A simple macroscopic theory of elastic rods is presented in which all assumptions but one are consistent with both Newtonian mechanics and special relativity. The one distinguishing assumption is the inertial equivalence of energy. While invariance of the theory under Lorentz transformations is proved, all physical consequences—including the stress and velocity dependence of the length and inertial mass of a rod as well as the velocities of sound through it—are derived and can be tested in any one inertial frame. Exact wave solutions of the basic equations are obtained for an idealized elastic material in which the velocity of sound is independent of amplitude. These solutions are used to account for the kinematics and dynamics of accelerated rods, including the time‐dependent processes which result in their overall Lorentz contraction.