PSI - Issue 13

Ramdane Boukellif et al. / Procedia Structural Integrity 13 (2018) 85–90

90

6

Boukellif et al. / Structural Integrity Procedia 00 (2018) 000–000

3.2.4. Parameter identification depending on strain measurements with presumed errors In order to investigate the sensitivity of the results to disturbances in the measured strains, random errors in the ”measured” strain components have been simulated. An inclined center crack in a finite plate (20 mm x 20 mm) is used for this investigation, see Fig. 2 (c) . Tab. 10 shows the results of the sensitivity analysis based on presumed measurement errors and demonstrates the sensitivity towards inaccurate strain determination. The results show that the external loads are not sensitive to inaccurate strain measurements. In contrast, crack length, inclination and position show non-negligible deviations starting from ± 5% assumed error in strains. Stress intensity factors are well reproduced up to ± 10% of assumed errors, then particularly K II becomes useless.

Table 10: Results of parameter identification depending on strain measurement with presumed errors.

parameters

given

identified presumed error in measured strain

0%

± 1%

± 3%

± 5%

± 7%

± 9%

± 11%

± 13%

( x c , y c ) = (20; 20) mm

¯ σ xx [MPa] ¯ σ yy [MPa] ¯ σ xy [MPa]

30 90 20 60

30 90

29 , 87 89 , 83 20 , 03 59 , 30 10 , 05 10 , 04 96 , 90 55 , 90 3 , 06

30 , 32 90 , 81 19 , 98 61 , 18

29 , 38 88 , 83 19 , 65 61 , 47

29 , 25 87 , 62 19 , 61

28 , 89 87 , 09 19 , 46 59 , 17

28 , 83 85 , 48 19 , 35 67 , 39

30 , 41 89 , 14 19 , 96 92 , 60

19 , 999 59 , 999

α [ ◦ ]

62 , 8 3 , 42

a [mm] x 0 [mm] y 0 [mm]

3

2 , 99 9 , 99

3 , 03 9 , 99

3 , 5

3 , 38

4 , 07

0 , 52 5 . 97

10 10

10 , 85

11 , 52

12 , 35

11 , 35 10 , 04

10

10 , 46 95 , 01 49 , 18

9 , 21

9 , 62

9 , 21

6 , 0

K I ( + ) [%] K I I ( + ) [%]

94 , 63 52 , 60

94 , 63 52 , 59

101 , 42

97 , 55 46 , 26

102 , 94

103 , 97

41 , 61

52 , 55

60 , 14

27 , 19

− 28 , 96

4. Conclusions

In this work, the concept of distributed dislocations is applied for the detection of cracks and the calculation of SIFs in finite and semi-infinite plane structures. The crack and loading parameters could be successfully determined by the solution of the inverse problem. The sensitivity analysis has shown that the positions of the strain gauges have no significant influence on the solution of the inverse problem, as long as they are not too close to the crack. The deviation of the results when the strain gauges are close to the crack may be due i.a. to the genetic algorithm, since the number of collocation points in the genetic algorithm is chosen as small as possible to minimize the time for identification. On the other hand, increasing the number of measurement points could improve the results. The assumed errors in the strains had almost no influence on the identified boundary loads. In contrast, the crack parameters are more sensitive. The quality of parameter identification thus depends on the accuracy of the measured strains where errors should not considerably exceed ± 5%. For this reason it is very important in real experiments to reduce errors in measurements of strains. It is interesting to see in the presented approach that even very small cracks in large plates with remote gauges can be identified accurately. Bilby, B. A. , Eshelby, J. D., 1968. Dislocations and the Theory of Fracture, in Fracture, Ed. H. Liebowitz. Academic Press, New York, 1968. Dundurs, J., Mura, T., 1964. Interaction between an edge dislocation and a circular inclusion. Journal of the Mechanics and Physics of Solids 12, 177–189. Erdogan, F., Gupta, G. D., Cook, T. S., 1973. Numerical solution of singular integral equations. In Methods of Analysis and Solutions of Crack Problems, (1973, Ed. G.C. Sih). Noordho ff , Leyden, 1973. Hills, D. A., Kelly, P. A., Dai, D. N., Korsunsky, A. M., 1996. Solution of Crack Problems. Kluwer Academic, Netherlands, 1996. Holland, J. H., 1992, Adaptation in Natural and Artificial Systems. MIT Press, 1992. Metropolis, N. Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., Teller, E., 1953. Equation of state calculations by fast computing machines, Journal of Chemical Physics, 21(6), 1087–1092. Weertman, J., 1996. Dislocation based fracture mechanics. World scientific publishing Co. Pte. Ltd, 1996. References