The stress intensity factor (K) is crucial to fracture me- chanics. It is calculated by the formula [7]:
K = O & B where: CJ = stress
a = crack length B = a geometry factor
The parameter K defines the stress field directly ahead of the crack tip, perpendicular to the crack plane (Figwe 12).
6, =- K
G
This equation is only valid a short distance ahead of the crack. Two cracks with very different geometries and very different loadings will have similar stress fields near their crack tips if they have the same stress intensity fac- tors. The stress intensity factor also determines the be- havior of the crack:
If & exceeds the fracture toughness, fast fracture oc- If AK is less than the A& (threshold stress intensity fac- tor), no crack growth occurs.
Lf AK is greater than A&, but L i s less than the frac- ture toughness, stable crack growth under cyclic load- ing occurs.
curs.
t
StraSS [Y)
Di
I Load I
\ c
L
ance Ahead of Crack Tip (x)
Figure 12. Stress distribution ahead of crack tip.
The stress intensity factor K should not be confused with the unrelated stress concentration factor K,. The units for K are:
(stress) x
For the English system, this is typically (hi) (inches).j. For the metric system, it is typically (mpa) meter^).^. The con- version factor from the metric to the English system is:
1 (mpa) (meters).j = .91 (hi) (iizches).j
340 Rules of Thumb for Mechanical Engineers
Fast Fracture
As stated earlier, fast fracture will occur when the stress intensity factor exceeds the fracture toughness. Fracture toughness is a material property that is dependent on tem- perature and component thickness.
Figure 13 shows that decreasing temperature decreases the fracture toughness. This is due to the lower ductility at lower temperatures. Figure 14 shows that increasing thick- ness decreases the fracture toughness. This is because thin specimens are subject to plane stress, which allows much more yielding at the crack tip. As specimen thickness is in-
-200 F ROOm
Temperalure Tempemre
Figure 13. Effect of decreasing temperature on fracture toughness.
Plane
e
U $
I I I I I I
10 20 30
0 0
Thickness le), mm
Figure 14. Effect of specimen thickness on fracture toughness p 41. (Reprinted with permission of John Wiley
& Sons, Inc.)
creased, it asymptotically approaches a minimum value.
This is the value that is quoted as the fracture toughness.
The actual fracture toughness for thin specimens may be somewhat greater than KIC.
Figure 15 shows how fracture toughness varies with yield strength for aluminum, titanium, and steels. Note that for all three, alloys which had very high yield strengths had relatively low values of hcture toughness. Also, be- cause the stress intensity factor is a function of crack length to the .5 power, reducing the fracture toughness by a fac- tor of 2 will reduce the the critical crack size by a factor of 4. Therefore, a designer who considers specifying an alloy with a high yield strengh should realize he may be sig- nificantly reducing the critical crack size. In field service, this could result in sudden failures instead of components being replaced after cracks were discovered.
Threshold Stress Intensity Factor
If the stress intensity factor range does not exceed the threshold value, then a crack will not propagate. ASTM de- fines the threshold value to be where the crack growth per cycle(da/dn) drops below 3 x lW9 inches per cycle. Unlike fracture toughness, the threshold stress intensity factor, AK,, is dependent upon the R ratio.
Threshold behavior is very important for components which must endure millions of cycles. In analysis, it can be used to make very conservative estimates. If the applied stress is known, a flaw size can be assumed, and the stress intensity factor can be calculated. As long as the threshold value exceeds AK, no crack growth can occur. If the stress
MPa
240
200
160
120 E I
80
40
0
0 40 EO 120 160 200 240 280 320 360
Yield strength kri
Figure 15. Fracture toughness versus yield strength for various classes of materials [18].
Fatigue 341
*
0 .
I
2
4
intensity factor range exceeds AK,, then more detailed analysis must be made to determine the number of cycles the component can be expected to last in service.
Sometimes, after many parts have been released for field service, a problem will be discovered in the manufacturing process. The question is then raised, “Do we need to replace these parts immediately, or can we safely wait and replace them during the next overhaul?’ Obviously, the first approach is safer, but may be extremely costly. However, if the parts are left in service and fail, the consequences can be devas- tating. Therefore, any analysis which is done must err on the conservative side. In these instances, the stress intensity fac- tor is often calculated and compared to the A&.
Estimates of threshold stress intensity factor. Testing to ob- tain AKth is quite time-consuming and, consequently, quite expensive. Cracks have to be propagated in a specimen, and then the load is slowly decreased until the crack stops propagating. Fortunately, there are some ranges available for different classes of materials at room temperature.
These are shown in Figures 16 through 20. It may be nec- essary to adjust these values to use them at elevated tem- peratures, which may be done by scaling the threshold value based on the ratio of Young’s modulus. For example, a threshold value for steel is needed at:
- I
8 -
1 i
6 -
An R ratio of .4
A temperature of 500 degrees
From Figure 16, the minimum threshold stress intensity fac- tor is 4.0 at room temperature. If Young’s modulus is 25 x 106 at 500 degrees and 30 x lo6 at room temperature, the min- imum stress intensity factor at 500 degrees should be 4.0 x (25/30) = 3.33.
R-ab
Figure 17. Relationship between threshold stress in- tensity range and R ratio in aluminum alloys [lq. (With
permission of Elsevier Science Ltd.)
342 Rules of Thumb for Mechanical Engineers
o 02 0 4 06 a ) L O
R-ntb
Figure 19. Relationship between threshold stress in- tensity range and R ratio in nickel alloys [lq. (With per- mission of Elsevier Science Ltd.)
o a2 0.0 0.6 a8 LO
R-ntio
Figure 20. Relationship between threshold stress in- tensity range and R ratio in titanium alloys [lq. (With per- mission of Elsevier Science Ltd.)
Crack Propagation Calculations
If the stress intensity factor is below the fracture tough- ness, and the range of the stress intensity factor is greater than the threshold value, the crack will grow in a stable man- ner. This is often referred to as “Subcritical crack growth.”
Three approaches are commonly used to relate the crack growth rate to the stress intensity factor (C, indicates the material constant):
Paris law [8]:
This law assumes that the data can be fit as a straight line on a log-log plot. This usually gives a fair defini- tion of the curve. It usually does not model the crack growth rate well at low and high values of AK.
Modified Paris law: This model seeks to overcome the limitations of the Paris law by using three sets of coefficients which are used over three ranges of stress intensity factors.
Hyperbolic sine model: This model strives to use one relationship that is applicable over the entire range of stress intensity factors, and accurately models the crack growth rate at low and high values of AK.
log (3 - = C1 sinh (C2[log(AK) + C,]) + C4
The analyst should realize that because cracks tend to grow at a continually increasing rate, most of the life oc- curs when the cracks are quite small. Therefore, it is im- portant to accurately model the crack growth rate at small values of AK, but usually not important at near-fracture val- ues. The exponent of the Paris law can be quite useful for determining the effect of a change in stress on crack growth life. Since AK is proportional to the applied stress range, The change in crack growth life may be estimated by:
If the stress is increased 1096, and the Paris law exponent is 4, the crack growth rate will be increased by (1.
which means it will grow 1.46 times faster. Therefore, the crack propagation life will be reduced by a factor of (U1.46) to 68% of its previous value.
Crack growth under cyclic loading for a given material is dependent upon three variables:
Fatigue 343
1. AK
2. R ratio (K~*/Kmm)
3. temperature
The crack growth data is typically shown on a log-log plot, such as shown in Figure 21. The crack propagation rate increases with increased stress intensity factor range and higher R ratios.
In the absence of more specific data, Barsom [9] rec- ommends using these rather conservative equations:
Ferritic-pearlitic steels:
- da (in./cycle) = 3.6 x lo-’’ (AK)”O0 dn
Martensitic steels:
- da (in. /cycle) = 6.6 x lo4 (AK)2.25 dn
When the minimum stress intensity factor is negative, a value of 0 should be used to calculate AK. This is because crack propagation does not occur unless the crack is open at the tip. If the amount of compression is small (R > -3,
then crack growth data at R = 0 (or .05) may be used with no significant loss in accuracy. If the amount of compres- sion is large, (R e -1), it may be wise to obtain data at the appropriate R ratio. Crack propagation at compressive R ra- tios is seldom done because the most commonly used test specimen, known as the compact tension (CT) specimen, can only be tested in tension.
Estimation of K
For a fairly uniform stress field, the analyst can estimate
K quite easily. Common crack types are shown in Figure 22. Approximate values of p are:
Austentitic stainless steels: P Crack mpe
- da (in./cycle) = 3.0 x lo-‘’ (AK)3.25 dn
0.71 corner or surface cracks 1.00 center cracked panels 1.12 through cracks Stress intensity factor range, AK, ksi 6
These values will change when the crack approaches a free surface. Typically, this is not a significant effect until the crack is about 40% of the way through the section for corner and surface cracks. It is much more significant for center and through cracks because of the loss of mss-sec- tional area.
1 LTJ
Q
\ 0 104 g
.- z^ 2
10-6 +j 6
E
10-8 g
.I-
2
5
rn
Y 0
1 ~7
10-8
(a) Surface Crack (b) Corner Crack
Figure 21. Increasing the R ratio increases the crack growth rate [14]. (Reprinted with permission of John Wiley & Sons, Inc.)
(c) Center Cracked Panel (d) Edge Crack
Figure 22. Common crack types.
344 Rules of Thumb for Mechanical Engineers
Computer codes which calculate crack growth use two approaches:
1. The cycle-by-cycle approach is the simplest, but it can be excessively time-consuming for slow-growing cracks. With this method, the stress intensity factors and crack growth for one cycle is calculated. The crack length is then increased by this amount and the process is repeated until the desired crack length is reached or the fracture toughness is exceeded.
2. With the step method, the number of cycles to grow the crack a certain distance (or step) is calculated after the crack growth rate is determined. This method generally requires significantly fewer iterations than the cycle-by-cycle method. Care must be taken in se- lecting the step size. If the step is too large, accuracy can be lost. If the step is too small, too much computer time will be required. For initial crack sizes around .015 inches, .001 makes a good step size.
Any engineer can write a simple computer code to per- form these iterations. All that is required is a stress inten- sity factor solution and a relationship between the stress in- tensity factor range and crack growth rate.
For simple cases, the crack growth life can be calculat- ed by simple integration. For example:
If da/dn = 2.0 x 1 0 - 1 2 ( ~ ~ ) 5
and K = 5 6 . 7 1
Solving for dn and integrating from initial crack size Ai and final crack size Af gives:
(Af’.’ - N = 1
(-1.5) (2 x (50 fi .71)
For an initial crack size of .015 inches and a final crack size of .050 inches, the crack growth life would be 153,733 cycles.
Plastic &ne Size
Because a crack is assumed to be infinitely sharp, the elas tic stresses are always infinite at the tip, but drop off very quickly. Yielding always occurs in the region ahead of the crack tip, which is referred to as the plastic zone. The size of the plastic zone under plane stress conditions can be es- timated by:
rp=-[-] 1 A K 2
2n Qy
Under plane strain, the plastic zone is approximately one- fourth as large. The plastic zone is important for a number of reasons:
For LEFM calculations to be valid, the crack length should be at least 10 times the length of the plastic zone.
Anyone testing to determine crack growth properties should realize that large and sudden changes in the loads can affect the plastic zone ahead of the crack and significantly alter the crack growth properties.
It is possible with a single overload to significantly re- duce the crack growth rate, or even arrest the crack. This can occur because the overload causes additional yield- ing in front ofthe crack, which inhibits its future growth through the region. Keep in mind that during the single overload, the crack grew at a greater rate, so it is not cer- tain what the net effect of the overload will be.
Creep Crack Growth
Creep crack growth (dddt) occurs when a tensile stress is applied for an extended time at a high temperature. This process should not be confused with conventional creep, which is an inelastic straining of material over time. Creep crack growth can be detected by metallurgical investiga- tion of the crack surfaces:
When cyclic crack growth dominates, the crack grows When creep crack growth dominates, the crack grows across the grains (transgranular).
along the grain boundaries (intergranular).
Several points should be made comparing creep and cyclic crack growth:
The rate of creep crack growth is related to the steady- state K, while cyclic crack growth rate is based on AK.
Creep crack growth is time dependent, while cyclic crack growth is not.
The threshold value for creep crack growth is much higher than it is for cyclic crack growth.
Temperature has a much greater influence on creep crack growth than on cyclic crack growth.
Fatigue 345