during the compression wave front. Furthermore, the electron viscosity of the mobile edge and screw dislocations was obtained in theory. And the order of calculated magnitude in terms of the mechanical-electric coupling strength that can be determined by the attenuation coefficient of the longitudinal and shear wave agrees with experimental results. Overall, the revealed important effects of the electron viscosity for the dynamic deformations of metals were investigated and the obtained findings may aid in understanding the related phenomena deeply.