The dynamic response of a long viscoelastic bar due to a step displacement at the end is considered. Neglecting geometric dispersion, the effect of realistic viscoelastic material properties is studied theoretically. The solution is obtained in the form of a Fourier sine integral, the convergence of which is studied numerically by piecewise integration to produce an alternating series. It is found that the initial step wave propagates with a high velocity corresponding to the glassy modulus of the material and its amplitude decays with time and distance along the rod. From a practical viewpoint the wave front may decay to immeasurable proportions and any measurable disturbance appears to travel thereafter, with a velocity which is smaller than the glassy wave speed. The effect of initial temperature is discussed. It is shown for thermorheologically simple materials that both the time and spatial variable are scaled by the same temperature dependent (shift) factor. As a consequence, the difference of wave propagation in hard and viscoelastic polymers is illustrated. It is also shown that limited material characterization is sufficient for certain dynamic problems. Comparison of the exact solution with two approximations is made.
Viscoelastic Material Characterization: A Realistic Approach to Stress Analysis