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.     

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.     

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 $\text{ΔK}$.     

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.     

Crack closure in 2219-T851 aluminum alloy

The effects of specimen thickness, stress ratio ($\text{R}$) and maximum stress intensity factor ($\text{K}{}_{\max }$) on crack closure (or opening) were studied using a 2219-T851 aluminum alloy. The crack length and the occurrence of crack closure were measured by an electrical potential method. The experimental work was carried out within the framework of linear-elastic fracture mechanics.The experimental results show that the onset of crack closure (or opening) dependes on $\text{R, K}{}_{\mathrm{<mtext>max</mtext>}}$), and specimen thickness. In terms of the “effective stress intensity range ratio” ($\text{U}$), as defined by Elber, the results show that $\text{U}$ tends to increase for increasing $\text{R}$, decrease for increasing $\text{K}{}_{\max }$, and decrease with increasing specimen thickness. From these trends, it is shown that the “effective stress intensity range” ($\text{ΔK}{}_{\mathrm{<mtext>eff</mtext>}}$) does not always increase with increasing stress intensity range ($\text{ΔK}$).The experimental results show that crack closure cannot fully account for the effects of stress ratio, specimen thickness and $\text{K}{}_{\max }$ on fatigue crack growth. The use of $\text{ΔK}{}_{\mathrm{<mtext>eff</mtext>}}$ as a parameter for characterizing the mechanical driving force for fatigue crack growth is questioned.     

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.     

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.     

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.     

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. $\text{m}\text{= 0}$.     

Prior-to-failure extension of flaws under monotonic and pulsating loadings

An equation governing the prior to failure crack propagation is proposed. For a rate-sensitive solid containing two-dimensional crack and subject to the tensile mode of fracture the differential equations are integrated numerically for the loads increasing monotonically in time. The resulting integral curves gs = σ(l) and l= l(t), i.e. load vs crack length and length vs time, indicate that the growth of cracks in the subcritical range is strongly rate dependent.The fatigue growth, viewed as a sequence of slow growth periods, is simulated on EAI 380 analogue computer. The fourth power law proposed by Paris is confirmed only within certain range of high-cycle fatigue propagation and for a rate-insensitive solid. Otherwise, that is for a more pronounced rate dependency induced by viscosity of a solid and/or in the proximity of the final instability point the growth is markedly enhanced. For sufficiently small ratios of the applied stress intensity range ΔK to the toughness K_c, the suggested fatigue growth law consists of two terms, i.e.

$\text{d}\text{l}\text{d}\text{n}\text{=}\text{l}{}_1\text{12}\text{4}\text{ΔK}\text{K}{}_{\mathrm{c}}{}^4\text{+Cf}{}^{\mathrm{?1}}\text{ΔK}\text{K}{}_{\mathrm{c}}{}^2\text{, l}{}_1\text{=}\text{πK}{}^2{}_{\mathrm{c}}\text{8Y}{}^2$

First term is the familiar Paris expression while the second one accounts for the rate-dependent contribution; f denotes frequency and Y is the yield strength. Rate-sensitivity C is defined by eq. (1.13).     

A new parameter for prediction of creep crack growth rate at high temperature

By analysis based on a series of experimental data obtained by continuous observations using high temperature microscope during the creep test without interruption in vacuum of 10^?5 mm Hg for the purpose of the crack length measurements, a new mathematical equation for prediction of high temperature creep crack growth rate has been proposed in terms of disposable parameters, that is $\text{α}\text{a}{}_{\mathrm{<mtext>eff</mtext>}}\text{σ}{}_{\mathrm{g}}\text{,σ}{}_{\mathrm{g}}$ and temperature for 304 stainless steel within the range of $\text{α}{}_{\mathrm{g}}$ and temperature concerned. It can be seen that it is the best one to fit the experimental data among any other formula proposed hitherto.The new parameter proposed herein

$\text{8.48 × 10}{}^3\text{t}\text{log}{}_{10}\text{α}\text{α}{}_{\mathrm{<mtext>eff</mtext>}}\text{α}{}_{\mathrm{<mtext>g</mtext>}}\text{4.66 × 10}{}^2\text{+ 5.46}\text{log}{}_{10}\text{α}{}_{\mathrm{<mtext>g</mtext>}}$

where

$\text{α = 1.98 +0.36}\text{a}\text{w}\text{? 2.12}\text{a}\text{w}{}^2\text{+ 3.42}\text{a}\text{w}{}^3\text{, a≦0.7w}$

may be used for characterizing the creep crack growth rate just similar as Larson-Miller parameter for the creep life.     

On the cyclic behavior of cast and extruded aluminum alloys. Part A: Fatigue crack propagation

The fatigue crack propagation (FCP) response of a cast and extruded aluminum alloy was examined as a function of mean stress and specimen orientation. The extruded alloy was tested in both the longitudinal and transverse orientation and no difference in FCP response was noted. FCP tests were conducted at R ratios of 0.1, 0.5, 0.65 and 0.8. In the threshold regime, it was seen that as R ratio increased, $\text{Δ}\text{K}{}_{\mathrm{TH}}$ decreased. In addition, ΔK_TH values determined for the cast alloy were superior to those determined for the extruded alloy at all R ratios examined. The threshold regime was also shown to be K_MAX rather than $\text{Δ}\text{K}$ dependent. At intermediate ΔK levels, a mean stress effect was seen for both alloys at R ratios less than 0.5. Crack closure was monitored during testing so that ΔK_EFF values could be determined. $\text{Δ}\text{K}{}_{\mathrm{EFF}}$ was seen to explain mean stress effects at intermediate $\text{Δ}\text{K}$ levels.     

Studies on crack growth rate under high temperature creep,fatigue and creep-fatigue interaction—II: On the role of fracture mechanics parameter aeffσg

For high temperature creep, fatigue and creep-fatigue interaction, several authors have recently attempted to express crack growth rate in terms of stress intensity factor $\text{K}{}_{\mathrm{I}}\text{= α}\text{aσ}{}_{\mathrm{g}}$, where a is the equivalent crack length as the sum of the initial notch length a₀ and the actual crack length $\text{a}{}^1$, that is, $\text{a = a}{}_0\text{+ a}{}^1$. On the other hand, it has been shown by Yokobori and Konosu that under the large scale yielding condition, the local stress distribution near the notch tip is given by the fracture mechanics parameter of $\text{aσ}{}_{\mathrm{g}}\text{?(σ}{}_{\mathrm{g}}$), where a is the cycloidal notch length, σ_g is the gross section stress and $\text{?(σ}{}_{\mathrm{g}}$) is a function of σ_g. Furthermore, when the crack growth from the initial notch is concerned, it is more reasonable to use the effective crack length a_eff taking into account of the effect of the initial notch instead of the equivalent crack length a. Thus we believe mathematical formula for the crack growth rate under high temperature creep, fatigue and creep-fatigue interaction conditions may be expressed at least in principle as function of $\text{a}{}_{\mathrm{eff}}\text{σ}{}_{\mathrm{g}}\text{, σ}{}_{\mathrm{g}}$ and temperature.In the present paper, the geometrical change of notch shape from the instant of load application was continuously observed during the tests without interruption under high temperature creep, fatigue and creep-fatigue interaction conditions. Also, the effective crack length a_eff was calculated by the finite element method for the accurate estimation of local stress distribution near the tip of the crack initiated from the initial notch root. Furthermore, experimental data on crack growth rates previously obtained are analysed in terms of the parameter of $\text{a}{}_{\mathrm{eff}}\text{σ}{}_{\mathrm{g}}$ with gross section stresses and temperatures as parameters, respectively.     

On the cyclic behavior of cast and extruded aluminum alloys. Part B: Fractography

In a prior study [1], the fatigue crack propagation (FCP) response of a cast and an extruded aluminum alloy was examined as a function of mean stress and specimen orientation while crack closure data were collected. In this work, extensive electron fractographic studies were conducted on the previously generated fatigue fracture surfaces using both scanning and transmission electron microscopy. The threshold micromorphology revealed crisp, cleavage-like facets. Striation spacing measurements at intermediate and high $\text{Δ}\text{K}$ levels were obtained to determine microscopic growth rates; these measurements were seen to vary with $\text{R}$ ratio and were best correlated with $\text{ΔK}{}_{\mathrm{EFF}}$ rather than $\text{Δ}\text{K}{}_{\mathrm{APP}}$. Slope changes in the $\text{da/da-}\text{Δ}\text{K}$ plots were identified and attempts made to establish correlations between the associated plastic zone sizes and microstructural dimensions. Of particular note, a stage IIa to IIb transition in the extruded material was found to correspond to a micromechanism change from faceted growth to striated growth when the reversed plastic zone size was similar to the subgrain dimension.     

Growth of fatigue cracks in metals and polymers

Empirical data on the propagation of tensile fatigue cracks in metals and thermoplastics have been examined. It was found that a cyclic crack propagation relationship, based on the stress intensity factor concept, exists which can be successfully utilised for both types of materials.The proposed equation has a form $\text{/.a}{}_{\mathrm{x}}\text{= MA}{}^{\mathrm{n}}$ where $\text{A}$ is a function of$\text{ΔK}$ and mean $\text{K}$. The analysis of results suggests that this equation incorporating the influence of mean stress intensity factor provides an excellent fit to the investigated data. The possible modified forms of such a relationship in terms of strain energy release rate, the crack tip yielding and the crack opening displacement concepts are also indicated.     

Models for the effect of chemisorbed gases on the electrical resistance of metal films

Three hypotheses are considered for the proportionate change of resistance $\text{ΔR}\text{R}{}_1$ when a continuous metal film of resistance R₁ adsorbs gas: (a) a loss of metallic properties at the surface; (b) a change in the electron scattering parameter at the surface; and (c) a change in the electron concentration throughout the film. Two models of the film structure are used to compare the predictions of these hypothesis with our experimental data for the systems $\text{O}{}_2\text{Er}$, $\text{CO}\text{Er}$, $\text{CO}\text{Ti}$ and $\text{O}{}_2\text{Ti}$. The flat plate model accounts for neither R₁ nor $\text{ΔR}\text{R}{}_1$. A simple version of the columnar model accounts for $\text{ΔR}\text{R}{}_1$ by either (a) or (b) for the first three systems and by (c) for the fourth, but it does not explain all the properties of the clean films. The linear approximation given by Mola and Heras for the columnar model of Mayadas and Shatzkes accounts satisfactorily for R₁ and $\text{ΔR}\text{R}{}_1$ for $\text{O}{}_2\text{Er}$, $\text{CO}\text{Er}$ and $\text{CO}\text{Ti}$ by increases in the electron scattering coefficient at the grain boundaries. It does not account for $\text{ΔR}\text{R}{}_1$ with $\text{O}{}_2\text{Ti}$ because that system incorporates gas into the metal lattice.     

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.     

Chromium-doped CdF2 crystals by ESR spectroscopy

Single crystals of Cr-doped CdF₂ in which only Cr²⁺ ions were present were obtained by heating CdF₂:CrF₃ samples in Cd vapors. The crystals produced were studied by ESR spectroscopy at X band at 4.2 ^oK. The spectrum obtained is described by the effective spin Hamiltonian:

g⁽²⁾_eff = 4g_zcosΘ, g⁽¹⁾_eff = 2g_zcosΘ, where i = 1 for the doublet of the spin S = 1, and i = 2 for the doublet of the spin S = 2; net effective spin $\text{S}\text{=}\text{1}\text{2}$; g_z=1.85±0.03; |Δ₍₂₎|=3.08±0.07 GHz; |Δ₍₁₎|=5.85±0.07 GHz; $\text{1}\text{6}\text{(Δ}{}^{\mathrm{o}}{}_{\mathrm{(2)}}\text{? Δ}{}^{\mathrm{o}}{}_{\mathrm{(1)}}\text{)=?11.01±0.02}\text{GHz}$.     

The plastic zone and growth of fatigue cracks in magnesium MA12 alloy at room and low temperatures

The plastic zone formed at the fatigue crack tip and the fracture topography in MA12 magnesium alloy samples, tested at 293 and 140 K in air and in vacuum, were analysed. It was found that the plastic zone formed in vacuum is characterized by a greater size ($\text{h}$) and degree of plastic strain that in air, and the crack growth rate ($\text{d}\text{l/}\text{d}\text{N}$) is lower. Temperature reduction leads to a decrease in $\text{h}$, while $\text{d}\text{l/}\text{d}\text{N}$ and the fracture mechanism are affected by temperature ambiguously, depending on the alloy microstructure and the $\text{K}{}_{\mathrm{<mtext>max</mtext>}}$ value. It was established that the size of the plastic zone can be described by the equation:

$\text{h=A}\text{(}\text{K}{}_{\max }\text{σ}{}_{\mathrm{0.2ps}}\text{)}{}^2$

where $\text{A}$ is a coefficient dependent on the alloy structural state, environment and test temperature. Evaluation of the cyclic plastic zone size at $\text{K}{}_{\mathrm{<mtext>max</mtext>}}$, corresponding to the transition from a low temperature region to a ‘Paris’ region, showed that this transition occurred when the cyclic plastic zon reached the structural parameter of the material.