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

1040

4

It is too complicated to measure the parameters in formula(7-9), therefore Neal and Jing developed a method to determine these parameters by transfer the relationship between pressure and specific volume to vN particle velocity.

1 ( ) v v dp dv =

v = −

(7)

c

1 1

2

1 2 ( / ) v d p dp

(8)

= −

a

v v =

1

1

dv

4

dv

2 3 v d p dp

1 1 1 [ ( c

2 3 ] a 1

=

−

(9)

b

/ )

v v =

1

1

3

dv

2 6

dv

1 a , 1 b and 1 c are calibrated from formula (4) and formulas (7-9), then shock-polar for the second shock is obtained. In fact, the vN particle velocity for the second shock could be seen as the two order approximation of Grüneisen EOS. 2 1 1 2 2 nr nr p p D u  = = (10) 2 2 2 1 2 2 2 1 2 2 tan nr nr nr nr u q D q u D  − = − (11) • 1Du2Gamma The relationships between incident shock and the polar are the same as formula(2) and (3). And the EOS of stiffed gas is taken for the second shock in formula(12).  is the adiabatic exponent (or adiabatic gamma) and considered as a constant 2 2 1 (2 1) 1    = − + − − .

2 1 1 q

p p −



2

2 1

(12)



=

−

tan

1

2

2

2

1 1 q p p  − − 2 1 (

) ( 1) 

( 1)  + + − + p p



c

2

2

1

0 0

• 1Gamma2Gamma For the incident shock and Mach stem the shock-polar could be written in formula(13). And the shock-polar for the second shock is the same as formula(12). 2 1 0 0 0 1 2 2 0 0 1 0 1 0 0 0 2 tan 1 ( ) ( 1) ( 1) 2 p p q q p p p p c       − = − − − + + − + (13)

3. Results and Discussion 3.1. Influence of different EOS on Shock-Polar

Beryllium(shorted as Be, initial density 0 1.85 / g cc  = , ( / ) 0.799 1.13 s p u cm s u  = + ,pressure after incident shock 1 1.0 p Mbar = ) is chosen as the interested metal in our study. Shock-polar theory expressions are derived from the pressure vs. streamline turning angel measured in the moving reference frame of the intersection point. Figure 2(a) shows the shock-polar diagram for the case of incident angle 0 35  = .It is regular reflection in this case, and result with 1Du2Du is much closer to that of 1Du2Grüneisen compared to the results of 1Du2Gama. For the case of 0 45  = in Figure 2(b), shock-polar diagram presents four types of intersection points by the Mach stem-polar and the reflected shock-polar. It is irregular reflection in this case. For the results taking EOS combinations of 1Du2Gama and 1Gama2Gama, the intersection points are on the left, and those are Mach reflections. For the results taking EOS combinations of 1Du2Du and 1Du2Grüneisen, the intersection points are on the right, so these are Von Neumann reflections.