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.     

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.     

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.     

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.     

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.     

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.     

Fatigue analysis of an edge crack specimen: Hysteresis strain energy density

Accumulative damage model based on the hysteresis strain energy density is proposed for predicting fatigue crack growth. Investigated is the application of sinusoidal loading on an edge crack whose growth rates are obtained by specifying the number of cycles, Δ$\text{N}$, for each growth step. The corresponding increment of crack growth, Δ$\text{a}$, is calculated by having the accumulated local strain energy density to reach certain critical value, $\text{(dW/dV)}{}_{\mathrm{c}}$. As it is to be expected, each growth increment Δ$\text{a}$ increases up to the point of unstable rapid fracture. The growth rate $\text{da/dN}$ versus a data are generated from the nonlinear incremental theory of plasticity. Because of the complexities involved in the stress and subcritical crack growth analysis, the finite element procedure is adopted such that the grid pattern is readjusted for each step of crack growth. Results for the edge crack specimen are displayed graphically and compared with those for the center cracked specimen made of the same material. The different growth characteristics are discussed and expected because material damage by fatigue is sensitive to changes in load history, specimen geometry and crack configuration. Insight into these nonlinear effects provides a means for establishing the range of applicability of the linear fatigue growth models. Discussed in particular are the $\text{da/dN}$ vs δ$\text{k}{}_1$ and AS relations where the linear theory of elasticity is used to calculate Δ$\text{K}{}_1$ and Δ$\text{S}$.     

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.     

Experimental research on surface crack propagation laws for low alloy steel

Surface crack propagation experiments were performed for low alloy steel. The testing result shows that the data for the crack propagation rate in the surface in the direction of the width may be treated by using Paris and Erdogan formula [1] for the crack propagation rate, and Shah and Kobayashi's formula for the stress intensity factor[2]. For the crack propagation rate in the direction of the depth, the data obtained cannot be treated in this way.It has been found that the data can also be treated by using the experimental formula suggested by Kawahara et al.[3].In addition, a test for investigating through-thickness crack propagation was made. It was found that the propagation rate of the through-thickness crack is much greater than those of the surface crack both in the direction of the width and in the direction of the depth. When $\text{ΔK(= 100 Kg/mm}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$) is the same, the propagation rate of through-thickness crack da/dN is five times as great as that of surface crack in the direction of the width.During the propagation of the crack, the relationship between the crack length b and the crack depth a is $\text{a}\text{b}\text{= A?B}\text{a}\text{h}$, where A = 0.97, B = 1.29.With db/dN determined, the prediction of fatigue life can be calculated by

$\text{N=}\text{∫}\text{a}{}_0\text{a}\text{A}\text{A?B}\text{a}\text{h}{}^2\text{d}\text{N}\text{d}\text{b}\text{d}\text{a}$

.     

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.     

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 in steels

From previous investigations of the mechanisms of both fracture and fatigue crack propagation, the static fracture model proposed by Lal and Weiss may be thought as reasonable for describing fatigue crack propagation in metals at both low and intermediate stress intensity factor ranges $\text{ΔK}$. Recent progress in fatigue crack propagation indicates that it is not only possible, but also necessary, to modify this static fracture model. Based on the modified static fracture model, the effective stress intensity factor range $\text{ΔK}{}_{\mathrm{eff}}$, which is defined as the difference between $\text{ΔK}$ and the fatigue crack propagation threshold value $\text{Δ}{}_{\mathrm{th}}$, is taken as the governing parameter for fatigue crack propagation. Utilising the estimates of the theoretical strengths of metals employed in industry, a new expression for fatigue crack propagation, which may be predicted from the tensile properties of the metals, has been derived. The correlation between the fatigue crack propagation rate and the tensile properties is thus revealed. The new expression fits the test results of fatigue crack propagation of steels below 10^?3 mm/cycle and indicates well the effect of stress ratio on the fatigue crack propagation rate.     

The effects of load ratio on environmentally assisted fatigue crack growth

A modification to the model of Weir et al. for surface reaction and transport controlled fatigue crack growth has been developed to explicitly account for the effect of load ratio on environmentally assisted fatigue crack growth. Load ratio was found to affect principally gas transport to the crack tip, and therefore affected only transport controlled crack growth response. Experimental verification of the modified model was made by studying the room temperature fatigue crack growth responses at different load ratios for a 2219-T851 aluminum alloy exposed to water vapor.The results show that the effects of load ratio can be attributed to two different sources—one relating to its effect on local deformation at the crack tip and is reflected through the mechanical component, (d$\text{a}$/d$\text{N}$)₀ and the other on its role in modifying environmental effect and is manifested through the corrosion fatigue component, $\text{(da/dN)}{}_{\mathrm{cf}}$ Furthermore, the results show that the saturation value of corrosion fatigue component, $\text{(da/dN)}{}_{\mathrm{cf,s}}$, is essentially independent of $\text{R}$, and that the exposure needed to produce “saturation response” ($\text{P}{}_0\text{/2f)}{}_{\mathrm{s}}$, as a function of load ratio can be predicted from the modified model. The modified model, therefore, allows one to predict the corrosion fatigue crack growth response for any load ratio on the basis of measurements made at a single load ratio, provided that the values of ($\text{da/dN}$), are known.     

Fracture toughness and fatigue crack propagation in high strength steel from room temperature to −180°c

Fracture toughness under tensile test and fatigue test on high strength steel at temperature ranging from room temperature to ?180°C were experimentally studied. The value of fracture toughness under fatigue test is considerably tower than that obtained under tensile test.Within the range from room temperature to ?100°C the following results were obtained: the power coefficient δ of the fatigue crack propagation rate $\text{[(dc)/(dN)] = AΔK}{}^5$ is related with [(1)/($\text{T}$)] as: $\text{δ = b}{}_1\text{+ [(a}{}_1\text{)/(kT)]}$. $\text{[(dc)/(dN)]}$ shows Arrhenius type, and, however, different equation from usual stress dependent rate process equation. The trend is in good agreement with the dislocation dynamics theory of fatigue crack propagation.     

The condition of fatigue crack growth in mixed mode condition

The condition of the initiation of fatigue crack growth in mixed mode conditions has been investigated by using precracked low carbon steel specimens.It is pointed out that, firstly, the critical condition of crack growth should be defined with regard to the modes of fatigue crack growth, i.e. shear mode and tensile mode. Secondly, it is proposed that the critical condition of fatigue crack growth is given by the local tensile stress and shearing stress at the notch tip determined by stress intensity factors $\text{K}{}_{\mathrm{I}}$ and $\text{K}{}_{\mathrm{II}}$, and that this criterion is generally applicable to in-plane-loading conditions, i.e. Mode $\text{I}$, Mode $\text{II}$ and Mixed Mode conditions.     

Mechanism of corrosion fatigue crack propagation in high strength steels

The crack propagation velocity in corrosion fatigue $\text{(d a/d N)}{}_{\mathrm{c}}$ were measured on the Ni-Cr-Mo steel quenched and tempered at 473 or 773 K.The steel with high sensitivity to delayed failure reveals the largest $\text{(d a/d N)}{}_{\mathrm{c}}$ under square load and the smaller $\text{(d a/d N)}{}_{\mathrm{c}}$ under positive saw tooth load. The frequency dependency of crack propagation characteristics indicates that the interaction between hydrogen atoms and the cyclic moving of triaxial position at crack tip acts an important role in the crack propagation mechanism, i.e. hydrogen concentration process controls the crack propagation of the steel.The steel with low susceptibility to delayed failure reveals, on the other hand, the largest $\text{(d a/d N)}{}_{\mathrm{c}}$ under the positive saw tooth load but the smallest $\text{(d a/d N)}{}_{\mathrm{c}}$ under the square load, i.e. the stress increasing time is important and the hydrogen invasion process is the controlling factor for the crack propagation.     

On crack opening stress level in fatigue by Eddy current technique

Linear elastic fracture mechanics relates fatigue crack growth with the stress intensity factor at the crack tip. Presence of residual deformations at the tip of a fatigue crack reduces the crack tip stress intensification such that effective stress intensity range $\text{ΔK}{}_{\mathrm{e}}\text{= U · ΔK}$. In this paper use of eddy current technique is exhibited to find the values of test value of effective stress range factor $\text{U}{}_{\mathrm{test}}$. A reasonable comparison between computed and experimental results of $\text{U}{}_1$ and $\text{U}{}_{\mathrm{test}}$ on two Al alloys 6061-T6 and 6063-T6 has recommended the Eddy Current Technology for finding out the values of crack opening stress level under given loading conditions.