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Comparative analysis of the fatigue short crack growth on Al 6061-T6 alloy by the exponential crack growth equation and a proposed empirical model
Accepted Manuscript Comparative analysis of the fatigue short crack growth on Al 6061-T6 alloy by the exponential crack growth equation and a proposed empirical model Jan Mayén, Arturo Abúndez, Isa Pereyra, Jorge Colín, Andres Blanco, S. Serna PII: DOI: Reference:
S0013-7944(16)30354-X http://dx.doi.org/10.1016/j.engfracmech.2017.03.036 EFM 5461
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
2 September 2016 25 March 2017 26 March 2017
Please cite this article as: Mayén, J., Abúndez, A., Pereyra, I., Colín, J., Blanco, A., Serna, S., Comparative analysis of the fatigue short crack growth on Al 6061-T6 alloy by the exponential crack growth equation and a proposed empirical model, Engineering Fracture Mechanics (2017), doi: http://dx.doi.org/10.1016/j.engfracmech. 2017.03.036
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Comparative analysis of the fatigue short crack growth on Al 6061-T6 alloy by the exponential crack growth equation and a proposed empirical model 1 1
Jan Mayén*, 1Arturo Abúndez, 2Isa Pereyra, 1Jorge Colín, 1Andres Blanco, 2S.Serna.
Tecnológico Nacional de México-Centro Nacional de Investigación y Desarrollo Tecnológico (CENIDET), Prolongación Palmira s/n esq Apatzingan, Col. Palmira. Cuernavaca, Morelos, México
2
CIICAp-FCQeI-Universidad Autónoma del Estado de Morelos Av. Universidad 1001 Col. Chamilpa C.P. 62209-Cuernavaca, Mor. México. *E-mail: [email protected], Phone/fax: Tel./fax: +52 777 362-7770 ext. 1408
Abstract In this work, the short crack propagation and growth rate nonlinear behavior of an Al 6061 alloy in T6 condition, were assessed. The specimens were fatigue tested at 20 Hz in a rotating bending fatigue machine with constant amplitude loading of 88%, 61% and 38% of yield strength, and the short crack initiation and propagation were followed up by optical microscopy. Interaction between the surface cracks and aluminum microstructure were identified: persistent slip marks (PSMs) can be observed on the surface specimen, where persistent slip bands (PSBs) emerge from the surface as a result of accumulated damage by intrusion-extrusion mechanism, which eventually leads to the accumulation of dislocations within the grain as a consequence of grain border barriers and stress concentration at second phase particles. The recorded length crack paths were later analyzed to study the crack growth by Frost and Dugdale exponential crack growth equation (ECG) and its comparison with a proposed model (PM) developed by multiple linear regression. The results showed that the crack growth rate is affected by microstructural features in the early growth stage, which is assumed to be from the nucleation up to 150 µm of crack length. The ECG is reliable only for the stage where the crack growth is mainly a consequence of stress concentration at crack tip; in contrast, the PM do properly describes the nonlinearity behavior between short and long crack growth.
Keywords: Fatigue crack propagation; aluminum alloy; exponential crack growth equation; multiple linear regression.
1. Introduction The aluminium alloys are the preferred material choice for numerous structural applications, specifically for the aerospace and automotive industries, since they exhibit benefits such as the ease of manufacture, corrosion resistance, weldability, along with lightweight over those of structural steels. The Al 6061 is well known as heat treatable alloy that exhibits high strength–weight ratio coupled with excellent corrosion resistance. Its extrudability is a unique feature that make it suitable for a broad usage in welded structural parts for vessels, aircrafts, and automotive body panels and pipelines. All these structural parts are subjected to high cycle loading which causes fatigue. The fatigue damage of materials has been studied since the discovery of the fatigue occurrence. Ewing et al. [1] studied the Swedish iron during rotating bending fatigue surface and focused on the surface damage. By the Usage of optical microscope, they found and studied the formation of surface markings that occurred during cyclic loading, where fatigue cracks developed later in fatigue life. Today, problems on crack propagation in aluminium and its alloys, are of great interest to the scientific community, where the vast accumulated experimental results and theoretical methods have been used to acquire a deep knowledge of crack propagation over the past few years. However, additional experimental and theoretical work is required in order to achieve a deeper understanding related to crack initiation (known as stage I) of fatigue on structural parts, which is a crucial constraint for structural design. Some researchers as Zhuang et al. [2] and Jones et al. [3], have commented on the lesser effect of the final crack size on fatigue crack life, mainly because the crack growth rate is much faster during the later stage of growth, where the crack growth rate is relatively slow in the early stage. This makes the total life strongly dependent on the initial crack size [2].
The fatigue crack propagation has three well-known stages: the stage I occurs when the crack has been nucleated and initiates its propagation, which is related to several microstructural features. Stage II, corresponds to an increase of the stress intensity factor as a consequence of an increase of the crack length, along with the activation of deformation systems. Stage III is related to a critical crack length and an unsteady behavior [4].
There are several studies reported in literature [5], [6] and [7], which are related to the crack propagation in stage I. For the aluminium alloys, the crack propagation in Stage I acquired more importance when two Comet jet airplanes crashed in 1954. These airplanes, were manufactured with 7075 aluminium components; the examination on the airplane remains showed that the crash was related to a fatigue failure initiated in the fuselage, along with a stress concentration at the windows and hatches [8]. However, the key research on fatigue crack propagation in the past decades have focused on Linear Elastic Fracture Mechanics approach (LEFM) rather than short crack or small crack issues [4], where it was originally thought that, for any given material and thickness, da/dN versus ΔK relationship was unique. Pearson [9], conducted a research related to the behavior of short cracks in commercial aluminium alloys, and he observed an atypical propagation phenomenon: short
cracks propagate below the ΔKth threshold. Also, Meggiolaro et al. [10] and Nalla et al. [11] stated that short cracks propagate below the effective stress intensity threshold ΔKth,eff. It has been verified that the crack nucleation and crack propagation in Al-Si cast alloys occur through the α-aluminum matrix, and is due to the stress intensity increase at the crack tip; when the crack is long enough, the stress intensity factor at the crack tip becomes the mechanical driving force for the crack propagation [12].
Some techniques, which are focused on the crack propagation characterization have been proposed over the years, where the most common technique used is the Linear Elastic Fracture Mechanics (LEFM). This technique has been effectively used to calculate the crack propagation from a flaw, defect or initial crack size a, related to fatigue cycles N. The crack propagation behavior usually follows the Paris equation [4]; therefore, the Paris equation continues to be the accepted standardized method worldwide to investigate the fatigue crack growth. However, several scientists [13], [14] and [15], have found that short fatigue cracks showed irregular behavior, which has been characterized as atypical when compared to longer cracks under nominally elastic cycling, for a diversity of commercial alloys employed in structural parts [16], whose behavior cannot be described by the stress intensity factor threshold (Kth) from the LEFM [17], suggesting that the crack growth and evaluation of failure requires that the crack tip stress field has to be evaluated at a length scale r0 in front of the crack. The characteristic length scale r0 has been widely used in the assessment of fatigue crack growth by the fracture mechanics [17], [18], [19], [20], [21]. Therefore, the evaluation of long cracks propagated under Mode I uniaxial loading, where the ratio r0/a have a tendency to be quite small and, as a consequence, the error in the usage of ΔK and Kmax to characterize the crack tip stress field is also quite small. However, for short cracks with length scales of the order of 10 µm, the ratio r0/a will not be small [22], consequently increasing the error in the usage of ΔK and Kmax. Subsequently, it has been confirmed by several authors [22] and [23] that the conclusions reached by [17], were correctly assessed. Therefore, it can be concluded that K dominance is lost for short cracks. Consequently, the similitude hypothesis is also invalid for short cracks. In this context, it was clear that the generalized Frost and Dugdale [24] model was a potential alternative. They studied the crack growth of fatigue cracks on huge thin panels of steel and light alloys, including aluminium plates with a central notch as crack starters; they related the fatigue crack growth rate to the crack length and to the third power of the stress range in a proportional way. Later, Ritchie [25] studied the nucleation and growth of fatigue cracks on aluminium and found that the propagation behavior was crack length dependent, and that the crack growth was dependent to native crystallographic characteristics as the direction of neighboring grains and grain border structure. This was also demonstrated by Molent et al. [26], Zhai et al. [27]and Polák et al. [28].
In this work, the crack length a, and fatigue cycles N values obtained by rotating bending fatigue testing of an aluminium 6061-T6 temper using a slightly modified notched specimen, were assessed.
The crack length and fatigue cycles values were the input parameters for the exponential crack growth equation (ECG) proposed by Frost and Dugdale, in order to characterize the short crack propagation behavior. In addition, a proposed robust multiple linear regression (PM) on crack length and fatigue cycles values was done in order to describe the correlation among stress amplitude, crack length and number of cycles during the fatigue tests. The results obtained by the PM showed an initial extrapolated crack ai value similar to those values obtained by the ECG. Furthermore, the PM allows to directly estimate the crack growth rate as a function of the stress amplitude for aluminium alloys. Finally, a microstructure relation to the crack growth initiation and early propagation of the cracks was studied to identify the possible microstructural features affecting the crack growth behavior.
2. Experimental Procedure The specimens were obtained from an aluminium 6061-T6 alloy plate with 12.5 mm thickness and machined to obtain longitudinal specimens along the extruded axis direction (T-L), as depicted in Fig. 1. The reported commercial chemical composition in wt% is: Al-Balance, Cr-0.1%max, Cu0.1%max, Fe-0.35%max, Mg-0.45-0.9%, Mn-0.1%max, Si-0.2-0.6%, Zn-0.1%max, Ti-0.1%max. The mechanical properties, experimentally obtained, were: 249 MPa for yield strength (YS), 69 GPa for Young’s modulus (E), 281 MPa for ultimate tensile strength (UTS), and a 13% for ultimate strain (ε).
Fig. 1. Extruded Axis Direction.
Jan Mayén et al. [29], following the research reported by Narasaiah and Ray [30], developed a new subsized specimen geometry, where regions “A” and “B” were modified in order to induce a crack initiation, as depicted in Fig. 2. The specimens used for crack growth tests were prepared as described elsewhere. [29]. The fatigue crack growth experiments were performed using a rotating bending fatigue machine by Fatigue Dynamics, Model RBF 200.
Fig. 2. Specimen proposed geometry (dimensions in mm).
The loading conditions for fatigue testing were: 242 MPa, 169 MPa and 106 MPa of stress amplitude, corresponding to 88%, 61% and 38% of yield strength of the aluminium alloy, respectively. These loads were obtained according to the Finite Element Analysis reported elsewhere [29], where the higher stress values in the specimen subjected to bending loads are located at regions A and B, including the notch effect. The experimental conditions for the crack growth by fatigue were: a) 20 Hz at room temperature, b) an initial fatigue test was performed in order to induce a 15-50 µm short crack at regions A and B, c) the crack paths were monitored by an optical microscope along the test, d) the tests were arrested once crack length was approximately 1x10 3 µm long. The recorded crack paths were posteriorly analyzed in order to obtain the short crack length, which were fitted by Eq. (1) [24].
(1)
Where, N is the number of accumulated cycles, kg is a coefficient that characterizes the crack growth rate, which is directly related to the geometry specimen, material and load [24]; a is the crack length and ai is the initial crack length related to a materials defect, a second phase particle or other microstructural features. In order to observe the PSB’s density along the crack path, the specimens were analyzed by optical microscope. One specimen, for each of the loading conditions was etched by immersion in Kellers etch (190 ml of distilled water, 5 ml of nitric acid, 3 ml of hydrochloric acid and 2 ml of hydrofluoric acid) for 10 seconds lapse of time, rinsed in distilled water followed by immersion in acetone and rapidly dried by hot air, in order to allow a slight revealed of the microstructure to observe the PSB’s by scanning electron microscope (SEM JSM 5900-LV) at 20 kV and 6 nA for the beam current.
2.1 Multiple linear regression methodology for crack length, stress amplitude and fatigue cycles number correlation. The main aim of this work, was the development of an empirical model that correlates the stress amplitude during fatigue cyclic loading with the crack length. The empirical model was developed by Multiple Linear Regression (MLR) technique, since there are more than two independent variables (predictors) in the experimental conditions. The model, using population information, is described by Eq. 2 [31],
(2)
This method requires the usage of some assumptions. These are necessary for the mathematical process, in order to properly find the optimal empirical equation that better fits the experimental data. The predictors x1=N1, x2= σ2, …xkt represents the control parameters for the experimental conditions, and are: a) number of fatigue cycles N, and b) stress amplitude σ. These assumptions are: 1. The error terms εi are normally distributed and equal to 0. 2. The error terms are independent of past error terms, which is
(3)
3. All the population have equal variances, S1= S2=S3=··· 4. The predictors are not correlated with each other. The Eq. (2) can be represented as the matrix form, where:
(4)
Then,
(5)
The least squares method is then applied to Eq. (5) in order to find the estimates of β that minimizes the sum of squares of the residual, (6)
Since ε=Y-Xβ. Expanding S(β) it is then obtained,
(7) Subsequently, (8) As it can be done in calculus to find the minimum of a function, the least squares estimate β can be obtained by solving the Eq. (8) based on the first derivative of S(β),
(9)
Then, (10) And thus β is obtained,
(11)
3. Experimental results and discussion
3.1 Relation between short crack propagation and microstructural features. In the Fig. 3, the final crack path on the specimen surface, which is located in the notch at region A of the specimen according to FEA results [29], which corresponds to the loading conditions at 88%, 61% and 38% of yield strength of the aluminium alloy.
a) 106 MPa.
b) 168 MPa
c) 242 MPa. Fig. 3 Crack initiation for the three different loading conditions.
The predominant crack growth mechanism observed at a detail by SEM observations (see Fig. 5) was crack propagation due to the intrusion-extrusion mechanism, which is described in Fig. 4.
Fig. 4 Intrusion – extrusion mechanism [32], [33].
This intrusion-extrusion mechanism, has been considered as the earliest evidence of metal fatigue, which is related to the development of surface deformation in the form of PSB’s that emerge inside the grains of the metal matrix [34]. These microstructural features are visible by their profile, as can be seen in the Fig. 5 (a). These PSB’s are the evidence of a severe plastic deformation localized within the grain that leads to the formation of the fatigue crack and its propagation.
The SEM micrographs depicted in Fig. 5, show evidence of PSB’s (a), as the main crack growth mechanism. In Fig. 5 I, it can be observed that the crack is propagating from the PSB’s (a). Also, decohesion of second phase particles (b) can be observed (see Fig. II). Another important observation is that second phase particles act as crack starters (c) (see Fig. 5 II) or propagation barriers (d) (see Fig. 5 I and II).
I)106 MPa
II) 168 MPa
III) 242 MPa Fig. 5 Microstructural features at the crack tip for the corresponding experimental loading conditions.
The aluminium 6061 alloy, contains intermetallic compounds of a very complex structure; these compounds are nucleated by the casting process and may act as crack starters or crack propagation barriers, depending on its chemical composition and morphology. Furthermore, the intermetallic compounds reported for this aluminum alloy are (Fe, Mn, Cu) 3SiAl12 and Mg2Si, being the latest the main phase reported [35]. In addition, the intrusion-extrusion mechanism leads to the accumulation of dislocations within the grain, which is due to grain border barrier; this mechanism allows the crack to propagate within the material matrix.
3.2 Determination of fatigue short crack growth by Frost and Dugdale exponential crack growth equation (ECG). The short crack length was inspected during the fatigue tests, which was carried out in batches, by acquiring images by optical microscopy from notched area, at each test stop. Once the tests were over, principal crack lengths were assessed as a function of loading cycles for the three loading
conditions mentioned above. The short crack path usually follows the slip plane or grain boundaries (see Fig. 5), which corresponds to the crystallographic orientation with the highest Schmidt factor [32].
The crack growth is depicted in Fig. 6 for the three cyclic loading conditions. As it can be observed, the crack for the specimens subjected to 242 MPa nucleated at a lower cycling than those specimens subjected to 168 MPa and 106 MPa, this is due to high localized plastic deformation at the notched region. As can be observed in Fig. 6, the slope of the crack growth curves represent the crack growth rate, which is related to the loading test condition, which implies that propagation rate is directly proportional to the stress amplitude applied. At the early stage of fatigue tests, for 242 MPa and 106 MPa a crack arrest can be observed (see enclosed points); in consequence, this instability of the crack growth has to be taken into account when estimating the crack growth rate.
Fig. 6. Crack growth for three different loading conditions.
In Fig. 6, the crack length as a function of fatigue cycles for the three experimental loading conditions is depicted. As can be observed, the cracks of the specimens subjected to 242 MPa and 106 MPa were nucleated earlier than those of the specimens subjected to 169 MPa, this maybe is due to the microstructural features acting as propagation barriers in the notched zone for the 106 MPa condition. The crack growth curve of the specimens subjected to 242 MPa depicts a linear behavior, which is related to a faster crack propagation due to a lack of interaction between crack and microstructural features due to the high localized plastic deformation at the crack tip. As can be seen in Fig. 6, the crack growth for the specimen subjected to 106 MPa is arrested from 103 up to 105 cycles, around to 33 µm long; after that, the crack grows at a constant rate during the test remaining.
This behavior can be related to the interaction between the crack path and the microstructural features, as stated by Polák et al. [28].
The Eq. (1) [24] relates the crack length a and the fatigue cycles N during fatigue tests, and indicates that the crack growth rate
directly depends on the crack length a, which represents the cumulative
crack length during the experimental fatigue test. The equivalent crack growth for the three experimental loading conditions were obtained according with Eq. (1), which can be rewritten as:
(12) And in the matrix form: (13)
Where ai is the initial crack length or the initial flaw size, and kg characterizes the crack growth rate at the applied stress amplitude, n is the amount of experimental data. The input information of the Eq. (13) are obtained from the experimental data depicted in Fig. 6 and, when solved, the values of kg and ai are obtained and, therefore the exponential equation that describes the behavior of crack growth is obtained. The Table 1 contains the values of the crack length obtained by solving Eq. (13), as a function of the testing loading conditions and the number of fatigue cycles.
Table 1. Crack growth as a function of loading condition. loading 242 MPa Test 1 242 MPa Test 2 169 MPa Test 1 169 MPa Test 2 106 MPa Test 1 106 MPa Test 1
Crack length
(µm)
R-square 0.98 0.97 0.89 0.93 0.89 0.87
The extrapolated crack values for the initial crack length at 0 cycles is in the range between 82 µm and 34 µm for 242 MPa and 106 MPa, respectively. This extrapolated crack is assumed as the initial defect that propagates the crack during the fatigue testing.
For the three tested conditions, an accelerated growth can be observed up to 100 µm, and the crack growth rate changes approximately between 100 µm and 200 µm, this is due to the influence of the microstructure at this early stage (indicated by the lines in Fig. 7, where two slopes can be approximated). The influence of microstructural features decreases inversely to crack length, and stress concentration at the crack tip, becomes the main mechanical driving force for crack growing,
as depicted in Fig. 7. The crack growth becomes steady above 150 µm length; this behavior may be explained by plastic strain field localized in front of crack tip.
Fig. 7. Unsteady behavior below the 150 µm length.
Following Eq. (14) [24], an analysis of the crack growth rate was performed for the three different loading conditions. The results of the analysis are depicted in Fig. 8, where is showed that for 242 MPa loading condition, the crack growth rate is higher than the two other loading conditions.
(14)
Fig. 8. Crack growth rate exponential equation approach.
The identification of crack nucleation and short crack growth depends strongly on detection method employed, whether it is optical or scanning electron microscopy. With the first, the detection can be performed about 20% of fatigue life, while with the second one, about 5% of fatigue life [36].
However, for low cost and simplicity testing, optical microscopy monitoring method by [30] for crack growth was used. The size of the initial crack depends on the microstructural dimensions of the specimen. Thus, if crack is nucleated by the development of PSB’s, the initial crack will be the same size as its containing grain; in contrast, if neighboring grains have same crystallographic orientation among them, it is possible that the cracks nucleate [37] throughout more than one neighboring grains; crack nucleation is also possible for second phase particles, as can be seen in Fig. 5 (c). Additionally, the linkage between the principal and secondary cracks as a crack growth mechanism, has been observed and this behavior contributes to an accelerated crack growth, as observed for 242 MPa loading condition. It is well known that the crack growth rate decreases as the crack tip crosses grain boundaries, including matrix/precipitate interfaces [37].
3.3 Proposed model (PM) for the estimation of the short crack growth. A similar short crack growth behavior analysis as described in the section above was studied, for the Al 6061-T6 alloy cycled with constant stress amplitude, using the multiple linear regression approach as described in section 2.1. A correlation was established between the measured crack lengths a and their corresponding given cycles N, as a function of the stress amplitude σa during the tests. The empirical model obtained, which describes the behavior of the crack growth as a function of stress amplitude and number of cycles, is given in Eq. (15):
(15)
Where N is the number of fatigue cycles, σa is the stress amplitude during the test and aσaN is the crack length as a function of stress amplitude, and P1 and P2 are defined by.
(16) (17) The corresponding coefficients βi are shown in Table 2. Table 2. Coefficient βi values obtained for the PM. Coefficients
Values -56.5927 4.7347x10-4 3.8157x10-11 0.7897 -4.3638x10-6 -2.2725x10-3 -5.0491x10-7 4.7080x10-13 -1.1745x10-2
1.2736x10-9 3.1364x10-4 0.98 It is of great importance to mention that the PM takes the stress amplitude conditions into account, which indicates that the interpolation in the tested range is possible for crack length prediction at other stress amplitude levels, in contrast to the ECG, where the stress amplitude variable is implicit in the equation, therefore an equation is needed for each condition to correlate the Kg as a function of stress amplitude, as can be seen in Table 1. The ECG equation proposed by Frost and Dugdale [24] in the late 60’s, and employed more recently by some researchers as Polák et al. [28] and [37], characterizes accurately the long crack growth behavior for a given stress amplitude; however, the relation of crack length with different stress amplitude ranges cannot be described by one single equation, and the ECG shows R2=< 0.9 for some crack propagation data, which is due to high data dispersion during the early crack propagation, a<150 µm, depicted in Fig. 8.
The standardized residuals values obtained from the multiple linear regression are plotted in Fig. 9. The distribution of the residuals showed in the plot confirms that: a) the data values are symmetrically distributed, tending to cluster towards the middle of the plot (horizontal line), b) the data values are clustered around the lower single digits of the y-axis and, c) the data values plotted exhibits an unpattern behavior, which indicates that the Eq. (15) describes accurately the crack growth behavior.
Fig. 9. Standardized residuals of “a”.
In Fig. 10, the estimated values of crack length versus the experimental values are plotted, where observable behavior between the experimental and estimated crack length values are linearly correlated.
Fig. 10. Linear correlation between experimental and predicted crack length data with a R2=0.98.
The length of the cracks for the experimental loading conditions can be approximately estimated by Eq. (15) for a valid interval from zero to Nf cycles, where is expected that an hypothetical crack grows from ai up to af. Then, the period where crack initiation occurred is substituted by the length ai of an imaginary crack at zero cycles and zero stress amplitude, which has a hypothetical length of 0 µm for the loading conditions according to Eq. (15).
The confidence intervals, obtained by the multiple linear regression methodology from the crack length of the specimens subjected to rotating bending fatigue at 242 MPa, 169 MPa and 106 MPa, are depicted in Fig. 11, Fig. 12, and Fig. 13, respectively. This estimated range of values, which is likely to include the unknown population for the crack length was calculated for a 95% confidence interval.
Fig. 11. 106 MPa 95% Confidence Interval.
Fig. 12. 169 MPa 95% confidence interval.
Fig. 13. 242 MPa 95% confidence interval.
In the figures above, it is observable that the proposed model takes into account the high dispersion data below 150 µm, due to the strong interactions between the fatigue crack path and the microstructural features. Additionally, the 106 MPa stress amplitude condition exhibits the higher data dispersion due to the low stress concentration in regions A and B in the fatigue specimen, which causes a chaotic behavior due to a slower crack growth that increases the interaction between the crack and the microstructural features of the Al 6061 alloy.
3.4 Comparison between the ECG and the PM for short crack growth estimation.
The PM obtained by multiple linear regression has been compared with the ECG [24]; the results for the Kg coefficient for the Al 6061 alloy obtained by the ECG are in good agreement with contemporary researchers as Polák et al. [28]. In the Fig. 14, a comparison is made among the experimental data (ED), the exponential equation (ECG) and the proposed empirical model (PM); the curves depicted in the Fig. 14 shows that the ECG equations describe the general behavior for the short crack, however it fails to describe the early stage of the crack, below 150 µm in length, where the highly data dispersion is due to the strong interactions between the crack path and the microstructural features. In contrast, the PM accurately describes the short crack growth and its transition to long crack even for the early stage of the crack, below to 150 µm length, which allows to estimate the fatigue life Nf.
Fig. 14. Model comparison for the fatigue crack length. In Table 3, the R2 for the ECG, and the PM are shown. It is clear that the correlation coefficient is directly proportional to the stress amplitude, this indicates that for lower stress amplitude values, the data dispersion increases, and the ECG is unable to predict the behavior accurately, while the PM has the same R2 for the stress amplitude tested ranges.
Table 3. Correlation coefficients for the models comparison. Model ECG ECG ECG PM
Stress Amplitude 106 169 242 Any stress amplitude
R2 0.88 0.91 0.97 0.98
In the Fig. 15, the crack growth rate is depicted for the three experimental loading conditions, where it is observable that the ECG generally describes the crack growth rate by the Kg coefficient, as described by Eq. (14), with a straight line for each stress amplitude during testing. In contrast, the PM involves the interaction between the crack and the microstructural features and takes into account the stress amplitude conditions during the fatigue testing. Hence, the PM can be used for crack length and crack growth rate prediction by interpolation in the tested stress amplitude range for the studied alloy, which is the range between the yield stress and the apparent fatigue limit of the Al 6061 alloy.
2x10-1 1x10-1
da/dN (m/cycle)
5x10-2 2x10-2 1x10-2
106 MPa ED 106 MPa ECG 106 MPa PM 169 MPa ED 169 MPa ECG 169 MPa PM 242 MPa ED 242 MPa ECG 242 MPa PM
5x10-3 2x10-3 1x10-3 5x10-4 2x10-4 1x10-4 5x10-5 10
20
30
40 50 60 7080 100
200
Crack length "a" (m)
300 400 500
700
1000
Fig. 15. Crack growth rate models comparison.
The correlation between the experimental data and the estimated values by ECG proposed by Frost and Dugdale, which has been used by several contemporary authors, and the PM is compared by the R2. The results of the correlation are plotted In Fig. 16 and Fig. 17 for the ECG and the PM, respectively. The ECG shows an R2=0.94 and the PM a R2=0.98, these results shows that the PM has competitive advantages against the ECG, as a more accurate estimation of the crack growth in early stages and, therefore, estimate the remaining life of the material.
Fig. 16. Experimental crack length vs.
Fig. 17. Experimental crack length vs
predicted values by ECG.
predicted values by PM.
4. Conclusions
The modification of the specimen geometry allows the observation of the short crack growth by both optical and electron microscopy, since the crack grows over a flat surface, during the rotating
bending fatigue test. Therefore, specimen geometry employed in this work has a lot of potential benefits for the study of initiation and propagation of short cracks, i.e. low cost test or simplified crack growth monitoring.
Experimental data obtained from fatigue testing showed that crack nucleation takes the highest fraction of the fatigue lifetime, where this period took about 6x105 and 3x105 cycles for 106 MPa and 164 MPa loading condition, respectively. It is assumed that the crack nucleation stage involves a short crack length up to 103 µm, which means that the highest fraction of the fatigue lifetime follows the behavior described by the PM, instead of the fracture mechanics approach.
The responsible mechanisms for crack nucleation and propagation found in this work were: grain boundaries as crack barriers, second phase particles as crack nucleation sites and barriers, and the PSB’s over the specimen surface as the primary mechanism for crack growth due to extrusionintrusion process.
The crack growth a was fitted by the ECG, and a crack growth coefficient kg for each experimental loading condition was obtained from the experimental data, which represents the crack growth resistance and is inversely proportional to the crack growth rate. Also, the crack growth a was fitted by multiple linear regression and an empirical equation was proposed, which showed good correlation between crack growth rate and crack length at any given loading condition and cycling. The PM describes the behavior for the stress amplitude conditions tested, which indicates the possibility to interpolate between the tested range of stress amplitude, in order to predict the crack length and the crack growth rate at any other stress amplitudes. On the other hand, the Frost and Dugdale ECG describes the whole crack growth by a single slope Kg; however, this equation has not the sensitivity required to properly describe the slope change on the transition between short and long crack growth.
Acknowledgements Funding supported by CONACyT through agreement Number 291113-CENIDET We acknowledge to Dr. Iván Puente-Lee for the assistance on the scanning electron microscopy.
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August, 2016
Highlights
Interaction between the surface cracks and aluminum microstructure were identified.
Recorded length crack paths were later analyzed to study the crack growth by Frost and Dugdale exponential crack growth equation (ECG) and its comparison with a proposed model (PM) developed by multiple linear regression.
The ECG is reliable only for the stage where the crack growth is mainly a consequence of stress concentration at crack tip; in contrast, the PM do properly describes the nonlinearity behavior between short and long crack growth.
S0013-7944(16)30354-X http://dx.doi.org/10.1016/j.engfracmech.2017.03.036 EFM 5461
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
2 September 2016 25 March 2017 26 March 2017
Please cite this article as: Mayén, J., Abúndez, A., Pereyra, I., Colín, J., Blanco, A., Serna, S., Comparative analysis of the fatigue short crack growth on Al 6061-T6 alloy by the exponential crack growth equation and a proposed empirical model, Engineering Fracture Mechanics (2017), doi: http://dx.doi.org/10.1016/j.engfracmech. 2017.03.036
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Comparative analysis of the fatigue short crack growth on Al 6061-T6 alloy by the exponential crack growth equation and a proposed empirical model 1 1
Jan Mayén*, 1Arturo Abúndez, 2Isa Pereyra, 1Jorge Colín, 1Andres Blanco, 2S.Serna.
Tecnológico Nacional de México-Centro Nacional de Investigación y Desarrollo Tecnológico (CENIDET), Prolongación Palmira s/n esq Apatzingan, Col. Palmira. Cuernavaca, Morelos, México
2
CIICAp-FCQeI-Universidad Autónoma del Estado de Morelos Av. Universidad 1001 Col. Chamilpa C.P. 62209-Cuernavaca, Mor. México. *E-mail: [email protected], Phone/fax: Tel./fax: +52 777 362-7770 ext. 1408
Abstract In this work, the short crack propagation and growth rate nonlinear behavior of an Al 6061 alloy in T6 condition, were assessed. The specimens were fatigue tested at 20 Hz in a rotating bending fatigue machine with constant amplitude loading of 88%, 61% and 38% of yield strength, and the short crack initiation and propagation were followed up by optical microscopy. Interaction between the surface cracks and aluminum microstructure were identified: persistent slip marks (PSMs) can be observed on the surface specimen, where persistent slip bands (PSBs) emerge from the surface as a result of accumulated damage by intrusion-extrusion mechanism, which eventually leads to the accumulation of dislocations within the grain as a consequence of grain border barriers and stress concentration at second phase particles. The recorded length crack paths were later analyzed to study the crack growth by Frost and Dugdale exponential crack growth equation (ECG) and its comparison with a proposed model (PM) developed by multiple linear regression. The results showed that the crack growth rate is affected by microstructural features in the early growth stage, which is assumed to be from the nucleation up to 150 µm of crack length. The ECG is reliable only for the stage where the crack growth is mainly a consequence of stress concentration at crack tip; in contrast, the PM do properly describes the nonlinearity behavior between short and long crack growth.
Keywords: Fatigue crack propagation; aluminum alloy; exponential crack growth equation; multiple linear regression.
1. Introduction The aluminium alloys are the preferred material choice for numerous structural applications, specifically for the aerospace and automotive industries, since they exhibit benefits such as the ease of manufacture, corrosion resistance, weldability, along with lightweight over those of structural steels. The Al 6061 is well known as heat treatable alloy that exhibits high strength–weight ratio coupled with excellent corrosion resistance. Its extrudability is a unique feature that make it suitable for a broad usage in welded structural parts for vessels, aircrafts, and automotive body panels and pipelines. All these structural parts are subjected to high cycle loading which causes fatigue. The fatigue damage of materials has been studied since the discovery of the fatigue occurrence. Ewing et al. [1] studied the Swedish iron during rotating bending fatigue surface and focused on the surface damage. By the Usage of optical microscope, they found and studied the formation of surface markings that occurred during cyclic loading, where fatigue cracks developed later in fatigue life. Today, problems on crack propagation in aluminium and its alloys, are of great interest to the scientific community, where the vast accumulated experimental results and theoretical methods have been used to acquire a deep knowledge of crack propagation over the past few years. However, additional experimental and theoretical work is required in order to achieve a deeper understanding related to crack initiation (known as stage I) of fatigue on structural parts, which is a crucial constraint for structural design. Some researchers as Zhuang et al. [2] and Jones et al. [3], have commented on the lesser effect of the final crack size on fatigue crack life, mainly because the crack growth rate is much faster during the later stage of growth, where the crack growth rate is relatively slow in the early stage. This makes the total life strongly dependent on the initial crack size [2].
The fatigue crack propagation has three well-known stages: the stage I occurs when the crack has been nucleated and initiates its propagation, which is related to several microstructural features. Stage II, corresponds to an increase of the stress intensity factor as a consequence of an increase of the crack length, along with the activation of deformation systems. Stage III is related to a critical crack length and an unsteady behavior [4].
There are several studies reported in literature [5], [6] and [7], which are related to the crack propagation in stage I. For the aluminium alloys, the crack propagation in Stage I acquired more importance when two Comet jet airplanes crashed in 1954. These airplanes, were manufactured with 7075 aluminium components; the examination on the airplane remains showed that the crash was related to a fatigue failure initiated in the fuselage, along with a stress concentration at the windows and hatches [8]. However, the key research on fatigue crack propagation in the past decades have focused on Linear Elastic Fracture Mechanics approach (LEFM) rather than short crack or small crack issues [4], where it was originally thought that, for any given material and thickness, da/dN versus ΔK relationship was unique. Pearson [9], conducted a research related to the behavior of short cracks in commercial aluminium alloys, and he observed an atypical propagation phenomenon: short
cracks propagate below the ΔKth threshold. Also, Meggiolaro et al. [10] and Nalla et al. [11] stated that short cracks propagate below the effective stress intensity threshold ΔKth,eff. It has been verified that the crack nucleation and crack propagation in Al-Si cast alloys occur through the α-aluminum matrix, and is due to the stress intensity increase at the crack tip; when the crack is long enough, the stress intensity factor at the crack tip becomes the mechanical driving force for the crack propagation [12].
Some techniques, which are focused on the crack propagation characterization have been proposed over the years, where the most common technique used is the Linear Elastic Fracture Mechanics (LEFM). This technique has been effectively used to calculate the crack propagation from a flaw, defect or initial crack size a, related to fatigue cycles N. The crack propagation behavior usually follows the Paris equation [4]; therefore, the Paris equation continues to be the accepted standardized method worldwide to investigate the fatigue crack growth. However, several scientists [13], [14] and [15], have found that short fatigue cracks showed irregular behavior, which has been characterized as atypical when compared to longer cracks under nominally elastic cycling, for a diversity of commercial alloys employed in structural parts [16], whose behavior cannot be described by the stress intensity factor threshold (Kth) from the LEFM [17], suggesting that the crack growth and evaluation of failure requires that the crack tip stress field has to be evaluated at a length scale r0 in front of the crack. The characteristic length scale r0 has been widely used in the assessment of fatigue crack growth by the fracture mechanics [17], [18], [19], [20], [21]. Therefore, the evaluation of long cracks propagated under Mode I uniaxial loading, where the ratio r0/a have a tendency to be quite small and, as a consequence, the error in the usage of ΔK and Kmax to characterize the crack tip stress field is also quite small. However, for short cracks with length scales of the order of 10 µm, the ratio r0/a will not be small [22], consequently increasing the error in the usage of ΔK and Kmax. Subsequently, it has been confirmed by several authors [22] and [23] that the conclusions reached by [17], were correctly assessed. Therefore, it can be concluded that K dominance is lost for short cracks. Consequently, the similitude hypothesis is also invalid for short cracks. In this context, it was clear that the generalized Frost and Dugdale [24] model was a potential alternative. They studied the crack growth of fatigue cracks on huge thin panels of steel and light alloys, including aluminium plates with a central notch as crack starters; they related the fatigue crack growth rate to the crack length and to the third power of the stress range in a proportional way. Later, Ritchie [25] studied the nucleation and growth of fatigue cracks on aluminium and found that the propagation behavior was crack length dependent, and that the crack growth was dependent to native crystallographic characteristics as the direction of neighboring grains and grain border structure. This was also demonstrated by Molent et al. [26], Zhai et al. [27]and Polák et al. [28].
In this work, the crack length a, and fatigue cycles N values obtained by rotating bending fatigue testing of an aluminium 6061-T6 temper using a slightly modified notched specimen, were assessed.
The crack length and fatigue cycles values were the input parameters for the exponential crack growth equation (ECG) proposed by Frost and Dugdale, in order to characterize the short crack propagation behavior. In addition, a proposed robust multiple linear regression (PM) on crack length and fatigue cycles values was done in order to describe the correlation among stress amplitude, crack length and number of cycles during the fatigue tests. The results obtained by the PM showed an initial extrapolated crack ai value similar to those values obtained by the ECG. Furthermore, the PM allows to directly estimate the crack growth rate as a function of the stress amplitude for aluminium alloys. Finally, a microstructure relation to the crack growth initiation and early propagation of the cracks was studied to identify the possible microstructural features affecting the crack growth behavior.
2. Experimental Procedure The specimens were obtained from an aluminium 6061-T6 alloy plate with 12.5 mm thickness and machined to obtain longitudinal specimens along the extruded axis direction (T-L), as depicted in Fig. 1. The reported commercial chemical composition in wt% is: Al-Balance, Cr-0.1%max, Cu0.1%max, Fe-0.35%max, Mg-0.45-0.9%, Mn-0.1%max, Si-0.2-0.6%, Zn-0.1%max, Ti-0.1%max. The mechanical properties, experimentally obtained, were: 249 MPa for yield strength (YS), 69 GPa for Young’s modulus (E), 281 MPa for ultimate tensile strength (UTS), and a 13% for ultimate strain (ε).
Fig. 1. Extruded Axis Direction.
Jan Mayén et al. [29], following the research reported by Narasaiah and Ray [30], developed a new subsized specimen geometry, where regions “A” and “B” were modified in order to induce a crack initiation, as depicted in Fig. 2. The specimens used for crack growth tests were prepared as described elsewhere. [29]. The fatigue crack growth experiments were performed using a rotating bending fatigue machine by Fatigue Dynamics, Model RBF 200.
Fig. 2. Specimen proposed geometry (dimensions in mm).
The loading conditions for fatigue testing were: 242 MPa, 169 MPa and 106 MPa of stress amplitude, corresponding to 88%, 61% and 38% of yield strength of the aluminium alloy, respectively. These loads were obtained according to the Finite Element Analysis reported elsewhere [29], where the higher stress values in the specimen subjected to bending loads are located at regions A and B, including the notch effect. The experimental conditions for the crack growth by fatigue were: a) 20 Hz at room temperature, b) an initial fatigue test was performed in order to induce a 15-50 µm short crack at regions A and B, c) the crack paths were monitored by an optical microscope along the test, d) the tests were arrested once crack length was approximately 1x10 3 µm long. The recorded crack paths were posteriorly analyzed in order to obtain the short crack length, which were fitted by Eq. (1) [24].
(1)
Where, N is the number of accumulated cycles, kg is a coefficient that characterizes the crack growth rate, which is directly related to the geometry specimen, material and load [24]; a is the crack length and ai is the initial crack length related to a materials defect, a second phase particle or other microstructural features. In order to observe the PSB’s density along the crack path, the specimens were analyzed by optical microscope. One specimen, for each of the loading conditions was etched by immersion in Kellers etch (190 ml of distilled water, 5 ml of nitric acid, 3 ml of hydrochloric acid and 2 ml of hydrofluoric acid) for 10 seconds lapse of time, rinsed in distilled water followed by immersion in acetone and rapidly dried by hot air, in order to allow a slight revealed of the microstructure to observe the PSB’s by scanning electron microscope (SEM JSM 5900-LV) at 20 kV and 6 nA for the beam current.
2.1 Multiple linear regression methodology for crack length, stress amplitude and fatigue cycles number correlation. The main aim of this work, was the development of an empirical model that correlates the stress amplitude during fatigue cyclic loading with the crack length. The empirical model was developed by Multiple Linear Regression (MLR) technique, since there are more than two independent variables (predictors) in the experimental conditions. The model, using population information, is described by Eq. 2 [31],
(2)
This method requires the usage of some assumptions. These are necessary for the mathematical process, in order to properly find the optimal empirical equation that better fits the experimental data. The predictors x1=N1, x2= σ2, …xkt represents the control parameters for the experimental conditions, and are: a) number of fatigue cycles N, and b) stress amplitude σ. These assumptions are: 1. The error terms εi are normally distributed and equal to 0. 2. The error terms are independent of past error terms, which is
(3)
3. All the population have equal variances, S1= S2=S3=··· 4. The predictors are not correlated with each other. The Eq. (2) can be represented as the matrix form, where:
(4)
Then,
(5)
The least squares method is then applied to Eq. (5) in order to find the estimates of β that minimizes the sum of squares of the residual, (6)
Since ε=Y-Xβ. Expanding S(β) it is then obtained,
(7) Subsequently, (8) As it can be done in calculus to find the minimum of a function, the least squares estimate β can be obtained by solving the Eq. (8) based on the first derivative of S(β),
(9)
Then, (10) And thus β is obtained,
(11)
3. Experimental results and discussion
3.1 Relation between short crack propagation and microstructural features. In the Fig. 3, the final crack path on the specimen surface, which is located in the notch at region A of the specimen according to FEA results [29], which corresponds to the loading conditions at 88%, 61% and 38% of yield strength of the aluminium alloy.
a) 106 MPa.
b) 168 MPa
c) 242 MPa. Fig. 3 Crack initiation for the three different loading conditions.
The predominant crack growth mechanism observed at a detail by SEM observations (see Fig. 5) was crack propagation due to the intrusion-extrusion mechanism, which is described in Fig. 4.
Fig. 4 Intrusion – extrusion mechanism [32], [33].
This intrusion-extrusion mechanism, has been considered as the earliest evidence of metal fatigue, which is related to the development of surface deformation in the form of PSB’s that emerge inside the grains of the metal matrix [34]. These microstructural features are visible by their profile, as can be seen in the Fig. 5 (a). These PSB’s are the evidence of a severe plastic deformation localized within the grain that leads to the formation of the fatigue crack and its propagation.
The SEM micrographs depicted in Fig. 5, show evidence of PSB’s (a), as the main crack growth mechanism. In Fig. 5 I, it can be observed that the crack is propagating from the PSB’s (a). Also, decohesion of second phase particles (b) can be observed (see Fig. II). Another important observation is that second phase particles act as crack starters (c) (see Fig. 5 II) or propagation barriers (d) (see Fig. 5 I and II).
I)106 MPa
II) 168 MPa
III) 242 MPa Fig. 5 Microstructural features at the crack tip for the corresponding experimental loading conditions.
The aluminium 6061 alloy, contains intermetallic compounds of a very complex structure; these compounds are nucleated by the casting process and may act as crack starters or crack propagation barriers, depending on its chemical composition and morphology. Furthermore, the intermetallic compounds reported for this aluminum alloy are (Fe, Mn, Cu) 3SiAl12 and Mg2Si, being the latest the main phase reported [35]. In addition, the intrusion-extrusion mechanism leads to the accumulation of dislocations within the grain, which is due to grain border barrier; this mechanism allows the crack to propagate within the material matrix.
3.2 Determination of fatigue short crack growth by Frost and Dugdale exponential crack growth equation (ECG). The short crack length was inspected during the fatigue tests, which was carried out in batches, by acquiring images by optical microscopy from notched area, at each test stop. Once the tests were over, principal crack lengths were assessed as a function of loading cycles for the three loading
conditions mentioned above. The short crack path usually follows the slip plane or grain boundaries (see Fig. 5), which corresponds to the crystallographic orientation with the highest Schmidt factor [32].
The crack growth is depicted in Fig. 6 for the three cyclic loading conditions. As it can be observed, the crack for the specimens subjected to 242 MPa nucleated at a lower cycling than those specimens subjected to 168 MPa and 106 MPa, this is due to high localized plastic deformation at the notched region. As can be observed in Fig. 6, the slope of the crack growth curves represent the crack growth rate, which is related to the loading test condition, which implies that propagation rate is directly proportional to the stress amplitude applied. At the early stage of fatigue tests, for 242 MPa and 106 MPa a crack arrest can be observed (see enclosed points); in consequence, this instability of the crack growth has to be taken into account when estimating the crack growth rate.
Fig. 6. Crack growth for three different loading conditions.
In Fig. 6, the crack length as a function of fatigue cycles for the three experimental loading conditions is depicted. As can be observed, the cracks of the specimens subjected to 242 MPa and 106 MPa were nucleated earlier than those of the specimens subjected to 169 MPa, this maybe is due to the microstructural features acting as propagation barriers in the notched zone for the 106 MPa condition. The crack growth curve of the specimens subjected to 242 MPa depicts a linear behavior, which is related to a faster crack propagation due to a lack of interaction between crack and microstructural features due to the high localized plastic deformation at the crack tip. As can be seen in Fig. 6, the crack growth for the specimen subjected to 106 MPa is arrested from 103 up to 105 cycles, around to 33 µm long; after that, the crack grows at a constant rate during the test remaining.
This behavior can be related to the interaction between the crack path and the microstructural features, as stated by Polák et al. [28].
The Eq. (1) [24] relates the crack length a and the fatigue cycles N during fatigue tests, and indicates that the crack growth rate
directly depends on the crack length a, which represents the cumulative
crack length during the experimental fatigue test. The equivalent crack growth for the three experimental loading conditions were obtained according with Eq. (1), which can be rewritten as:
(12) And in the matrix form: (13)
Where ai is the initial crack length or the initial flaw size, and kg characterizes the crack growth rate at the applied stress amplitude, n is the amount of experimental data. The input information of the Eq. (13) are obtained from the experimental data depicted in Fig. 6 and, when solved, the values of kg and ai are obtained and, therefore the exponential equation that describes the behavior of crack growth is obtained. The Table 1 contains the values of the crack length obtained by solving Eq. (13), as a function of the testing loading conditions and the number of fatigue cycles.
Table 1. Crack growth as a function of loading condition. loading 242 MPa Test 1 242 MPa Test 2 169 MPa Test 1 169 MPa Test 2 106 MPa Test 1 106 MPa Test 1
Crack length
(µm)
R-square 0.98 0.97 0.89 0.93 0.89 0.87
The extrapolated crack values for the initial crack length at 0 cycles is in the range between 82 µm and 34 µm for 242 MPa and 106 MPa, respectively. This extrapolated crack is assumed as the initial defect that propagates the crack during the fatigue testing.
For the three tested conditions, an accelerated growth can be observed up to 100 µm, and the crack growth rate changes approximately between 100 µm and 200 µm, this is due to the influence of the microstructure at this early stage (indicated by the lines in Fig. 7, where two slopes can be approximated). The influence of microstructural features decreases inversely to crack length, and stress concentration at the crack tip, becomes the main mechanical driving force for crack growing,
as depicted in Fig. 7. The crack growth becomes steady above 150 µm length; this behavior may be explained by plastic strain field localized in front of crack tip.
Fig. 7. Unsteady behavior below the 150 µm length.
Following Eq. (14) [24], an analysis of the crack growth rate was performed for the three different loading conditions. The results of the analysis are depicted in Fig. 8, where is showed that for 242 MPa loading condition, the crack growth rate is higher than the two other loading conditions.
(14)
Fig. 8. Crack growth rate exponential equation approach.
The identification of crack nucleation and short crack growth depends strongly on detection method employed, whether it is optical or scanning electron microscopy. With the first, the detection can be performed about 20% of fatigue life, while with the second one, about 5% of fatigue life [36].
However, for low cost and simplicity testing, optical microscopy monitoring method by [30] for crack growth was used. The size of the initial crack depends on the microstructural dimensions of the specimen. Thus, if crack is nucleated by the development of PSB’s, the initial crack will be the same size as its containing grain; in contrast, if neighboring grains have same crystallographic orientation among them, it is possible that the cracks nucleate [37] throughout more than one neighboring grains; crack nucleation is also possible for second phase particles, as can be seen in Fig. 5 (c). Additionally, the linkage between the principal and secondary cracks as a crack growth mechanism, has been observed and this behavior contributes to an accelerated crack growth, as observed for 242 MPa loading condition. It is well known that the crack growth rate decreases as the crack tip crosses grain boundaries, including matrix/precipitate interfaces [37].
3.3 Proposed model (PM) for the estimation of the short crack growth. A similar short crack growth behavior analysis as described in the section above was studied, for the Al 6061-T6 alloy cycled with constant stress amplitude, using the multiple linear regression approach as described in section 2.1. A correlation was established between the measured crack lengths a and their corresponding given cycles N, as a function of the stress amplitude σa during the tests. The empirical model obtained, which describes the behavior of the crack growth as a function of stress amplitude and number of cycles, is given in Eq. (15):
(15)
Where N is the number of fatigue cycles, σa is the stress amplitude during the test and aσaN is the crack length as a function of stress amplitude, and P1 and P2 are defined by.
(16) (17) The corresponding coefficients βi are shown in Table 2. Table 2. Coefficient βi values obtained for the PM. Coefficients
Values -56.5927 4.7347x10-4 3.8157x10-11 0.7897 -4.3638x10-6 -2.2725x10-3 -5.0491x10-7 4.7080x10-13 -1.1745x10-2
1.2736x10-9 3.1364x10-4 0.98 It is of great importance to mention that the PM takes the stress amplitude conditions into account, which indicates that the interpolation in the tested range is possible for crack length prediction at other stress amplitude levels, in contrast to the ECG, where the stress amplitude variable is implicit in the equation, therefore an equation is needed for each condition to correlate the Kg as a function of stress amplitude, as can be seen in Table 1. The ECG equation proposed by Frost and Dugdale [24] in the late 60’s, and employed more recently by some researchers as Polák et al. [28] and [37], characterizes accurately the long crack growth behavior for a given stress amplitude; however, the relation of crack length with different stress amplitude ranges cannot be described by one single equation, and the ECG shows R2=< 0.9 for some crack propagation data, which is due to high data dispersion during the early crack propagation, a<150 µm, depicted in Fig. 8.
The standardized residuals values obtained from the multiple linear regression are plotted in Fig. 9. The distribution of the residuals showed in the plot confirms that: a) the data values are symmetrically distributed, tending to cluster towards the middle of the plot (horizontal line), b) the data values are clustered around the lower single digits of the y-axis and, c) the data values plotted exhibits an unpattern behavior, which indicates that the Eq. (15) describes accurately the crack growth behavior.
Fig. 9. Standardized residuals of “a”.
In Fig. 10, the estimated values of crack length versus the experimental values are plotted, where observable behavior between the experimental and estimated crack length values are linearly correlated.
Fig. 10. Linear correlation between experimental and predicted crack length data with a R2=0.98.
The length of the cracks for the experimental loading conditions can be approximately estimated by Eq. (15) for a valid interval from zero to Nf cycles, where is expected that an hypothetical crack grows from ai up to af. Then, the period where crack initiation occurred is substituted by the length ai of an imaginary crack at zero cycles and zero stress amplitude, which has a hypothetical length of 0 µm for the loading conditions according to Eq. (15).
The confidence intervals, obtained by the multiple linear regression methodology from the crack length of the specimens subjected to rotating bending fatigue at 242 MPa, 169 MPa and 106 MPa, are depicted in Fig. 11, Fig. 12, and Fig. 13, respectively. This estimated range of values, which is likely to include the unknown population for the crack length was calculated for a 95% confidence interval.
Fig. 11. 106 MPa 95% Confidence Interval.
Fig. 12. 169 MPa 95% confidence interval.
Fig. 13. 242 MPa 95% confidence interval.
In the figures above, it is observable that the proposed model takes into account the high dispersion data below 150 µm, due to the strong interactions between the fatigue crack path and the microstructural features. Additionally, the 106 MPa stress amplitude condition exhibits the higher data dispersion due to the low stress concentration in regions A and B in the fatigue specimen, which causes a chaotic behavior due to a slower crack growth that increases the interaction between the crack and the microstructural features of the Al 6061 alloy.
3.4 Comparison between the ECG and the PM for short crack growth estimation.
The PM obtained by multiple linear regression has been compared with the ECG [24]; the results for the Kg coefficient for the Al 6061 alloy obtained by the ECG are in good agreement with contemporary researchers as Polák et al. [28]. In the Fig. 14, a comparison is made among the experimental data (ED), the exponential equation (ECG) and the proposed empirical model (PM); the curves depicted in the Fig. 14 shows that the ECG equations describe the general behavior for the short crack, however it fails to describe the early stage of the crack, below 150 µm in length, where the highly data dispersion is due to the strong interactions between the crack path and the microstructural features. In contrast, the PM accurately describes the short crack growth and its transition to long crack even for the early stage of the crack, below to 150 µm length, which allows to estimate the fatigue life Nf.
Fig. 14. Model comparison for the fatigue crack length. In Table 3, the R2 for the ECG, and the PM are shown. It is clear that the correlation coefficient is directly proportional to the stress amplitude, this indicates that for lower stress amplitude values, the data dispersion increases, and the ECG is unable to predict the behavior accurately, while the PM has the same R2 for the stress amplitude tested ranges.
Table 3. Correlation coefficients for the models comparison. Model ECG ECG ECG PM
Stress Amplitude 106 169 242 Any stress amplitude
R2 0.88 0.91 0.97 0.98
In the Fig. 15, the crack growth rate is depicted for the three experimental loading conditions, where it is observable that the ECG generally describes the crack growth rate by the Kg coefficient, as described by Eq. (14), with a straight line for each stress amplitude during testing. In contrast, the PM involves the interaction between the crack and the microstructural features and takes into account the stress amplitude conditions during the fatigue testing. Hence, the PM can be used for crack length and crack growth rate prediction by interpolation in the tested stress amplitude range for the studied alloy, which is the range between the yield stress and the apparent fatigue limit of the Al 6061 alloy.
2x10-1 1x10-1
da/dN (m/cycle)
5x10-2 2x10-2 1x10-2
106 MPa ED 106 MPa ECG 106 MPa PM 169 MPa ED 169 MPa ECG 169 MPa PM 242 MPa ED 242 MPa ECG 242 MPa PM
5x10-3 2x10-3 1x10-3 5x10-4 2x10-4 1x10-4 5x10-5 10
20
30
40 50 60 7080 100
200
Crack length "a" (m)
300 400 500
700
1000
Fig. 15. Crack growth rate models comparison.
The correlation between the experimental data and the estimated values by ECG proposed by Frost and Dugdale, which has been used by several contemporary authors, and the PM is compared by the R2. The results of the correlation are plotted In Fig. 16 and Fig. 17 for the ECG and the PM, respectively. The ECG shows an R2=0.94 and the PM a R2=0.98, these results shows that the PM has competitive advantages against the ECG, as a more accurate estimation of the crack growth in early stages and, therefore, estimate the remaining life of the material.
Fig. 16. Experimental crack length vs.
Fig. 17. Experimental crack length vs
predicted values by ECG.
predicted values by PM.
4. Conclusions
The modification of the specimen geometry allows the observation of the short crack growth by both optical and electron microscopy, since the crack grows over a flat surface, during the rotating
bending fatigue test. Therefore, specimen geometry employed in this work has a lot of potential benefits for the study of initiation and propagation of short cracks, i.e. low cost test or simplified crack growth monitoring.
Experimental data obtained from fatigue testing showed that crack nucleation takes the highest fraction of the fatigue lifetime, where this period took about 6x105 and 3x105 cycles for 106 MPa and 164 MPa loading condition, respectively. It is assumed that the crack nucleation stage involves a short crack length up to 103 µm, which means that the highest fraction of the fatigue lifetime follows the behavior described by the PM, instead of the fracture mechanics approach.
The responsible mechanisms for crack nucleation and propagation found in this work were: grain boundaries as crack barriers, second phase particles as crack nucleation sites and barriers, and the PSB’s over the specimen surface as the primary mechanism for crack growth due to extrusionintrusion process.
The crack growth a was fitted by the ECG, and a crack growth coefficient kg for each experimental loading condition was obtained from the experimental data, which represents the crack growth resistance and is inversely proportional to the crack growth rate. Also, the crack growth a was fitted by multiple linear regression and an empirical equation was proposed, which showed good correlation between crack growth rate and crack length at any given loading condition and cycling. The PM describes the behavior for the stress amplitude conditions tested, which indicates the possibility to interpolate between the tested range of stress amplitude, in order to predict the crack length and the crack growth rate at any other stress amplitudes. On the other hand, the Frost and Dugdale ECG describes the whole crack growth by a single slope Kg; however, this equation has not the sensitivity required to properly describe the slope change on the transition between short and long crack growth.
Acknowledgements Funding supported by CONACyT through agreement Number 291113-CENIDET We acknowledge to Dr. Iván Puente-Lee for the assistance on the scanning electron microscopy.
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August, 2016
Highlights
Interaction between the surface cracks and aluminum microstructure were identified.
Recorded length crack paths were later analyzed to study the crack growth by Frost and Dugdale exponential crack growth equation (ECG) and its comparison with a proposed model (PM) developed by multiple linear regression.
The ECG is reliable only for the stage where the crack growth is mainly a consequence of stress concentration at crack tip; in contrast, the PM do properly describes the nonlinearity behavior between short and long crack growth.