PSI - Issue 59

Olena Romashko-Maistruk et al. / Procedia Structural Integrity 59 (2024) 352–359 O. Romashko-Maistruk and V. Romashko / Structural Integrity Procedia 00 (2023) 000 – 000

356

5

Thus, under a standardized (quasi-static) loading mode (Fig. 3), this energy is proposed to be calculated on the basis of an incorrect fractional-rational function according to expression (4):

   

   

 o c 

   

   

   

   

   

f

dV dU



1

1

 o  c





2

2 c ck 1

c

1

2

u

d

E

  k

d









 







1

c c

co c

c

c





1

c



2

   

   

 2

f

2 1

1

2 1

k

k

 







  

 

1

ck c

ln( 1) ,  k



 



 







2

2

k

k

k





(4)

where c  - current strains of compressed concrete; ck f and corresponding concrete critical strains under the standardized static loads action;

1 c  - the strength of compressed concrete and the

ck со с f k E 1    - characteristics

of compressed concrete deformability. The hypothesis of the invariance of the concrete ultimate deformation (destruction) potential energy and its independence from the concrete loading mode (Romashko and Romashko (2019), Romashko and Romashko Maistruk (2022)) allows to equate expressions (3) and (4).

 с

f с,du

2

u 2

1

f ck

u 1

 с

 с1

 сe,u

Fig. 3. Diagrams of the concrete potential energy under loading ultimate deformation of: 1 – standardized short-term; 2 - instantaneous dynamic.

Based on formula (3), the limit values of the compressed concrete under instantaneous application of dynamic loads strength can be calculated using the expression:

2 E u co  

f



(5)

.

, c du

2

Then the function of the limit values of the compressed concrete dynamic increase factor, taking into account dependence (4), will take the following form:





2

   

   

 2

f

2

2 1

1

2 1

k

k

k

 





  

  

, ck c du f

.

(6)

 ln( 1) k

DIF

 

 









u

2

2

k

k

k





1  DIF at

6 1 10    s   ;

Assuming the following boundary conditions for the concrete dynamic increase factor:

3 1 10   s   , based on the numerical analysis of the experimental studies results Cowell (1966) and

u DIF DIF  at

Kono et al. (2001), the dependence of DIF on the strain rate of compressed concrete was obtained: