PSI - Issue 6

Y.V. Petrov et al. / Procedia Structural Integrity 6 (2017) 134–139 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

137

4



c



T

,





   

2





  

T T  

1

* x c T    

T

T

(6)

,



  





2

T T

2

 



   

2





c T T 

1

T





2

2

Thus, for the pulse under consideration, expression (5) determines the smallest time at which the criterion condition (1) is violated, and in (6) the coordinate of the corresponding cross section is given. At subsequent times, the region where this condition is violated will propagate along the material from the point with the * x with the wave speed c in the direction of decreasing x to the section with the coordinate min / 2 x cT  , if T   , and   min 1 2 / 2 x c T T T      if 2 T   . In the case when 2 T T    the condition of destruction in the direction of decrease x will not spread. In the direction of increase x , the destruction condition will propagate for all durations of the applied pulse. This advance will be limited by the size of the sample and the conditions on the right boundary. One of the features of failure under dynamic loading is the possibility of applying a load exceeding the critical load, i.e. minimally necessary for tearing material. Exceeding the applied load above the critical one can be interpreted as overload. Destruction in this case can already occur with different values of the time of discontinuity and the coordinates of the spall section. In the case of an overload, analytic expressions similar to (5) and (6) can also be obtained for the trapezoidal profile of the initial pulse under consideration, but they are too cumbersome to bring in this note. Fig. 1 a-c shows the corresponding dependence of the fracture time on the coordinate of the sample cross section for various pulse durations and overloads. In this case, for a solid line, there is no overload (threshold pulse), dotted - there is an overload of 2 times, a point-by-turn - 10 times. In Fig. 1a T   ( 3 / 4 T   ), in Fig. 1b 2 T T    ( 2 5 / 4, 5 /8 T T T    ) and finally, in Fig, 1c - 2 T   ( 2 9 / 4, 5 /8 T T T    ).The coordinate of the section and the time are normalized. It can be seen that the action of an "overloaded" pulse leads to the fact that the destruction condition occurs simultaneously in some zone of the sample. This zone is wider the higher the overload. Thus, it can be expected that under real pulse loading during experiment in the material may arise an area in which simultaneously (or almost simultaneously) the destruction condition is realized.

4. Comparison with the experiment on the formation of spall sections

In the work of Kubota et al. 2008, experiments were conducted to study the fracture zone in the case of spallation. Sand material (Kimachi sandstone) of cylindrical shape with a diameter of 60 mm and a length of 300 mm was used for samples material. Its density was  =2018 kg/m 3 , longitudinal wave velocity was c  2742 m/s,

Fig. 1. Dependence of the fracture time on the coordinate of the section for trapezoidal pulses.