## Stress Intensity Factors

12 As shown in the preceding chapter, analytic derivation of the stress intensity factors of even the simplest problem can be quite challenging. This explain the interest developed by some mathematician in solving fracture related problems. Fortunately, a number of simple problems have been solved and their analytic solution is found in stress intensity factor handbooks. The most commonly referenced ones are Tada, Paris and Irwin's (Tada et al. 1973), and Roorke and Cartwright, (Rooke and Cartwright 1976), and Murakami (Murakami 1987)

13 In addition, increasingly computer software with pre-programmed analytical solutions are becoming available, specially in conjunction with fatigue life predictions.

14 Because of their importance, expressions of SIF of commonly encountered geometries will be listed below:

Middle Tension Panel (MT), Fig. 7.1

1.152

1.152

We note that for w very large with respect to a, = 1 as anticipated.

Figure 7.2: Single Edge Notch Tension Panel

Figure 7.3: Double Edge Notch Tension Panel

Single Edge Notch Tension Panel (SENT) for jfr = 2, Fig. 7.2

We observe that here the (3 factor for small crack (-^ <C 1) is grater than one and is approximately 1.12.

Double Edge Notch Tension Panel (DENT), Fig. 7.3

15Miw)

bw 2

Compact Tension Specimen (CTS), Fig. 7.5 used in ASTM E-399 (Anon. n.d.) Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials

p bw

"NT"

We note that this is not exactly the equation found in the ASTM standard, but rather an equivalent one written in the standard form. Figure 7.4: Three Point Bend Beam Figure 7.5: Compact Tension Specimen Figure 7.6: Approximate Solutions for Two Opposite Short Cracks Radiating from a Circular Hole in an Infinite Plate under Tension

Figure 7.6: Approximate Solutions for Two Opposite Short Cracks Radiating from a Circular Hole in an Infinite Plate under Tension

Circular Holes: First let us consider the approximate solution of this problem, Fig. 7.6, then we will present the exact one:

Approximate: For a plate with a far field uniform stress a, we know that there is a stress concentration factor of 3. for a crack radiating from this hole, we consider two cases

Short Crack: jj —0, and thus we have an approximate far field stress of 3cr, and for an edge crack ¡3 = 1.12, Fig. 7.6 thus

Long Crack D ^ 2a + D, in this case, we can for all practical purposes ignore the presence of the hole, and assume that we have a central crack with an effective length aef f = 2a^D, thus

Similarly, if we had only one single crack radiating from a hole, for short crack, ß remains equal to 3.36, whereas for long crack, Fig. 7.7 we obtain: Figure 7.7: Approximate Solutions for Long Cracks Radiating from a Circular Hole in an Infinite Plate under Tension Figure 7.8: Radiating Cracks from a Circular Hole in an Infinite Plate under Biaxial Stress

Exact: Whereas the preceding equations give accurate results for both very short and very large cracks, in the intermediary stage an exact numerical solution was derived by Newman (Newman 1971), Fig. 7.8

where, using Newman's solution ¡3 is given in Table 7.1

Pressurized Hole with Radiating Cracks: Again we will use Newman's solution for this problem, and distinguish two cases:

Pressurized Hole and Crack: or A = 1

For both cases, 3 is given in Table 7.1. We note that for the pressurized hole only, Ki decreases with crack length, hence we would have a stable crack growth. We also note that Ki would be the same for a pressurized crack and borehole, as it would have been for an unpressurized hole but an identical far field stress. (WHY?)

 a R ß Biaxial Stress ß Pressurized Hole A = -1 A = 1 A = 0 A = 1 A = 0 1.01 0.4325 0.3256 0.2188 .2188 .1725 1.02 .5971 .4514 .3058 .3058 .2319 1.04 .7981 .6082 .4183 .4183 .3334 1.06 .9250 .7104 .4958 .4958 .3979 1.08 1.0135 .7843 .5551 .5551 .4485 1.10 1.0775 .8400 .6025 .6025 .4897 1.15 1.1746 .9322 .6898 .6898 .5688 1.20 1.2208 .9851 .7494 .7494 .6262 1.25 1.2405 1.0168 .7929 .7929 .6701 1.30 1.2457 1.0358 .8259 .8259 .7053 1.40 1.2350 1.0536 .8723 .8723 .7585 1.50 1.2134 1.0582 .9029 .9029 .7971 1.60 1.1899 1.0571 .9242 .9242 .8264 1.80 1.1476 1.0495 .9513 .9513 .8677 2.00 1.1149 1.0409 .9670 .9670 .8957 2.20 1.0904 1.0336 .9768 .9768 .9154 2.50 1.0649 1.0252 .9855 .9855 .9358 3.00 1.0395 1.0161 .99267 .99267 .9566 4.00 1.0178 1.0077 .9976 .9976 .9764

Table 7.1: Newman's Solution for Circular Hole in an Infinite Plate subjected to Biaxial Loading, and Internal Pressure

Table 7.1: Newman's Solution for Circular Hole in an Infinite Plate subjected to Biaxial Loading, and Internal Pressure Figure 7.9: Pressurized Hole with Radiating Cracks Figure 7.10: Two Opposite Point Loads acting on the Surface of an Embedded Crack Figure 7.11: Two Opposite Point Loads acting on the Surface of an Edge Crack

Point Load Acting on Crack Surfaces of an Embedded Crack: The solution of this problem, Fig. 7.10 and the subsequent one, is of great practical importance, as it provides the Green's function for numerous other ones.

Point Load Acting on Crack Surfaces of an Edge Crack: The solution of this problem, Fig. 7.11 is

0 0