PSI - Issue 59

Svitlana Fedorova et al. / Procedia Structural Integrity 59 (2024) 279–284 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

283

5

 the parameters 2 ( 2,3,...) k c k  do not in any way affect the behavior of the model (16), (5) under stationary loading and should be used to approximate experimental data under two-stage loading: exp s R R [ ( )] x X L   ;  the free parameter does not in any way affect the behavior of model (16), (5) under stationary and stepwise loading and can be used to adapt this model to experimental moments of failure under other types of unsteady loading or to the experimentally observed evolution of damage variable. 4. Natural restrictions In order for the evolutionary equation (16) not to contradict the meaning of the process of damage, the potential must satisfy natural restrictions

2     2 0 dD dF D F

(17)

.

0,

which for potential 2 F (8) taking into account (15) lead to the condition

(18)

2 (1 ...) 1 c c L c L     .

21

22

23

Since the parameters 2 ( 2,3,...) k c k  have already been determined at this stage, this inequality limits the values of the free parameter 21 c and the load L . 5. Application to experimental data Application of this theory to experimental data on low-cycle strength from Golos and Ellyin (1988) (tests 1-24 from Table 3) for a linear neutral function gave 22 . Fig. 1 show the dependences of the PMR defect on the dimensionless load jump moment exp s s R 1 ( ) x X L   for different groups of minimum and maximum load values under two-stage loading. The results for increasing load are denoted by O  , and for decreasing load O  . It can be seen that the experimental values of the PMR defect differ significantly and systematically from zero, which cannot be reflected by factorized damageability functions. These results show that at low load values (Fig. 1a), the linear neutral function (19) underestimates the deviation from the Palmgren-Miner rule. At higher load values (Fig. 1c), the picture is opposite. The quadratic neutral function can improve the result. Natural constraint (18) for function (19) can be formulated as 84,14 c  2 2 21 22 ( , ) (1 ) L c c L    c (19)

min{ (1 84,14 )} min(1 84,14 ) c L c L c   

(1 84,14max{ }) 1 L 

(20)

 

21

21

21

In these experiments max{ } 0,008 L  . Therefore, from the last inequality we obtain a restriction on the free parameter: 21 3,06 c  .