Chapter 1

1 Overview

Introduction

This Chapter provides an overview of the notions used in this Thesis, starting with the derivation of stress vs. number of cycles (S/N) curve models for short cracks, since the work of these years is relying on the stress-life approach. Then, an overview of crack propagation models is given, in fact power law type crack propagation equations are integrated to recharacterize the concept of S/N curve for long cracks. Subsequently, the effect of notches in the stress-life formulation is treated mostly by means of the theory of the critical distances. Besides providing a variety of fatigue S/N curve models, a brief overview of damage accumulation rules for fatigue prediction deriving from variable amplitude loading is given. The Chapter concludes with a brief overview of the regulatory aspects of fatigue tolerance evaluation for rotorcrafts.

1.1 The SN curve

The advent of the First Industrial Revolution in the 18th century caused a dramatic increase of the frequency of work of the machines; this event could be identified as the trigger for the widespread study of fatigue in the last two centuries. Indeed, even if it has been known for centuries that repeatedly applying a load causes early ruptures, as it used to happen to long distance travelling boats, it was only around 1830s that engineers and scientists started investigating how a load much lower than the material strength can cause failure if applied many times. The first reference to the fatigue of metals dates to 1837 when Albert [1], a German mining administrator, published the first document in history relating to a fatigue test was published. The test was aimed to understand the causes of the failure of the conveyor chains in the Clausthal mines in 1829. The test involved the entire component, not only a representative specimen of material. Therefore, since the replacement hemp rope was an expensive good, Albert invented the wire rope which certainly is the innovation he is remembered for. Independently, in 1839 Poncelet [2], a French military engineer, used the adjective “tired” (in French, fatigué) to describe steels under cyclic stress and assumed that steel components experience a decrease of durability when they undergo repeated variable loads. In 1843 Rankine [3] and York [4], [5] focused their attention on railway axles thanks to the establishment of the Her Majesty’s Railway Inspectorate instituted due to the increasing number of accidents, amongst which the so-called Versailles disaster where at least 55 people lost their lives due to the failure of the axle tree of the first engine on the 5th October 1842. Anyway, the term fatigue was coined only in 1854 by the Englishman Braithwaite [6], who discussed the fatigue failures of multiple components as water pumps, brewery equipment and, of course, railway axles. Many other English and German studies on the deterioration of railway axles were conducted in those years [7]–[10], but the work of Wöhler, Royal “Obermaschinenmeister” of the “Niederschlesisch-Mährische Railways in Frankfurt an der Oder”, was the milestone paving the way for the modern conception of fatigue testing and interpretation of results. In 1858 [11] and 1860 [12] August Wöhler measured for 22,000 km the service loads of railway axles with deflection gages personally developed and from his studies concluded that “If we estimate the durability of the axles to be 200,000 miles with respect to wear of the journal bearings, it is therefore only necessary that it withstands 200,000 bending cycles of the magnitude measured without failure”. Such statement represents the first suggestion for a safe life design philosophy with retirement time (or distance travelled). Wöhler then calculated the stresses deriving from service loads and concluded that the higher the stress amplitude is, the more detrimental influence on the axle will be, plus a tensile mean stress anticipates the rupture. Furthermore, he even stated that a replacement of the axle would have been necessary if radial flaws propagated up to 20 mm into the material, and this procedure could be interpreted as an ancestor of the flaw tolerant safe life methodology [13]. Notwithstanding, Wöhler’s test results were tabulated and not plotted until 1875, when Spangenberg [14] adopted unusual linear axes to present these data. Furthermore, stress vs. number of cycles (S/N) curves were addressed as Wöhler curves only in 1936 by Kloth and Stroppel [15]. The idea to plot many fatigue test data, including the 60 years old Wöhler’s experiments, in logarithmic axes and interpolate them with a power law is from 1910 by Basquin [16]. The equation relating maximum or alternate stress with the number of cycles is

S = b̅ · N^ā

(BASQUIN - 1.1)

Where S denotes the stress, alternate or maximum, N the number of cycles and and are constants depending on the material. Later such law took his name. The proof of existence of a fatigue limit was given four years later by Stromeyer [17]. In 1914 he conducted tests on small scale specimens in order to reduce to a minimum the difference in terms of chemical composition and mechanical properties in the component under test. The specimens were loaded in bending and twisting moment and the stress plotted against the empirical formula (10⁶/N)^1/4 resulted in a straight line corresponding to the fatigue limit. Stromeyer also theorized that if the maximum stress would have exceeded the yielding, the constant slope ¼ would have certainly changed. The final form of Stromeyer’s law simply adds a constant term to (1.1).

S = b̅ · N^ā + S_e

(STROMEYER - 1.2)

Where S_e is the fatigue limit and for the same material and defined in (1.2) have different values from the corresponding quantities defined in (1.1). In 1924 Palmgren [18] in his studies for the ball bearings life estimation introduced a term B to (1.2), i.e.

S = b̅ · (N+B)^a + S_e

(WEIBULL - 1.3)

The B introduces an inflection point in the equation both when plotted in S/log(N) and when plotted in log(S)/log(N) axes, consequently low cycle fatigue data are better fitted by this model. Indeed, as suggested by Shanley [18] in the 1956 “Colloquium on Fatigue” [20], Basquin’s law fails to model low cycle fatigue since it does not predict correctly the strain at high stresses, while by using (1.3) one supposes that a stress close to the ultimate tensile strength causes much lower strain, hence it can be applied a certain number of times without failure. This is also confirmed by Epremian and Mehl [21] that in 1952 showed that an S/N diagram, when the alternate strength is close to the ultimate tensile strength, can be fit with very good agreement by a probability scale instead of a logarithmic scale and this suggests that at high stresses alternating plastic strain dependence on stress amplitude is primarily of statistic nature. Nevertheless, since not including physically based assumptions on the plasticization effect, an S/N curve cannot replace a strain-life approach[22], [23] which has been specifically introduced for life prediction in the low-cycle fatigue regime. It rather aims providing a qualitative description of the low cycle fatigue regime, as the Basquin’s law when truncated at low number of cycles. The model found by Palmgren has been widely used by Weibull since 1949 [24]. For this reason, (1.3) will be addressed here as Weibull’s law. Engineer and mathematician, Ernst Hjalmar Waloddi Weibull (1887-1979) gave a huge contribution to material science and statistics in his prolific scientific career. Concerning fatigue, there are tens of documents, many of which are ICAF (International Committee on Aeronautical Fatigue and Structural Integrity) proceedings [25]–[28] and reports to the Aeronautical Research Institute of Sweden (FFA) [29]–[33]. His contribution to the field is principally, but not only, related with the statistical aspects of fatigue. Weibull, in Sweden, approached material science by developing a theory of static strength [34], then his interest extended to fatigue [24]–[28], [30]–[33]. (1.3) has been used by the author in Chapter 5. It is worth of mention that Yokobori [35] and Shanley [19] developed independently theories for interpreting the parameters b and a from physical quantities as absolute temperature, loading frequency, number of preferred nucleation sites per unit volume, etc. As stated by Weibull, (1.3) is the most realistic way of describing S/N data pretty much all over the domain, nevertheless the model has not been extensively used in history and usually Basquin’s law is preferred because of its simplicity. Indeed, most usually the simplified power law is preferred to the four parameters law, and for this reason most of the databases in Literature are available as Basquin’s constants b, a. This is also due to some generalizations and links that can be derived quite naturally starting from a pure power law fatigue behavior.

1.2 The crack growth curves

Crack growth curve equations have been extensively studied since the ‘50s of the last century, almost in conjunction with the first catastrophic accident of the first jet transportation aircraft, the De Havilland Comet 1 [36]–[38] (1954), where the propagation of fatigue cracks in the upper fuselage panels, starting from the sharp corner of the top rectangular windows brought 35 deaths. As Frost and Dugdale [39] already pointed out in 1958, the cylindrical specimens were the protagonists of the first half of the 20th century in the panorama of fatigue, and this typology of specimens makes really complex the study of crack growth, meaning that no special attention up to that moment was given to crack growth testing of sheet specimens. Secondly, Frost and Dugdale observed that “In aircraft structures the ‘fail-safe’ design philosophy requires the structure to be constructed in such a way that fatigue cracks do not cause catastrophic failure before corrective measures can be taken”. The first scientist that emphasized the necessity to have polished surfaces to minimize the hotspots from where a crack can propagate is Griffith [40] who extended the theorem of minimum potential energy to the phenomena of rupture of elastic solids. Griffith’s work was motivated by Inglis’ [41] linear elastic solution for the stress around an elliptical hole asymptotically loaded in tension, from which he predicted that the stress would go to infinite as the ratio between the minor and major axis goes to zero. Griffith’s theory provides correct predictions as long as brittle materials, such as glass or ceramics, are considered. Starting from the pioneering work of Griffith, Berto and Lazzarin [42] provided an exhaustive overview of local approaches for the description of brittle and quasi-brittle fracture of engineering materials. Anyway, since in structural materials there is almost always some inelastic deformations around the crack faces, Griffith’s hypothesis of linear elastic medium in structural metals application becomes highly unrealistic. For this reason, the first crack growth equation relating the stress with the crack growth rate (i.e. the crack length increment per cycle) did not make use of the elastic energy approach. The formulation dates 1953 and has been proposed by Head [43]; it is based on Inglis’ [41] solution and the final simplified form of the crack growth equation is

da/dN = φ(S_a, S_y, S_f') · a^3/2 · t_p^-½

(HEAD - 1.4)

Where N is the number of cycles, φ(S) is (asymptotically) a linear function of the stress, yielding, and strength, a is the half crack size, and t_p is the thickness of the plastic zone ahead of the crack tip. Frost and Dugdale [39] argued that t_p is not a constant independent of crack length and derived the exponential model for crack propagation, seldom used up to nowadays

da/dN = K_FD · S_a³ · a	(a)	(FROST-DUGDALE - 1.5)
ln(a/a_i)= K_FD · S_a³ · N	(b)	(FROST-DUGDALE - 1.5)

In which a_i is the initial size of the crack and K_FD is an experimental quantity depending on the cubic power of the remote alternating stress S_a. During WWII, a group of researchers of the U.S. Naval Research Labs headed by George Rankine Irwin realized that plasticity plays an important role in [44] in fracture mechanics. On this purpose, Griffith’s energy formulation was modified in order to make it account for plasticity, too, i.e. the energy release was redefined by adding a plastic dissipation term. Another major achievement of Irwin’s work is certainly the relation between the energy release rate G and the stress intensity factor in opening mode K_I:

G = K_I²/E^*

(GRIFFITH - 1.6)

Where E^*=E for plane stress or E/(1-ν)² for plane strain. The critical stress intensity factor is the value of K beyond which a crack starts to propagate and is addressed as fracture toughness K_C. Namely, the toughness is the resistance to fracture of a material, it is a material property and is defined as the stress intensity factor required for a crack to advance from length a to aC. The fracture toughness values have been grouped by material family by Ashby [45], [46] and are shown in Figure 1.1.

Figure 1.1 - Fracture toughness against yield strength (from Ashby [45], [46]). The dotted lines are the value of K_Ic/(πS_y²), i.e. approximately the diameter of the process zone

Few years later, in 1963, Paris and Erdogan [47] published a work substantiated by many experimental tests where they postulated, differently from Head or Frost and Dugdale, that the crack growth is described by a power law of the stress intensity factor, viz.

da/dN = C · ΔK^m

(PARIS-ERDOGAN - 1.7)

At the time of publication, the authors were uncertain on the value of the exponent m, in fact there is a famous statement in their paper saying: “The authors are hesitant but cannot resist the temptation to draw the straight line slope 1/4 through the data...”. Therefore, the so-called Paris’ law, or Paris-Erdogan law, has been formulated in principle with fixed m=4. Indeed, as evident from Figure 1.2 (taken from Ashby [45], [46]), the majority of engineering alloys is concentrated in the neighborhood of m=4, and this may have tricked the authors.

Figure 1.2 - Ashby map for Paris’ slope against toughness ratio (from [45], [46]). Many engineering alloys are concentrated in the range (ΔΚ_th/K_Ic=0.1, m=4)

Paris’ law is considered valid within the range ΔK_th<ΔK<ΔK_cr, where ΔK_cr is the critical stress intensity factor range which depends on the toughness as ΔK_cr=(1-R)K_Ic, with R load ratio, and ΔK_th is the threshold value below which the crack should not propagate¹. In the following years there have been many attempts to generalize Paris’ law, mainly to account for mean stress effect, crack closure and near threshold/near failure modelling. The simplest model of Paris’ law for mean stress effect has been proposed in 1970 by Walker [49]:

da/dN = C₀ · ΔK^m / (1-R)^(1-γ)·m

(WALKER - 1.8)

With γ being Walker exponent and R the load ratio. A more general form of Walker equation had already been given in 1967 by Forman et al. [50]:

da/dN = C · ΔK^m · [K_c - ΔK/(1-R)]^-1

(FORMAN ET AL. - 1.9)

Forman equation is, to some extent, the ancestor of the most sophisticated crack propagation equation available in Literature: the NASGROTM equation

da/dN = C · [(1-φ)/(1-R) · ΔK]^m · (1-ΔK_th/ΔK)^p / (1-K_max/K_c) ^q

(NASGRO - 1.10)

Where the mean stress effect is accounted for through (1–R), and the crack closure effect through (1–φ). The equation is shown in Figure 1.3 for a sample ASTM A579 Grade 75 steel, forged. Three stages of the crack propagation are easily identified: (i) in the near threshold regime (so-called stage I) slight changes in the microstructure imply high changes in the crack growth rate; (ii) in the stable propagation the crack grows as a power law (Paris’ law); (iii) when the crack reaches a length close to the critical ac, the propagation becomes unstable. Generally, Stage I and II take almost the 90% of the entire life.

Figure 1.3 - Example crack growth curves according to NASGRO 3.0 equation for an ASTM A579 steel. The mean stress effect is clearly visible from the figure

The quantity f in Equation is called Newman [51] opening function and is the ratio between the opening and maximum stress (intensity factor). i.e.:

φ = K_op/K_max =	max(R, Σ_j=0³ A_j · R^j)	R ≥ 0	(1.11)
	Σ_j=0¹ A_j · R^j	-2 ≤ R < 0

Where A_j are some empirical constants depending on a value α ranging from 1 (plane stress) to 3 (plane strain), the maximum stress S_max and the flow stress S₀ = ½·(S_U+S_y) (average between the yielding and ultimate tensile strength).

A₀ =	(0.825 - 0.34·α + 0.05·α₂) · ^α√cos(π/2·S_maxS₀)	(1.12)
A₁ =	(0.415 - 0.071·α)·S_maxS₀
A₂ =	1 - A₀ - A₁ - A₃
A₃ =	2A₀ + A₁ - 1

The near threshold and near failure areas are dealt through the exponents p and q. As regards the calculation of the ΔKth, the NASGRO 3.0 [52]² equation defines

ΔK_th = ΔK₀ · [a/(a + a₀]^½ / [(1 - φ)/(1 - A₀)/(1 - R)]^{1 + C_thR}

(1.13)

Being Cth a constant to be calibrated and ΔK0 the threshold stress intensity factor for long cracks calculated at R=0. Equation (1.13) has been derived to take into consideration the small crack effect demonstrated by Tanaka et al. [53]. The value a0 is a quantity of paramount importance: the intrinsic crack size. It was defined for the first time in 1980 by El Haddad [54]. El Haddad’s constant is, by definition, proportional to the square of the ration between the fatigue threshold and the fatigue limit (at fixed load ratio), i.e.

a₀ = 1/π ·(ΔK_th/ΔS_e)²

(1.14)

Cracks smaller than El Haddad intrinsic size do not follow Paris’ law even for ΔK>ΔKth, whereas the fatigue behavior in this range of crack size is ruled by the fatigue limit. The quantity a0 is also used in the famous interpolating equation proposed by El Haddad-Dowling-Topper-Smith [54] and modified by Atzori and Lazzarin [55]–[58] in the following form:

ΔS_e = ΔK_th/ √[π(f·a + a₀)]

(1.15)

This equation fits the data of the Kitagawa-Takahashi [59] diagram³ first proposed in 1976:

Figure 1.4 - Kitagawa-Takahashi diagram for some material data taken from [53], [54], [59]. The red line is El Haddad equation, while the grey line is the Kitagawa-Takahashi criterion. zone

1.3 Notch fatigue

The idea behind Equation (1.14) is one aspect of the “Theory of the Critical Distances” (TCD) whose ancestor, as summarized by Taylor [62], and Yao et al. [63], can be identified in the effective stress concentration factor Kf first proposed by Neuber [64] in 1946, then picked up by Kuhn and Hardrath [65] who in the early ‘50s assumed that the notched specimen fails if the averaged stress over the distance AKH ahead of the notch root is equal to the fatigue limit Se of the plain specimen. From this hypothesis, Kf was calculated as

K_f = 1 + (K_t - 1)·[1 + π/(π - ω)·√(Α_ΚΗ/ρ)]^-1

(1.16)

Where K_t is the theoretical stress concentration factor (e.g. 3 for a circular hole in an infinite plate, from Kirsch [66] solution), ω is the notch flank angle, ρ is the notch root radius and the distance A_KH is a material constant. The stress concentration factor equation was later modified by Neuber as

K_f = 1 + (K_t - 1)·[1 + √a_N/ρ)]^-1

(1.17)

In 1949 Peterson [67] derived his version of the stress concentration equation based on the hypotheses that (i) the notched material fails if the point stress at a distance d0 away from the notch root is at least equal to the fatigue strength of the plain specimen and (ii) the stress ahead of the notch root drops linearly up to d₀ obtaining

K_f = 1 + (K_t - 1)·[1 + √a_P/ρ)]^-1

(1.18)

Where a_P is Peterson’s material constant. Nevertheless, as also confirmed by Topper et al. [68], as ρ increases the fatigue limit is actually fully controlled by the theoretical stress concentration factor, thus K_f→K_t since a_P/ρ→0 and the notch is addressed as blunt notch. From a mathematical point of view, a notch can be effectively be addressed as blunt when its characteristic size a* excedes [56]–[58], [69]

a* ≥ K_t²·a₀

(1.19)

Hence, for instance for a hole in an infinite plate this occurs when the circle radius is almost one order of magnitude larger than a₀. For aluminum alloys and steels, typically 10 μm < a₀ < 100 μm respectively implying 0.1 mm < a* < 1 mm respectively. Therefore, Atzori and Lazzarin suggested the following infinite life design criterion for notched components

K_f = min⁡[√(1 + a/a₀), K_t]

(1.20)

The combination of Equations (1.19) and (1.20) implies that below a* notches behave similarly to cracks, and above a* they behave as blunt notches. In the last decades, tens of notch sensitivity estimation models for infinite life design have been proposed. Most of them have been collected in 2004 by Ciavarella and Meneghetti [70] who reviewed a series of classical and modern approaches to the stress concentration factor estimate concluding that Neuber’s method [64] is the most conservative and accurate among the “classical” approaches whilst the Atzori-Lazzarin criterion is the most conservative yet easy-to-use between the “modern” ones. Therefore, they proposed their personal modification to the Atzori-Lazzarin criterion to make it consistent with Lukáš and Klesnil [71] discussion which can be interpreted as a modification of Neuber’s rule including the effect of cyclic plasticity. The Ciavarella-Meneghetti criterion for infinite life design is:

K_f = min⁡{[1 + (a/a₀)^r]^1/(2r), K_t} 0 < r ≤ 1

(1.21)

Equation (1.21) for r = 1 obviously returns the Atzori-Lazzarin criterion, while for r = ½ gives Lukáš-Klesnil criterion. Bazant [72] has shown in detail that the expression [1 + (a/a₀)^r]^1/(2r) is an asymptotic matching with truncation at the first order between the large-size (a≫a0) and the short-size (a≪a0) expansions of the crack propagation criterion in terms of stress intensity factor, concluding that El Haddad equation can be interpreted as a “matching asymptotics” solution for the transition between fatigue endurance towards fatigue threshold dominated threshold. The first modern reformulation of Neubers’ idea is attributed to Tanaka [73]. The formulation is based on the assumption, confirmed by experimental evidence, that the stress that can be withstood at the notch root/crack tip without causing defect can be higher than K_t·S_nom. Thus, Tanaka averaged the local stress ahead of the notch root/crack tip up to a distance l₀ = 2a₀

S_l₀ = 1/l₀ · ∫_a^a+l₀ S(x) dx

(1.22)

From this assumption, the effective stress intensity factor according to the TCD is

K_f = S_l₀/S_nom

(1.23)

Tanaka’s model has been picked up and extended to other variants by Taylor who formulated “a unifying theoretical model” in 1999. Indeed, Taylor’s extension of the TCD defines three variants: (i) point, (ii) line, and (iii) area. Considering the system as described in Figure 1.5, the stress according to the TCD is:

S_l₀ = S(l₀) = S(½a₀)

(POINT - 1.24)

S_l₀ = 1/l₀ · ∫_a^a+l₀ S(x) dx = 1/(2a₀) · ∫₀^2a₀ S(x + a) dx

(LINE - 1.25)

S_l₀ = 2/(πa₀²) · ∫_-½π^½π∫₀^a₀ S(r)·φ(r) rdrdθ

(AREA - 1.26)

Figure 1.5 - Stress field ahead of a crack (red) and of a circular notch(blue) for r, θ=0°. The green point, line and circle express the TCD process zones in its variants

The design criterion just mentioned can be easily extended to finite life. This has been done by Susmel and Taylor [74]–[77] and by Ciavarella et al. [78] through similar, yet different approaches. The key assumption is postulating that the intrinsic crack size follows a power law of the number of cycles up to a critical value a0u defined as a function of the toughness and the fatigue strength at one cycle (or the ultimate tensile strength), i.e.

a₀^u = 1/π · (K_Ic/S'_f)²

(1.27)

Through this assumption the Atzori-Lazzarin diagram can be extended to finite life and a general S/N curve model valid for both crack and notches can be obtained. The model is described in Chapter 2. The validation has been done with experimental data available in Literature (from the SAE Keyhole test program [79]) both considering constant and variable amplitude loading.

1.4 Damage accumulation rules

The majority of the mechanical components undergo complex load histories in their operating life, called variable-amplitude (VA) loading. For this reason, VA life prediction still attracts the attention of engineers and researchers, indeed multiple damage models keep being proposed until very recently. Especially the most recent models tend to be more and more sophisticated; for example, in 2019 Susmel et al. [80] proposed a strain energy density based model to predict VA life of notched components. Another energy based method was recently (2018) proposed by Braccesi et al. [81] to predict VA fatigue life in multiaxial stress state. The model is formulated in the frequency domain and converts the multiaxial stress state into an equivalent uniaxial Von Mises stress that is then used to perform the life prediction calculation. Going back to the basics, the simplest VA fatigue prediction rule has been proposed about a century ago (1924) by Palmgren [18] for the fatigue calculation of ball-bearings. Supposing that the load history consists of N_H cycles that have been counted through one of the multiple cycle-counting algorithms available in Literature. The counted cycles are then grouped into N_B load blocks, each one containing n_j cycles at the stress amplitude S_{a_j} and the corresponding fatigue strength N(S_{a_j})=N_j. Therefore, the damage rule is expressed as

D = Σ_j=1^N_B n_j/N_j

(1.28)

In other words, Palmgren postulated the linear accumulation of the fatigue damage, postulating that the failure occurs as the damage goes to unity without providing a derivation for the rule, and the same holds for Langer [82] that in 1937 postulated the same rule applied separately to the crack initiation and to the crack propagation phases. The first derivation of the linear damage accumulation rule has been formulated by Miner [83]. His hypothesis was that the work that can be adsorbed until failure is a constant value and that the amount of work adsorbed during n_j is directly proportional to n_j. Thus, said W the total work and w_j the work adsorbed during the block n_j, the criterion is Σ_jw_j = W. The use of Miner hypothesis (n_j/N_j = w_j/W) leads immediately to Equation (1.28). Miner conducted a series of tests on smooth and riveted 2024-T3 aluminum alloy sheet specimens by applying load histories having 2 ≤ N_N ≤ 4 and found 0.61 ≤ Σ_jn_j/N_j ≤ 1.45, very close to 1 on average. Since then the linear damage accumulation rule has been addressed very often as Miner’s rule, but probably Palmgren-Miner’s (PM) rule, is the more corrected form and it is how the rule will be called in this work. Since that time many works have been published to verify the PM rule and to find its limits of validity. For example, also the author, together with Ciavarella and Papangelo [84] has co-authored a work which will be dealt with in Chapter 4 where they show that the limit values of PM rule range from 0.001 to 10. Several theories that tried to overcome this limit and generalize the rule have been proposed. The most comprehensive overview of cumulative fatigue damage theories was published by Fatemi and Yang [85] in 1997. The authors identified eight categories of damage rules that they grouped in as many tables, some of which are listed hereafter:

Phenomenologically based damage theories (work before 1970)
To this period belong theories categorizable into five groups: the damage curve approach, endurance limit-based approach, S/N curve modification approach, two stage damage approach, and crack growth-based approach. The damage curve approach defines the damage curve by plotting damage D vs. cycle ratio n_j/N_j. Therefore, the damage curve of the PM rule simply corresponds to the bisector of the first quadrant. The major limitations of PM rule are: (i) no load level dependence, (ii) no load sequence dependence, (iii) no load interaction accountable. Marco and Starkey [86] proposed a load level dependent damage theory that modifies the damage curves at each level j, i.e.

D = Σ_j=1^N_B (n_j/N_j)^ζ_j (1.29)

Where ζj varies at each level j.
Concerning endurance limit reduction theories, they have been introduced to model the effect of residual stresses on the fatigue behavior, as stated by Kommers [87], [88]. The S/N curve modification has been used by Corten and Dolon [89] and by Freudenthal [90] to include load interaction effects. Basically, these methods correspond to a rotation of the power law around a point at low cycles. The first damage theory based on crack growth concept was presented by Valluri [91], [92]. The damage model is based on dislocation theory and elastoplastic fracture mechanics.

da/dN = C_V φ(S) a (1.30)

Refined double linear damage rule and refined damage curve approach
The double linear damage rule (DLDR) basically defines a knee in the damage curve. In the refined version (R-DLDR) the knee point is derived from a first order series expansion of the damage curve approach (DCA) which was empirically formulated by Manson and Halford [93], [94]. Said ai and af the initial and final crack length, the damage curve approach defines an effective crack growth depending on some material parameters β1 and β2, i.e.:

D_j = a_j/a_f = a₀/a_f + (1 - a₀/a_f)⋅(n_j/N_j)^{β₁⋅N_j^β₂}

(1.31)

Where ζ_j varies at each level j.
A comparison between some damage curves is provided Figure 1.6. It can be seen how the slope of the R-DLDR equals the first derivative of the DCA at the extremes.

Figure 1.6 - Comparison between the damage curves for linear damage rule (PM rule), damage curve approach (DCA) and refined double linear damage rule (R-DLDR)

Theories using the crack growth concept
Crack growth approaches have met a wide approval among the damage calculation theories because crack length is the simplest measure of damage. One of the most famous approaches in this direction was proposed by Barsom [95] and translates the VA load history in an equivalent CA load by calculating the root-mean-square of the stress intensity factor range, i.e. ΔK_rms

ΔK_rms = 1/N_B √(∑_j=1^N_H ΔK_j²) (1.32)

Where N_H is the total number of cycles in the load history. It is noteworthy that this empirical method does not require the cycles to be counted, and that it does not account for load sequence effects.

Tens of other cumulative fatigue damage theories have been described by Fatemi and Yang, based on the S/N curve modification approach [96]–[98], or energy-based [99], [100] or even continuum damage mechanics-based [101], [102]. Despite hundreds of damage accumulation theories have been proposed in the last decades, the PM rule remains by far the most used damage accumulation method. Maybe because of its simplicity, or maybe because no other simple rule has demonstrated to be more accurate without proper calibration or even because it has been demonstrated that applying the PM rule corresponds to the integration of the Paris’ law, which for its part still is widely used in crack propagation calculations. As regards this Thesis, the PM rule has been chosen because its application led to some interesting generalizations shown in Chapters 3 and 4.

1.5 Brief outline of regulatory aspects in rotorcrafts Notch fatigue

The oldest method to perform fatigue tolerance evaluation of metallic rotorcraft structures is addressed as safe life. The expression is derived from the concept of safety by retirement, which means that the component is not allowed to show any defect that may weaken it below the design value during its entire lifespan. Safe life breaks down when loads are too high (usually high altitudes/speeds), when fatigue lives shall be extended (for economic reasons), or when stronger materials with poorer fatigue properties shall be used. A graphical representation of the safe life philosophy is given in Figure 1.7. Figure 1.7: Strength vs. life plot showing the strength trend along the component lifespan and the safe life which must be lower than the “minimum expected life”. After the abovementioned Comet accidents, safe life design philosophy was integrated with a safe by design approach called fail-safe. According to this design philosophy, the structure must safely withstand the maximum load without catastrophic failure for all the time between two inspections, even after partial or total failure of one of its principal structural elements. Fail-safe structures must have some redundancies to do so. Obviously, redundancies imply higher weight, which makes the fail-safe design not optimal where not strictly necessary. For this reason, fail-safe has been integrated with the most modern design philosophy called damage tolerance, defined as the ability of the structure to withstand fatigue loads, corrosion or accidental damage until such damage is detected through inspections or malfunctions and it is repaired. A damage tolerant structure is assumed to be flawed and the initial dimension of this defect corresponds conventionally to the maximum non-detectable flaw size. Such sizes are defined in the certification process and are listed, for each component, in a document called threat assessment. A hybrid philosophy between damage tolerance and safe life is addressed as flaw-tolerance safe life design [103], [104]. A flaw-tolerant structure is a safe life structure which can withstand fatigue loads for its entire lifetime even if a flaw (introduced by manufacturing, or inspection) is present. The Authorities (FAA, EASA) clearly state that the safe life design can be applied only after demonstrating that damage tolerance cannot be applied. Typically, in rotorcrafts this happens for the landing gears, main and tail rotor shafts, etc. Therefore, damage tolerance design philosophy is becoming progressively of widespread application, also under the pressure of the Airworthiness Authorities that, after the successful application to the military and commercial fixed wing world as well as to engines, are convinced that this is the gold standard to ensure safety against fatigue cracking and accidental damages. All the concepts just given, together with the duties regarding fatigue substantiation of a rotorcraft principal structural element (PSE), are detailly described in the Federal Aviation Administration (FAA), DOT rules 29.571 (for metals) and 29.573 (for damage tolerance and composites).

Figure 1.7 - Strength vs Life plot showing the strength trend along the component lifespan and the safe life which must be lower than the "minimum expected life"

After the abovementioned Comet accidents, safe life design philosophy was integrated with a safe by design approach called fail-safe. According to this design philosophy, the structure must safely withstand the maximum load without catastrophic failure for all the time between two inspections, even after partial or total failure of one of its principal structural elements. Fail-safe structures must have some redundancies to do so. Obviously, redundancies imply higher weight, which makes the fail-safe design not optimal where not strictly necessary. For this reason, fail-safe has been integrated with the most modern design philosophy called damage tolerance, defined as the ability of the structure to withstand fatigue loads, corrosion or accidental damage until such damage is detected through inspections or malfunctions and it is repaired. A damage tolerant structure is assumed to be flawed and the initial dimension of this defect corresponds conventionally to the maximum non-detectable flaw size. Such sizes are defined in the certification process and are listed, for each component, in a document called threat assessment. A hybrid philosophy between damage tolerance and safe life is addressed as flaw-tolerance safe life design [103], [104]. A flaw-tolerant structure is a safe life structure which can withstand fatigue loads for its entire lifetime even if a flaw (introduced by manufacturing, or inspection) is present. The Authorities (FAA, EASA) clearly state that the safe life design can be applied only after demonstrating that damage tolerance cannot be applied. Typically, in rotorcrafts this happens for the landing gears, main and tail rotor shafts, etc. Therefore, damage tolerance design philosophy is becoming progressively of widespread application, also under the pressure of the Airworthiness Authorities that, after the successful application to the military and commercial fixed wing world as well as to engines, are convinced that this is the gold standard to ensure safety against fatigue cracking and accidental damages. All the concepts just given, together with the duties regarding fatigue substantiation of a rotorcraft principal structural element (PSE), are detailly described in the Federal Aviation Administration (FAA), DOT rules 29.571 (for metals) and 29.573 (for damage tolerance and composites).

“§ 29.571 Fatigue Tolerance Evaluation of Metallic Structure.

A fatigue tolerance evaluation of each principal structural element (PSE) must be performed, and appropriate inspections and retirement time or approved equivalent means must be established to avoid catastrophic failure during the operational life of the rotorcraft. The fatigue tolerance evaluation must consider the effects of both fatigue and the damage determined under paragraph (e)(4) of this section. Parts to be evaluated include PSEs of the rotors, rotor drive systems between the engines and rotor hubs, controls, fuselage, fixed and movable control surfaces, engine and transmission mountings, landing gear, and their related primary attachments.
For the purposes of this section, the term:

Catastrophic failure means an event that could prevent continued safe flight and landing
Principal structural element (PSE) means a structural element that contributes significantly to the carriage of flight or ground loads, and the fatigue failure of that structural element could result in catastrophic failure of the aircraft.

The methodology used to establish compliance with this section must be submitted to and approved by the Administrator.
Considering all rotorcraft structure, structural elements, and assemblies, each PSE must be identified.
Each fatigue tolerance evaluation required by this section must include:
1. In-flight measurements to determine the fatigue loads or stresses for the PSEs identified in paragraph (d) of this section in all critical conditions throughout the range of design limitations required by § 29.309 (including altitude effects), except that maneuvering load factors need not exceed the maximum values expected in operations.
2. The loading spectra as severe as those expected in operations based on loads or stresses determined under paragraph (e)(1) of this section, including external load operations, if applicable, and other high frequency power-cycle operations.
3. Takeoff, landing, and taxi loads when evaluating the landing gear and other affected PSEs.
4. For each PSE identified in paragraph (d) of this section, a threat assessment which includes a determination of the probable locations, types, and sizes of damage, taking into account fatigue, environmental effects, intrinsic and discrete flaws, or accidental damage that may occur during manufacture or operation.
5. A determination of the fatigue tolerance characteristics for the PSE with the damage identified in paragraph (e)(4) of this section that supports the inspection and retirement times, or other approved equivalent means.
6. Analyses supported by test evidence and, if available, service experience.
A residual strength determination is required that substantiates the maximum damage size assumed in the fatigue tolerance evaluation. In determining inspection intervals based on damage growth, the residual strength evaluation must show that the remaining structure, after damage growth, is able to withstand design limit loads without failure.
The effect of damage on stiffness, dynamic behavior, loads, and functional performance must be considered.
Based on the requirements of this section, inspections and retirement times or approved equivalent means must be established to avoid catastrophic failure. The inspections and retirement times or approved equivalent means must be included in the Airworthiness Limitations Section of the Instructions for Continued Airworthiness required by Section 29.1529 and Section A29.4 of Appendix A of this part.
If inspections for any of the damage types identified in paragraph (e)(4) of this section cannot be established within the limitations of geometry, inspectability, or good design practice, then supplemental procedures, in conjunction with the PSE retirement time, must be established to minimize the risk of occurrence of these types of damage that could result in a catastrophic failure during the operational life of the rotorcraft.

[Doc. No. FAA-2009-0413, Amdt. 29-55, 76 FR 75442, Dec. 2, 2011]”

“§ 29.573 Damage Tolerance and Fatigue Evaluation of Composite Rotorcraft Structures.

Each applicant must evaluate the composite rotorcraft structure under the damage tolerance standards of paragraph (d) of this section unless the applicant establishes that a damage tolerance evaluation is impractical within the limits of geometry, inspectability, and good design practice. If an applicant establishes that it is impractical within the limits of geometry, inspectability, and good design practice, the applicant must do a fatigue evaluation in accordance with paragraph (e) of this section.
The methodology used to establish compliance with this section must be submitted to and approved by the Administrator.
Definitions:

Catastrophic failure is an event that could prevent continued safe flight and landing.
Principal Structural Elements (PSEs) are structural elements that contribute significantly to the carrying of flight or ground loads, the failure of which could result in catastrophic failure of the rotorcraft.
Threat Assessment is an assessment that specifies the locations, types, and sizes of damage, considering fatigue, environmental effects, intrinsic and discrete flaws, and impact or other accidental damage (including the discrete source of the accidental damage) that may occur during manufacture or operation.

Damage Tolerance Evaluation:
1. Each applicant must show that catastrophic failure due to static and fatigue loads, considering the intrinsic or discrete manufacturing defects or accidental damage, is avoided throughout the operational life or prescribed inspection intervals of the rotorcraft by performing damage tolerance evaluations of the strength of composite PSEs and other parts, detail design points, and fabrication techniques. Each applicant must account for the effects of material and process variability along with environmental conditions in the strength and fatigue evaluations. Each applicant must evaluate parts that include PSEs of the airframe, main and tail rotor drive systems, main and tail rotor blades and hubs, rotor controls, fixed and movable control surfaces, engine and transmission mountings, landing gear, other parts, detail design points, and fabrication techniques deemed critical by the FAA. Each damage tolerance evaluation must include:
  1. The identification of all PSEs;
  2. In-flight and ground measurements for determining the loads or stresses for all PSEs for all critical conditions throughout the range of limits in § 29.309 (including altitude effects), except that maneuvering load factors need not exceed the maximum values expected in service;
  3. The loading spectra as severe as those expected in service based on loads or stresses determined under paragraph (d)(1)(ii) of this section, including external load operations, if applicable, and other operations including high-torque events;
  4. A threat assessment for all PSEs that specifies the locations, types, and sizes of damage, considering fatigue, environmental effects, intrinsic and discrete flaws, and impact or other accidental damage (including the discrete source of the accidental damage) that may occur during manufacture or operation; and
  5. An assessment of the residual strength and fatigue characteristics of all PSEs that supports the replacement times and inspection intervals established under paragraph (d)(2) of this section.
2. Each applicant must establish replacement times, inspections, or other procedures for all PSEs to require the repair or replacement of damaged parts before a catastrophic failure. These replacement times, inspections, or other procedures must be included in the Airworthiness Limitations Section of the Instructions for Continued Airworthiness required by § 29.1529.
  1. Replacement times for PSEs must be determined by tests, or by analysis supported by tests, and must show that the structure is able to withstand the repeated loads of variable magnitude expected in-service. In establishing these replacement times, the following items must be considered:
    1. Damage identified in the threat assessment required by paragraph (d)(1)(iv) of this section;
    2. Maximum acceptable manufacturing defects and in-service damage (i.e., those that do not lower the residual strength below ultimate design loads and those that can be repaired to restore ultimate strength); and
    3. Ultimate load strength capability after applying repeated loads.
  2. Inspection intervals for PSEs must be established to reveal any damage identified in the threat assessment required by paragraph (d)(1)(iv) of this section that may occur from fatigue or other in-service causes before such damage has grown to the extent that the component cannot sustain the required residual strength capability. In establishing these inspection intervals, the following items must be considered:
    1. The growth rate, including no-growth, of the damage under the repeated loads expected in-service determined by tests or analysis supported by tests;
    2. The required residual strength for the assumed damage established after considering the damage type, inspection interval, detectability of damage, and the techniques adopted for damage detection. The minimum required residual strength is limit load; and
    3. Whether the inspection will detect the damage growth before the minimum residual strength is reached and restored to ultimate load capability, or whether the component will require replacement.
3. Each applicant must consider the effects of damage on stiffness, dynamic behavior, loads, and functional performance on all PSEs when substantiating the maximum assumed damage size and inspection interval.
Fatigue Evaluation: If an applicant establishes that the damage tolerance evaluation described in paragraph (d) of this section is impractical within the limits of geometry, inspectability, or good design practice, the applicant must do a fatigue evaluation of the particular composite rotorcraft structure and:
1. Identify all PSEs considered in the fatigue evaluation;
2. Identify the types of damage for all PSEs considered in the fatigue evaluation;
3. Establish supplemental procedures to minimize the risk of catastrophic failure associated with the damages identified in paragraph (d) of this section; and
4. Include these supplemental procedures in the Airworthiness Limitations section of the Instructions for Continued Airworthiness required by § 29.1529.

[Doc. No. FAA-2009-0660, Amdt. 29-59, 76 FR 74664, Dec. 1, 2011]”

Conclusion

All the basic concepts regarding the topics covered in this work have been given, alongside with a brief overview of regulatory aspects in rotorcrafts fatigue tolerance evaluation. Specifically, the reader now has a deeper insight of (i) two and four parameters S/N curve, (ii) constant and variable amplitude fatigue, (iii) damage accumulation rules, (iv) theory of the critical distances for crack/notch stress-life evaluation, (v) crack propagation, (vi) design philosophies.

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¹ The crack behavior below threshold is still a quite challenging quantity to predict and measure. The reader is advised to read the work of Zerbst et al. [48] to have a better insight of the topic.

² NASGRO 3.0 is the last free version of the equation. Indeed, since the version 4.0 the equation has been amended considerably especially in the calculation of the threshold stress intensity factor.

³ Data have been collected in the form of Kitagawa-Takahashi diagram already in 1973 by Sprowls et al. [60], as also pointed out by Sadananda and Sarkar [61], but Kitagawa and Takahashi were the first observing that below a0 the threshold stress remains constant and propagation is ruled by the fatigue limit.

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