Area of isostress near the notch root.
This study proposes a quantitative method for predicting fatigue limit reliability of a notched metal containing inhomogeneities. Since the fatigue fracture origin of the notched metal cannot be determined in advance because of stress nonuniformity, randomly distributed particles, and scatter of a matrix, it is difficult to predict the fatigue limit. The present method utilizes a stress-strength model incorporating the “statistical hardness characteristics of a matrix under small indentation loads” and the “statistical hardness characteristics required for non-propagation of fatigue cracks from microstructural defects”. The notch root is subdivided into small elements to eliminate the stress nonuniformity. The fatigue limit reliability is predicted by unifying the survival rates of the elements obtained by the stress-strength model according to the weakest link model. The method is applied to notched specimens of aluminum cast alloy JIS AC4B-T6 containing eutectic Si, Fe compounds and porosity. The fatigue strength reliability at 107 cycles, which corresponds to the fatigue limit reliability, is predicted. The fatigue limits of notch root radius ρ = 2, 1, 0.3, and 0.1 mm are obtained by rotating-bending fatigue tests. It is shown that the fatigue limits predicted by the present method are in good agreement with the experimental ones.
- metal fatigue
- fatigue limit reliability
- notch effect
- aluminum cast alloy
Aluminum cast alloys are widely applied, for example, in motor vehicles, ships, aircraft, machines, and structures, owing to the high cast ability and high specific strength [1, 2, 3]. They can be improved so as to meet specific mechanical properties by tuning the casting method, the alloying elements, and the cooling and heat treatment conditions [4, 5, 6]. Generally, precipitation hardening, also called age-hardening, is used to strengthen the aluminum cast alloys, which brings the dense precipitate of particles such as eutectic Si. The precipitations form fine microstructures such as dendrites, which significantly improve the mechanical properties. However, the resultant stress concentrations by the precipitations further to fatigue fracture unfortunately [7, 8, 9]. Moreover, the possibility of the fatigue fracture increases more and more if microstructural flaws such as porosity are created in the casting process [10, 11, 12, 13, 14, 15]. Because the precipitate particles and the microstructural defects are unique, the fatigue strength of the aluminum cast alloys is obliged to treat statistically.
Statistical fatigue test methods [16, 17] are standardized to determine the reliability of the fatigue strength. However, because they require many fatigue tests, it is time-consuming to determine the fatigue strength reliability at 107 stress cycles. Moreover, because the weakest region which controls the fatigue strength of the specimen is not known, the present materials cannot be improved rationally. Hence, a faster, rational method for quantitatively and nondestructively predicting the effect of inhomogeneities on fatigue strength is necessary for safe and reliable machine designs and for economical and quick material developments.
Several methods for predicting the fatigue strength at 107 stress cycles, which are equivalent to the statistically determined fatigue limit of aluminum cast alloys, have been proposed [18, 19, 20, 21, 22, 23, 24]. Through a series of stress analyses and fatigue experiments, Murakami et al. [18, 19, 20] clarified the non-propagation limit of a fatigue crack initiated by a microstructural defect and proposed a simple formula for predicting the fatigue limit of a plain specimen containing defects [18, 19, 20]. The non-propagation limit of a fatigue crack initiated by microstructural defect is determined by the defect size and mechanical characteristics of the matrix near the defect. The maximum defect, which is often estimated by extreme statistics, is therefore assumed to be the origin of the fatigue fracture. Most of the methods are based on the assumption that fatigue fracture begins at the maximum defects, and they often do not consider the interference effects of inhomogeneities and the scatter of the hardness of the matrix . Because aluminum cast alloys have much higher densities of inhomogeneities, it is presumed that the interference effect is not negligible and the maximum inhomogeneity is not in the severest mechanical state necessarily. Additionally, in the case of a notched specimen, the stress varies significantly. The most severe mechanical defect should be used for prediction, even if it is not maximal. Generally, the fatigue limit of a notched specimen of a homogeneous metal in which microstructural defect is not the origin of the fatigue fracture consists of the microcrack and macrocrack non-propagation limits [26, 27, 28, 29, 30, 31, 32]. This fact is widely used in predicting fatigue limit. However, since microstructural defects act as crack initiation sites, the fatigue limit of an inhomogeneous metal also cannot be predicted by these two types of crack non-propagation limits.
In this study, a quantitative method for predicting the fatigue limit reliability of a notched metal containing inhomogeneous particles is proposed. The present method is also based on the stress-strength model and is applied to notched specimens of an Al-Si-Cu alloy (JIS AC4B). The inhomogeneous particle in the alloy comprises eutectic Si and Fe compounds and porosity in the matrix. Rotating-bending fatigue tests are performed on the notched specimens of AC4B-T6 by changing notch root radius variously. The validity of the present method is examined by comparing its numerical prediction with experimental results.
2. Crack non-propagation limits for predicting fatigue limit of notched specimen
|notch root radius|
|limit notch root radius|
|fatigue limit of notched specimen|
|fatigue limit of plain specimen|
|microcrack non-propagation limit|
|long macrocrack non-propagation limit|
|small macrocrack non-propagation limit|
Generally, when fatigue tests are performed on a notched specimen by changing the notch root radius for a given notch depth , the typical relationship between the fatigue limit and is as shown in Figure 1; here, is the fatigue limit of the plain specimen, is the microcrack non-propagation limit, is the macrocrack non-propagation limit, and is a material property known as the branch point, the critical value of which determines whether the non-propagating crack exists along the notch root [26, 27]. If the notch is sufficiently deep, is constant .
If , is the fatigue limit . can be predicted from the mechanical characteristics of the microstructure. Conversely, if , is the fatigue limit . is constant and independent of . This means that the is equal to the fatigue limit of the cracked specimen as . That is, the notch can be assumed to be a crack and can be predicted by the fracture mechanics.
In the case of metals containing microstructural defects, the non-propagation limit of the fatigue crack that originates from the microstructural defect may be the fatigue limit. Because the defect is categorized as a macrocrack, the low macrocrack non-propagation limit is differentiated from . The threshold stress intensity factor range determines whether the fatigue crack originating from the macrocrack is arrested. The value of is an indication of the dependency of the different crack lengths . In this study, a crack for which is constant irrespective of its length, and which exhibits the small-scale yielding (SSY), is defined as a long macrocrack. Conversely, a crack for which is dependent on the length, and which exhibits the large-scale yielding (LSY), is defined as a small macrocrack [33, 34]. The three following types of crack non-propagation limits are introduced and defined to predict the fatigue limit of a notched specimen of aluminum cast alloy :
|:||This is the non-propagation limit of a microcrack that is initiated by repeated irreversible plastic strains in a homogeneous notch stress field without microstructural and structural stress concentrations|
|:||This is the non-propagation limit of a three-dimensional fatigue crack that originates from microstructural defects such as nonmetallic inclusions and pits in a homogeneous notch stress field without other microstructural and structural stress concentrations|
|:||This is the non-propagation limit of structural long macrocracks such as deep notches with|
Figure 2 is a schematic illustration of the relationships between and each of , , and . Further, and are, respectively, the branch point and limit notch root radius, which determines whether the fatigue limit is affected by the microstructural defects. and decrease as decreases, whereas attains a constant value and becomes independent of . If , is equal to the fatigue limit . If , is equal to . If , and are cut off by , and is equal to .
Because the hardness is locally scattered and numerous defects are distributed through the material, the microcrack and defect that determine the fatigue fracture cannot be determined in advance. In this situation, the probabilities of the arrest of the microcrack and the fatigue crack originating from the defect are, respectively, determined by the statistical characteristics of the hardness and the statistical characteristics of the defect. That is, and are, respectively, described by probability distributions.
3. Method for predicting fatigue limit reliability of notched metal containing inhomogeneous particles
|size of th surface element|
|size of th region where the relative first principal stress corresponds to|
|, ,||region required for the non-propagation of fatigue crack|
|size of surface defect|
|lower limit size of small surface crack|
|size of internal defect|
|, ,||geometric correction factor|
|fatigue limit reliability of notched specimen|
|, , ,||distribution|
|, ,||limit hardness|
|stress relaxation effect|
|, ,||stress intensity factor|
|stress concentration factor|
|threshold stress intensity factor range|
|lower limit value of|
|upper limit value of|
|the number of surface cracks with in a unit area|
|the number of particles with in a unit volume|
|types of inhomogeneous particles|
|the number of particles in a unit volume|
|the number of surface elements|
|the number of solid elements|
|existence probability of particles with|
|limit size of small interior crack|
|survival rate of notched specimen|
|survival rate of surface element with microcracks|
|, ,||survival rate of solid element with microstructural defects|
|population of distribution|
|size of th solid element|
|mean of distribution|
|,||first principal stress|
|,||relative first principal stress|
|,||stress produced by the spherical particle in the infinite body under and|
|stress gradient of first principal stress|
This section presents a method for predicting the fatigue limit reliability of a notched specimen with stress concentration factor , notch depth , and notch root radius under zero mean stress. The control volume is actually divided into surface and solid elements so that the stresses applied to the elements can be assumed to be constant. The fatigue strengths of all the elements are then stochastically evaluated by the stress-strength model on the mesoscale. The fatigue limit reliability is also predicted by assembling the fatigue strengths using the weakest link model .
3.1 Stress relaxation effect of interference of inhomogeneous particles
Figure 3 is a schematic illustration of the analytical model of a metal containing inhomogeneous particles. The metal is approximated by a cubic lattice model to determine the stress relaxation effect of the interference of the particles .
3.1.1 Statistical characteristics of inhomogeneous particles
The probability of existence of such particles is given by the following equation :
Here, and are material constants, is the particle radius, and is the probability of the existence of particles with radii greater than .
The total number of particles in a unit volume is denoted by . The average number of particles with radii greater than in a unit volume, , is given by the following equation :
A particle cross-sectioned by the specimen surface is projected onto a plane perpendicular to the first principal stress. The projected area is then modified as shown in Figure 4 by considering the mechanics. The modified area is denoted by . The average number of cross-sectioned particles with areas larger than in a unit area, , is given by the following equation [12, 24]:
Here, is a gamma function of the second kind.
3.1.2 Average radius and distance
The average particle radius is evaluated by the following equation:
If particles are regularly arranged in a unit volume as shown in Figure 3, the average distance between the particles is evaluated by the following equation:
3.1.3 Stress relaxation effect
Nisitani  proposed a method for approximately solving the interference problem of notches by superposing simple basic solutions to satisfy the equilibrium conditions at the stress concentration point.
When the uniform tensile stress at infinity, , is applied to an infinite body, it is supposed that a stress field composed of and is formed around the particle. and are set to satisfy the equilibrium condition at point (0, 0, ). Because the stress acting on a single particle in the -direction, , is composed of and the stresses due to the other particles, the stress equilibrium condition in the -direction is as follows:
Here, is the stress in the -direction at (0, 0, ) produced by the spherical particle located at (, , ) in the infinite body under and .
The stress equilibrium condition in the -direction is also given by
Here, is the stress in the -direction at (0, 0, ) produced by the spherical particle located at (, , ) in the infinite body under and .
3.1.4 Characteristics of elastic stress field near notch root
If the notch is sufficiently deep, a unique stress field determined by the maximum stress and is formed near the notch root . The first principal stress normalized by the maximum stress at the notch root is denoted by . Figure 5 shows a contour map of near the notch root in a semi-infinite plate under tensile stress . The value of is independent of the notch shape . If the notch root is divided as shown in Figure 5, the length of the notch edge and size of -th region in which the relative first principal stress is are denoted by and (), respectively. The values of and are given in Table 1 .
To predict the fatigue limit reliability, the control volume is set at the notch root and divided into surface and solid elements. The sizes of the solid and surface elements are denoted by and (), respectively. In the case of a typical notch, as in the notched specimen, the control volume can be divided as shown in Figure 5. For example, when a circular bar with a circumferential notch is divided, and are approximated as follows :
3.2 Fatigue survival rate of surface element containing microcracks
3.2.1 Statistical characteristics of Vickers hardness
The authors proposed a virtual small cell model for predicting the statistical characteristics of the Vickers hardness in a small region [25, 35]. If the population of the virtual small cells is described by an arbitrary distribution of the mean and variance , the statistical characteristics of the Vickers hardness can be described by the normal distribution of the mean and the variance , based on the central limit theory, where is the number of virtual small cells in the indentation area.
Here, is the Vickers hardness of the matrix that does not contain microstructural defects, is the sample mean, and is the sample variance.
Based on the central limit theorem, the relationship between the sample mean and the population mean is . Further, the relationship between the sample variance and the population variance is described by the distribution with the freedom degree of [25, 35]:
3.2.2 Fatigue survival rate of surface element containing microcracks
(, , and are in MPa, is in kgf/mm2, and are in mm.)
If the stress relaxation effect is considered, is applied to -th surface element. Because fatigue fracture occurs when is greater than , the limit hardness that determines the occurrence is given by the following equation:
Here, is a function obtained by solving Eq. (18) on .
It is supposed that exhibits the microcrack non-propagation limit characteristics in Eqs. (16) and (17). If of the matrix is greater than in , fatigue fracture will not occur in . Therefore, the probabilitythat fatigue fracture does not occur in below is given by [25, 35]
Here, is the normal distribution of and .
If fatigue fracture does not occur in all the surface elements, the notched specimen would not be broken by the microcrack. Therefore, the fatigue survival rate of a surface element containing microcracks, , is obtained by multiplying the fatigue survival rates of all the surface elements as follows [25, 35]:
Here, is the number of surface elements.
3.3 Fatigue survival rate of surface element containing microstructural defects
The authors  proposed a method for predicting the reliability of the small macrocrack non-propagation limit for a nonzero stress gradient using the “statistical hardness characteristics of a matrix under small indentation loads” and the “statistical hardness characteristics required for non-propagation of fatigue cracks originating from microstructural defects in a material” . The stress relaxation effect was introduced into the method to make it applicable to a metal containing dense inhomogeneous particles.
is divided into two crack non-propagation limits, namely, the non-propagation limit of the small crack originating from the interior defect and the non-propagation limit of the small crack originating from the surface defect.
3.3.1 Fatigue survival rate of solid element containing interior microstructural defects
Because the fatigue crack that originates from a defect propagates on the plane perpendicular to the first principal radial stress, a spherical particle of radius is projected onto this plane and assumed to be a penny-shaped crack. If the projected area is denoted by , its square root is related to as follows:
(is in MPa, is in kgf/mm2, and is in μm).
(is in MPa, is in kgf/mm2, and is in μm.)
Here, , , and are the loads used to create the indentation for obtaining the Vickers hardness and the indentation area .
If the fatigue fracture does not occur in all the solid elements, the notched specimen would not be broken by the small interior defect. Therefore, the fatigue survival rate of a solid element containing interior microstructural defects is obtained by multiplying the fatigue survival rates of all the solid elements as follows [25, 35]:
Here, is the number of solid elements.
3.3.2 Fatigue survival rate of surface element containing surface microstructural cracks
(is in MPa, is in kgf/mm2, and is in μm.)
The fatigue survival rate of -th surface element containing surface microstructural cracks, , is given by
3.3.3 Fatigue survival rate of element with microstructural defects
Because the material contains types of inhomogeneous particles, the fatigue survival rate is given by the following equation:
3.4 Prediction of long macrocrack non-propagation limit
of the notched specimen with is equal to the fatigue limit of the cracked specimen obtained by . is the upper limit of and is constant regardless of the crack length. can be obtained as follows :
3.5 Prediction of fatigue limit reliability
The probability that fatigue fracture is caused by microcracks or microstructural defects is obtained by the complementary event defined by the product of and . Because the fatigue limit of a notched specimen, cannot be lower than , and its fatigue limit reliability is obtained as follows :
4. Fatigue experiment
4.1 Material, shape of specimen, and experimental procedure
The material used for the experiment was Al-Si-Cu alloy (JIS AC4B). The age-hardened aluminum cast alloy is identified as AC4B-T6. Table 2 shows its mechanical properties.
|9.8 N, 30 sec||29.4 mN, 30 sec|
|: Young’s modulus (GPa)||: 0.2% proof stress (MPa)|
|: ultimate tensile strength (MPa)||: elongation (%)|
|: Vickers hardness of matrix with inhomogeneous particles (kgf/mm2)|
|: Vickers hardness of matrix without inhomogeneous particles (kgf/mm2)|
Figure 6 shows the configurations of the specimens. The notch depth and opening angle were set at 0.5 mm and 60°, respectively. The notch root radii were set at 2, 1, 0.3, and 0.1 mm, respectively. All the specimens were machined; polished with fine emery paper, alumina (3 μm), and diamond paste (1 μm); and also chemically polished. Rotating-bending fatigue tests were carried out according to JIS Z2274 and the earlier studies [27, 28, 29, 30, 31] under stress amplitude – MPa and frequency Hz. An Ono-type rotating-bending fatigue machine of a capacity 15 Nm was used for the tests. The nominal stress used for the analyses of the experimental results was the stress at the minimum cross section, where the diameter was 5 mm. The fatigue life was defined as the total number of stress cycles to failure.
4.2 Experimental results
Figure 7 shows -curves obtained from the results of the tests. Figure 8 shows optical micrographs of a specimen when = 0.1 mm for = 90 MPa. Fatigue limit when mm. = 95 when and mm. = 90 when = 0.1 mm. Since the non-propagating macrocrack was not observed at the fatigue limit when and 2 mm, it can be said that the microcrack non-propagating limit or the small macrocrack non-propagating limit appears as the fatigue limit. When = 0.1 mm for = 90 MPa and = 0.3 mm for = 95 MPa, the non-propagating macrocracks were observed along the notch root. Therefore, the long macrocrack non-propagating limit = 90 MPa.
5. Examination of validity of prediction method
5.1 Notch sensitivity to crack initiation limit in age-hardened aluminum alloy
Figure 9 shows the relationship between and using early fatigue data of previous studies [28, 29, 30]. values of the curves are shown in Figure 10. When values were approximated with the lines by the least squares method, the following equation was obtained:
(are in mm, and is in kgf/mm2.)
Once has been predicted, can be used instead of .
5.2 of age-hardened aluminum alloy
Figure 11 shows the values of obtained from the early fatigue data of for different Al-Si-X alloys, where X is a transition element [28, 29, 31]. An approximation of obtained from the lines by the least squares method was used to derive the following equation:
(and are in MPa, and is in kgf/mm2.)
Here, is the lower limit of and = 0.5 MPafor the Al-Si-X alloys.
5.3 Evaluation of statistical characteristics of inhomogeneous particles
The present aluminum cast alloy AC4B-T6 contains three main types of inhomogeneous particles, namely, eutectic Si and Fe compounds and porosity. Surrounding an irregular cross section with a smooth convex curve as shown in Figure 12, the area is defined as . The values of are obtained from by the following equation [12, 37]:
Here, is the number of cross-sectional particles in a unit area for which on a unit area. Considering the assumption that is given by Eqs. (1) and (2), the asymptotic characteristics of Eq. (42) are expressed by the following equation [12, 37]:
5.4 Evaluation of statistical characteristics of Vickers hardness of matrix without inhomogeneous particles
In this study, the Vickers hardness was measured at the position of 2.5–3.0 mm from the center on the circular cross section obtained by cutting the specimen grip under indentation load mN in consideration with the stress distribution at the notch root. Figure 16 shows the results plotted on a normal probability paper. The sample mean and sample variance were 91.8 kgf/mm2 and 486.0 (kgf/mm2)2, respectively.
5.5 Evaluation of value
Because the values of for the eutectic Si were much greater than those of the Fe compounds and porosity, as shown in Table 3, was calculated using only the eutectic Si. The eutectic Si was assumed to be a rigid body , and the following values were used for the calculation: = 68 GPa, = 105 GPa, and = = 0.3 . Using = 0.192, was determined to be 1.055.
|Eutectic Si||8.73 × 106||1.6||1.04||0.932||4.86|
|Fe compound||2.20 × 107||0.5||0.10||0.200||3.57|
|Porosity||(m)||1.20 × 102||0.3||0.180||0.167||21.5|
|(m)||1.00 × 105||0.3||0.0180|
5.6 Evaluation of , , and values
5.6.1 Evaluation of value
Because a microcrack often grows radially, it is approximated by the semielliptical crack shown in Figure 17.
in Eq. (20) can also be roughly evaluated. The approximation of the microcrack by the semielliptical macrocrack is such that the crack non-propagation limits are equal. If the macrocrack is located near the notch root of radius and it is sufficiently small, its non-propagation limit is given by the following equation:
(is in MPa, is in kgf/mm2 is in μm, and is in mm.)
Conversely, when the macrocrack is sufficiently large, is greater than . The macrocrack is thus treated as being large, and its non-propagation limit is categorized as , which is given by the following equation:
(is in MPa, is in kgf/mm2, and : in μm.)
is approximated to be 1. Because the average Vickers hardness of the matrix of the present AC4B-T6 is about 91.8 kgf/mm2, the microcrack non-propagation limit was estimated to be 160 MPa using Eq. (17). The non-propagating crack length of the present AC4B-T6 for = 20 mm was about 60 μm when = 120 MPa. From these experimental results, the of the present study was assumed to be 70 μm.
was set to achieve = 0.4. Using = 2.5 mm, = 70 μm is equivalent to of 39.2 μm. Figure 18 shows the relationship between and .
5.7 Comparison and examination of predicted and experimental results
The fatigue limit reliability of the notched specimen shown in Figure 6 was predicted by the present method. The region in which the first principal stress is within the range of = [0.95, 1] at the center of the specimen was adopted as the control volume. In this case, the region was ring-like.
When = 152 kgf/mm2 is used, the = 5.06 MPais predicted from Eq. (41). Because the value of for the present specimen is 0.167 (i.e., using = 5 mm and = 0.5 mm, as in Section 4.1), is 0.754 . In this case, the predicted value of is 84.7 MPa. Considering that the experimentally determined value of is 90 MPa, the prediction is confirmed to be good.
Figure 19 shows the fatigue limit reliability . The thick solid line represents the case of = 2 mm, whereas the thin solid line represents the case of = 0.3 mm. The value of for = 0.3 mm suddenly changes from 0 to 1 when = 84.7 MPa, which is due to and being cut off by . In other words, the inhomogeneous particles have almost no effect on the fatigue limit reliability in terms of initiating a fatigue crack. Instead, the eutectic Si actually strengthens the matrix.
Figure 20 shows the relationship between and . The solid line represents 50% reliability, the broken line represents 90% reliability, the single-dotted chain line represents 99% reliability, and the open marks represent the experimental results. Because the fatigue limit obtained by the ordinary fatigue test is equivalent to 50% fatigue limit reliability, the solid line agrees well with the open marks. The little differences between the open marks of = 0.3 and 0.1 mm and the solid line can be attributed to the fact that of the present AC4B-T6 was unknown and the corresponding value for the of Al-Si-X alloy was used for predicting . It is expected that an even better prediction accuracy would be achieved by using the true Nevertheless, was well predicted, which validated the proposed method for notched AC4B-T6 specimens.
This study proposed a nondestructive method for predicting the fatigue limit reliability of notched specimens of a metal containing inhomogeneous particles densely. The method was applied to aluminum cast alloy JIS-AC4B-T6. Rotating-bending fatigue tests were performed on the notched specimens of AC4B-T6 with notch root radius = 2, 1, 0.3, and 0.1 in order to examine the validity of the present method. Since the non-propagating macrocracks were observed along the notch root, the long macrocrack non-propagating limit appears as the fatigue limit when = 0.1 and0.3 mm. On the other hand, since the non-propagating macrocrack was not observed when and 2 mm, it can be said that the microcrack non-propagating limit or the small macrocrack non-propagating limit appears as the fatigue limit. The fatigue limits predicted by the present method were in good agreement with the experimental ones.
The method is not only convenient for use in predicting fatigue strength reliability for the reliable design of machine and structures, but it is also time effective and can be applied to the economic development of materials.