On one- and two-parameter analyses of short fatigue crack growth

Abstract

Recent data on short fatigue crack growth in two cast and hot isostatically pressed (hipped) aluminum alloys obtained by Shyam, Allison and Jones have been analyzed in terms of a previously proposed one-parameter short crack model which includes consideration of elastic–plastic effects, the Kitagawa effect and the development of crack closure in the wake of a newly formed crack. The material constants obtained in a prior investigation of short crack growth behavior in a cast aluminum alloy tested under fully reversed loading were used as a basis for the present analysis. The predicted rates of fatigue crack propagation are in accord with the experimental results. In the discussion, aspects of the two-parameter approach presented by Shyam et al. are compared with those of the one-parameter method of analysis used herein.

Introduction

In contrast to the extent of experimental information concerning the behavior of long fatigue cracks there is much less experimental information concerning the behavior of short fatigue cracks, despite the fact that most of the fatigue lifetime is spent in the short crack range. Therefore a welcome addition to the fatigue literature on short cracks has recently been provided by Shyam et al. [1] who examined short fatigue crack propagation in two cast aluminum alloys as a function of R ratio (−1, 0.1 and 0.3) and temperature (20–250 °C). In their paper they also presented a method of analysis based upon a two-parameter approach to fatigue crack growth which was related to the following equation

$$\frac{\mathrm{d}a}{\mathrm{d}N}=C\mathrm{\Delta }{K}^m{K}_{\max }^n$$

where a is the crack length, N is the number of cycles, and C, m and n are empirical constants. (The Paris law, da/dN = C(ΔK)^m, is an example of a one-parameter relation.)

The specific expression developed by Shyam et al. for the rate of fatigue crack growth is

$$\frac{\mathrm{d}a}{\mathrm{d}N}=\kappa {\phi }_{\text{c}}{\phi }_{\text{m}}$$

where κ is an empirical constant, φ_c is the cyclic crack tip displacement

$${\phi }_{\text{c}}=\frac{16{\sigma }_{\text{Y}}(1-{\nu }^2)a}{\mathrm{\pi }E}\ln \left\{\sec \left(\frac{\mathrm{\pi }{\sigma }_{\max }[1-R]}{4{\sigma }_{\text{Y}}}\right)\right\}$$

corresponding to ΔK in Eq. (1), and φ_m is the monotonic crack tip displacement

$${\phi }_{\text{m}}=\frac{8{\sigma }_{\text{Y}}(1-{\nu }^2)a}{\mathrm{\pi }E}\ln \left\{\sec \left[\frac{\mathrm{\pi }{\sigma }_{\max }}{2{\sigma }_{\text{Y}}}\right]\right\}$$

corresponding to K_max in Eq. (1). The results obtained by Shyam et al. for all test conditions are shown in Fig. 1.

In the long crack regime where closure is fully developed and small scale yielding conditions are satisfied, Eq. (2) reduces to the following one-parameter equation

$$\frac{\mathrm{d}a}{\mathrm{d}N}=\frac{\kappa (1-{\nu }^2)}{2}\frac{\mathrm{\Delta }{K}^4}{(E{\sigma }_{\text{Y}}{)}^2}$$

Therefore the two-parameter approach proposed by Shyam et al. relates only to the short crack regime.

In their experimental program two cast and hot-isostatically pressed (HIP) aluminum alloys were tested, W319–T7 and A356–T6. Cylindrical specimens, 5.1 mm in diameter in the test section, were used. Two opposing notches, 28 mm in diameter and 1 mm in depth, were machined tangent to the specimen length to facilitate the laser drilling of surface notches and the subsequent observation of short cracks. Laser-drilled notches which simulated near-surface porosity were 100 μm or greater in depth, and their lengths, 2a₀, ranged from 258 μm to 574 μm. The specimens were tested under axial loading at 30 Hz. The materials, yield strengths and test conditions are given in Table 1.

The acronyms MSDAS and LSDAS stand for medium and low secondary dendrite arm spacing, respectively.

In the present paper an alternative approach will be utilized in the analysis of the data obtained by Shyam et al. This approach is based upon crack tip opening considerations and the single-parameter relationship [2].

$$\frac{\mathrm{d}a}{\mathrm{d}N}=\frac{A^{\prime }}{{\sigma }_{\text{Y}}E}(\mathrm{\Delta }{K}_{\text{eff}}-\mathrm{\Delta }{K}_{\text{effth}}{)}^2=A(\mathrm{\Delta }{K}_{\text{eff}}-\mathrm{\Delta }{K}_{\text{effth}}{)}^2$$

where A′ is an empirical dimensionless constant and A is an empirical constant of units (MPa)⁻². ΔK_eff is equal to K_max − K_op, K_op being the stress intensity factor at the crack opening level. ΔK_effth is the effective range of the stress intensity factor at the threshold level. The threshold level is taken to correspond to a rate of crack growth of 10⁻¹¹ m/cycle.

In order to make use of Eq. (6) in the short crack range, modification is necessary since short fatigue cracks differ from large cracks in the following three aspects [3]:

• Crack growth can be elastic–plastic in nature rather than linear-elastic because of a high ratio of the fatigue strength to the yield strength and the consequent large ratio of the plastic zone size to the crack length.

• Crack closure is a function of the crack length. In the wake of a crack of some microns in length the crack closure level is zero, but as the crack grows to a length of a millimeter or so the crack closure level rises to that of a large crack.

• In the very short crack range the rate of crack growth is determined by the range of cyclic stress rather than the range of the stress intensity factor (Kitagawa effect [4]).

Irwin [5] proposed that the linear-elastic approach could be extended to include elastic–plastic behaviour, i.e., those cases where the crack-tip plastic zone size is large with respect to the crack length, by increasing the actual crack length, a, by one-half of the plastic zone size. If the plastic zone size is taken to be that as defined by Dugdale [6], then the modified crack length, a_mod, is given as:

$$a_{\text{mod}}=a+\frac{1}{2}\left(-1+\sec \frac{\mathrm{\pi }}{2}\frac{{\sigma }_{\max }}{{\sigma }_Y}\right)·a=a·F$$

where σ_max is the maximum stress in a loading cycle, σ_Y is the monotonic yield strength, and F, the elastic–plastic correction factor, is given by

$$F=\frac{1}{2}\left(1+\sec \frac{\mathrm{\pi }}{2}\frac{{\sigma }_{\max }}{{\sigma }_Y}\right)$$

The level of crack closure developed in the wake of a crack varies from zero for a newly formed crack up to K_opmax for a macroscopic crack. The following expression has been proposed [7] to describe this transient in crack closure behavior:

$$\mathrm{\Delta }{K}_{\text{op}}=(1-{\mathrm{e}}^{-k\lambda })(K_{\text{opmax}}-K_{\min })$$

where ΔK_op is the value of K_op − K_min in the transient range, k is a material constant (dimension of length, m⁻¹) which determines the rate of crack closure development, λ is the length of the newly formed crack (units m), and K_opmax is the magnitude of the crack opening level associated with completion of the transient period of growth. The value of λ at the end of the transient period is generally less than a millimeter.

In order to achieve a smooth transition from ΔK control of the rate of fatigue crack growth for cracks of macroscopic size to Δσ control for cracks of microscopic size, (the Kitagawa effect [4]), El Haddad et al. [8] added a constant, b, to the actual crack length. When a crack was large with respect to b, b could be neglected. When a crack was short with respect to b, then b became the controlling parameter determining the stress intensity factor. In the limit, as zero crack length is approached, the stress intensity factor becomes $\mathrm{\Delta }\sigma \sqrt{\pi b}$. The constant b was evaluated by equating this quantity to ΔK_th, with Δσ set equal to the range of stress at the endurance limit. A procedure somewhat similar to that of El Haddad and Topper’s will be used in this analysis to deal with the Kitagawa transition.

Irwin [9] has shown that the stress intensity factor, K, is related to the stress concentration factor, K_T, by the following equation:

$$K=\underset{\rho \to 0}{\lim }{\sigma }_{\text{m}}\sqrt{\frac{\mathrm{\pi }\rho }{4}}=\underset{\rho \to 0}{\lim }{K}_{\text{T}}\sigma \sqrt{\frac{\mathrm{\pi }\rho }{4}}$$

where ρ is the tip radius of the stress concentrator, σ_m is the maximum stress at the tip of the stress concentrator, and σ is the remote stress. In order to achieve the desired transition between the threshold level for fatigue crack growth and the fatigue strength, Eq. (10) is modified as follows:

$$K=\underset{\rho \to {\rho }_{\text{e}}}{\lim }{K}_{\text{T}}\sigma \sqrt{\frac{\mathrm{\pi }\rho }{4}}$$

where ρ_e is a material constant.

In the case of a panel containing a central crack under Mode I loading, K_T is equal to $K_{\text{T}}=(1+2\sqrt{a/\rho })$, and Eq. (11) becomes,

$$K=\left(\sqrt{\frac{\mathrm{\pi }{\rho }_{\text{e}}}{4}}+\sqrt{\mathrm{\pi }a}\right)\sigma$$

The material constant ρ_e can be converted to an effective length dimension, r_e, by equating the following relations for the stress intensity factor

$${\sigma }_{\max }\sqrt{\frac{\mathrm{\pi }{\rho }_{\text{e}}}{4}}={\sigma }_{\text{yy}}\sqrt{2\mathrm{\pi }{r}_{\text{e}}}$$

so that r_e is equal to ρ_e/8 in magnitude. Eq. (13) is obtained by equating the stress intensity factor $\sigma \sqrt{\mathrm{\pi }{\rho }_{\text{e}}/4}$ to the LEFM stress intensity factor, $\sigma \sqrt{2\mathrm{\pi }r}$. Therefore, r_e is the distance from the crack tip to the point ahead of the crack tip where the LEFM value of the stress intensity factor is equal to ${\sigma }_{\max }\sqrt{\mathrm{\pi }{\rho }_{\text{e}}/4}$. In this modified approach r_e is considered to be the effective length of an inherent flaw. In this interpretation of r_e, a newly formed crack is only significant when its length exceeds r_e, since for crack lengths less than r_e, the stress intensity factor associated with r_e will be larger. It is pointed out that there is no relationship between r_e and an actual defect. It is merely an adjustable parameter introduced, as in El Haddad, Topper and Smith’s case [8], in order to deal with the Kitagawa effect in a quantitative manner.

The driving force for fatigue crack growth, ΔK, is generalized to take into account in geometries other than just the center-cracked panel as

$$\mathrm{\Delta }K=\left(\sqrt{2\mathrm{\pi }{r}_{\text{e}}F}+Y\sqrt{\mathrm{\pi }\mathit{aF}}\right)\mathrm{\Delta }\sigma$$

where the value of Y depends upon the crack shape. It is assumed that the initial crack shape in an unnotched specimen is semi-circular, then the value of Y is 0.73 [10]. The magnitude of r_e is of the order of 1 μm. Its value is determined by setting a equal to r_e, ΔK equal to the effective range of the stress intensity factor at the threshold level, ΔK_effth corresponding to da/dN = 10⁻¹¹ m/cycle, and Δσ equal to the stress range at the fatigue strength level, Δσ_EL (10⁷ cycles), i.e.,

$$r_{\text{e}}=\frac{1}{4.6\mathrm{\pi }F}{\left(\frac{\mathrm{\Delta }{K}_{\text{effth}}}{\mathrm{\Delta }{\sigma }_{\text{EL}}}\right)}^2$$

Upon taking into account elastic–plastic behavior, crack closure and the Kitagawa effect, Eq. (6) becomes:

$$\frac{\mathrm{d}a}{\mathrm{d}N}=A{\left[\left(\sqrt{2\mathrm{\pi }{r}_{\text{e}}F}+Y\sqrt{\mathrm{\pi }\mathit{aF}}\right)\mathrm{\Delta }\sigma -\left(1-{\mathrm{e}}^{-k\lambda }\right)\left(K_{\text{opmax}}-K_{\min }\right)-\mathrm{\Delta }{K}_{\text{effth}}\right]}^2$$

The use of Eq. (16) requires that the following independent material constants be known or estimated: A, σ_Y, k, K_opmax and ΔK_effth.

Eq. (16) can be expressed in a more compact form as

$$\frac{\mathrm{d}a}{\mathrm{d}N}={\mathit{AM}}^2$$

where M, the net driving force for fatigue crack propagation, is the quantity in brackets in Eq. (16).

In analyzing the data of Shyam et al., we will use the same material constants as were used in a previous analysis of short fatigue crack growth at room temperature under fully reversed loading in a cast and hipped aluminum alloy, AC4C–T6 (σ_Y = 241 MPa) [11]. These constants are listed in Table 2.

Eq. (16) can then be written as

$$\frac{\mathrm{d}a}{\mathrm{d}N}=A{\left[\sqrt{\mathrm{\pi }F}(0.0014+0.73\sqrt{a})\mathrm{\Delta }\sigma -(1-{\mathrm{e}}^{-15\text{,}000\lambda })(1.5-K_{\min })-1.0\right]}^2$$