Crack Tip Plastic Zone Effect on Fatigue Crack Propagation

Abstract

The Paris-Erdogan equation can predict fatigue crack propagation only under particular conditions. It fails when R≠0 or under variable amplitude loads or with small cracks and cannot predict overload retardation etc. in all these cases fatigue crack propagation depends on what is actually happening at the crack tip in the plastic zone. Therefore, the study of the plastic zone behavior is fundamental to address the fatigue crack propagation issue and adjourn the Paris-Erdogan equation.

11.1 The Fatigue Mechanism

It has been shown in the previous section that the fatigue crack propagation in metallic materials follows a trend that can be described by the Paris-Erdogan Eq. (10.10). However, the Paris postulate is valid only under particular circumstances as constant load cycling in Region II of fatigue, long cracks and stress ratio R equal zero. If these circumstances are not met the Paris-Erdogan equation fails to represent the FCGR of the material. The fundamental point to understand is that the crack propagation is controlled by what is actually happening ahead of the crack in the plastic zone. Therefore, the particular circumstances of validity of the Paris postulate must be seen in terms of plastic zone size and behavior during the cycling. But fatigue itself depends on the plastic zone behavior so that to answer the fundamental question of why a crack grows by fatigue, before even asking whether the Paris postulate is valid or not, it is necessary to address the issue of the crack tip plastic zone. The equilibrium of a crack under a stress state was first addressed by the Englishman Griffith in 1920 [1]. He considered a fixed ends infinite plate of ideally elastic material carrying a central transverse crack of initial length 2a _o . The plate, of unit thickness, is loaded at its ends by a stress σ, as schematized in Fig. 11.1.

Fig. 11.1

Schematic instability condition for a crack of length 2a in an ideally elastic material

If the crack extends over a length da a certain amount of elastic energy G = dU/da is released. Since the plate is fixed at its ends, so that no kinetic energy can be introduced and its behavior is ideally elastic, this released elastic energy will be converted into the only possible available energy that is the surface energy of the two new surfaces created 2Γda, Γ being the surface energy of the metal per unit area A. Under the stress state σ the crack will become unstable and propagate without arresting if the elastic energy release rate G, also called driving force, will equal the surface energy absorption rate d(2Γda)/dA that represents the resistance R of the material to crack propagation. Therefore, the instability or critical condition is

$$ G_{c} \, \ge \, R $$

(11.1)

where the subscript c denotes criticality. Griffith evaluated the energy release rate G of the geometry of Fig. 11.1 as πσ ² a/E, with E being the Young’s modulus, therefore Eq. (11.1) becomes

$$ \frac{{\pi \sigma_{c}^{2} a}}{E} = 2 \cdot \Upgamma . $$

(11.2)

In a G-R versus crack length a space, the driving force is a straight line with angular coefficient given by σ ² and R is a constant equal to 2Γ independent of the crack length, as schematized in Fig. 11.1. It is an in–out situation in that a crack under a stress σ either becomes unstable when σ = σ _c and propagates indefinitely, line 3 in Fig. 11.1, or stays there as it is, line 1 or 2. Griffith demonstrated the validity of his theory making experiments on glass that he considered a material most likely similar to an ideally brittle (linear elastic) material. Real materials, though brittle can be, always present a certain degree of plasticity that make it possible to generate a plastic zone ahead of the crack tip. Therefore, in real materials plastic energy released or stored as plastic deformation on the crack surface and, in particular, in the plastic zone takes the dominant role in the equilibrium of the crack under stress. Plastic energy, in fact, even in very brittle materials can be thousands of times larger than the surface energy that can be neglected completely. Accordingly, in real materials the driving force G and the resistance R are no longer linear and constant, respectively, as in ideally brittle materials, but depend on the crack length a since the size of the plastic zone depends on the crack length, see Eq. (10.6 and 10.5). Now, they follow a power law relationship, as schematized in Fig. 11.2. The consequence is that now the critical condition becomes the tangency between G and R, as shown in Fig. 11.2. But there is another extremely important consequence from the fatigue stand point. The new situation is no longer the in–out situation of the ideally elastic case of Fig. 11.1. In real materials, even though a stress state and crack length combination doesn’t result in a critical state, that is to say G < G _c , there is enough energy into the system to drive the crack up to point A, as shown in Fig. 11.2, where the equilibrium between G and R is reached. The process is called stable tearing because the crack tip tears a little bit without breaking through catastrophically. Upon unloading, a new crack is left into the system whose length is a _o  + Δa, as shown in Fig. 11.2. Actually, this Δa is not a fatigue crack growth since it is much larger than a single cycle growth or striation, but it is the precursor of that. When the system is reloaded it would be expected that the equilibrium between the driving force G and the resistance R already reached in the previous cycle still holds and nothing would happen. But experience indicates the crack would grow further by fatigue. The author [2, 3] supposed that if this happens it is because during reloading the plastic zone ahead of the crack tip can no longer absorb energy with the same rate as before so that the crack has to grow further to widen the initial plastic zone into new virgin material and regain the energy lost. This is fatigue. He also made the hypothesis that the loss of energy release rate was due to a shake-down effect (see Sect. 5.4.2) that in the reloading prevents the material in the plastic zone from absorbing the same amount of energy as before. An amount of energy proportional to the elastic energy, about 10–30 %, is lost. Based on these assumptions, Milella [2, 3] derived an equation to assess the fatigue crack growth per cycle

$$ \Updelta a = \frac{{61.5 \cdot 10^{ - 3} \varphi }}{{\alpha^{2} }}\frac{{\Updelta K^{{2\left( {n + 1} \right)}} }}{{\sigma_{y} }}\left[ {\frac{{\alpha \sigma_{y}^{2} - \sigma^{2} }}{{2K_{Ic}^{2} \left( {\alpha \sigma_{y}^{2} - \sigma_{c}^{2} } \right) - \Updelta K^{2} \left( {\alpha \sigma_{y}^{2} - \sigma_{{}}^{2} } \right)}}} \right]^{n} $$

(11.3)

where σ _y is the yield strength, n the exponent of the power low representing the R curve of the material shown in Fig. 11.2

$$ R = C \cdot \Updelta a^{n} $$

(11.4)

with C characteristic of the material, K _Ic is the toughness of the material, σ _c the stress at fracture that results in K _Ic , α is related to the plastic constraint factor pcf by the relationship

$$ \alpha = 2\left( {pcf} \right)^{2} = 2\left( {\frac{{\sigma_{e} }}{{\sigma_{y} }}} \right)^{2} $$

(11.5)

where σ _e is the equivalent yield strength of the material under multiaxial stress state ahead of the crack tip, that can be as high as 2.5. σ _y (G. Irwin suggested to adopt the value of 6 for α), and φ a material factor related to the stretched-zone size (SZ)

$$ SZ = 10^{{ - 3}} \varphi \left( {\frac{{K_{{Ic}} }}{{\sigma _{y} }}} \right)^{2} . $$

(11.6)

Fig. 11.2

Schematic of the driving force G and resistance R in real elastic–plastic materials

Fractography shows that between fatigue striations and final dimple or cleavage fracture by overload there is a featureless zone known as stretched-zone. This zone must necessarily be the fractographic evidence of the maximum fatigue crack growth Δa _max that can be possible in the material before fracture occurs considering the final overload as ¼ cycle. Fig. 11.3 shows a stretched-zone in aluminum 7178-T651 alloy [4]. Table 11.1 presents the values of the φ factor found by Milella [3] for several alloys.

Fig. 11.3

Fractograph of stretched-zone observed in 7178-T651 aluminum alloy [4]

Table 11.1 φ factors for the evaluation of the stretched-zone

It interesting to note how Eq. (11.3) contains the fracture toughness of the material that was found to have an important role on the FCGR of materials (see Sect. 10.8).

11.2 Fatigue Threshold

In the study of fatigue through the S–N curves or Wöhler curves it was identified a Region III, see Fig. 1.5, of the fatigue limit where the material is no longer subjected to fatigue fracture. This is because the fatigue limit σ _f represents the highest stress amplitude that cannot propagate an existing micro defect possibly existing in the material. Below σ _f , in fact, persistent slip bands (see Sect. 1.4.1) are either not produced or, if they are, the associated microscopic defect cannot be driven by the applied stress through adjacent grains, remaining confined. Something analogous happens also as far as fatigue crack propagation is concerned. Below a certain value of the stress intensity factor, called threshold factor or K _th , Fig. 10.6, cracks remain dormant precisely as the micro cracks were dormant below the fatigue limit σ _f (see Sect. 1.5.1). Measurements of ΔK _th were not that easy since it is in the range of 10⁻⁸ mm/cycle and to produce a crack growth of just one millimeter at that ΔK requires 10⁸ cycles and times extremely long with the usual cycling frequencies. To-day the introduction of resonant-type machines, see Fig. 3.3, working at 100–150 Hz has made things simpler reducing that time down to just few days. Results so far obtained judge the fatigue threshold to be located between 4 and 10 MPa√m for most materials. At these low levels of ΔK _I there is a sharp break in the FCGR, da/dN, versus ΔK _I relationship. The existence of the fatigue threshold produces the characteristic sigmoidal shape of the FCGR, da/dN, curve shown in Fig. 10.6. In the vicinity of ΔK _th cracks grow with a much lower rate but much higher dependency on the ΔK _I variation than in Region II of the Paris-Erdogan equation. However, fatigue threshold is not a unique characteristic of the materials, but depends on many factors. Among these, certainly the most important are the R-ratio defined as the ratio between the minimum, K _min and maximum, K _max, applied stress intensity factor

$$ \begin{gathered} \begin{array}{*{20}c} {\Updelta K} & = & {K_{\max } - K_{\min } } \\ \end{array} \hfill \\ \begin{array}{*{20}c} R & = & {\frac{{K_{\min } }}{{K_{\max } }}} \\ \end{array} \hfill \\ \end{gathered} $$

(11.7)

and the metallurgical effects. Figure 11.4 is an interesting example of ΔK _I,th variation with R. In particular, Fig. 11.4a [5] refers to a low-carbon Ni-base Astroloy and shows how the ΔK _I,th variation as function of both the R-ratio and the grain size (GS), at ambient temperature and in dry air. Samples with different grain size were obtained by altering the hot isostatic pressing and solution treatment procedures. The yield strength was varying from a minimum of 910 MPa for the 50 μm GS material to a maximum of 1110 MPa for the 5 μm GS material and the ultimate strength from 1350 MPa for the 50 μm GS material to a maximum of 1550 MPa for the 5 μm GS material. Fig. 11.4b [6] shows the ΔK _I,th variation for an aluminum alloy type 2618 as a function of R at ambient temperature and in dry air. Some characteristic features must be caught. First, the R-ratio has a large influence on the threshold value ΔK _I,th . In the Astroloy, the ΔK _I,th almost reduces to a half going from R = 0.1 to R = 0.5. Furthermore, the metallurgical effect of GS that is not apprized at R = 0.1, becomes well evident at R = 0.5. The response of the 5 and 13 μm GS alloys separates completely from that of the 26 and 50 μm GS alloys. Hence, the metallurgical effect gets out at higher R-ratios. But there is one more feature that must be stressed; even though the factor R and the metallurgy influence the threshold value, the different FCGR curves seem to merge in Region II of the da/dn versus ΔK _I diagram. The same metallurgical effect, in terms of different ultimate strength, can be seen in the titanium alloy type Ti-6Al-4 V, as shown in Fig. 11.5 [7].

Fig. 11.4

a FCGR in Astroloy as function of R and GS (data from [5]). b in 2618 aluminum alloy [6]

Fig. 11.5

Separation of FCGR curves with strength in the threshold region for Ti-6Al-4 V alloy [7]

An overall view of the impact of the R-ratio is offered in Fig. 11.6 that show the effect of the R-ratio on the FCGR, da/dN, behavior observed at very low ΔK _I levels for low-alloyed carbon steel type A 533 B Cl1 and A 508 Cl2 used in the construction of nuclear pressure vessels [8]. The various curves are given at constant rate Δa/ΔN in the vicinity of the threshold and at the threshold ΔK _I,th . Note that as the R-ratio increases from 0.1 to 0.8, the threshold level decreases passing from about 6–7 MPa√m to just 3 MPa√m. It is also interesting to note that the R-ratio effect reduces as the FCGR, da/dN, increases. Figure 11.7 present another very interesting point that is the effect of temperature on the low FCGR behavior of the same steels of Fig. 11.6 at an R-ratio of 0.1. Note that the threshold level appears to reach a minimum at about 100–120 °C. Recalling Fig. 10.37, in that interval of temperatures the toughness K _Ic of the material reaches its maximum, then it drops at both lower and higher temperatures. This actually means that despite the apparent concave trend, the ratio ΔK _I,th /K _Ic remains the same at all temperatures. This further confirms what has been said in Sect. 10.8 about the role of toughness on FCGR of materials, i.e., that the effective value of a ΔK _I excursion on the FCGR, da/dN, behavior of a material should be assessed by indexing ΔK _I to the relative K _Ic . Tanaka [9], studying a large number of Fe-alloys with yield strength varying from 255 to 1590 MPa, found that the threshold stress intensity factor ΔK _I,th varied with both the R-ratio and the yield strength of the material. The results of his study are shown in Fig. 11.8 [9]. The dependence of ΔK _I,th on the R-ratio appears to be linear for all steels considered and all curves converge theoretically towards zero. In practice, saturation occurs at about 2–4 MPa√m for R-ratio higher than 0.7–0.8. This saturation effect was already observed by Barsom and Rolfe [10] on another series of steels that, anyhow, included also stainless steel, type 18/8, and mild steel considered by Tanaka. Characteristic, in Fig. 11.8, is the behavior of high strength steel SNCM439 of 1590 yield strength, whose response appears to be independent of the R-ratio and fixes the lower bound of ΔK _I,th . This dependency of the threshold stress intensity factor on the R-ratio induce to believe that ΔK _I,th is composed of two components, one is constant and is an intrinsic characteristic of the material ΔK _th(in) the other, instead, is dependent on external factor among which the R-ratio is the most effective, ΔK _th,R

$$ \begin{array}{*{20}c} {\Updelta K_{th} } & = & {\Updelta K_{th,in} + \Updelta K_{th,R} } \\ \end{array} . $$

(11.8)

Fig. 11.6

∆K _I range variations vs. R-ratio for constant crack propagation rates of A 533 and A 508 steels in air at room temperature (modified from [8])

Fig. 11.7

∆K _I range variations vs. temperature for constant crack propagation rates of A 533 and A 508 steels (modified from [8])

Fig. 11.8

R-ratio effect on the ∆K _I,th threshold for 7 different steels [9]

Klesnil e Lukas [11] proposed an empirical relationship between and the R-ratio analogous to that proposed by Walker between ΔK and R (see Sect. 11.4)

$$ \begin{array}{*{20}c} {\Updelta K_{th} } & = & {\Updelta K_{th,o} \cdot \left( {1 - R} \right)^{s} } \\ \end{array} $$

(11.9)

in which ΔK _th,o is the threshold value that corresponds to R = 0 and s a fitting exponent.

11.3 Plastic Zone and R-Ratio Effect on Fatigue Threshold

The R-ratio effect on the threshold stress intensity factor can be explained by the residual stress system set up ahead of the crack tip during the unloading phase. What is actually happening is schematized in Fig. 11.9. During the load application phase a plastic zone develops at the crack tip, Fig. 11.9a, which allows the tip to open and blunt. Beyond this plastic enclave, assumed as a first approximation to be circular in shape (G. Irwin assumption), there is the elastic high amplitude stress field that reach the yield strength and go higher in strain hardening materials. On the crack flanks the so called plastic wake of residual deformations remains as memory of previous deformation cycles. During unloading, crack closes, but because of the shake-down effect (see Sect. 5.4.2) most of the material ahead of the plastic zone will go into residual compression and will be balanced by a system of tensile residual stresses, Fig. 11.9b.

Fig. 11.9

Schematic of the crack tip plastic zone and residual stress state

The compressive residual stress state certainly exerts a beneficial effect on the fatigue behavior since in the following reloading it will be necessary to overcome it before the crack opens and restarts to grow. This actually means that it will not be the entire load excursion Δσ to cause fatigue, but rather only the part of it that will open the crack, Δσ _op . This will reduce the effective applied ΔK. Let’s follow what happens in the entire plastic zone ahead of the crack tip. This is schematized in Fig. 10.10. Under the load history (a), point A, right at the tip of the crack, during unloading will undergo the largest hysteresis loop between A and A′ on the stress–strain curves relative to ΔK = K _I,max  − K _I,min with K _I,min  = 0. On the border of the plastic zone opposite to point A, the corresponding point E moves on the elastic straight line 0E. Between those two extreme points behavior there is any other point of the plastic zone that, upon unloading, enters into e residual compressive stress state, as B′, C′ and D′, for example. As said before, the consequence is that upon reloading the effective ΔK _eff that will challenge the crack will not be given by K _I,max  − K _I,min , but will be ΔK _eff  = K _I,max  − K _I,op  < ΔK. Note that if the K _I,min increases, load history (b) of Fig. 11.10, which means that the R-ratio increases, the residual compressive stress state reduces, points A″, B″, C″ and D″, so that also the difference between ΔK and ΔK _eff reduces up to a point when K _I,min reaches K _I,op , as shown by the time history (c) in Fig. 11.10, when the entire ΔK will be effective, ΔK = ΔK _eff , because the residual compressive stress state disappears completely, points A‴, B‴, C‴ and D‴. Now, as the load increases the crack opens immediately. If this simple schematization of the crack behavior can explain why a minimum K _I,min below K _I,op cannot be effective, it cannot account for any reduction of fatigue strength when R-ratio further increases and it becomes K _I,min  < K _I,op . To fully understand this point it is necessary to go back to the fatigue mechanism explained in Sect. 11.1. The crack grows, it was said, because at any reloading the material in the plastic zone ahead of the crack tip is no longer able to absorb energy with the same rate as in the previous cycle. This forces the crack to grow further and regain the losses thorough the extension of the plastic zone into new virgin material. As clearly explained in [3] this capability to absorb new additional energy to compensate the loss of energy absorption rate decreases more and more as the K _I,min increases because the area under the curve of Fig. 11.10 reduces more and more, so that the crack has to grow at increasing rates with increasing R-ratios.

Fig. 11.10

Increasing the K _I,min to K _I,op reduces the residual compressive stress field to zero

11.4 R-Ratio Effect on the FCCG

It has been shown in the previous section that the progressive elevation of the K _I,min value and, consequently, of the ratio R = K _min /K _max leads to a parallel reduction of the threshold stress intensity factor ΔK _th . The R-ratio effect persists, though in a less fashion, in Region II of the FCGR, see Fig. 10.6, where the Paris-Erdogan Eq. (10.10) holds and is further emphasized in Region III. As to Region II, Fig. 11.11 shows this effect in a titanium alloy type Ti-6Al-4 V [12]. The R-ratio is varying from a rather high 0.7, representative of a very high mean stress σ _m  > 0, down to negative values representative of a stress cycle with negative mean stress, σ _m  < 0. The FCGR, da/dN, varies by a stunning factor of one hundred moving from R = − 5.0 to R = 0.7 to which, by the way, it corresponds a very low threshold value of just 4 MPa√m. Walker [13] proposed an empirical relationship to correct the Paris-Erdogan equation to take into account the R-ratio effect

$$ \begin{array}{*{20}c} \frac{da}{dN} & = & {C \cdot \Updelta K_{eff}^{n} } \\ \end{array} $$

(11.10)

where ΔK _eff is given by

$$ \begin{array}{*{20}c} {\Updelta K_{eff} } & = & {K_{max } \left( {1 - R} \right)^{m} } \\ \end{array} $$

(11.11)

in which m is a material characteristic to be experimentally determined since it is

$$ \begin{array}{*{20}c} {K_{max } } & = & {\frac{{K_{min } }}{R}} \\ \end{array} $$

(11.12)

so that it is

$$ \begin{gathered} \begin{array}{*{20}c} {\Updelta K} & = & {\begin{array}{*{20}c} {K_{max } - K_{min } } & = & {K_{min } \left( {\frac{1 - R}{R}} \right)} \\ \end{array} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {K_{max } } & = & {\frac{\Updelta K}{{\left( {1 - R} \right)}}} \\ \end{array} \hfill \\ \end{gathered} $$

(11.13)

and Eq. (11.11) can be written as

$$ \begin{array}{*{20}c} {\Updelta K_{eff} } & = & {\Updelta K \cdot \left( {1 - R} \right)^{p} } \\ \end{array} $$

(11.14)

with p = m − 1. Therefore, Eq. (11.10) becomes

$$ \begin{array}{*{20}c} \frac{da}{dN} & = & {C \cdot \Updelta K_{{}}^{n} } \\ \end{array} \left( {1 - R} \right)^{s} $$

(11.15)

with s = n (m − 1). The experimental points of Fig. 11.11 have been retraced as function of ΔK _eff given by Eq. (11.14), with p = − 0.4. Results are shown in Fig. 11.12. It can be seen the strong compacting action that has led, in practice, to the superposition of all experimental data obtained for R varying from 0.7 to −1.0. Only data relative to 0 > R ≠ − 1 remain out of the common trend even though they compacted in a separate group. This deficiency of the Walker ΔK _eff to represent data for negative R-ratios appears to be a characteristic common to all corrective equations, including the Elber ΔK _eff , see Fig. 11.12 however, it has been shown how the compression cycle doesn’t really contribute to the crack opening and, therefore, to the crack growth. It can be convenient, then, to revisit the experimental data of Fig. 11.11 in terms of K _I,max rather than of the entire ΔK _I . This can be seen in the diagram of Fig. 11.13 that appears to upturn results of Fig. 11.12. This time, the compacting action applies only to negative R-ratios or R = 0, at most, while data relative to positive R-ratios seem to spread and deviate, resulting in lower growth under higher ΔK _I . Another interesting example of FCGR dependency on R-ratio is shown in Fig. 11.14 [8] for type 7075-T6 clad aluminum alloy fatigued in dry air at ambient temperature at 3400 cpm. Introducing the Walker correction given by Eq. (11.14) with p = − 0.5, the experimental data of Fig. 11.14 rearrange as shown in Fig. 11.15. As for the titanium alloy, also for the aluminum clad there is a remarkable compatting. Crooker et al. [14, 15]. studying the fatigue behavior of a martensitic alloy type HY-80 of 965 MPa yield strength (see Fig. 10.26) in the range −2 ≤ R ≤ 0.75 found that the Walker Eq. (11.14) was well representing the FCGR using p = − 0.5. But as the R-ratio increases, the ΔK needed to produce the unstable crack growth decreases, which actually means that the da/dN curves in Region III of the FCGR diagram, see Fig. 10.6, diverge as it happened in Region I. this can be seen in Fig. 11.16 [16] relative to an aluminum alloy type 2219-T851. This raveling in Region I and III, respectively, of the da/dN curves that in Region II get closer is schematized in Fig. 11.17.

Fig. 11.11

FCGR, da/dN, vs. ∆K _I diagram for a titanium alloy type Ti-6Al-4 V, as function of the R-ratio [12]

Fig. 11.12

FCGR, da/dN, vs. ∆K _eff diagram for a titanium alloy type Ti-6Al-4 V, as function of the R-ratio [12]

Fig. 11.13

FCGR, da/dN, vs. ∆K _I,max diagram for a titanium alloy type Ti-6Al-4 V, as function of the R-ratio (data from [12])

Fig. 11.14

FCGR, da/dN, vs. ∆K _I diagram for 7075-T6 clad aluminum alloy, as function of the R-ratio [8]

Fig. 11.15

FCGR, da/dN, vs. ∆K _eff diagram for 7075-T6 clad aluminum alloy, as function of the R-ratio [8]

Fig. 11.16

FCGR, da/dN, vs. ∆K _I diagram for 2219-T851 aluminum alloy, as function of the R-ratio [16]

Fig. 11.17

Schematic of the continuous K _I,th reduction and instability advance due to increasing R-ratio

11.5 Crack Closure

It has been shown, in Sect. 11.3, how the R-ratio effect can be explained in terms of ΔK _eff that represents that part of the entire stress intensity factor excursion ΔK active in producing the growth of the crack. This ΔK _eff is, in practice, the component that effectively opens the crack. Another approach to define ΔK _eff , that has gained wide acceptance, it that proposed by Elber in 1970 [17], referred to as crack closure. Working with some types of fracture mechanics specimens, Elber observed that during unloading the crack was already closed before the tension stress became zero. In other words, during unloading the compliance of the specimens carrying a crack, C _cr , was equal that of the loading phase, predicted by theoretical formulas, but only up to a certain load level, generally high, below which it decreased reaching the value that competed to the non-cracked specimens. This is schematically shown in Fig. 11.18a; if a specimen containing a crack is loaded up to B on its, F-δ, load–deflection curve and then unloaded, it starts moving on the line BC parallel to the initial load line that represents the stiffness k _cr  = F/δ _cr of the cracked specimen (k _cr  = 1/C _cr ), but upon reaching point C instead of continuing on the same course CD it deviates along line CE that represents the stiffness of the non-cracked specimen k = F/δ. Elber ascribed such deviation from the initial course to the rough and uneven nature of the crack surfaces that during unloading did not match perfectly, but get in contact well before the load went to zero. Therefore, the two crack surfaces were able to exert a reaction F _op opposing the complete closure of the crack or, from a different point of view, the crack closed before the load went to zero. In terms of ΔK this effect of crack closure results in the specimen being not cycled between K _min and K _max because it will never return in the state of K _min , but it will remain in a rather higher K _op that corresponds to the force F _op acting at the moment when the crack surfaces get in contact, as schematized in Fig. 11.18b. Therefore, the effective ΔK _eff challenging the crack will no longer be equal to K _max  − K _min , but to

$$ \begin{gathered} \Updelta K_{eff} \, = \,K_{max } - K_{op} . \hfill \\ \hfill \\ \end{gathered} $$

(11.16)

Fig. 11.18

a Schematic of the loading–unloading behavior of a cracked specimen that shows crack closure. b effective cycle

This, in practice, is worth a reduction of the RCCF, da/dN. As such, Elber proposed a modified Paris-Erdogan equation

$$ \begin{array}{*{20}c} \frac{da}{dN} & = & {C \cdot \Updelta K_{eff}^{n} } \\ \end{array} . $$

(11.17)

It must be underlined that the K _op is not a material characteristic, since it is the results of a crack closure produced by the roughness of the crack surfaces. As matter of fact, besides the material characteristics the crack roughness depends also on the load level or K _max ; to different K _max is likely that different crack roughness correspond. An interesting example is provided by the experimental observation of the surface roughness, shown in Fig. 11.19 [7], produced by fatiguing a titanium alloy, type Ti-6Al-4 V, thermally aged to three different ultimate strength already considered in Sect. 10.7.5. Indeed, there is a material dependency of the surface roughness that, ceteris paribus, results more pronounced in the lower strength alloy that allows larger plastic deformation, but it is worth noting that at higher FCGR associated to higher ΔK, the surface roughness is lower. In the three cases of Fig. 11.19 the R-ratio was kept constant and equal to 0.05, therefore the largest roughness is associated to the lowest K _max . Elber also verified the dependency of the K _eff on the R-ratio. He introduced an non dimensional parameter U as ratio between ΔK _eff and ΔK

$$ \begin{gathered} \begin{array}{*{20}c} U & \equiv & {\frac{{\Updelta K_{eff} }}{\Updelta K}} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {} & = & {\frac{{K_{max } - K_{op} }}{{K_{max } - K_{min } }}} \\ \end{array} \hfill \\ \end{gathered}$$

(11.18)

so that Eq. (11.18) becomes

$$ \begin{array}{*{20}c} \frac{da}{dN} & = & {C \cdot \left( {U \cdot \Updelta K} \right)^{n} } \\ \end{array} . $$

(11.19)

Fig. 11.19

Experimental measurements of the surface roughness in a Ti-6Al-4 V alloy heat hardened to three different strength and subjected to different ΔK that result in different FCGR, da/dN (reproduced with permission of [7])

Working with an aluminum alloy type 2023-T3, Elber found that the parameter U was not constant, but could be linked to the R-ratio by a relationship of the type

$$ \begin{array}{*{20}c} U & = & {0.5 + 0.4 \cdot R} \\ \end{array} . $$

(11.20)

Other researchers have confirmed the results obtained by Elber [18–21]. An interesting example regarding the Elber ΔK _eff is provided in Fig. 11.20 that reports the experimental data obtained by Zhang et al. [22] on aluminum alloy type 7475-T7351. It can be observed the same R-ratio effect already seen in Fig. 11.11 for the titanium alloy and Fig. 11.16 for the aluminum alloy type 2219-T851. Zhang et al. Found that the parameter U was effectively depending on the R-ratio through an expression of the type

$$ \begin{array}{*{20}c} {\begin{array}{*{20}c} U & = & {\frac{{\Updelta K_{eff} }}{\Updelta K}} \\ \end{array} } & = & {0.618 + 0.365 \cdot R + 0.139 \cdot R^{2} } \\ \end{array} . $$

(11.21)

Fig. 11.20

FCGR,da/dN, vs. ΔK observed by Zhang et al. [22] for Al 7475-T7351 aluminum alloy as a function of the R-ratio

Reanalyzing the data of Fig. 11.20 as function of the ΔK _eff given by Eq. (11.22) it is obtained the diagram of Fig. 11.21. The result offers interesting considerations. First, the compacting action operated by the Elber ΔK _eff exists, but not for the data obtained at high negative R-ratio that stand apart. This failure of the ΔK _eff corrective action was evidenced also for the Walker ΔK _eff in Sect. 11.4, see Fig. 11.12. Secondly, since it is ΔK = K _max (1 − R), for R > 0 at equal ΔK the corresponding K _max will increase with increasing R. therefore, if in terms of ΔK _eff the experimental data come closer it means that K _op is a function of K _max and increases as the K _max decreases, according to the reasoning about the surface roughness. This actually means that for high R-ratio, let’s say R > 0.7, the corresponding K _op is so small that, in practice, the ΔK _eff coincides with ΔK. An analogous effect has already been observed for the threshold stress intensity factor ΔK _th that for R > 0.7 practically saturated, as shown in Fig. 11.8. An experimental verification of this circumstance was given by Newman [23, 24] working with an aluminum alloy, type 2219-T851. Note that this coincidence between ΔK _eff and ΔK for R ≥ 0.7 can be used for the experimental measurement of K _op . In fact, working at high R-ratio (R ≥ 0.7) it is possible to infer the values of constants C and n in Eq. (11.18) simply assuming that ΔK is equal to ΔK _eff . Once these values are known, interpolating the experimental results obtained at lower R-ratio it will be possible to infer the applied ΔK _eff and, therefore, the K _op value via Eq. (11.17). As to the first consideration about the failure of ΔK _eff to compact the FCGR, da/dN, in the negative R field, the data of Fig. 11.20 can be reanalyzed as function of the applied K _I,max , as already done in Sect. 11.4 and shown in Fig. 11.13 for the titanium alloy. Results are presented in Fig. 11.22. Again, the new diagram presents a superposition of experimental data obtained at R < 0 or positive but close to zero, but those relative to R > 0 remain apart. About the surface roughness, it must be said that it certainly plays an important role, but it is not the only factor affecting the crack closure. Suresh e Ritchie [25] have identified five different causes affecting crack closure that are schematized in Fig. 11.23. The first, Fig. 1.23a, is the surface roughness already analyzed aggravated by a though minimum Mode II aperture that can be activated on a microscopic scale and inhibits the perfect merging of crack surfaces at unloading. Another important crack closure effect is due to the wake of residual local deformations left behind the crack tip as it propagates, Fig. 11.23b. Inside the monotonic plastic zone formed at maximum load, there is the cyclic plastic zone in which reversed plasticity occurs. According to the shake-down model explained in Sect. 5.4.2, reversed plasticity requires that the stress exceeds twice the yield stress. This was also pointed out by Rice [26]. Based on Eq. (10.6), this implies that the reversed plastic zone should be ¼ of the monotonic plastic zone. Also this zone leaves a wake along the growth of the crack, but much smaller than the monotonic plastic wakes. Because of the difference in the process volumes of the small plastic wake on the rim of the crack and the large monotonic plastic wake this latter prevails. This phenomenon is referred to as plastic induced crack closure. Moreover, if the cycling is given in a potentially aggressive environment, even humid air, the resulting oxide formation will act as a wedge inside the crack anticipating its closure, Fig. 11.23c. Also a possible internal pressure exerted by a fluid can affect the crack closure, as schematized in Fig. 11.23d. Finally, Fig. 11.23e, a possible metallurgical transformation operated by the plastic deformation of the crack tip, which is equivalent to a cold-working effect, can result in a residual opening or squeezing of the crack tip, depending whether the new phase has a larger or smaller structure. For instance, the transformation of metastable austenite into martensite is not resulting into any significant cell size modification. At variance, the transformation of ferrite or perlite into martensite happens with volume enlargement of the elemental cell structure. It is possible to conclude that if from the one end the crack closure effect can explain the R-ratio effect on the FCGR, da/dN, the empirical dependency of the ΔK _eff on R makes its use not dissimilar from any other empirical relationship.

Fig. 11.21

FCGR, da/dN, of Fig. 11.20 reanalyzed in terms of Elber ΔK _eff

Fig. 11.22

Reanalysis of data of Fig. 11.20 by Zhang et al. [22] as function of the K _I,max

Fig. 11.23

Possible causes for crack closure identified by Suresh e Ritchie [25]

11.6 Overload Retardation

It has been long observed by many researchers [27–35] that any single overload imposed during a constant amplitude cycling results in a retardation of the crack growth in the following cycles. Fig. 11.24 [33] is a significant example of this phenomenon known as overload retardation. A series of specimens of type 2024-T3 aluminum alloy have been subjected to different load histories indicated as A, B and C in Fig. 11.24. If the fatigue cracks growth is measured versus the elapsed number of applied cycles N for each load history, the corresponding curves A, B and C of Fig. 11.24 are obtained. For the constant load history A the crack growth has a regular trend, but if the constant amplitude load cycling is regularly interrupted by a single overload (load history B) it will be observed a period of reduced crack growth rate after each overload. It will take many cycles before the FCGR returns to be the previous one.

Fig. 11.24

Crack growth dependency on the load history in 2024-T3 aluminum alloy [33]. Load histories A and B result in growth retardation

This delay depends also on the type of overload. In the sequence B of Fig. 11.24 the overload was fully reversed, as the constant load cycling. But if the overload is pulsating, as in load history C, the delaying effect will last longer, as shown in Fig. 11.24. In all cases, however, the crack will return to grow at the same rate as before the overload application. The retardation effect can be explained by considering the residual compressive stress state set up ahead of the crack tip, already discussed in Sect. 11.3 and shown in Fig. 11.9. The overload, in fact, generates a plastic zone that is larger than that associated to the smaller constant amplitude cycling. Therefore, when the smaller amplitude loads are applied they have to overcome the high compressive residual stresses left ahead of the crack tip. With reference to Fig. 11.10 let’s assume that the overload has produced the hysteresis loop AA′ while the lower amplitude cycles result in the DD′ one. The consequence is that, starting from A′, the effective ΔK _eff associated to cycle DD′ is enormously reduced. In the limit, if the maximum ΔK is smaller than the ΔK _op associated to the overload the crack will not grow anymore. Wheeler [31, 32] has ascribed this retardation effect precisely to the residual stress state and it evolution during the cycling subsequent to the overload. He supposed that the retardation effect could persist as long as the new and smaller plastic zone remained confined within the larger one, as schematized in Fig. 11.25. Figure 11.25a refers to the first cycle after overload; the plastic zone produced by normal cycles is smaller than the plastic enclave created by the overload. In (b) the plastic zone of the running crack is still within the overload plastic enclave; retardation persists. Finally, Fig. 11.25c, the plastic zone has reached the border of the larger overload plastic enclave; retardation is about to come to an end. Recalling that the crack tip plastic zone radius r _p is given by

$$ \begin{array}{*{20}c} {r_{p} } & = & {\frac{1}{\alpha \pi }\left( {\frac{{K_{max } }}{{\sigma_{y} }}} \right)^{2} } \\ \end{array} $$

(11.22)

with σ _y being the yield strength of the material and α equal to 2 for plain stress conditions and 6 for plain strain conditions, it is likely to assume that the retardation effect will last up to the moment when the crack growth Δa will equal the overload plastic zone size 2·r _p,ovrld

$$ \begin{array}{*{20}c} {\Updelta a} & \cong & {2r_{p,ovrld} } \\ \end{array} . $$

(11.23)

Fig. 11.25

Wheeler retardation model

Figure 11.26 [34] presents the crack growth measurements done for a prismatic specimen type SEN (see Appendix A) 100 × 20 × 10 mm of 9Cr1Mo steel, after a series of applied cycles with K _a equal to 12 MPa√m and R = 0.1. The overload was equal to three times the normal cyclic load and resulted in an applied 3·K _a  = 36 MPa√m. It is interesting to see how from the moment of overload application the crack has grown of about 1.8 mm before returning to grow with the same rate as before. The alloy considered has yield strength of about 450 MPa. Considering that the growth measurement were done on the surface where triaxiality is practically null and the plain stress conditions prevail, Eq. (11.22) yields the value r _p,ovrld  ~ 1 mm and, therefore, the plastic zone size is about 2 mm wide, very close to the 1.8 mm crack growth. Wheeler introduced an empirical retardation factor Φ _R

$$ \begin{array}{*{20}c} {\Upphi_{R} } & = & {\left( {\frac{{\Updelta a + r_{p} }}{{r_{p,ovrld} }}} \right)^{\gamma } } \\ \end{array} $$

(11.24)

in which γ is a material constant that can be inferred from the interpolation of experimental data. The Φ _R factor corrects the FCGR, da/dN, according to the relationship

$$ \begin{array}{*{20}c} {\left( \frac{da}{dN} \right)_{R} } & = & {\Upphi_{R} \frac{da}{dN}} \\ \end{array} $$

(11.25)

from which it can be assessed the number of cycles N _R the crack needs to extend through the plastic zone generated by the overload

$$ \begin{array}{*{20}c} {N_{R} } & = & {\frac{1}{C}\int\limits_{{a_{o} }}^{{a_{f} }} {\frac{da}{{\Upphi_{R} \cdot \Updelta K_{eff}^{n} }}} } \\ \end{array} $$

(11.26)

with a _o being the initial crack length at the moment of overload imposition and a _f the final one after the growth through the overload plastic zone r _p,ovrld when the retardation effect vanishes

$$ \begin{array}{*{20}c} {a_{f} } & = & {a_{o} } \\ \end{array} + r_{p,ovrld} - r_{p} $$

(11.27)

where r _p is the radius of the crack tip plastic zone produced by the constant load cycling. Note that in Eq. (11.26) the Paris-Erdogan equation takes into consideration the possibly existing R-ratio effect through the use of the ΔK _eff . Actually, things can be more complicated than so far supposed. In fact, von Euw et al. [30] observed that retardation can be, in turn, delayed, as schematically shown in Fig. 11.27.

Fig. 11.26

FCGR, da/dN, before and after the imposition of an overload (reproduced with permission of [34])

Fig. 11.27

Delayed retardation after overload imposition

This delayed retardation doesn’t occur immediately after the overload imposition. The crack does not follow path AC of Fig. 11.27, but rather an initial phase AB in which the FCGR is still high, dough lower than before overload, followed by the real retardation BC. Elber [35] and Bernard et al. [36] ascribed the overload retardation effect to the system of residual stresses created in the plastic zone invoking a crack closure effect, such that illustrated in Fig. 11.23b. Also the sequence of different load histories application can have an impact on the retardation effect. If, for example, two load histories are applied to the system, each having a constant amplitude equal to σ _a1 and σ _a2 , respectively, with σ _a1  ≫ σ _a2 , the sequence of application σ _a1  → σ _a2 will give rise to a fatigue crack growth retardation that, on the contrary, will not be seen at all in the opposite sequence σ _a2  → σ _a1 . This is schematized in Fig. 11.28 [37] in which AA′ and BB′ are the crack growth curves under the two loading sequences, respectively. More recently, Kim et al. [38] working with specimens of AISI 4340 steel found that the fatigue crack after retardation bifurcated. Bifurcation leads to a reduction of the stress intensity factor K _I that has to distribute upon two crack tips, not necessarily in equal part. Therefore, bifurcation produces a FCGR reduction that persist up to the moment when one of the two crack tips, the one having the higher ΔK _I that advances faster, takes the lead and eventually obscures the other rebuilding the single crack ΔK _I . However, this circumstance seems not be confirmed by other investigators so that bifurcation must be associated to that single experience. Temperature can strongly reduce the overload retardation effect. To this end, referring to the previously mentioned experience of Cotterill and Knott [34] shown in Fig. 11.26, while at room temperature the fatigue crack growth Δa preceding the overload imposition was restored after a crack growth approximately equal to the overload plastic zone width at 525 °C this recovery happened before, after about 10–20 % of the plastic zone size. This recovery must be attributed to the overload residual stress relaxation operated by temperature that eventually, cancel out completely the residual stress state, depending on the temperature. It is worth noting that a dwell period, after overload is imposed, before restarting the cycling at constant load reduced dramatically the retardation severity at 525 °C but not at RT or 225 °C. This confirms that the retardation is attributed to thermally-activated time-dependent process, which cancels the effect of overload due to plasticity-induced closure.

Fig. 11.28

Load sequence effect on overload retardation [37]

11.7 Growth of Short Cracks

The definition of micro-crack and macro-crack has already been introduced in Sect. 1.5. Micro-cracks, also called MSC or mechanically small cracks, are defects generated during Stage I of fatigue, see Sect. 1.5.2 and Fig. 1.64, within a grain or few grains and have a dimension varying from just 3 μm to about 300 μm. We shall consider MSC as small cracks, just increasing the size a little bit up to 1 mm. Long cracks go beyond 1 mm length. Such a distinction is not just heuristic or related to the validity of continuum mechanics or fracture mechanics. Point is that during the last decades it has been found that small or short cracks grow earlier (smaller ΔK _th ) and at faster rate than long cracks [39–44]. Worth mentioning are results shown in Fig. 11.29 obtained by Pineau et al. [44] on a ductile or nodular cast iron 3.8 % C, 3.21 Si, 0.27 Mn, 0.04 Mg, 0.01 P, with yield strength equal to 270 MPa, ultimate of 450 MPa, HV30 hardness equal to 185 and fatigue limit at 10⁷ cycles of 205 MPa. The technique to obtain small cracks is that of using notched specimens fatigue pre-cracked that are machined to remove a surface layer up to leave a crack of wanted size, in any case smaller than one millimeter. The diagram of Fig. 11.29 [44] shows the comparison of FCGR, da/dN, obtained with specimens carrying long cracks (10 mm) and specimens with short cracks of 240, 330 and 550 μm, respectively. Some features are worth noting.

Fig. 11.29

FCGR, da/dN, obtained by Pineau et al. [44] for specimens of ductile cast iron carrying long cracks (10 mm) and short cracks

First, it can be seen that in Region I of fatigue the threshold stress intensity factor K _th depends on the crack size; about 5 MPa√m for the 240 μm crack, about 6 MPa√m for the 330 μm crack and about 7 MPa√m for the 550 μm crack, versus the 8,2 MPa√m of long cracks. Second, the FCGR, da/dN, of short cracks is Region I of fatigue depends on the crack length and is higher for smaller cracks. Finally, data coming from all types of cracks merge in Region II of fatigue, as already seen for the R-ratio effect. However, in the case of short cracks this findings may have other reasons. One of the conditions of validity of linear elastic fracture mechanics (LEFM) and K _I is that the crack tip plastic zone size be small, negligible with respect to all other dimensions. Now, since the plastic zone radius given by Eq. (11.22) already at ΔK _I  = 10 MPa√m is of the order of 300/400 μm, i.e., equal or larger than the same crack size one may argue that LEFM is not valid any longer. On the contrary, data coming from long cracks are valid in terms of ΔK _I , since the crack size is at least an order of magnitude larger than the crack tip plastic zone size. Moreover, in Region I a crack closure effect can be responsible of the larger FCGR, dough with the limitation already evidenced about the applicability of LEFM. Measurements of ΔK _op for the various crack sizes have yielded the results of Fig. 11.30 [44].

Fig. 11.30

Measurements of ΔK _op as function of crack size [44]

It can be seen that below one millimeter of crack length, the ΔK _op reduces progressively, whilst for cracks longer than one millimeter it remains constant at about 4 MPa√m. On the base of Eq. (11.17) this finding results in a parallel increase of ΔK _eff for short cracks that could explain the larger FCGR in Rsegion I of fatigue. Reanalyzing the experimental data of Fig. 11.29 with respect to ΔK _eff it is obtained the diagram of Fig. 11.31 [44]. The discrepancy among the various FCGR reduced very much.

Fig. 11.31

Reanalysis of data of Fig. 11.29 with respect to ΔK _eff [44]

However, this behavior considered characteristic of small cracks is not always confirmed by experimental data. Several researchers [45–48] reports episodes of complete independency of FCGR of the crack length. An example of such a behavior is offered in Fig. 11.32 for quenched and tempered 4340 steel, 1187 MPa yield strength, whose FCGR of small cracks is practically the same of long cracks.

Fig. 11.32

FCGR, da/dN, observed for small and long cracks in 4340 steel [48]

Small cracks were penny shaped with depth a from 15 to 200 μm. Fatigue loads were pulsating with R = 0.1. A comprehensive review of existing FCGR data for short and long cracks has been done by Lankford [48] in order to understand differences in the fatigue behavior of short and long cracks. Examining 11 different alloys for which LEFM was applicable to the FCGR measurement, Lankford tried to establish a relationship between the short crack capability to grow faster than long cracks and a metallurgical characteristic M that could be the grain size or, in the case of martensitic steels, the martensite lamellae. He found that the discrepancy between small and long cracks held when the radius r _p of the crack tip plastic zone was smaller than this metallurgical characteristic M that is to say when the plastic zone was well within the grain. Table 11.2 provides a synoptic picture of Lankford results [49]. It is shown the minimum crack size, a _min , of small cracks and the FCGR with respect to long cracks; yes means equal behavior, whilst no indicates different behavior. Characteristic, to this end, are results obtained for aluminum alloys, practically identical, DTD 5050 and 7075-T651 [39–48] shown in Fig. 11.33 [49]. Some small cracks arrested completely within the grain (curve A), others, after some growth delay with a minimum about the average grain size, restarted to grow (curves B, C and D). Others, finally, grew with a rate higher than that of long cracks up to merge into Region II of the da/dN versus ΔK _I diagram (curve E). The different behavior was ascribed to metallurgical barriers, as the grain boundaries, on cracks too short, as already discussed in Sect. 1.5. Table 11.2

Table 11.2 Materials, RCCF and mechanical-metallurgical parameters [49]

Fig. 11.33

FCGR of small and long cracks in aluminum alloys type DTD 5050 and 7075-T651 [49]

11.8 Variable-Amplitude Load Fluctuation

It has been considered, in Sect. 11.6, the case of single or multiple overloads imposed to constant load cycling. We shall now examine the more general case of a spectrum of variable-amplitude load fluctuation. Many attempts of addressing the issue and many models have been proposed with varying degrees of success [50–53]. A rather interesting approach has been proposed by Barsom [50] and is referred to as the root-mean-square (rms) model. The model is based on the usual approach of defining a single stress intensity factor ΔK _eff to be used in a Paris-Erdogan expression that results in the same FCGR as that produced by the actual load time history.

$$ \frac{da}{dN} = A\left( {\Updelta K_{eff} } \right)^{n} . $$

(11.28)

The Barsom approach is based on the root-mean-square

$$ \Updelta K_{eff} = \Updelta K_{rms} = \sqrt {\frac{{\sum\nolimits_{i = 1}^{k} {\Updelta K_{I}^{2} } }}{n}} $$

(11.29)

where ΔK _I is the stress intensity factor excursion relative to each peak of the loading sequence. The method is similar to that successfully used in seismic engineering when it must be assessed the stress response of a complex system, such as a long and branched piping, anchored to many different points of different buildings or structures having different individual floor response spectra not-in-phase in case of a seismic event. The same method has been inspiring the approach suggested in Sect. 8.5.4 referred to as equivalent spectrum. In his original work Barson considered two different loading spectra as reference spectra. These to spectra are shown in Fig. 11.34 [50]. They are characterized by a non-symmetric probability density function (pdf, see Chap. 4) with a modal value or peak value P _mod (see Fig. 4.3) different from the mean one. The first spectrum, Fig. 11.34a, is characterized by a ratio of the standard deviation Δ to peak value, Δ/P _mod, equal 1. The second, Fig. 11.34b, by ∆/P _mod = 0.5. Specimens of WOL type (see Appendix A) were repeatedly subjected to load time history (a) and (b), respectively, and the crack length was measured as function of the elapsed number of cycles N. results are shown in Fig. 11.35 [50]. The stress excursion ∆σ indicated in Fig. 11.35 is that relative to the peak or modal value σ _mod of each spectrum. Figure 11.35 also reports the results obtained with a constant load spectrum, σ _a  = σ _mod , for which the standard deviation is zero. As it can be seen, the fastest FCGR is that pertaining to the load spectrum (a) of Fig. 11.34 whilst the lowest FCGR competes to the constant load spectrum. The tangent at any point of the curves provides the FCGR, da/dN, relative to the selected crack length. To this length a ∆K _mod can be associated through the calibration function f(a/w) that for the specimen considered is known. The result is a FCGR versus ∆K for all three curves of Fig. 11.35, as shown in Fig. 11.36a [50]. If instead of the ∆K _mod , the ∆K _rms is used the three different curves of Fig. 11.36a merge into almost a single one, as shown in Fig. 11.36b [50]. Barsom completed his experimental campaign using different load spectra as those shown in Fig. 11.37 [50]. Spectra were always including also the constant load time history as reference. Results in terms of FCGR versus ∆K _rms are shown in Fig. 11.38 [50]. The give credit to the method proposed.

Fig. 11.34

Two variable-amplitude random-sequence load fluctuation investigated by Barsom [50]

Fig. 11.35

Fatigue crack growth under spectra loads of Fig. 11.34 [50]

Fig. 11.36

a FCGR, da/dN, as a function of the modal stress intensity factor ΔK _mod . b of the ΔK _rms [50]

Fig. 11.37

Various random sequence used by Barsom [50]

Fig. 11.38

Summary of FCGR, da/dN, under the load fluctuations sequences of Fig. 11.37 [50]

11.9 Sample Problems

11.9.1 FCGR in Helicopter Blade

The blade of a helicopter rotor during a scheduled inspection has been found affected by a tiny surface crack on the side put in traction the wing lift, as schematized in Fig. 11.39. Its dimensions are a _o  = 0.3 mm depth and c _o  = 1.4·a = 0.42 mm length. The thickness of the metal, an aluminum alloy type 6061-T651, is t = 4.5 mm. The alloy has a yield strength of 266 MPa. The blade has been instrumented with strain gauges that found a tension stress variable from 26 to 75 MPa. Knowing that the rotor has 200 rpm calculate how many life hours are left before the crack becomes a through-wall-crack, assuming that the crack grows maintaining always the same aspect ratio 1:1.4. The blade section can be approximated by a long and thin plate. For this latter geometry the expression of the stress intensity factor K _I is (see Appendix A)

$$ K_{I} = \sigma \sqrt {\frac{\pi a}{Q}} \cdot F\left( {\frac{a}{c};\frac{a}{s};\frac{a}{w};\theta } \right). $$

(11.30)

Fig. 11.39

Blade section and crack detail

The two non-dimensional functions F and Q, whose expression are given in Appendix A, depend on the crack size a as shown in the diagram of Fig. 11.40. This means the explicit integration of the Paris-Erdogan equation is rather difficult. It will be used the finite difference integration approach.

Fig. 11.40

F and Q functions vs. crack depth for c/a = 1.4

Since the stress is varying from a minimum of 26 MPa to a maximum of 75 MPa, the stress amplitude ∆σ = 75 − 26 = 49 MPa is applied with an R-ratio of 0.35. Its effect will be affecting the FCGR, da/dN. To consider this effect, the Walker correction will be used, see Eq. (11.10) and the ∆K replaced by the ∆K _eff

$$ \begin{gathered} \frac{da}{dN} = C \cdot \Updelta K_{eff}^{n} \hfill \\ \Updelta K_{eff}^{{}} = \Updelta K\left( {1 - R} \right)^{p} . \hfill \\ \end{gathered} $$

(11.31)

For the aluminum alloy used for the blade, the constant C, n and p are

$$ C = 7,7 \cdot 10^{{ - 8}} ,\,n = 3,45\,and\,p = - 0,5 $$

The integration of the Paris-Erdogan Eq. (10.10) by the finite differenced method (see example 1 of Sect. 10.10) is done dividing the possible crack growth interval Δa = 4.5 − 0.3 = 4.2 mm into 30 equal parts in each of which the ΔK _eff and the F and Q factors can be considered constant. Results are shown in Table 11.3. To become a through-wall-crack there will be necessary 1.7·10⁷ cycles. Considering that the rotor angular velocity is 200 rpm, it will take

$$ t = \frac{{1.7 \cdot 10^{7} }}{200 \cdot 60} = 1416.67\,{\text{h}} . $$

Table 11.3 Results of finite difference integration of the Paris-Erdogan equation