5 The solution considered in the previous section for dynamic tensile along the center-line of the sphere can be used to estimate the dynamic mode I stress intensity factor of a vertical microcrack as shown in Fig. 1 (Deng and Nemat-Nasser [13]; Freund [8]): 1 2 1 ( ) 1 - ÷ ÷ ø ö ç ç è æ ÷ - ÷ ø ö ç ç è æ p - = s R R hoop ID c l c l t a K & & (17) where ID K is the dynamic stress intensity factor, ( )t hoop s is the maximum hoop stress along the center-line of the axis of compression obtained in the previous section, l& is the velocity of crack growth, a is the half size of the microcrack and Rc is the Rayleigh wave speed. It has been assumed that the microcrack is relatively small comparing to the size of the sphere such that local tensile field can be considered as a far field uniform stress. The speed of crack growth can be non-uniform. We should also emphasized that the speed of crack growth should not exceed the Rayleigh wave speed. It can be shown that the speed of crack growth can be determined as (Deng and Nemat-Nasser, [13]) IC IS IC IS R K K K K l c 2 1 - - =& (18) where IS K is the static mode I stress intensity factor and IC K is the dynamic fracture toughness. Figure 1: Dynamic growth of vertical microcrack in sphere under double impact test. Chau et al. (2000) showed that the dynamic energy required for fragmentation can be estimated as 1.5 times of that required for static compression. Figure 2 was extracted from Chau et al. (2000). FUTURE WORK TO BE DONE In the case of double impact test, the present approach described needs to be combined with the dynamic motion of the drop weight attached to the upper platen. The application of the contact force during the dynamic impact needs to be evaluated by applying Newton second law to the falling rigid upper platen. (t) P(t)
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