PSI - Issue 13

Mina Iskander et al. / Procedia Structural Integrity 13 (2018) 976–981 Iskander and Shrive/ Structural Integrity Procedia 00 (2018) 000 – 000

980

5

Table 1. Peak K I values and corresponding crack lengths

Model

C5

C10

C20

E10x5

E10x20

Peak K I (N/mm

3/2 )

49.37

70.18

99.27

82.72

56.17

Crack length at peak K I (mm)

0.45

0.90

1.55

1.00

0.60

K I (N/mm

3/2 ) at 0.45 mm crack

-

65.78

76.49

72.25

55.61

To begin to understand the effect of flaw height (initial flaw length parallel to the compression), C10 is compared to E10x20 and E10x5. The three models have the same width of void (10 mm) but differing heights (5, 10 and 20 mm). One might think that the vertical ellipse (20 mm height) would be more critical than the horizontal ellipse, but the opposite appears from the plots in Fig. 4. So, the flatter the void is, the more critical it would appear to be, assuming the width is kept constant. The same conclusion may be drawn from comparing C5 and E10x5. Again, the voids have essentially the same stress at their top and bottom tips but differing peak stress intensities. However, a very wide, flat crack would simply close under the compression, so further study on the effect of flaw shape in relation to the direction of applied compression is clearly required.

4. Results and Discussion: strain energy and end reaction

The total strain energy and the end reaction loads were recorded for models C10 and C20 and compared for crack lengths up to 6 mm propagating in 0.5 mm increments. The results are presented in Table 2. The total strain energy with no void present was also calculated so that we can examine the effect of the presence of the void in a solid medium on the total strain energy.

Table 2. Strain energy and end reaction versus crack length

Crack length (mm)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

C10 SE (N.mm) 414808 414806 414801 414797 414794 414791 414789 414787 414786 414785 414784 414783 414783 Reaction (kN) 1329.05 1329.04 1329.03 1329.01 1329.00 1328.99 1328.99 1328.98 1328.98 1328.97 1328.97 1328.97 1328.97 C20 SE (N.mm) 413342 413338 413331 413322 413314 413305 413297 413290 413283 413277 413272 413267 413263 Reaction (kN) 1324.35 1324.34 1324.31 1324.28 1324.26 1324.23 1324.20 1324.18 1324.16 1324.14 1324.12 1324.11 1324.09 It can be observed from Table 2 that the total strain energy of the non-cracked models depends on the size of the void; the bigger the void, the lower the total strain energy in the model, but also the lower the amount of material. The end reaction is also less with the bigger void (the element will be more flexible with the bigger void and thus less force is needed to impose the same displacement). The total strain energy of a solid model with no voids in it is 415298 N.mm. Thus, for C10, the total stain energy is reduced by 0.118% due to the presence of the void while for C20, the reduction is 0.471%, 4-fold that of C10. This matches exactly the increase in the volume of the voids; C20 has four times the volume of C10. At the same time, the peak value of K I for C20 is increased by the square root of 2 over that of C10, although this is not quite the relationship between the K I values for C5 and C10. Thus doubling the size of the void quadruples the energy reduction due to the presence of the void and increases the peak stress intensity by a factor of root two. The strain energy can also be seen to change at different rates during crack propagation depending on the void size. For C10 the maximum decrement is 5 N.mm when the crack grows from 0.5 mm to 1 mm. For C20 the maximum decrease is 9 N.mm as the crack extends from 1.0 to 1.5 mm in length. From Table 1, the peak values for K I occurred at crack lengths of 0.9 mm and 1.55 mm for C10 and C20, respectively. Therefore, the maximum change in strain energy occurs at the peak K I for the models studied. The total change in strain energy for a 6 mm crack for C10 is 25 N.mm and 79 N.mm for C20: consequently, the bigger void is more critical in that more energy will be released to drive crack extension. In contradistinction to the tension case, the percentage of the energy released for a 6 mm crack extension is only 0.019% of the total in the specimen for C20 and 0.006% for C10. This again emphasizes the