PSI - Issue 13

Xin Yu et al. / Procedia Structural Integrity 13 (2018) 1037–1042 X. Yu, X. Huang and M. Zheng / Structural Integrity Procedia 00 (2018) 000 – 000

1039

3

There is no simple or experimental or empirical formula to describe the relationship of the second shock or the reflected shock. Four different combinations of EOS are discussed here, of which 1Du2Grüneisen inidicates that the principle shock wave taking vN particle velocity relationship and the second shock taking the EOS of Grüneisen, 1Du2Du indicates both the principle and the second shock taking vN particle velocity relationships, 1Du2Gamma indicates that the principle shock taking vN particle velocity and the second shock taking the EOS of stiffed gas, and 1Gamma2Gamma indicates that the principle and the second shock are both taking the EOS of stiffed gas. It is known that the regular parameters of solids are measured on the theoretical basis of Grüneisen. So, the results from 1Du2Grüneisen are thought to be consistent to the experimental results and are used as reference values to compare the difference with the other three combinations of EOS in the subsequent theoretical analysis. 2.2. Shock-polar Diagram • 1Du2Grüneisen The relationship between incident shock and the polar is in formula(2) and (3). Here 0 p and 1 p is the pressure before and after the incident shock, and 0 0 p = . 0  and 0 q are density and the incoming velocity before shock in a reference system consolidated on the shock wave. 1 D and 1 u is the incident shock velocity and the particle velocity after the shock. 1  is the turning angle between incoming flow and the streamlines after shock. 1 0 0 1 1 p p D u  = + (2) 2 2 1 0 1 1 2 0 1 1 tan u q D q u D  − − (3) For the second shock, Brown and Ravichandran derived the relationship between the pressure and the specific volume as seen in formula(4). 0 v is the initial specific volume, 2 v and 2 p are the specific volume and the pressure after the shock, and the  is the material parameter of Grüneisen  . 0 0 / / v v  =  and 0 2 2/3   = − are considered here for convenience to deduce the turning angle between the streamlines before and after the reflection shock 2  in formula(5). 1 v and 1 q is the specific volume and the pressure before the reflected shock in a reference system consolidated on it. Then formula(4) and (5) compose the shock-polar of the reflected shock.

2

c





v v

v v

1 2

1 2

0

2 0

2 0

)]/[1 (1 )] 

− −

v v −

− − +

( p v v −

(1 )[1

(

)

0 2

1 0 2

v

v

v

0

=

p

(4)

2



1 1 ( −

v v −

)

1 2

v

2

p p −

2 2 1

)( − − − ) (

2 1 1 2 p p v v

v

(

)

1

q

1

(5)

=



tan

2

2 1 p p q v q − − 1 1

1

• 1Du2Du The relationships between incident shock and the polar are the same as formula(2) and (3). And that for the second shock can be written in formula(6). Here 1 a , 1 b and 1 c are under determination parameters. 2 nr D and 2 nr u are the reflected shock speed and the particle velocity relative to 1 n u , and 1 n u is the component particle velocity in the direction perpendicular to the reflected shock, so 2 2 1 nr n n D D u = − and 2 2 1 nr n n u u u = − . 2 2 1 1 2 1 2 nr nr nr D c a u b u = + + (6)