PSI - Issue 28

A.M. Bragov et al. / Procedia Structural Integrity 28 (2020) 2174–2180 Author name / Structural Integrity Procedia 00 (2019) 000–000

2176

3

From the one-dimensional theory of propagation of elastic waves in semi-infinite bars, it is known that the deformation in the wave  (t) is related to the mass velocity dU/dt by a simple relation: ( ) 1 t dU C dt    , (1) where the displacement U(t) of particles in the wave:

( ) t U t C t dt     . ( )

(2)

0

The scheme of specimen loading in the SHPB set-up is shown in figure 1. Based on formula (2), it can record the movement of the ends of the bars 1 and 2 adjacent to the specimen. The movement of the left end face U 1 (t) is the sum of the movement U I 1 (t) caused by the propagation of the impulse ε I (t) and the movement U R 1 (t) caused by the propagation of the impulse ε R (t) :   1 0 0 0 ( ) ( ) ( ) ( ) ( ) t t t I R I R U t C t dt C t dt C t t dt                      . (3) When writing formulas for compressive pulses (incident  I (t) and transmitted  T (t) ), for which the particle mass velocity vector coincides with the direction of wave propagation, a plus sign is assigned. For the reflected tensile pulse  R (t) , the vectors of the particle mass velocity and the wave propagation velocity are oppositely directed and, when calculating the displacements, the reflected momentum must be taken with a minus sign. The movement of the right end face U 2 (t) is caused by the propagation of the impulse  T (t) :

2 t T U t C t dt     . ( ) ( )

(4)

0

The average engineering strain of the specimen of length L 0 will be equal to:

( ) U t U t  ( )

(5)

( ) t

s 



1

2

L

0

or, if it is expressed through impulses in the bars:

t

C t





I

R

T



.

(6)

( ) t

( )

( ) t    ( )  

t dt

s 





L

0 0

dt t d s s / ( )     :

Hence, the strain rate of the specimen

I C t





.

(7)

( )

( )

( ) t   R 

( ) t

T



s t



 





L

0

To find stress in the specimen, the forces at its ends should be considered. The force at the left end face Р 1 (t) is the sum of the compressive force P I 1 (t) caused by the impulse  I (t) and the force P R 1 (t) caused by the impulse  R (t) , and