Open access peer-reviewed chapter

# Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense Inhomogeneities Using Fracture Mechanics

By Tatsujiro Miyazaki, Shigeru Hamada and Hiroshi Noguchi

Submitted: May 10th 2019Reviewed: July 5th 2019Published: August 22nd 2019

DOI: 10.5772/intechopen.88413

## Abstract

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.

### Keywords

• metal fatigue
• fatigue limit reliability
• notch effect
• aluminum cast alloy
• inhomogeneity

## 1. Introduction

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 [25]. 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

 Nomenclature t notch depth ρ notch root radius ρ0 branch point ρd limit notch root radius σw fatigue limit of notched specimen σw0 fatigue limit of plain specimen σw1 microcrack non-propagation limit σw2 long macrocrack non-propagation limit σwd small macrocrack non-propagation limit

Generally, when fatigue tests are performed on a notched specimen by changing the notch root radius ρfor a given notch depth t, the typical relationship between the fatigue limit σwand ρis as shown in Figure 1; here, σw0is the fatigue limit of the plain specimen, σw1is the microcrack non-propagation limit, σw2is the macrocrack non-propagation limit, and ρ0is 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, ρ0is constant [27].

If ρ>ρ0, σw1is the fatigue limit [33]. σw1can be predicted from the mechanical characteristics of the microstructure. Conversely, if ρρ0, σw2is the fatigue limit [33]. σw2is constant and independent of ρ. This means that the σw2is equal to the fatigue limit of the cracked specimen as ρ0. That is, the notch can be assumed to be a crack and σw2can 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 σw2. The threshold stress intensity factor range ΔKthdetermines whether the fatigue crack originating from the macrocrack is arrested. The value of ΔKthis an indication of the dependency of the different crack lengths [20]. In this study, a crack for which ΔKthis constant irrespective of its length, and which exhibits the small-scale yielding (SSY), is defined as a long macrocrack. Conversely, a crack for which ΔKthis 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 [35]:

 σw1: 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 σwd: 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 σw2: This is the non-propagation limit of structural long macrocracks such as deep notches with ρ<ρ0

Figure 2 is a schematic illustration of the relationships between ρand each of σw1, σwd, and σw2. Further, ρ0and ρdare, respectively, the branch point and limit notch root radius, which determines whether the fatigue limit is affected by the microstructural defects. σw1and σwddecrease as ρdecreases, whereas σw2attains a constant value and becomes independent of ρ. If ρρd, σwdis equal to the fatigue limit σw. If ρ0<ρ<ρd, σw1is equal to σw. If ρρ0, σw1and σwdare cut off by σw2, and σw2is equal to σw.

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, σw1and σwdare, respectively, described by probability distributions.

## 3. Method for predicting fatigue limit reliability of notched metal containing inhomogeneous particles

 Nomenclature Aj size of jth surface element Aj∗ size of jth region where the relative first principal stress corresponds to σ1,j∗ Anpc, AnpcR, AnpcP region required for the non-propagation of fatigue crack areaP size of surface defect areaP1 lower limit size of small surface crack areaR size of internal defect F, FP, FR geometric correction factor Fσw fatigue limit reliability of notched specimen fHVM1, fHVMS, fHVMR, fHVMP HVMdistribution fχ2 χ2distribution gP, gR,gS limit hardness γm stress relaxation effect HV, HVM Vickers hardness KIn, KIP,KIR stress intensity factor Kt stress concentration factor ΔKw threshold stress intensity factor range ΔKwLL lower limit value of ΔKw ΔKwUL upper limit value of ΔKw MS0areaP0 the number of surface cracks with areaP≥areaP0in a unit area MV0R0 the number of particles with R≥R0in a unit volume Md types of inhomogeneous particles N¯V0 the number of particles in a unit volume nS the number of surface elements nV the number of solid elements P, PR indentation load PVR0 existence probability of particles with R≥R0 Rc limit size of small interior crack Sσw survival rate of notched specimen Sσw1 survival rate of surface element with microcracks Sσwd, SσwdI,SσwdS survival rate of solid element with microstructural defects sHVM2 population of HVMdistribution Vj size of jth solid element λ, ν material constant μHVM mean of HVMdistribution σ1, σ1,j first principal stress σ1∗, σ1,j∗ relative first principal stress σymTmZm, σzmTmZm stress produced by the spherical particle in the infinite body under σz=Zmand σx=σy=σz=Tm σm mean stress σn stress amplitude χσ1 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 Kt, notch depth t, and notch root radius ρunder 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 [25].

### 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 [36].

#### 3.1.1 Statistical characteristics of inhomogeneous particles

The probability of existence of such particles is given by the following equation [37]:

PVR0=expR0λν.E1

Here, νand λare material constants, R0is the particle radius, and PVR0is the probability of the existence of particles with radii greater than R0.

The total number of particles in a unit volume is denoted by N¯V0. The average number of particles with radii greater than R0in a unit volume, MV0R0, is given by the following equation [37]:

MV0R0=N¯V0·PVR0.E2

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 areaP. The average number of cross-sectioned particles with areas larger than areaP0in a unit area, MS0areaP0, is given by the following equation [12, 24]:

MS0areaP0=λN¯V001t1t2Γ1+1νareaP0λθν
+Γ1+1νareaP0λθ+νdt,E3
θ+=π2+21t2,E4
θ=sin1tt1t2.E5

Here, Γis a gamma function of the second kind.

#### 3.1.2 Average radius and distance

The average particle radius is evaluated by the following equation:

Rm=0RdPVRdR=λΓ1+1ν.E6

If N¯V0particles are regularly arranged in a unit volume as shown in Figure 3, the average distance between the particles is evaluated by the following equation:

pm=1N¯V01/3E7

#### 3.1.3 Stress relaxation effect

Nisitani [38] 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, σz=1, is applied to an infinite body, it is supposed that a stress field composed of σz=Zmand σx=σy=σz=Tmis formed around the particle. Tmand Zmare set to satisfy the equilibrium condition at point (0, 0, Rm). Because the stress acting on a single particle in the z-direction, Tm+Zm, is composed of σz=1and the stresses due to the other particles, the stress equilibrium condition in the z-direction is as follows:

Tm+Zm=1+ijk000i,j,k=σzmTmZmxi,j,k=ipmyi,j,k=jpmzi,j,k=Rmkpm.E8

Here, σzmTmZmis the stress in the z-direction at (0, 0, Rm) produced by the spherical particle located at (ipm, jpm, kpm) in the infinite body under σz=Zmand σx=σy=σz=Tm.

The stress equilibrium condition in the y-direction is also given by

Tm=ijk000i,j,k=σymTmZmxi,j,k=ipmyi,j,k=jpmzi,j,k=Rmkpm.E9

Here, σymTmZmis the stress in the y-direction at (0, 0, Rm) produced by the spherical particle located at (ipm, jpm, kpm) in the infinite body under σz=Zmand σx=σy=σz=Tm.

Tmand Zmare obtained by solving the simultaneous linear Eqs. (8) and (9). In this study, the stress relaxation effect γmof the interference of the particles is assumed to be

γm=Tm+Zm.E10

#### 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 ρis formed near the notch root [27]. The first principal stress normalized by the maximum stress at the notch root is denoted by σ1. Figure 5 shows a contour map of σ1=0.41near the notch root in a semi-infinite plate under tensile stress [35]. The value of σ1is independent of the notch shape [27]. If the notch root is divided as shown in Figure 5, the length of the notch edge and size of j-th region in which the relative first principal stress is σ1,jare denoted by ljand Aj(j=1,), respectively. The values of ljand Ajare given in Table 1 [35].

j123456
σ1,j1–0.950.95–0.90.9–0.80.8–0.70.7–0.60.6–0.5
lj/ρ0.4630.2160.3610.3580.4020.496
Aj/ρ0.00830.01810.07030.1440.2960.657

### Table 1.

Area of isostress near the notch root.

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 Vjand Aj(j=1,), 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, Vjand Ajare approximated as follows [35]:

VjAj×average diameter ofjthregion×π,E11
Ajlj×average diameter ofjthregion×π.E12

### 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 μand variance s2, the statistical characteristics of the Vickers hardness can be described by the normal distribution of the mean μand the variance s2/nc, based on the central limit theory, where ncis the number of virtual small cells in the indentation area.

If m1Vickers hardness values are measured in this way using an indentation load P, their statistical characteristics are described by the following normal distribution [25, 35]:

fHVM1HVM=12πsHVM12expHVMμHVM122sHVM12.E13

Here, HVMis the Vickers hardness of the matrix that does not contain microstructural defects, μHVM1is the sample mean, and sHVM12is the sample variance.

Based on the central limit theorem, the relationship between the sample mean μHVM1and the population mean μHVM0is μHVM1=μHVM0. Further, the relationship between the sample variance sHVM12and the population variance sHVM02is described by the χ2distribution with the freedom degree of n=m11[25, 35]:

fχ2χ2=12Γn/2χ22n21expχ22E14
Γt=0xt1exdx,χ2=m1sHVM12sHVM02E15

#### 3.2.2 Fatigue survival rate of surface element containing microcracks

The microcrack non-propagation limit σw0is determined by the average characteristics of the material properties around the microcrack. σw0can be empirically predicted by the following equations [20, 26]:

σw0σm=0=1.6HVM,E16
σw1σm=0=σw0σm=0Kt1+4.5ε0σm=0/ρ=fHVMρσm=0Kt.E17

(σw0, σw1, and σmare in MPa, HVMis in kgf/mm2, and ρandε0are in mm.)

If the stress relaxation effect γmis considered, γmσ1,jis applied to j-th surface element. Because fatigue fracture occurs when γmσ1,j/Ktis greater than σw1, the limit hardness gSσ1,jthat determines the occurrence is given by the following equation:

gS=f1HVMρσm=0Ktσw1=γmσ1,j/Kt.E18

Here, f1is a function obtained by solving Eq. (18) on HVM.

It is supposed that AnpcSexhibits the microcrack non-propagation limit characteristics in Eqs. (16) and (17). If HVMof the matrix is greater than gSin AnpcS, fatigue fracture will not occur in AnpcS. Therefore, the probabilitySσw10,jthat fatigue fracture does not occur in AnpcSbelow σ1,jis given by [25, 35]

Sσw10,j=gSfHVMShvmdhvm.E19

Here, fHVMSis the normal distribution of μHVMSand sHVMS2.

The fatigue survival rate Sσw1,jof jth surface element containing microcracks is given by the following equation [25, 35]:

Sσw1,j=0fχ2·Sσw10,jAjAnpcSdχ2.E20

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, Sσw1, is obtained by multiplying the fatigue survival rates of all the surface elements as follows [25, 35]:

Sσw1=j=1nSSσw1,j.E21

Here, nSis the number of surface elements.

### 3.3 Fatigue survival rate of surface element containing microstructural defects

The authors [25] 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” [25]. The stress relaxation effect was introduced into the method to make it applicable to a metal containing dense inhomogeneous particles.

σwdis divided into two crack non-propagation limits, namely, the non-propagation limit σwdIof the small crack originating from the interior defect and the non-propagation limit σwdSof 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 Ris projected onto this plane and assumed to be a penny-shaped crack. If the projected area is denoted by areaR, its square root is related to Ras follows:

areaR=πR.E22

The stress intensity factor KIRof the small interior crack is given by [20, 35]

KIR=0.5FRγmσ1,jπareaR,E23
FR=4π5/412π43πareaRρ.E24

Moreover, the threshold stress intensity factor range ΔKwof the small surface defect of size areaPin the metal with Vickers hardness HVMis given by the following equation [33, 34]:

ΔKwσm=0=2αβareaP1/3ln2β/HVM+1,E25

α=3.3×103and β=120.

(ΔKwis in MPam, HVMis in kgf/mm2, and areaPis in μm).

The limit hardness that determines whether the fatigue crack originating from the interior microstructural crack is arrested, gRσ1,jareaR, is given based on the relationship areaR=1.7areaPby the following equation [25, 35]:

gR=240/exp1.56×240FRγmσ1,jareaR1/61E26

(σ1,jis in MPa, gRis in kgf/mm2, and areaRis in μm.)

The relationship between μHVMRand μHVM0can be expressed as follows [25, 35]:

μHVMR=μHVM0=μHVM1.E27

Moreover, the relationship between sHVMR2and sHVM02can be expressed as follows [25, 35]:

AnpcR·sHVMR2=AHVM0·sHVM02.E28

Here, AHVM0=P/μHVM0, AnpcR=PR/gR, and PRare the loads used to create the indentation for obtaining the Vickers hardness gRand the indentation area AnpcR.

The fatigue survival rate of j-th solid element containing interior defects, SσwdI,jm, is given by the following equation [25, 35]:

SσwdI,jm=0fχ2expRc0gRpdImdMV,jmdRfHVMRdhvmdRdχ2.E29

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 SσwdImof a solid element containing interior microstructural defects is obtained by multiplying the fatigue survival rates of all the solid elements as follows [25, 35]:

SσwdIm=j=1nVSσwdI,jm.E30

Here, nVis the number of solid elements.

#### 3.3.2 Fatigue survival rate of surface element containing surface microstructural cracks

The stress intensity factor KIPof a small surface crack of size areaPis given by the following equation [20, 35]:

KIP=0.65FRγmσ1,jπareaP,E31
FP=0.9681.021areaPρ.E32

Further, the limit hardness gPσ1,jareaPthat determines whether the small surface crack is arrested is given by the following equation [25, 35]:

gP=240/exp1.43×240FRγmσ1,jareaP1/61.E33

(σ1,jis in MPa, gPis in kgf/mm2, and areaPis in μm.)

The fatigue survival rate of j-th surface element containing surface microstructural cracks, SσwdS,jm, is given by

SσwdS,jm=0fχ2expareaPc0gPpdSmdMS,jmdareaPfHVMPdhvmdareaPdχ2E34

The fatigue survival rate SσwdSmof a surface element containing surface microstructural cracks is obtained by multiplying the fatigue survival rates of all the surface elements as follows [25, 35]:

SσwdSm=j=1nSSσwdS,jm.E35

#### 3.3.3 Fatigue survival rate of element with microstructural defects

The fatigue survival rate Sσwdmof an element containing microstructural defects is obtained by multiplying SσwdImand SσwdSmas follows [25, 35]:

Sσwdm=SσwdIm×SσwdSm.E36

Because the material contains Mdtypes of inhomogeneous particles, the fatigue survival rate Sσwdis given by the following equation:

Sσwd=m=1MdSσwdm.E37

### 3.4 Prediction of long macrocrack non-propagation limit σw2

σw2of the notched specimen with ρρ0is equal to the fatigue limit of the cracked specimen obtained by ρ0. ΔKwULis the upper limit of ΔKwand is constant regardless of the crack length. σw2σm=0can be obtained as follows [35]:

σw2σm=0=ΔKwULσm=02Fπt.E38

### 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 Sσw1and Sσwd. Because the fatigue limit of a notched specimen, σwcannot be lower than σw2, and its fatigue limit reliability Fσwis obtained as follows [35]:

Fσw=0σwσw21Sσw1×Sσwdσw>σw2E39

## 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.

Eσ0.2σBδHVHVM
9.8 N, 30 sec29.4 mN, 30 sec
742923491.515292
E: Young’s modulus (GPa)σ0.2: 0.2% proof stress (MPa)
σB: ultimate tensile strength (MPa)δ: elongation (%)
HV: Vickers hardness of matrix with inhomogeneous particles (kgf/mm2)
HVM: Vickers hardness of matrix without inhomogeneous particles (kgf/mm2)

### Table 2.

Mechanical properties.

Figure 6 shows the configurations of the specimens. The notch depth tand opening angle θwere set at 0.5 mm and 60°, respectively. The notch root radii ρwere 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 σa=60110 MPa and frequency f=50Hz. An Ono-type rotating-bending fatigue machine of a capacity 15 Nm was used for the tests. The nominal stress σnused for the analyses of the experimental results was the stress at the minimum cross section, where the diameter dwas 5 mm. The fatigue life Nfwas defined as the total number of stress cycles to failure.

### 4.2 Experimental results

Figure 7 shows S-Ncurves obtained from the results of the tests. Figure 8 shows optical micrographs of a specimen when ρ = 0.1 mm for σw = 90 MPa. Fatigue limit σw=105when ρ=2 mm. σw = 95 when ρ=1and 0.3 mm. σw = 90 when ρ = 0.1 mm. Since the non-propagating macrocrack was not observed at the fatigue limit when ρ=1and 2 mm, it can be said that the microcrack non-propagating limit σw1or the small macrocrack non-propagating limit σwdappears as the fatigue limit. When ρ = 0.1 mm for σw = 90 MPa and ρ = 0.3 mm for σw = 95 MPa, the non-propagating macrocracks were observed along the notch root. Therefore, the long macrocrack non-propagating limit σw2 = 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 Ktσw1/σw0and ρusing early fatigue data of previous studies [28, 29, 30]. ε0values of the curves are shown in Figure 10. When ε0values were approximated with the lines by the least squares method, the following equation was obtained:

ε0σm=0=5.0×104HB0.0164,E40
ρ0.5,97HB207.

(ε0andρare in mm, and HBis in kgf/mm2.)

Once σw1has been predicted, HVMcan be used instead of HB.

### 5.2 ΔKwULof age-hardened aluminum alloy

Figure 11 shows the values of ΔKwULobtained from the early fatigue data of σw2for different Al-Si-X alloys, where X is a transition element [28, 29, 31]. An approximation of ΔKwULobtained from the lines by the least squares method was used to derive the following equation:

ΔKwULσm=0=ΔKwLL+0.03HB,E41
40HB100.

(ΔKwULand ΔKwLLare in MPam, and HBis in kgf/mm2.)

Here, ΔKwLLis the lower limit of ΔKwand ΔKwLL= 0.5 MPamfor 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 areaA. The values of rare obtained from areaAby the following equation [12, 37]:

r=areaAπ.E42

Figure 13 shows the measured MA0values of eutectic Si and Fe compounds and porosity. The relationship between MV0and MA0is as follows [12, 37]:

MA0r0=2r0R2r2dMV0dRdR.E43

Here, MA0r0is the number of cross-sectional particles in a unit area for which rr0on a unit area. Considering the assumption that MV0R0is given by Eqs. (1) and (2), the asymptotic characteristics of Eq. (42) are expressed by the following equation [12, 37]:

MA0r02πνλr0λ1ν2N¯V0expr0λν.E44

The line of Eq. (44) is drawn to best fit the MA0values obtained by Eq. (43) to determine the values of N¯V0, ν, and λ.

Figure 14 shows the values of MV0for porosity, Fe compounds, and eutectic Si. Figure 15 shows the values of MS0for the porosity, Fe compounds, and eutectic Si.

### 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 P=29.4mN in consideration with the stress distribution at the notch root. Figure 16 shows the results plotted on a normal probability paper. The sample mean μHVM1and sample variance sHVM12were 91.8 kgf/mm2 and 486.0 (kgf/mm2)2, respectively.

### 5.5 Evaluation of γmvalue

Because the values of Rm/pmfor the eutectic Si were much greater than those of the Fe compounds and porosity, as shown in Table 3, γmwas calculated using only the eutectic Si. The eutectic Si was assumed to be a rigid body [7], and the following values were used for the calculation: EM = 68 GPa, EI = 105 GPa, and νM = νI = 0.3 [35]. Using Rm/pm = 0.192, γmwas determined to be 1.055.

Inhomogeneous particleN¯V0[1/mm3]νλ[μm]Rm[μm]pm[μm]
Eutectic Si8.73 × 1061.61.040.9324.86
Fe compound2.20 × 1070.50.100.2003.57
Porosity(R105μm)1.20 × 1020.30.1800.16721.5
(R<105μm)1.00 × 1050.30.0180

### Table 3.

Parameters of particle size distribution.

### 5.6 Evaluation of AnpcS, AnpcR, and AnpcPvalues

#### 5.6.1 Evaluation of AnpcSvalue

Because a microcrack often grows radially, it is approximated by the semielliptical crack shown in Figure 17.

AnpcSin 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 ρand it is sufficiently small, its non-propagation limit σwdSis given by the following equation:

σwdS=1.43HVM+120FPareaP1/6,E45
FP=0.9681.021×103areaPρ.E46

(σwdSis in MPa, HVMis in kgf/mm2 ,areaPis in μm, and ρis in mm.)

Conversely, when the macrocrack is sufficiently large, ΔKwis greater than ΔKwUL. The macrocrack is thus treated as being large, and its non-propagation limit is categorized as σw2, which is given by the following equation:

σw2=13.0HVM+16.7FPareaP1/2.E47

(σw2is in MPa, HVMis in kgf/mm2, and areaP: in μm.)

FPis 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 σw0was estimated to be 160 MPa using Eq. (17). The non-propagating crack length of the present AC4B-T6 for ρ = 20 mm was about 60 μm when σn = 120 MPa. From these experimental results, the lnpc0of the present study was assumed to be 70 μm.

lnpcwas set to achieve b/l = 0.4. Using c = 2.5 mm, lnpc0 = 70 μm is equivalent to areanpc0of 39.2 μm. Figure 18 shows the relationship between lnpcand 1/ρ.

#### 5.6.2 Evaluation of AnpcRand AnpcPvalues

AnpcRis a function of PRand gR; AnpcPis a function of PPand gP. Because gRand gPcan be calculated using Eqs. (26) and (33), respectively, only PRand PPneed to be further examined. In this study, it is assumed that PP = 0.3 kgf. Considering the difficulty in evaluating PR, it is also assumed that PR=PP.

### 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 σ1=σ1/σmax = [0.95, 1] at the center of the specimen was adopted as the control volume. In this case, the region was ring-like.

When HB = 152 kgf/mm2 is used, the ΔKwUL = 5.06 MPamis predicted from Eq. (41). Because the value of ξfor the present specimen is 0.167 (i.e., using d = 5 mm and t = 0.5 mm, as in Section 4.1), Fis 0.754 [39]. In this case, the predicted value of σw2is 84.7 MPa. Considering that the experimentally determined value of σw2is 90 MPa, the prediction is confirmed to be good.

Figure 19 shows the fatigue limit reliability Fσw. The thick solid line represents the case of ρ = 2 mm, whereas the thin solid line represents the case of ρ = 0.3 mm. The value of Fσwfor ρ = 0.3 mm suddenly changes from 0 to 1 when σw = 84.7 MPa, which is due to σw1and σwdbeing cut off by σw2. 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 σwand 1/ρ. 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 ρ = 0.3 and 0.1 mm and the solid line can be attributed to the fact that ΔKwULof the present AC4B-T6 was unknown and the corresponding value for the of Al-Si-X alloy was used for predicting σw2. It is expected that an even better prediction accuracy would be achieved by using the true ΔKwUL.Nevertheless, σwwas well predicted, which validated the proposed method for notched AC4B-T6 specimens.

## 6. Conclusions

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 ρ = 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 σw2appears as the fatigue limit when ρ = 0.1 and0.3 mm. On the other hand, since the non-propagating macrocrack was not observed when ρ=1and 2 mm, it can be said that the microcrack non-propagating limit σw1or the small macrocrack non-propagating limit σwdappears 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.

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Tatsujiro Miyazaki, Shigeru Hamada and Hiroshi Noguchi (August 22nd 2019). Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense Inhomogeneities Using Fracture Mechanics, Fracture Mechanics Applications, Hayri Baytan Ozmen and H. Ersen Balcioglu, IntechOpen, DOI: 10.5772/intechopen.88413. Available from:

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