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In this study, fatigue tests under different R ratios were conducted on the AZ61 Mg alloy to investigate its fatigue lifetimes and fatigue crack growth (FCG) behavior. The fracture surface of the failed specimens was investigated using a scanning electron microscope to study the size of the intermetallic compounds from which the pioneer fatigue crack initiated and led to the final failure of the specimen. To determine the maximum size of the intermetallic compounds existing within the cross section of the specimen at higher risk, Gumbel’s extreme-value statistics were utilized. In the present study, the intermetallic compounds contained within the specimen were assumed to be the initial cracks existing in the material before the fatigue tests. A modified linear elastic fracture-mechanics parameter, M, proposed by McEvily et al., was used to analyze the short FCG behavior under different stress ratios, R. The relation between the rate of FCG and M parameter was found to be useful and appropriate for predicting the fatigue lifetimes under different R ratios. Moreover, the probabilistic stress-fatigue life (P-S-N) curve of the material under different R ratios could be predicted with this method, which utilizes both the FCG law and a statistical distribution of sizes of the most dangerous intermetallic compounds. The evaluated results were in good agreement with the experimental ones. This correspondence indicates that the estimation method proposed in the present study is effective for evaluation of the probabilistic stress-fatigue life (P-S-N) curve of the material under different R ratios.

Magnesium alloys are attractive materials from the viewpoints of lightness, dimensional stability, cutting performance, specific strength, and recyclability. For these reasons, they are used in various industrial products, such as car components and structural materials. At present, the process of die-casting is widely used for these alloys. However, the use of extruded materials is increasing, and it is of importance to understand the fatigue performance of the extruded Mg alloys.

It is well known that defects and intermetallic compounds are contained in Mg alloys when they are processed. It is considered that these defects and intermetallic compounds strongly influence the statistical distribution of the fatigue lifetimes and fatigue crack growth (FCG) behavior. Several studies on the fatigue of extruded Mg alloys [

In this study, fatigue tests under different R ratios were conducted using an extruded Mg alloy to investigate the statistical distribution of fatigue lifetimes. Further, the FCG behavior of the material was also investigated. Then, the sizes of the intermetallic compounds, from which the main crack initiated and caused the final failure of the specimen, were investigated in detail by observations of the fractured surfaces of the specimens using a scanning electron microscope (SEM).

The statistical distribution of sizes of the most dangerous intermetallic compounds was investigated. For the data analysis, Gumbel’s extreme-value statistics [

To evaluate the FCG behavior and stress-fatigue life (S-N) curve at different stress ratios, the modified linear elastic fracture mechanics (LEFM) method proposed by McEvily et al. [

The modified LEFM approach [

d a d N = A ( Δ K eff − Δ K effth ) 2 , (1)

where A is a material constant; ΔK_{eff} is the range of the effective stress intensity factor and is defined as K_{max}, the maximum stress intensity factor in a loading cycle, minus K_{op}, the stress intensity factor at the crack-opening level; and ΔK_{effth} is the effective range of the stress intensity factor at the threshold level.

The growth behavior of short fatigue cracks differs from the growth of large fatigue cracks in the following three important aspects.

Irwin [_{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_{op}_{max} for a macroscopic crack. The following expression has been proposed [

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_{op}_{max} 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 [

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 [

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 (10^{7} 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 [

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_{op}_{max}, 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

For the fatigue tests in the present study, round bar specimens were used.

Al | Zn | Mn | Fe | Si | Cu | Ni | Mg |
---|---|---|---|---|---|---|---|

6.03 | 0.57 | 0.38 | 0.002 | 0.008 | 0.0016 | 0.0005 | Bal. |

Tensile Strength | 0.2% Proof stress | Elongation | Vickers Hardness |
---|---|---|---|

312 [MPa] | 285 [MPa] | 18.3[%] | 59.5 |

diameter of 12 mm. For the measurement of the crack opening stress intensity factor, a single edge notched (SEN) specimen, as shown in _{op}, was measured using the elastic compliance method that utilizes the relation of load-strain.

To study the fatigue lifetimes and FCG behavior, two types of fatigue tests, the rotating bending fatigue test at stress ratio of −1 (R = −1) and push-pull fatigue test at R = 0.1 and 0.5, were conducted under laboratory air conditions. The fatigue tests at R = −1 were carried out using the rotating bending fatigue test machine at 15 Hz using the round bar specimens (

The push-pull fatigue tests were conducted using a servo-hydraulic fatigue-testing machine at 15 Hz. The replica method was employed to investigate the short FCG behavior. The fatigue tests were periodically interrupted at a constant interval during the fatigue process to obtain replicas of the specimen surfaces. The crack lengths recorded on the replicas were observed with an optical microscope with a magnification of 200. For a calculation of the stress intensity factor, K, for the surface fatigue crack, the following expression was used:

K = Y σ π a , (8)

where σ is the applied stress amplitude, a is the half crack length, and Y is a crack-shape correction factor. A value of 0.73 for Y was used, assuming that the crack-shape is semi-circular.

To study the FCG behavior of the through thickness fatigue crack, three-point bending fatigue tests were conducted at R = 0.1 and at a frequency of 15 Hz, using the servo-hydraulic fatigue testing machine. The notched specimen, as shown in

K = 3 S P 2 t W 2 π a ⋅ F ( α ) , α = a W , (9)

F ( α ) = 1.99 − α ( 1 − α ) ( 2.15 − 3.93 α + 2.7 α 2 ) ( 1 + 2 α ) ( 1 − α ) 3 / 2

where P is the applied load, S is length of the supporting point, t is specimen thickness, W is specimen width, and a is the crack length. The crack opening stress-intensity factor, K_{op}, during a loading cycle was determined by affixing strain gauges ahead of the crack tip. At that time, the elastic compliance method [

To clarify the crack initiation sites in the eight fatigue-fractured specimens, the fracture surfaces of these specimens were examined using an SEM. It was found that the pioneer crack that caused the final failure of the specimen was initiated from the intermetallic compounds contained in the specimen. This result was common to the eight test pieces. Then, the sizes (areas) of the intermetallic compounds were measured for each specimen. The statistical distribution of the areas of the intermetallic compounds was utilized for estimation of the P-S-N curves of the Mg alloy used. To do this, the square roots of the areas of the crack initiation sites were plotted on a Gumbel’s extreme probability paper.

^{5} and 10^{6} 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.

probe micro-analyzer (EPMA), these black spots were identified as an intermetallic compound, Mn-Al. As shown in _{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. _{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.

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.

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 [

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)

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

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}).

K_{op}_{max} | K_{effth} | σ_{y} | k |
---|---|---|---|

2.0 [MPam^{1/2}] | 0.6 [MPam^{1/2}] | 285 [MPa] | 16000 [m^{−1}] |

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.

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 ^{1/2} and of the corresponding diameters for different values of cumulative probability are listed in

The estimated P-S-N curves are shown by the blue solid curves in _{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

Cumulative probability (%) F(area^{1/2}) | 1 | 50 | 90 |
---|---|---|---|

Square root of the area of the intermetallic compound | 6.83 μm | 12.9 μm | 26.4 μm |

Diameter of the intermetallic compound | 7.70 μm | 14.5 μm | 29.8 μm |

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.

In the present study, the fatigue lifetimes and distributions of the fatigue lifetimes of an extruded Mg alloy were investigated at different stress ratios. Then, the intermetallic compounds contained within the specimen were assumed to be the initial cracks existing in the material before the fatigue tests. A modified linear elastic fracture-mechanics parameter, M, proposed by McEvily et al. [

1) The stress amplitude, σ a , at constant fatigue lives increased with a decrease in the values of R, from 0.5 to −1. For R = −1; at high stress amplitudes of 180 and 170 MPa, the ranges of data variation for the fatigue lifetimes were almost the same. However, at a lower stress amplitude of 160 MPa, the extent of data variation for the fatigue lifetimes became larger than that at higher stress amplitudes, 180 and 170 MPa.

2) Many intermetallic compounds (Mn-Al) were contained in the extruded Mg alloy AZ61. The fatigue lifetimes of the Mg alloy AZ61 increased with a decrease in the density of intermetallic compounds. This result indicates that the density of intermetallic compounds is a critical factor that controls the fatigue-lifetime length.

3) The modified linear elastic fracture-mechanics parameter M, proposed by McEvily et al., was found to be effective to analyze the short FCG behavior of the extruded Mg alloy AZ61. The S-N diagrams at different stress ratios estimated by using the M parameter were in good agreement with the experimental results.

4) For evaluation of the distribution of fatigue lifetimes, the distribution of the extreme-value for the square root of the area of the intermetallic compound and the relation, da/dN vs. M, were used. Good agreement was confirmed between the experimental and calculated results.

The authors declare no conflicts of interest regarding the publication of this paper.

Masuda, K., Ishihara, S., Ishiguro, M. and Shibata, H. (2018) Study on Fatigue Lifetimes and Their Variation of Mg Alloy AZ61 at Various Stress Ratios. Materials Sciences and Applications, 9, 993-1007. https://doi.org/10.4236/msa.2018.913072