Irwin [8] proposed that the linear-elastic approach could be extended to include the elastic-plastic behavior, 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 assumed as that defined by Dugdale [9] , then the modified crack growth length, a_mod, is given as

where σ max is the maximum stress in a loading cycle, σ Y is the yield strength, and F, the elastic-plastic correction factor, is given by

F = 1 2 ( sec π 2 σ max σ Y + 1 ) .

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 [10] to describe this transient in crack closure behavior:

Δ K o p = ( 1 − e − k λ ) ( K o p max − K min ) , (3)

where ΔK_op is the value of K_op − K_min in the transient range; k is a material constant (units 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 a 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 [11] ). To achieve a smooth transition from ΔK control of the rate of FCG for cracks of macroscopic size to Δ σ control for cracks of microscopic size (the Kitagawa effect [11] ), the following driving force for FCG, ΔK, is now generalized [12] [13] [14] to take into account geometries other than just the center-cracked panel as

Δ K = ( 2 π r e F + Y π a F ) Δ σ (4)

where the value of Y depends upon the crack shape. If it is assumed that the initial crack shape in an unnotched specimen is semi-circular, then the value of Y is 0.73 [15] .

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 (da/dN = 10⁻¹¹ m/cycle), and Δ σ equal to the stress range at the fatigue strength level, Δ σ EL (10⁷ cycles), i.e.,

r e = 1 4.5 π F ( Δ K effth Δ σ EL ) 2 (5)

In this modified approach, r_e is considered 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, as 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, such as intermetallic compounds, in the actual material. It is merely an adjustable parameter introduced, as in the El Haddad, Topper and Smith’s case [16] , to deal with the Kitagawa effect [11] in a quantitative manner.

Considering these three aspects, i.e., the elastic-plastic behavior, crack closure, and Kitagawa effect, Equation (1) becomes

d a d N = A [ ( 2 π r e F + Y π a F ) Δ σ − ( 1 − e − k λ ) ( K o p max − K min ) − Δ K effth ] 2 (6)

The use of Equation (6) requires that the following independent material constants be known or estimated: A, Δ σ EL , σ Y , k, K_opmax, and ΔK_effth.

Equation (6) can be expressed in a more compact form as

d a d N = A M 2 , (7)

where M, the net driving force for FCG, is the quantity in brackets in Equation (6).

The material used was an extruded Mg alloy AZ61. Its chemical composition and mechanical properties are listed in Table 1 and Table 2, respectively. As can be seen in Table 1, the present magnesium alloy contains 6% Al, 0.6% Zn, and 0.4% Mn.

For the fatigue tests in the present study, round bar specimens were used. Figure 1(a) shows the shape and dimensions of the specimen used. As can be seen from the figure, the specimen has a gauge diameter of 6 mm and a gripping

diameter of 12 mm. For the measurement of the crack opening stress intensity factor, a single edge notched (SEN) specimen, as shown in Figure 1(b), was used. A strain gauge was attached at a point 1 to 2 mm in front of the crack tip, and the crack opening stress intensity factor, K_op, was measured using the elastic compliance method that utilizes the relation of load-strain.

Figure 2 shows S-N curves for the three different stress ratios, R = −1, 0.1, and 0.5. The data attached with the arrows in this figure indicate that, up to the plotted number of cycles, the specimen was not yet broken. For the stress ratio R = −1, the statistical distributions of fatigue lives are also shown in Figure 2. The statistical fatigue tests were conducted at three stress amplitudes, 180 MPa, 170 MPa, and 160 MPa, using fifteen to twenty specimens for each of them. As seen in the figure, the stress amplitude, σ a , at constant fatigue lives, such as 10⁵ and 10⁶ cycles, increases with a decrease in the values of R, from 0.5 to −1. For R = −1, the range of data variation at the high stress amplitudes, 180 and 170 MPa, is almost the same. However, at a lower stress amplitude of 160 MPa, the extent of variation in fatigue life becomes larger.

Figure 3 shows optical-microscope images of specimen surfaces that were fatigued at a constant stress amplitude of 170 MPa. As can be seen in these photographs, many black spots are distributed on the specimen surface. Using an electron

probe micro-analyzer (EPMA), these black spots were identified as an intermetallic compound, Mn-Al. As shown in Figure 3, the fatigue lifetimes, N_f, increased with a decrease in the intermetallic compound density. This result indicates that the density of the intermetallic compound is a critical factor that controls the fatigue lifetime length.

In these fatigue tests, conducted at a constant stress amplitude of 170 MPa, the fatigue lifetime, N_f, for each of the specimens increased with a decrease in the number of intermetallic compounds contained in the specimen.

To investigate crack initiation and FCG behavior, successive observations of the unnotched specimen surfaces during the fatigue process were conducted using the replica method. Figure 4 shows a typical FCG behavior, i.e., 2a vs. N relation, of the specimen tested at stress amplitude of 165 MPa and stress ratio of −1 (R = −1). Here, 2a is the crack length and N is the number of cycles of the repeated stress. The specimen failed at 49,496 cycles. The figure shows that the fatigue crack initiated at about 10% of the fatigue lifetime. This result means that the fatigue life N_f can be approximated by the crack growth life N_p, neglecting the crack initiation life N_i. The same trends were also confirmed in other similar tests.

Figure 5 shows the fracture surface of the specimen fatigue fractured at stress amplitude of σ a = 160 MPa under stress ratio, R = −1. As can be seen in the figure, the fatigue crack initiated from the intermetallic compound (Mn-Al), indicated by the white arrow. It can also be seen that the intermetallic compound existed just beneath the specimen surface. This finding was common with other observations.

To investigate the size of intermetallic compounds at the crack initiation site of the specimen, the fracture surfaces of the eight specimens fatigued at the same stress amplitude were investigated using SEM. From the SEM observations, although the amount if experimental data is small, the projected area of the intermetallic compound of each sample was measured. Figure 6 shows the distribution of the square roots of the intermetallic compound areas observed at the fracture origins in the eight specimens. The data were plotted on an extreme-value probability paper [6] . The horizontal and vertical axes of this figure show the square

roots of area, area^1/2, and cumulative probability of area^1/2, F(area^1/2), respectively. As can be seen, the experimental data are well approximated by the straight line drawn in the figure and are represented by the following double exponential function:

F ( area max 1 / 2 ) = exp [ − exp { − ( area max 1 / 2 − γ α ) } ] , (10)

where F(area_max^1/2) indicates the cumulative probability of the square root of the area, area^1/2, of the intermetallic compound at the fracture origin, and α and γ are material constants. Using the least-square method, the values of these constants, α and γ , were evaluated as 3.19 and 11.7, respectively.

The value of r_e in Equation (6) is less than 1 μm, approximately. When cracks are generated from large intermetallic compounds having a diameter of 20 μm or more, the length of the initial crack is longer than the value of r_e. In such a situation, we can ignore the Kitagawa effect [11] , and thus, the first term on the right side of Equation (6) can be deleted. Regarding the crack initiation mechanism, it is presumed that part of the interface between the intermetallic compound (Mn-Al) and the matrix Mg is loosened or broken by interfacial sliding or stress concentration owing to the fatigue load, and the crack is generated from the interface.

A detailed study on the crack initiation mechanism will be needed in the future. In the following discussion, for simplicity, we assume that the largest intermetallic compound is equivalent to the pioneer crack causing the final failure of the specimen. That is, the diameter of the largest intermetallic compound is assumed to be equal to the initial crack length of the pioneer crack.

d a d N = A [ Y Δ σ π a F − ( 1 − e − k a ) ( K o p max − K min ) − Δ K e f f t h ] 2 = A M 2 (11)

Figure 7 shows the log-log plot of the relation between the rate of fatigue crack propagation da/dN and the modified linear elastic fracture-mechanics parameter M. The values of the parameters used for calculation of M can be derived from Figure 8 by using the through crack as discussed in 3.2.2 Section. The values of the parameters are listed in Table 3. In Figure 7, the data for the short surface fatigue crack obtained from the rotating bending fatigue tests (R = −1) are plotted. In addition, the FCG data for the through thickness fatigue crack at R = 0.1 are also plotted in the figure.

As seen in the figure, the rate of FCG, da/dN, for both the surface crack and through thickness cracks can be expressed by using the M parameter (solid line in Figure 7). Therefore, the relationship between da/dN and M of the extruded Mg alloy can be expressed by Equation (12).

d a d N = 9.0 × 10 − 9 M 2 . (12)

It is possible to regard the FCG life, N_g, as equal to fatigue lifetimes, N_f, as the fatigue crack initiates early in the fatigue process. FCG life can be estimated by numerically integrating Equation (12), from an initial crack length (a_i) to the critical crack length (a_c).

with the assumption that crack size is equivalent to diameter of the intermetallic compound, an initial crack length, a_i, can be determined by using the extreme-value of maximum area of the intermetallic compound. In the present study, to evaluate the S-N curves, the average intermetallic-compound diameter, 14.5 μm, was used as the value of 2a_i. The critical crack length, a_c, should be determined from the value of the fatigue fracture toughness. However, for the sake of simplicity, in this study, a constant value of 4 mm was used regardless of stress amplitude. Figure 9 shows the comparison between the experimental data and calculation results. As can be seen in the figure, the calculated S-N curves for the three R values are in good agreement with the experimental values, except for R = 0.1.

The fatigue failures of the present Mg alloy specimens were caused by the crack that initiated from the most dangerous intermetallic compound in the specimen. Thus, the distribution of square root of the area of the intermetallic compound shown in Figure 6 was used for estimation of the P-S-N curve of the material. The square roots of the area of the intermetallic compound, with different values of cumulative probability such as 0.01, 0.5, 0.99, were determined from Figure 6 and Equation (10). These values of area^1/2 and of the corresponding diameters for different values of cumulative probability are listed in Table 4.

The estimated P-S-N curves are shown by the blue solid curves in Figure 10. Here, P(N_p) = 0.01, P(N_p) = 0.5, and P(N_p) = 0.99 indicate the cumulative probabilities of fatigue failure of the material, respectively. In the figure, the experimental fatigue data are also shown for a comparison purpose. From this figure, relatively good agreement can be seen between the estimated P-S-N curves and experimental data. Therefore, the distribution of the square roots of the maximum intermetallic-compound areas is convenient and useful for evaluation of the distribution of the alloy fatigue lifetimes.

In the present study, the distribution of fatigue lifetimes of the extruded Mg alloy was derived considering only the statistical distribution of the sizes of intermetallic compounds existing at the crack initiation site. To predict a more

accurate distribution of fatigue lifetimes, it is also necessary to consider the statistical variations in crack growth life.

In addition, for the sake of simplicity, the distribution of fatigue life was estimated by using the area distribution of intermetallic compounds at a small number of crack-occurrence sites under a constant stress amplitude. However, to obtain a more accurate fatigue life distribution, it is necessary to measure the area of more intermetallic compounds.