Dynamic response of an elastic sphere under diametral impacts

This paper presents a new analytical solution for an elastic sphere under a double impact load which applies as a Heaviside step function of time along a diameter. The suddenly applied loads are modeled as either uniform or Hertz contact stress and are applied through two patches on two ends of a diameter of the solid sphere. The method of solution uses decomposition theory, in which the final dynamic solution is decomposed into two auxiliary problems: (I) a static solution of the applied loading; and (II) the free vibration of the sphere subject to an initial deformed shape induced by the auxiliary problem I. Time evolutions of the stress field at selected points along the axis of compression agree with the solution by [Jingu, T., Nezu, K., 1985. Transient stress in an elastic sphere under diametrical concentrated impact loads. Bulletin JSME 28 (245), 2553–2561] when contact zone is very small (i.e. case of suddenly applied point loads). If energy loss is allowed at each boundary reflection, our long term solution converges to the static solution. The size and shape of the compressive cones under the applied loads are relatively insensitive to the size of the contact zone, while the magnitude of tensile stress along the axis of compression decreases with the contact area. The maximum tensile hoop stress along the axis of compression always appears at point r/a = 0.38 and at time of 2.61T1 (where 2T1 is the time for P-wave traveling across the sphere). Based on wave interference plots, inferred failure patterns are proposed and compared to those observed failure pattern in our double impact test.