PSI - Issue 28

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

2177

4

the force at the right end face Р 2 (t) is caused by the impulse 

T (t) . Then, taking into account Hooke's law (the bars

have a high elastic limit and deform elastically):   1 ( ) ( ) ( ) I R P t EA t t     ,

(8)

2 ( ) ( ) T P t EA t   ,

(9)

where E and A are Young's modulus and a cross-sectional area of the bars, respectively. The average force P average is:

1 2 P t P t  2 ( ) ( )

.

(10)

P



average

Hence the average value of the engineering stress in the specimen:

average P EA





.

(11)

( ) t

( ) t

( ) t   R 

( ) t

I 

T

s 







2 A A

0

0

Here А 0 – the initial cross-sectional area of the specimen. As already noted, the stress state of the specimen, due to its short length and long duration of the loading pulse, is almost uniform. Therefore, with sufficient accuracy, we can assume that the forces at the ends of the specimen are equal. From here follows the relation expressing the basic assumption of this method. ( ) ( ) ( ) I R T t t t      . (12) Substituting this expression in (6), (7) and (11), we obtain simple formulas for calculating the stress, strain, and strain rate in the specimen, which are usually used in research practice:

( ) T EA t A

,

(13)

( )

s t



  

0

t

0 0 C t dt L    2 ( ) R

,

(14)

( )

s t



 

2 ( ) R C t

.

(15)

( )



s t



   

L

0

The stress equilibrium condition evaluated by considering the stress histories at both ends of the specimen, assuming that both bars are made of the same material and have the same cross-sectional area (Zhang and Zhao (2014), Hassan and Wille (2017)):

    P t P t P t P t       1 2

    t t

    t t R     R  

    t t

 

I   I

T

P t



 

 

.

(16)

100% 2   

100% 2   

100%

R t





 

T

average P t

1

2

Better results are achieved the smaller R(t) is.