Prediction of threshold and ultra-low fatigue crack growth rates

A model was derived to predict the true threshold value for fatigue crack growth in the absence of crack closure. The model, based only on the tensile and cyclic properties of the material, was successfully verified against a set of experimental data on medium and high strength steels and one aluminium alloy. Good agreement with experimental results was also obtained for Region I of the $\text{d}\text{a/}\text{d}\text{N vs ΔK}$ curve using a fatigue crack growth rate equation based on the same model.Fatigue crack growth data obtained from the medium strength steel CK45 in the normalized state and two heat-treated conditions were analysed. Good data correlation was shown using a previously developed normalizing parameter, $\text{φ = (ΔK}{}^2\text{?ΔK}{}^2{}_{\mathrm{<mtext>th</mtext>}}\text{)/(K}{}^2{}_{\mathrm{<mtext>c</mtext>}}\text{?K}{}^2{}_{\mathrm{<mtext>max</mtext>}}\text{)}$, in the entire range of fatigue crack growth rates and for stress ratios ranging from 0.1 to 0.8.     

Fatigue crack propagation work coefficient—a material constant giving degree of resistance to fatigue crack growth

Study on fatigue crack growth in steels was carried out from energetic point of view, i.e. taking account of plastic work around the fatigue crack. Based on the examination of the relation between fatigue crack growth rate (da/dN) and the plastic work around the fatigue crack tip (W_0.02 in SUS304, Fe-3Si and HT 60 steels, a material constant-fatigue crack propagation work coefficient-Q_0.02 is proposed. It is the ratio of W_0.02 to da/dN and means the degree of the resistance to fatigue crack growth. Numerical expression of Q_0.02 by mechanical properties was derived, which is given by

$\text{Q}{}_{0.02}\text{=}\text{9.3x10}{}^1\text{(σ}{}_{\mathrm{y}}\text{+σ}{}_{0.2}\text{)}\text{σ}{}_{\mathrm{y}}{}^{1.3}$

Comparison of Q_0.02 of various steels showed that Q_0.02 of high strength steels is very small compared with that of low strength steels. Graphical representation of the relation between Q_0.02 and da/dN at various values of ΔK/σ_y for steels revealed that da/dN at given value of ΔK/σ_y increase with decreasing Q_0.02. It is shown that fatigue crack growth behaviour of a steel (da/dN-ΔK relation) can be obtained from the Q_0.02-da/dN diagram by knowing the mechanical properties. Discussion on design stress level of the steels is also given.     

Condition for stable growth of branched cracks

In order to clarify the reason why the stable growth of branched cracks occurs in delayed failure, while not in other subcritical crack propagation process such as fatigue, the stress intensity factor after crack branching in delayed failure was dropped to various values, and the propagation behavior of both cracks was investigated.The well balanced growth of branched cracks in delayed failure occurs only when the crack propagation velocity after crack branching belongs to the region II where the crack propagation velocity is constant independently of $\text{K}$. The fatigue cracks at the tips of artificially branched cracks, on the other hand, can not propagate stably, and only either crack propagates preferentially.The exponent in the crack propagation law ($\text{d}\text{a/}\text{d}\text{t =}\text{c}{}_1\text{K}{}^{\mathrm{<mtext>m</mtext>}}\text{or}\text{d}\text{a/}\text{d}\text{N =}\text{c}{}_2\text{(ΔK)}{}^{\mathrm{<mtext>m</mtext>}}$) expresses the degree of unbalance growth of branched cracks. The stable growth of branched cracks occurs only when the crack propagation velocity is constant independently of $\text{K}$ or $\text{ΔK}$, i.e. $\text{m}\text{= 0}$.     

Effect of mechanical properties of material on rate of fatigue crack propagation

Many experimental and analytical equations on a rate of a fatigue crack propagation have been proposed. However, it seems that they can not fully express its complex behavior. There are still many problems remaining to be solved in order to clarify its mechanism. One of them is to clarify the relation between the rate of the crack propagation and the mechanical properties of material. In this paper, the rate of the crack propagation is analysed to clarify this problem. This analysis is based on the observation results of the fatigue crack propagation behavior previously by the authors. The analytical result is compared with the experimental one to make sure that they agree with each other. The conclusion obtained is; the rate of fatigue crack propagation is expressed by using the stress intensity factors as

$\text{dl}\text{dN}\text{= {}\text{c}\text{[Y}{}^2\text{F}{}^{\mathrm{a}}\text{E}{}^{\mathrm{a(1?n)}}\text{]}\text{} (K}{}_{\max }\text{)}{}^2\text{(K}{}_{\mathrm{a}}\text{)}{}^{\mathrm{a(2?n)}}$

. where C is a constant; E, Young's modulus; F, plastic coefficient; Y, yield stress; K_max and K_a, maximum and amplitude of the stress intensity factor, and α and n, exponents of the Manson-Coffin's law and work-hardening.     

A model for fatigue crack growth in a threshold region

The prediction of fatigue crack growth at very low ΔK values, and in particular for the threshold region, is important in design and in many engineering applications. A simple model for cyclic crack propagation in ductile materials is discussed and the expression

$\text{da}\text{dN}\text{=}\text{2}{}^{\mathrm{1+n}}\text{(1?2v)(ΔK}{}^2{}_{\mathrm{eff}}\text{?ΔK}{}^2{}_{\mathrm{c,eff}}\text{)}\text{4(1+n)π σ}{}^{\mathrm{1?n}}{}_{\mathrm{yc}}\text{E}{}^{\mathrm{1+n}}\text{?}{}^{\mathrm{1+n}}{}_{\mathrm{f}}$

developed. Here, n is the cyclic strain hardening exponent, σ_yc is cyclic yield, and ε_f is the true fracture strain. The model is successfully used in the analysis of fatigue data BS 4360-50D steel.     

A cumulative model of fatigue crack growth and the crack closure effect

A model of fatigue crack growth based on an analysis of elastic/plastic stress and strain at the crack tip is presented. It is shown that the fatigue crack growth rate can be calculated using the local stress/strain at the crack tip by assuming that a small highly strained area $\text{x}{}^1$, existing at the crack tip, is responsible for the fatigue crack growth, and that the fatigue crack growth may be regarded as the cumulation of successive crack re-initiations over a distance $\text{x}{}^1$. It is shown that crack closure can be modelled using the effective contact zone g behind the crack tip. The model allows the fatigue crack growth rate over the near threshold and linear ranges of the general da/dN versus ΔK curve to be calculated. The fatigue crack growth retardation due to overload and fatigue crack arrest can also be analysed in terms of g and $\text{x}{}^1$.Calculated fatigue crack growth rates are compared with experimental ones for low and high strength steel.     

Fracture mechanics approach to fatigue crack initiation from deep notches

Fracture mechanics approach is applied to fatigue crack initiation at the tips of deep, blunt notches including those with very small notch-tip radius. The theoretical relations between the stress intensity range ΔK_ρ and the notch-tip radius ρ for a fixed life for crack initiation were derived based on the models of dislocation-dipole accumulation and blocked slip-band. Those are approximated by a simpler equation: $\text{ΔK}{}_{\mathrm{\rho }}\text{ΔK}{}_{\mathrm{o}}\text{= (1 + ρ/ρ0)}\text{1}\text{2}$ where ΔK₀ and ρ0 are material constants which are related to the fatigue strength of smooth specimens Δρ0 as $\text{Δρ0 =}\text{2ΔK}{}_0\text{(πρ0)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. The results of experiments done with bluntly notched compact tension specimens of a structural low-carbon steel agree with the above relation between $\text{ΔK}{}_{\mathrm{gr}}\text{ΔK}{}_{\mathrm{o}}$ and ρ/ρ_o. The method to predict ΔK_o, ρ_o and Δρ_o from the fatigue data of cracked and smooth specimens is proposed.     

The effect of temperature and R ratio on fatigue crack growth in A612 grade B steel

Fatigue-crack propagation rates in ASTM$\text{A612}$ Grade B steel were investigated at room temperature and ?100°F (?73°C) with $\text{R}\text{ratio}\text{= ?0.1}\text{and}\text{+0.67}$. The data were evaluated in terms of the crack propagation rates ($\text{d}\text{a/}\text{d}\text{N}$) as a function of the alternating stress intensition ($\text{ΔK}$), according to $\text{d}\text{a/}\text{d}\text{N = e+(v ? e)(? 1}\text{n}\text{(1 ?}\text{Δ}\text{K/K}{}_{\mathrm{b}}\text{))}{}^{\mathrm{t/k}}$. It was found that crack growth rates were increased due to increasing $\text{R}$ ratio. Also the dependence of crack growth rates on $\text{R}$ ratio is strongest at the lowest crack growth rates where a $\text{ΔK}$ fatigue threshold is established. Crack growth rates were decreased due to decreasing test temperature in the slow crack growth region. However, it was found that crack growth rates were increased due to decreasing test temperature in the fast crack growth region near the upper instability asymptote. Decreased test temperature and increased $\text{R}$ ratio interact synergistically to increase crack growth rates for the entire range of $\text{ΔK}$.     

An analysis of reported fatigue crack growth rate data with special reference to Ti-6A1-4V

A review of the published literature on fatigue crack growth suggests that power-law growth in Ti-6A1-4V is sensitive to microstnictural changes which result in variations in the fatigue mechanism. Microstnictures which promote secondary cracking along α/β interfaces display slow growth rates while microstructures which promote dimpled rupture display fast growth rates. Examples of similar effects are found in other alloy systems.Typically, the power-law growth are found in other alloy systems. It is also suggested that the power-law regime begins at $\text{ΔK ～- 13 MNm}{}^{\mathrm{<mtext>?</mtext><mtext>case3</mtext><mtext>2</mtext>}}$, coinciding with the lower limit of striation formation on the fracture surface. The upper limit occurs at about $\text{K}{}_{\max }\text{= 1/2K}{}_{\mathrm{c}}$. At higher growth rates, the Forman equation appears to be adequate.The normalized stress intensity factor, $\text{ΔK/E}$, required to produce a given growth rate in Ti-6A1-4V is on the order of that for other Ti-base alloys, ferritic steels, martensitic steels and aluminum alloys. Austenitic steels, which deform by planar slip are much more resistant to crack growth over much of the stress intensity range normally encountered.     

Some formulas for the crack opening stress level

Crack growth data for 2024-T3 sheet material were analysed with different formulas for $\text{ΔK}{}_{\mathrm{eff}}$ as a function fo the stress ratio $\text{R}$. The data covered $\text{R}$ values from ?1.0 to 0.54. A good correlation was obtained for $\text{ΔK}{}_{\mathrm{eff}}$/$\text{ΔK}$ = 0.55 + 0.33$\text{R}$ + 0.12$\text{R}{}^2$ The relation between log $\text{d}\text{a/}\text{d}\text{n}$ and log $\text{ΔK}{}_{\mathrm{eff}}$ was non-linear for high crack rates (> 1 μm/c).     

Fatigue crack propagation behaviour of rotor and wheel materials used in steam turbines

The fatigue crack propagation characteristics of several rotor and wheel materials that are commonly used in rotating components of steam turbines were investigated. Particular emphasis was placed on the behaviour at near-threshold growth rates, ie below 10^?5 mm/cycle, approaching the fatigue-crack propagation threshold, $\text{ΔK}{}_{\mathrm{<mtext>th</mtext>}}$. The lifetimes of the cracks of interest lie mostly in this region, and it is also the region where few data are available.The effects of load ratio on the fatigue crack growth rates were examined, as well as the tensile, Charpy V-notch and fracture toughness properties of the rotor and wheel materials. The relationship between fatigue crack propagation behaviour and fractographic features was examined. Fatigue crack growth rate data, $\text{da/d}\text{N}$ vs stress intesity range $\text{ΔK}$, were fitted with a four parameter Weibull survivorship function. This curve fitting can be used for life estimation and establishment of $\text{ΔK}{}_{\mathrm{<mtext>th</mtext>}}$. The results show that load ratio and microstructure play a role in determining the fatigue crack threshold and fatigue crack growth behaviour.     

Cyclic analysis of a propagating crack and its correlation with fatigue crack growth

Stress and strain field of a propagating fatigue crack and the resulting crack opening and closing behavior were analysed. It was found that a propagating fatigue crack was closed at tensile external loads due to the cyclically induced residual stresses. Strain range value $\text{Δ?}{}_{\mathrm{y}}$ in the vicinity of the crack tip was found to be closely related with the effective stress intensity factor range $\text{ΔK}{}_{\mathrm{eff}}$ which was determined on the basts of the analytical crack opening and closing behavior at its tip. Application of this analysis to the non-propagating fatigue crack problem and the fatigue crack propagation problems under variable stress amplitude conditions revealed that both $\text{Δ?}{}_{\mathrm{y}}$ and $\text{ΔK}{}_{\mathrm{eff}}$ were essential parameters governing fatigue crack growth rate.     

Mechanics of fatigue crack propagation by crack-tip plastic blunting

The power relation between the fatigue crack propagation rate $\text{d}\text{a/}\text{d}\text{N}$ and the $\text{J}$ integral range $\text{ΔJ}$ was obtained for OFHC copper, 0.04%C steel and stainless steel (Type 304). The physical meaning of the relation was investigated based on the observation of the crack opening behavior and fractography. The striations, whose spacing was equal to $\text{d}\text{a/}\text{d}\text{N}$, confirm crack-tip blunting as being an operating mechanism for crack growth. $\text{d}\text{a/}\text{d}\text{N}$ was expressed as a power function of the crack-tip opening displacement (Δ CTOD), $\text{d}\text{a/}\text{d}\text{N = A(Δ}\text{CTOD}\text{)}{}^{\mathrm{p}}$, where $\text{p}$ was larger than one. The major portion of Δ CTOD contributes to crack growth at high rates, while a considerable fraction of Δ CTOD occurs behind the crack tip at low rates. Δ CTOD is correlated to $\text{ΔJ}$ divided by the yield strength σ'_Y through the equation,$\text{Δ}\text{CTOD}\text{= B(ΔJ/σ}{}_{\mathrm{Y}}\text{′)}{}^{\mathrm{q}}$, where $\text{q}$ is nearly equal to one for 0.04%C steel and is larger than one for copper. The variation of $\text{q}$ with material was explained based on the observed distribution of the crack opening displacement. The final equation for fatigue crack growth is given as $\text{d}\text{a/}\text{d}$N = A · B^p(ΔJ/σ_Y′)^pq. When the shape of the crack tip opening is geometrically similar, both $\text{p}$ and $\text{q}$ are one. For a general case, both are larger than one, yielding the exponent of the $\text{d}\text{a/}\text{d}\text{N-ΔJ}$ relation deviating from 1 up to 2.3.     

Plastic work during fatigue crack propagation in a high strength low alloy steel and in 7050 Al-Alloy

Theoretically and empirically a fatigue crack propagation rate of the form

$\text{d}\text{c}\text{d}\text{N}\text{∞}\text{(ΔK)}{}^4\text{μσ}{}^2\text{U}$

is indicated where μ is the shear modulus, σ is an appropriate measure of the alloy's strength, and U is the energy to make a unit area of fatigue crack. The local stress-strain curves in the plastic zone around a propagating fatigue crack were determined using tiny foil strain gages. The areas in the hysteresis loops were integrated over the plastic zone for a unit area of crack advance to give an approximate value for U. The non-hysteretic plastic work was neglected in this calculation but its contribution to the total plastic work in the plastic zone near the crack tip is small.     

The influence of environment and stress ratio on fatigue crack growth at near threshold stress intensities in low-alloy steels

Fatigue crack propagation at low stress intensities has been studied in two low alloy steels in a variety of environments with particular emphasis being placed on the influence of stress ratio and strength level. It was found that fatigue crack growth rates are lower and threshold stress intensities ($\text{ΔK}{}_0$) are higher in vacuum than in humid, laboratory air but, in dry gaseous environments (argon, hydrogen and air) and at low stress ratio ($\text{R ~ 0.1}$), crack growth rates are faster and $\text{ΔK}{}_0$ values are lower than in laboratory air. However, the influence of stress ratio is considerably greater in laboratory air than in dry gaseous environments with the result that, at high stress ratio ($\text{R ~ 0.8}$) $\text{ΔK}{}_0$ values are similar in all environments examined. Increasing material strength level resulted in higher, near-threshold crack growth rates and a reduction in $\text{ΔK}{}_0$ in both dry and humid air environments. The results are discussed in terms of the influence of crack closure and environmental effects on fatigue crack growth behaviour. The importance of corrosion debris produced in fatigue cracks at low stress intensities is also discussed.     

Fatigue crack propagation from a crack inclined to the cyclic tensile axis

Cyclic stresses with stress ratio R = 0.65 were applied to sheet specimens of aluminium which have an initial crack inclined to the tensile axis at angles of 30°, 45°, 72° or 90°. The threshold condition for the non-propagation of the initial crack was found to be given by a quadratic form of the ranges of the stress intensity factors of modes I and II. The direction of fatigue crack extension from the inclined crack was roughly perpendicular to the tensile axis at stress ranges just above the threshold value for non-propagation. On the other hand, at stress ranges 1.6 times higher than the threshold values the crack grew in the direction of the initial crack. The rate of crack growth in the initial crack direction was found to be expressed by the following function of stress intensity factor ranges of mode I, K₁, and mode II, K₂: $\text{dc}\text{dN}\text{= C(K}{}_{\mathrm{eff}}\text{)sum}$, where $\text{K}{}_{\mathrm{eff}}\text{= [K}{}_1{}^4\text{+ 8K}{}_2{}^4\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$. This law was derived on the basis of the fatigue crack propagation model proposed by Weertman.     

Effect of loading history on stress corrosion cracking in high strength steel

The effect of preloading on crack nucleation time was examined with compact tension specimens having various notch radius in 0.1N-H²SO⁴ aqueous solution for 200°C tempered AISI 4340 steel. Crack nucleation time t_n increases by preloading for a given apparent stress intensity factor K_p2. The curve K_?2 vs. t_n deviates upward from the curve for the non preloading case. A linear relationship between the crack nucleation time and parameter is seen in semi-log diagram, where is taken as the value at t_n=α due to preloading. The apparent threshold stress intensity factor increases with K_?2 which is the apparent stress intensity factor of preloading. A detached crack is nucleated at some distance from the notch root and extends in a form of circle. This distance increases with increasing K_?2. The effect of load reduction during crack growth was examined. When the K-value was reduced from K₁ to K₂, an incubation time was observed before the crack started growing under the K₂-value. The incubation time t_m tends to increase with increasing ΔK = K₁-K₂. The threshold stress intensity factor was also found to increase for high load reduction.In order to explain these experimental results, a new dislocation model is proposed on the basis of stress induced diffusion of hydrogen in high stress region ahead of the notch root or a crack. This model suggests that the change in the crack nucleation time and the increase of the incubation time due to preloading or load reduction are caused by reducing the hydrostatic pressure and by spreading the hydrogen saturated region which requires more time for the hydrogen accumulation due to preloading or load reduction. The theory predicts the experimentally observed relations between and t_n and between log t_in and ΔK.     

A cumulative model of fatigue crack growth

A model of fatigue crack growth based on an analysis of elastic/plastic stress and strain at the crack tip is presented. It is shown that the fatigue crack growth rate can be calculated by means of the local stress/strain at the crack tip. The local stress and strain calculations are based on the general solutions given by Hutchinson, Rice and Rosengren. It is assumed that a small highly strained area existing at the crack tip is responsible for the fatigue crack growth. It is also assumed that the fatigue crack growth rate depends mainly on the width, x¹, of the highly strained zone and on the strain range, $\text{Δ?}{}^1$, within the zone. A relationship between stress intensity factor K and the local strain and stress has been developed. It is possible to calculate the local strain for a variety of crack problems. Then, the number of cycles N¹ required for material failure inside the highly strained zone is calculated. The fatigue crack growth rate is calculated as the ratio $\text{x}{}^1\text{N}{}^1$.The calculated fatigue crack growth rates were compared to the experimental ones. Two alloys steels and two aluminium alloys were analyzed. Good agreement between experimental and theoretical results is obtained.     

Crack tip closure and environmental crack propagation

Crack propagation rate, da/dN, and crack tip closure stress, σ_cc, in part-through crack fatigue specimens of aluminum alloys are drastically affected by gaseous environments. The present studies indicate that the crack closure reflects the influence of the environment on the plastic deformation at the crack tip, and, therefore, on the crack propagation rates. Postulating that da/dN is mainly determined by $\text{ΔK}{}_{\mathrm{eff}}\text{∝ (σ}{}_{\max }\text{?σ}{}_{\mathrm{cc}}\text{)}$ (instead of $\text{ΔK ∝ (σ}{}_{\max }\text{?σ}{}_{\min }\text{)}$, as is done traditionally) leads to the relationship $\text{da/dN = A(ΔK}{}_{\mathrm{eff}}\text{)}{}^{\mathrm{n}}$ in which $\text{A}$ and $\text{n}$ are virtually independent of the gaseous environment. The exponents are $\text{n ≈ 3.3}$ for Al 7075 T651 and $\text{n ≈ 3.1}$ for Al 2024 T351, respectively.