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

*Tatsujiro Miyazaki, Shigeru Hamada and Hiroshi Noguchi*

## **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 10<sup>7</sup> 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 [1–3]. 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 [4–6]. Generally, precipitation hardening, also called agehardening, 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 [7–9]. Moreover, the possibility of the fatigue fracture increases more and more if microstructural flaws such as porosity are created in the casting process [10–15]. 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 10<sup>7</sup> 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 10<sup>7</sup> stress cycles, which are equivalent to the statistically determined fatigue limit of aluminum cast alloys, have been proposed [18–24]. Through a series of stress analyses and fatigue experiments, Murakami et al. [18–20] 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 [18–20]. 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 [26–32]. 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. Rotatingbending 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.

plain specimen, *σ<sup>w</sup>*<sup>1</sup> is the microcrack non-propagation limit, *σ<sup>w</sup>*<sup>2</sup> is the macrocrack non-propagation limit, and *ρ*<sup>0</sup> is a material property known as the branch point, the critical value of which determines whether the non-propagating crack exists along

If *ρ*>*ρ*0, *σ<sup>w</sup>*<sup>1</sup> is the fatigue limit [33]. *σ<sup>w</sup>*<sup>1</sup> can be predicted from the mechanical characteristics of the microstructure. Conversely, if *ρ*≤*ρ*0, *σ<sup>w</sup>*<sup>2</sup> is the fatigue limit [33]. *σ<sup>w</sup>*<sup>2</sup> is constant and independent of *ρ*. This means that the *σ<sup>w</sup>*<sup>2</sup> is equal to the fatigue limit of the cracked specimen as *ρ* ! 0. That is, the notch can be assumed to

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

the notch root [26, 27]. If the notch is sufficiently deep, *ρ*<sup>0</sup> is constant [27].

be a crack and *σ<sup>w</sup>*<sup>2</sup> can be predicted by the fracture mechanics.

*Schematic illustration of fatigue limit of a notched structure without defects.*

**Figure 1.**

**39**

**Nomenclature**

*DOI: http://dx.doi.org/10.5772/intechopen.88413*

*t* notch depth *ρ* notch root radius *ρ*<sup>0</sup> branch point

*ρ<sup>d</sup>* limit notch root radius

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense…*

*σ<sup>w</sup>* fatigue limit of notched specimen *σ<sup>w</sup>*<sup>0</sup> fatigue limit of plain specimen *σ<sup>w</sup>*<sup>1</sup> microcrack non-propagation limit *σ<sup>w</sup>*<sup>2</sup> long macrocrack non-propagation limit *σwd* small macrocrack non-propagation limit

## **2. Crack non-propagation limits for predicting fatigue limit of notched specimen**

Generally, when fatigue tests are performed on a notched specimen by changing the notch root radius *ρ* for a given notch depth *t*, the typical relationship between the fatigue limit *σ<sup>w</sup>* and *ρ* is as shown in **Figure 1**; here, *σ<sup>w</sup>*<sup>0</sup> is the fatigue limit of the *Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*


#### **Figure 1.**

However, the resultant stress concentrations by the precipitations further to fatigue fracture unfortunately [7–9]. Moreover, the possibility of the fatigue fracture increases more and more if microstructural flaws such as porosity are created in the casting process [10–15]. Because the precipitate particles and the microstructural defects are unique, the fatigue strength of the aluminum cast alloys is obliged to

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 10<sup>7</sup> 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

Several methods for predicting the fatigue strength at 10<sup>7</sup> stress cycles, which are equivalent to the statistically determined fatigue limit of aluminum cast alloys, have been proposed [18–24]. Through a series of stress analyses and fatigue experiments, Murakami et al. [18–20] 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 [18–20]. 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 [26–32]. 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

machine designs and for economical and quick material developments.

cannot be predicted by these two types of crack non-propagation limits.

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. Rotatingbending 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**

Generally, when fatigue tests are performed on a notched specimen by changing the notch root radius *ρ* for a given notch depth *t*, the typical relationship between the fatigue limit *σ<sup>w</sup>* and *ρ* is as shown in **Figure 1**; here, *σ<sup>w</sup>*<sup>0</sup> is the fatigue limit of the

In this study, a quantitative method for predicting the fatigue limit reliability of

treat statistically.

*Fracture Mechanics Applications*

**specimen**

**38**

*Schematic illustration of fatigue limit of a notched structure without defects.*

plain specimen, *σ<sup>w</sup>*<sup>1</sup> is the microcrack non-propagation limit, *σ<sup>w</sup>*<sup>2</sup> is the macrocrack non-propagation limit, and *ρ*<sup>0</sup> 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, *ρ*<sup>0</sup> is constant [27].

If *ρ*>*ρ*0, *σ<sup>w</sup>*<sup>1</sup> is the fatigue limit [33]. *σ<sup>w</sup>*<sup>1</sup> can be predicted from the mechanical characteristics of the microstructure. Conversely, if *ρ*≤*ρ*0, *σ<sup>w</sup>*<sup>2</sup> is the fatigue limit [33]. *σ<sup>w</sup>*<sup>2</sup> is constant and independent of *ρ*. This means that the *σ<sup>w</sup>*<sup>2</sup> is equal to the fatigue limit of the cracked specimen as *ρ* ! 0. That is, the notch can be assumed to be a crack and *σ<sup>w</sup>*<sup>2</sup> 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

## *Fracture Mechanics Applications*

non-propagation limit is differentiated from *σw*2. The threshold stress intensity factor range *ΔKth* determines whether the fatigue crack originating from the macrocrack is arrested. The value of *ΔKth* is an indication of the dependency of the different crack lengths [20]. In this study, a crack for which *ΔKth* 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 *ΔKth* 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 [35]:

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, *σw*<sup>1</sup> and *σwd* are, respectively, described by proba-

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense…*

**3. Method for predicting fatigue limit reliability of notched metal**

*<sup>j</sup>* size of *j*th region where the relative first principal stress corresponds to *σ* <sup>∗</sup>

*Anpc*, *Anpc R*, *Anpc P* region required for the non-propagation of fatigue crack

1*,j*

**containing inhomogeneous particles**

*DOI: http://dx.doi.org/10.5772/intechopen.88413*

*Aj* size of *j*th surface element

p size of surface defect

p size of internal defect *F*, *FP*, *FR* geometric correction factor

*f <sup>χ</sup>*<sup>2</sup> *χ*<sup>2</sup> distribution *gP*, *gR*, *gS* limit hardness

*γ<sup>m</sup>* stress relaxation effect *HV*, *HVM* Vickers hardness *KIn*, *KIP*, *KIR* stress intensity factor *Kt* stress concentration factor

*ΔKwLL* lower limit value of *ΔKw ΔKwUL* upper limit value of *ΔKw*

<sup>p</sup> <sup>1</sup> lower limit size of small surface crack

*F<sup>σ</sup><sup>w</sup>* fatigue limit reliability of notched specimen

*HVM* distribution

*ΔKw* threshold stress intensity factor range

*Md* types of inhomogeneous particles *NV*<sup>0</sup> the number of particles in a unit volume

*nS* the number of surface elements *nV* the number of solid elements

*Rc* limit size of small interior crack *S<sup>σ</sup><sup>w</sup>* survival rate of notched specimen

*P*, *PR* indentation load

� � the number of surface cracks with ffiffiffiffiffiffiffiffiffiffiffi

*PV*ð Þ *R*<sup>0</sup> existence probability of particles with *R* ≥*R*<sup>0</sup>

*S<sup>σ</sup>w*<sup>1</sup> survival rate of surface element with microcracks

*MV*0ð Þ *R*<sup>0</sup> the number of particles with *R*≥*R*<sup>0</sup> in a unit volume

*areaP* p ≥ ffiffiffiffiffiffiffiffiffiffiffi *areaP*

<sup>p</sup> <sup>0</sup> in a unit area

bility distributions.

Nomenclature

*A*<sup>∗</sup>

ffiffiffiffiffiffiffiffiffiffiffi *areaP*

ffiffiffiffiffiffiffiffiffiffiffi *areaP*

ffiffiffiffiffiffiffiffiffiffiffi *areaR*

*f HVMP*

*MS*<sup>0</sup>

**41**

ffiffiffiffiffiffiffiffiffiffiffi *areaP* <sup>p</sup> <sup>0</sup>

*f HVM*1, *f HVMS*, *f HVMR*,


**Figure 2** is a schematic illustration of the relationships between *ρ* and each of *σw*1, *σwd*, and *σw*2. Further, *ρ*<sup>0</sup> and *ρ<sup>d</sup>* are, respectively, the branch point and limit notch root radius, which determines whether the fatigue limit is affected by the microstructural defects. *σ<sup>w</sup>*<sup>1</sup> and *σwd* decrease as *ρ* decreases, whereas *σ<sup>w</sup>*<sup>2</sup> attains a constant value and becomes independent of *ρ*. If *ρ*≥*ρd*, *σwd* is equal to the fatigue limit *σw*. If *ρ*<sup>0</sup> < *ρ* < *ρd*, *σ<sup>w</sup>*<sup>1</sup> is equal to *σw*. If *ρ*≤ *ρ*0, *σ<sup>w</sup>*<sup>1</sup> and *σwd* are cut off by *σw*2, and *σ<sup>w</sup>*<sup>2</sup> is equal to *σw*.

**Figure 2.** *Schematic illustration of fatigue limit of a notched structure with defects.*

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*

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, *σw*<sup>1</sup> and *σwd* are, respectively, described by probability distributions.


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

non-propagation limit is differentiated from *σw*2. The threshold stress intensity factor range *ΔKth* determines whether the fatigue crack originating from the macrocrack is arrested. The value of *ΔKth* is an indication of the dependency of the different crack lengths [20]. In this study, a crack for which *ΔKth* 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 *ΔKth* 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 alumi-

*σw*1: 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

*σw*2: This is the non-propagation limit of structural long macrocracks such as deep notches with *ρ* < *ρ*<sup>0</sup>

**Figure 2** is a schematic illustration of the relationships between *ρ* and each of *σw*1, *σwd*, and *σw*2. Further, *ρ*<sup>0</sup> and *ρ<sup>d</sup>* are, respectively, the branch point and limit notch root radius, which determines whether the fatigue limit is affected by the microstructural defects. *σ<sup>w</sup>*<sup>1</sup> and *σwd* decrease as *ρ* decreases, whereas *σ<sup>w</sup>*<sup>2</sup> attains a constant value and becomes independent of *ρ*. If *ρ*≥*ρd*, *σwd* is equal to the fatigue limit *σw*. If *ρ*<sup>0</sup> < *ρ* < *ρd*, *σ<sup>w</sup>*<sup>1</sup> is equal to *σw*. If *ρ*≤ *ρ*0, *σ<sup>w</sup>*<sup>1</sup> and *σwd* are cut off by *σw*2, and

*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

num cast alloy [35]:

*σ<sup>w</sup>*<sup>2</sup> is equal to *σw*.

**Figure 2.**

**40**

*Schematic illustration of fatigue limit of a notched structure with defects.*

concentrations

*Fracture Mechanics Applications*


*PV*ð Þ¼ *<sup>R</sup>*<sup>0</sup> exp � *<sup>R</sup>*<sup>0</sup>

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense…*

probability of the existence of particles with radii greater than *R*0.

ffiffiffiffiffiffiffiffiffiffiffi *areaP* <sup>p</sup> <sup>0</sup>

þ*Γ* 1 þ

Here, *Γ* is a gamma function of the second kind.

*Rm* ¼

ð1 0

1 *ν ;*

*<sup>θ</sup>*<sup>þ</sup> <sup>¼</sup> *<sup>π</sup>*

*<sup>θ</sup>*� <sup>¼</sup> sin �<sup>1</sup>

The average particle radius is evaluated by the following equation:

*<sup>R</sup> dPV*

ð<sup>∞</sup> 0

by the following equation [37]:

*DOI: http://dx.doi.org/10.5772/intechopen.88413*

<sup>p</sup> <sup>0</sup> in a unit area, *MS*<sup>0</sup>

*MS*<sup>0</sup>

*3.1.2 Average radius and distance*

ffiffiffiffiffiffiffiffiffiffiffi *areaP* <sup>p</sup> <sup>0</sup> � � <sup>¼</sup> *<sup>λ</sup> NV*<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffi *areaP*

**Figure 4.**

**43**

*Spheroidal particle cut by surface.*

Here, *ν* and *λ* are material constants, *R*<sup>0</sup> is the particle radius, and *PV*ð Þ *R*<sup>0</sup> is the

The total number of particles in a unit volume is denoted by *NV*0. The average number of particles with radii greater than *R*<sup>0</sup> in a unit volume, *MV*0ð Þ *R*<sup>0</sup> , is given

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

> *t* ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>t</sup>*<sup>2</sup> <sup>p</sup> *<sup>Γ</sup>* <sup>1</sup> <sup>þ</sup>

ffiffiffiffiffiffiffiffiffiffiffi *areaP* <sup>p</sup> <sup>0</sup> *λ θ*þ

<sup>2</sup> <sup>þ</sup> <sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>t</sup>* � *<sup>t</sup>* ffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>R</sup> dR* <sup>¼</sup> *λ Γ* <sup>1</sup> <sup>þ</sup>

*λ* � � � �*<sup>ν</sup>*

*MV*0ð Þ¼ *R*<sup>0</sup> *NV*<sup>0</sup> � *PV*ð Þ *R*<sup>0</sup> *:* (2)

� �, is given by the following equation [12, 24]:

1 *ν ;*

� � �*<sup>ν</sup>*�� *dt,* (3)

1 *ν*

� � � � �*<sup>ν</sup>*

ffiffiffiffiffiffiffiffiffiffiffi *areaP* <sup>p</sup> <sup>0</sup> *λ θ*�

<sup>1</sup> � *<sup>t</sup>*<sup>2</sup> <sup>p</sup> *,* (4)

<sup>1</sup> � *<sup>t</sup>*<sup>2</sup> <sup>p</sup> *:* (5)

� �*:* (6)

*:* (1)

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 *ρ* 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 [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]:

**Figure 3.** *Approximate model of metal with inhomogeneous particles.*

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*

$$P\_V(R\_0) = \exp\left\{-\left(\frac{R\_0}{\lambda}\right)^{\nu}\right\}.\tag{1}$$

Here, *ν* and *λ* are material constants, *R*<sup>0</sup> is the particle radius, and *PV*ð Þ *R*<sup>0</sup> is the probability of the existence of particles with radii greater than *R*0.

The total number of particles in a unit volume is denoted by *NV*0. The average number of particles with radii greater than *R*<sup>0</sup> in a unit volume, *MV*0ð Þ *R*<sup>0</sup> , is given by the following equation [37]:

$$M\_{V0}(R\_0) = \overline{N}\_{V0} \cdot P\_V(R\_0). \tag{2}$$

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 ffiffiffiffiffiffiffiffiffiffiffi *areaP* <sup>p</sup> <sup>0</sup> in a unit area, *MS*<sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffi *areaP* <sup>p</sup> <sup>0</sup> � �, is given by the following equation [12, 24]:

$$M\_{\rm S0} \left( \sqrt{area\_{P0}} \right) = \lambda \overline{N}\_{\rm V0} \int\_0^1 \frac{t}{\sqrt{\mathbf{1} - t^2}} \left\{ \Gamma \left( \mathbf{1} + \frac{\mathbf{1}}{\nu}, \left( \frac{\sqrt{area\_{P0}}}{\lambda \, \theta^-} \right)^{\nu} \right) \right\}$$

$$+ \Gamma \left( \mathbf{1} + \frac{\mathbf{1}}{\nu}, \left( \frac{\sqrt{area\_{P0}}}{\lambda \, \theta^+} \right)^{\nu} \right) \right\} dt,\tag{3}$$

$$
\theta^{+} = \frac{\pi}{2} + 2\sqrt{1 - t^2},
\tag{4}
$$

$$
\theta^- = \sin^{-1} t - t\sqrt{1 - t^2}.\tag{5}
$$

Here, *Γ* 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:

$$R\_m = \int\_0^\infty R \frac{dP\_V}{R} dR = \lambda \,\Gamma\left(\mathbf{1} + \frac{\mathbf{1}}{\nu}\right). \tag{6}$$

**Figure 4.** *Spheroidal particle cut by surface.*

*S<sup>σ</sup>wd* , *S<sup>σ</sup>wdI* ,*S<sup>σ</sup>wdS* survival rate of solid element with microstructural defects

stress produced by the spherical particle in the infinite body under *σ<sup>z</sup>* ¼ *Zm* and

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 *ρ* 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

**3.1 Stress relaxation effect of interference of inhomogeneous particles**

The probability of existence of such particles is given by the following

**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].

*HVM* population of *HVM* distribution

<sup>1</sup>*,j* relative first principal stress

*σ<sup>x</sup>* ¼ *σ<sup>y</sup>* ¼ *σ<sup>z</sup>* ¼ *Tm*

*χσ*<sup>1</sup> stress gradient of first principal stress

*3.1.1 Statistical characteristics of inhomogeneous particles*

*Approximate model of metal with inhomogeneous particles.*

*Vj* size of *j*th solid element *λ, ν* material constant

*Fracture Mechanics Applications*

*μHVM* mean of *HVM* distribution *σ*1, *σ*1*,j* first principal stress

*σ<sup>m</sup>* mean stress *σ<sup>n</sup>* stress amplitude

*s* 2

*σ* ∗ <sup>1</sup> , *σ* <sup>∗</sup>

*σym*ð Þ *Tm; Zm* , *σzm*ð Þ *Tm; Zm*

link model [25].

equation [37]:

**Figure 3.**

**42**

If *NV*<sup>0</sup> 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:

$$p\_m = \left(\frac{1}{\overline{N}\_{V0}}\right)^{1/3} \tag{7}$$

#### *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*<sup>∞</sup> ¼ 1, is applied to an infinite body, it is supposed that a stress field composed of *σ<sup>z</sup>* ¼ *Zm* and *σ<sup>x</sup>* ¼ *σ<sup>y</sup>* ¼ *σ<sup>z</sup>* ¼ *Tm* is formed around the particle. *Tm* and *Zm* are set to satisfy the equilibrium condition at point (0, 0, *Rm*). Because the stress acting on a single particle in the *z*-direction, *Tm* þ *Zm*, is composed of *σ<sup>z</sup>*<sup>∞</sup> ¼ 1 and the stresses due to the other particles, the stress equilibrium condition in the *z*-direction is as follows:

$$T\_m + Z\_m = 1 + \sum\_{(i,j,k)} \sum\_{\ne(0,0,0)} \stackrel{\infty}{\sim} \sum \sigma\_{zm}(T\_m, Z\_m)|\_{\mathcal{X}\_{i,j,k}} = -ip\_m \qquad . \tag{8}$$

$$\begin{array}{ll} \mathcal{Y}\_{i,j,k} = -jp\_m \\ \mathcal{Z}\_{i,j,k} = R\_m - kp\_m \end{array}$$

Here, *σzm*ð Þ *Tm; Zm* is 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 *σ<sup>z</sup>* ¼ *Zm* and *σ<sup>x</sup>* ¼ *σ<sup>y</sup>* ¼ *σ<sup>z</sup>* ¼ *Tm*.

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

$$T\_m = \sum\_{(i,j,k)\neq(0,0,0)} \sum\_{(0,0,0)} ^\infty \sum \sigma\_{ym}(T\_m, Z\_m) \Big|\_{\mathcal{K}\_{i,j,k}} = -ip\_m \qquad . \tag{9}$$

$$\begin{aligned} \mathcal{Y}\_{i,j,k} &= -ip\_m \\ z\_{i,j,k} &= R\_m - kp\_m \end{aligned}$$

Here, *σym*ð Þ *Tm; Zm* is 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 *σ<sup>z</sup>* ¼ *Zm* and *σ<sup>x</sup>* ¼ *σ<sup>y</sup>* ¼ *σ<sup>z</sup>* ¼ *Tm*.

*Tm* and *Zm* are obtained by solving the simultaneous linear Eqs. (8) and (9). In this study, the stress relaxation effect *γ<sup>m</sup>* of the interference of the particles is assumed to be

$$
\chi\_m = T\_m + Z\_m. \tag{10}
$$

elements are denoted by *Vj* and *Aj* ( *j* ¼ 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,

*j* **1 2 3456**

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense…*

<sup>1</sup>*,j* 1–0.95 0.95–0.9 0.9–0.8 0.8–0.7 0.7–0.6 0.6–0.5

*<sup>j</sup> =ρ* 0.463 0.216 0.361 0.358 0.402 0.496

*<sup>j</sup> =ρ* 0.0083 0.0181 0.0703 0.144 0.296 0.657

*<sup>1</sup>* ¼ ½ � *0:4; 1 near the notch root.*

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 *μ* and

limit theory, where *nc* is the number of virtual small cells in the indentation area. If *m*<sup>1</sup> Vickers hardness values are measured in this way using an indentation load *P*, their statistical characteristics are described by the following normal

2, the statistical characteristics of the Vickers hardness can be described by

**3.2 Fatigue survival rate of surface element containing microcracks**

*<sup>j</sup>* � ðaverage diameter of *j* th regionÞ � *π,* (11)

*<sup>j</sup>* � ðaverage diameter of *j* th regionÞ � *π:* (12)

<sup>2</sup>*=nc*, based on the central

*Vj* and *Aj* are approximated as follows [35]:

*Contour map of relative first principal stress σ* <sup>∗</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.88413*

*Area of isostress near the notch root.*

*Vj* ffi *<sup>A</sup>*<sup>∗</sup>

*Aj* ffi *l* ∗

variance *s*

**45**

**Figure 5.**

*σ* ∗

*l* ∗

**Table 1.**

*A*<sup>∗</sup>

distribution [25, 35]:

*3.2.1 Statistical characteristics of Vickers hardness*

the normal distribution of the mean *μ* and the variance *s*

#### *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 *ρ* is formed near the notch root [27]. The first principal stress normalized by the maximum stress at the notch root is denoted by *σ* <sup>∗</sup> <sup>1</sup> . **Figure 5** shows a contour map of *σ* <sup>∗</sup> <sup>1</sup> ¼ ½ � 0*:*4*;* 1 near the notch root in a semi-infinite plate under tensile stress [35]. The value of *σ* <sup>∗</sup> <sup>1</sup> is 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 *σ* <sup>∗</sup> <sup>1</sup>*,j* are denoted by *l* ∗ *<sup>j</sup>* and *A*<sup>∗</sup> *<sup>j</sup>* ( *j* ¼ 1*,* ⋯), respectively. The values of *l* ∗ *<sup>j</sup>* and *A*<sup>∗</sup> *<sup>j</sup>* are given in **Table 1** [35].

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

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*

**Figure 5.** *Contour map of relative first principal stress σ* <sup>∗</sup> *<sup>1</sup>* ¼ ½ � *0:4; 1 near the notch root.*


#### **Table 1.**

If *NV*<sup>0</sup> particles are regularly arranged in a unit volume as shown in **Figure 3**, the

*NV*<sup>0</sup> � �1*=*<sup>3</sup>

Nisitani [38] proposed a method for approximately solving the interference problem of notches by superposing simple basic solutions to satisfy the equilibrium

When the uniform tensile stress at infinity, *σz*<sup>∞</sup> ¼ 1, is applied to an infinite body, it is supposed that a stress field composed of *σ<sup>z</sup>* ¼ *Zm* and *σ<sup>x</sup>* ¼ *σ<sup>y</sup>* ¼ *σ<sup>z</sup>* ¼ *Tm* is formed around the particle. *Tm* and *Zm* are set to satisfy the equilibrium condition at point (0, 0, *Rm*). Because the stress acting on a single particle in the *z*-direction, *Tm* þ *Zm*, is composed of *σ<sup>z</sup>*<sup>∞</sup> ¼ 1 and the stresses due to the other particles, the

Here, *σzm*ð Þ *Tm; Zm* is 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 *σ<sup>z</sup>* ¼ *Zm* and

<sup>∞</sup>X*σym*ð Þ *Tm; Zm*

Here, *σym*ð Þ *Tm; Zm* is 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 *σ<sup>z</sup>* ¼ *Zm* and

*Tm* and *Zm* are obtained by solving the simultaneous linear Eqs. (8) and (9). In

If the notch is sufficiently deep, a unique stress field determined by the maximum stress and *ρ* is formed near the notch root [27]. The first principal stress

the notch root is divided as shown in **Figure 5**, the length of the notch edge and size

∗ *<sup>j</sup>* and *A*<sup>∗</sup>

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

normalized by the maximum stress at the notch root is denoted by *σ* <sup>∗</sup>

of *j*-th region in which the relative first principal stress is *σ* <sup>∗</sup>

this study, the stress relaxation effect *γ<sup>m</sup>* of the interference of the particles is

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

<sup>∞</sup>X*σzm*ð Þj *Tm; Zm xi,j,k* ¼ �*ipm*

� � *yi,j,k* ¼ �*jpm zi,j,k* ¼ *Rm* � *kpm*

*xi,j,k* ¼ �*ipm yi,j,k* ¼ �*jpm zi,j,k* ¼ *Rm* � *kpm*

*γ<sup>m</sup>* ¼ *Tm* þ *Zm:* (10)

<sup>1</sup> is independent of the notch shape [27]. If

<sup>1</sup> ¼ ½ � 0*:*4*;* 1 near the notch root in a semi-infinite plate

(7)

*:* (8)

*:* (9)

<sup>1</sup> . **Figure 5**

∗ *<sup>j</sup>* and

<sup>1</sup>*,j* are denoted by *l*

*<sup>j</sup>* are given in **Table 1** [35].

average distance between the particles is evaluated by the following equation:

*pm* <sup>¼</sup> <sup>1</sup>

*3.1.3 Stress relaxation effect*

*Fracture Mechanics Applications*

conditions at the stress concentration point.

*Tm* <sup>þ</sup> *Zm* <sup>¼</sup> <sup>1</sup> <sup>þ</sup>X X

*Tm* <sup>¼</sup> X X

ð Þ *i; j; k* 6¼ ð Þ 0*;* 0*;* 0 *i, j, k* ¼ �∞

*3.1.4 Characteristics of elastic stress field near notch root*

*σ<sup>x</sup>* ¼ *σ<sup>y</sup>* ¼ *σ<sup>z</sup>* ¼ *Tm*.

*σ<sup>x</sup>* ¼ *σ<sup>y</sup>* ¼ *σ<sup>z</sup>* ¼ *Tm*.

shows a contour map of *σ* <sup>∗</sup>

under tensile stress [35]. The value of *σ* <sup>∗</sup>

*<sup>j</sup>* ( *j* ¼ 1*,* ⋯), respectively. The values of *l*

assumed to be

*A*<sup>∗</sup>

**44**

stress equilibrium condition in the *z*-direction is as follows:

ð Þ *i; j; k* 6¼ ð Þ 0*;* 0*;* 0 *i, j, k* ¼ �∞

*Area of isostress near the notch root.*

elements are denoted by *Vj* and *Aj* ( *j* ¼ 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, *Vj* and *Aj* are approximated as follows [35]:

$$V\_j \cong A\_j^\* \times \text{(average diameter of } j \text{ th region)} \times \pi,\tag{11}$$

$$A\_j \cong l\_j^\* \times (\text{average diameter of } j \text{th region}) \times \pi. \tag{12}$$

#### **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 *μ* and variance *s* 2, the statistical characteristics of the Vickers hardness can be described by the normal distribution of the mean *μ* and the variance *s* <sup>2</sup>*=nc*, based on the central limit theory, where *nc* is the number of virtual small cells in the indentation area.

If *m*<sup>1</sup> Vickers hardness values are measured in this way using an indentation load *P*, their statistical characteristics are described by the following normal distribution [25, 35]:

$$f\_{H\_{\rm VM}}(H\_{\rm VM}) = \frac{1}{\sqrt{2\pi\mathfrak{s}\_{H\_{\rm VM}}^2}} \exp\left\{-\frac{\left(H\_{\rm VM} - \mu\_{H\_{\rm VM}}\right)^2}{2\mathfrak{s}\_{H\_{\rm VM}}^2}\right\}.\tag{13}$$

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, *S<sup>σ</sup>w*<sup>1</sup> , is obtained by multiplying the

fatigue survival rates of all the surface elements as follows [25, 35]:

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense…*

Here, *nS* is the number of surface elements.

*DOI: http://dx.doi.org/10.5772/intechopen.88413*

dense inhomogeneous particles.

defect of size ffiffiffiffiffiffiffiffiffiffiffi

following equation [33, 34]:

(*ΔKw* is in MPa ffiffiffiffi

relationship ffiffiffiffiffiffiffiffiffiffiffi

**47**

*areaP*

*<sup>α</sup>* <sup>¼</sup> <sup>3</sup>*:*<sup>3</sup> � <sup>10</sup>�3and *<sup>β</sup>* <sup>¼</sup> 120.

*areaR*

<sup>p</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>7</sup> ffiffiffiffiffiffiffiffiffiffiffi

*<sup>S</sup><sup>σ</sup>w*<sup>1</sup> <sup>¼</sup> <sup>Y</sup>*nS*

*j*¼1

**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

*σwd* is divided into two crack non-propagation limits, namely, the nonpropagation limit *σwdI* of the small crack originating from the interior defect and the non-propagation limit *σwdS* of the small crack originating from the surface defect.

*3.3.1 Fatigue survival rate of solid element containing interior microstructural defects*

ffiffiffiffiffiffiffiffiffiffiffi *areaR* <sup>p</sup> <sup>¼</sup> ffiffiffi

*KIR* ¼ 0*:*5 *FR γ<sup>m</sup> σ*1*,j*

*<sup>π</sup>*<sup>5</sup>*=*<sup>4</sup> <sup>1</sup> � <sup>2</sup>

*FR* <sup>¼</sup> <sup>4</sup>

<sup>m</sup><sup>p</sup> , *HVM* is in kgf/mm2

the interior microstructural crack is arrested, *gR σ*1*,j;* ffiffiffiffiffiffiffiffiffiffiffi

*areaP*

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

ffiffiffi *<sup>π</sup>* <sup>p</sup> � <sup>4</sup> 3 *π* � � ffiffiffiffiffiffiffiffiffiffiffi

Moreover, the threshold stress intensity factor range *ΔKw* of the small surface

The limit hardness that determines whether the fatigue crack originating from

<sup>Δ</sup>*Kw*j*<sup>σ</sup>m*¼<sup>0</sup> <sup>¼</sup> <sup>2</sup>*αβ* ffiffiffiffiffiffiffiffiffiffiffi

p in the metal with Vickers hardness *HVM* is given by the

*areaP* p <sup>1</sup>*=*<sup>3</sup>

, and ffiffiffiffiffiffiffiffiffiffiffi *areaP* p is in μm).

p by the following equation [25, 35]:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *π* ffiffiffiffiffiffiffiffiffiffiffi *areaR*

> *areaR* p

ln 2ð Þ *<sup>β</sup>=HVM* <sup>þ</sup> <sup>1</sup> *,* (25)

*areaR*

� � p , is given based on the

*ρ* � �*:* (24)

p q

area is denoted by *areaR*, its square root is related to *R* as follows:

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 *R* is projected onto this plane and assumed to be a penny-shaped crack. If the projected

*S<sup>σ</sup>w*1*,j:* (21)

*<sup>π</sup>* <sup>p</sup> *<sup>R</sup>:* (22)

*,* (23)

Here, *HVM* is the Vickers hardness of the matrix that does not contain microstructural defects, *μHVM*<sup>1</sup> is the sample mean, and *s* 2 *HVM*<sup>1</sup> is the sample variance.

Based on the central limit theorem, the relationship between the sample mean *μHVM*<sup>1</sup> and the population mean *μHVM*<sup>0</sup> is *μHVM*<sup>1</sup> ¼ *μHVM*0. Further, the relationship between the sample variance *s* 2 *HVM*<sup>1</sup> and the population variance *s* 2 *HVM*<sup>0</sup> is described by the *<sup>χ</sup>*<sup>2</sup> distribution with the freedom degree of *<sup>n</sup>* <sup>¼</sup> *<sup>m</sup>*<sup>1</sup> � 1 [25, 35]:

$$f\_{\chi^2}(\chi^2) = \frac{1}{2\Gamma(n/2)} \left(\frac{\chi^2}{2}\right)^{\frac{n}{2}-1} \exp\left(-\frac{\chi^2}{2}\right) \tag{14}$$

$$I(t) = \int\_0^\infty \varkappa^{t-1} e^{-\varkappa} d\varkappa, \chi^2 = m\_1 \frac{s\_{H\_{\text{VM}}1}^2}{s\_{H\_{\text{VM}}0}^2} \tag{15}$$

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

The microcrack non-propagation limit *σ<sup>w</sup>*<sup>0</sup> is determined by the average characteristics of the material properties around the microcrack. *σ<sup>w</sup>*<sup>0</sup> can be empirically predicted by the following equations [20, 26]:

$$
\left.\sigma\_{w0}\right|\_{\sigma\_m=0} = \mathbf{1}.\mathbf{6}\ H\_{\text{VM}},\tag{16}
$$

$$\left. \sigma\_{w1} \right|\_{\sigma\_m = 0} = \frac{\sigma\_{w0}|\_{\sigma\_m = 0}}{K\_l \sqrt{1 + 4.5 \varepsilon\_0 |\_{\sigma\_m = 0}/\rho}} = f(H\_{\rm VM}, \rho, \sigma\_m = 0, K\_t). \tag{17}$$

(*σw*0, *σw*1, and *σ<sup>m</sup>* are in MPa, *HVM* is in kgf/mm<sup>2</sup> , and *ρ and ε*<sup>0</sup> are in mm.)

If the stress relaxation effect *γ<sup>m</sup>* is considered, *γ<sup>m</sup> σ*1*,j* is applied to *j*-th surface element. Because fatigue fracture occurs when *γ<sup>m</sup> σ*1*,j=Kt* is greater than *σw*1, the limit hardness *gS σ*1*,j* � � that determines the occurrence is given by the following equation:

$$\mathbf{g}\_S = f^{-1}(H\_{\rm VM}, \rho, \sigma\_m = \mathbf{0}, K\_t)\big|\_{\sigma\_{\rm wt} = \mathbf{y}\_m \ \sigma\_{\mathbf{t}, \mathbf{j}}/K\_t}.\tag{18}$$

Here, *f* �<sup>1</sup> is a function obtained by solving Eq. (18) on *HVM*.

It is supposed that *Anpc S* exhibits the microcrack non-propagation limit characteristics in Eqs. (16) and (17). If *HVM* of the matrix is greater than *gS* in *Anpc S*, fatigue fracture will not occur in *Anpc S*. Therefore, the probability *S<sup>σ</sup>w*10*,j* that fatigue fracture does not occur in *Anpc S* below *σ*1*,j* is given by [25, 35]

$$S\_{\sigma\_{w1}0,j} = \int\_{\mathcal{S}\_{\mathcal{S}}}^{\infty} f\_{H\_{V\mathcal{M}}\mathbb{S}}(h\_{vm}) dh\_{vm}. \tag{19}$$

Here, *f HVMS* is the normal distribution of *μHVMS* and *s* 2 *HVMS*.

The fatigue survival rate *S<sup>σ</sup>w*1*,j* of *j*th surface element containing microcracks is given by the following equation [25, 35]:

$$\mathcal{S}\_{\sigma\_{\mathfrak{w}\mathfrak{z},j}} = \int\_0^\infty f\_{\chi^{\mathbb{Z}}} \cdot (\mathbb{S}\_{\sigma\_{\mathfrak{w}\mathfrak{z}},0,j})^{\frac{A\_j}{A\_{\mathfrak{w}\mathfrak{w}} \cdot S}} d\chi^2. \tag{20}$$

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*

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, *S<sup>σ</sup>w*<sup>1</sup> , is obtained by multiplying the fatigue survival rates of all the surface elements as follows [25, 35]:

$$\mathcal{S}\_{\sigma\_{w1}} = \prod\_{j=1}^{n\_S} \mathcal{S}\_{\sigma\_{w1},j}. \tag{21}$$

Here, *nS* is the number of surface elements.

*f HVM*1ð Þ¼ *HVM*

structural defects, *μHVM*<sup>1</sup> is the sample mean, and *s*

2

*<sup>f</sup> <sup>χ</sup>*<sup>2</sup> *<sup>χ</sup>*<sup>2</sup> � � <sup>¼</sup> <sup>1</sup>

*Γ*ðÞ¼ *t*

predicted by the following equations [20, 26]:

*<sup>σ</sup>m*¼<sup>0</sup> <sup>¼</sup> *<sup>σ</sup><sup>w</sup>*0<sup>j</sup>

*Kt*

*gS* ¼ *f* �1

(*σw*0, *σw*1, and *σ<sup>m</sup>* are in MPa, *HVM* is in kgf/mm<sup>2</sup>

the *<sup>χ</sup>*<sup>2</sup> distribution with the freedom degree of *<sup>n</sup>* <sup>¼</sup> *<sup>m</sup>*<sup>1</sup> � 1 [25, 35]:

ð<sup>∞</sup> 0 *xt*�<sup>1</sup> *e*

*3.2.2 Fatigue survival rate of surface element containing microcracks*

*σ<sup>w</sup>*0j

*σm*¼0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4*:*5*ε*0j

*<sup>σ</sup>m*¼<sup>0</sup>*=<sup>ρ</sup>*

ð Þ *HVM; ρ; σ<sup>m</sup>* ¼ 0*;Kt*

teristics in Eqs. (16) and (17). If *HVM* of the matrix is greater than *gS* in *Anpc S*, fatigue fracture will not occur in *Anpc S*. Therefore, the probability *S<sup>σ</sup>w*10*,j* that

> ð<sup>∞</sup> *gS*

It is supposed that *Anpc S* exhibits the microcrack non-propagation limit charac-

The fatigue survival rate *S<sup>σ</sup>w*1*,j* of *j*th surface element containing microcracks is

*f <sup>χ</sup>*<sup>2</sup> � *S<sup>σ</sup>w*10*,j* � � *Aj*

�<sup>1</sup> is a function obtained by solving Eq. (18) on *HVM*.

fatigue fracture does not occur in *Anpc S* below *σ*1*,j* is given by [25, 35]

ð<sup>∞</sup> 0

*S<sup>σ</sup>w*10*,j* ¼

Here, *f HVMS* is the normal distribution of *μHVMS* and *s*

*S<sup>σ</sup>w*1*,j* ¼

given by the following equation [25, 35]:

If the stress relaxation effect *γ<sup>m</sup>* is considered, *γ<sup>m</sup> σ*1*,j* is applied to *j*-th surface element. Because fatigue fracture occurs when *γ<sup>m</sup> σ*1*,j=Kt* is greater than *σw*1, the

� � that determines the occurrence is given by the following

� �

*σw*1¼*γ<sup>m</sup> σ*1*,j=Kt*

*f HVMS*ð Þ *hvm dhvm:* (19)

2 *HVMS*.

*Anpc Sdχ*<sup>2</sup>

2*Γ*ð Þ *n=*2

between the sample variance *s*

*Fracture Mechanics Applications*

*σ<sup>w</sup>*1j

limit hardness *gS σ*1*,j*

equation:

Here, *f*

**46**

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*πs* 2 *HVM*1

<sup>q</sup> exp � *HVM* � *<sup>μ</sup>HVM*<sup>1</sup>

*HVM*<sup>1</sup> and the population variance *s*

�*xdx, <sup>χ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>m</sup>*<sup>1</sup>

2

exp � *<sup>χ</sup>*<sup>2</sup> 2 � �

> *s* 2 *HVM*1 *s* 2 *HVM*0

*<sup>σ</sup>m*¼<sup>0</sup> ¼ 1*:*6 *HVM,* (16)

, and *ρ and ε*<sup>0</sup> are in mm.)

*:* (18)

*:* (20)

<sup>q</sup> <sup>¼</sup> *f H*ð Þ *VM; <sup>ρ</sup>; <sup>σ</sup><sup>m</sup>* <sup>¼</sup> <sup>0</sup>*;Kt :* (17)

Here, *HVM* is the Vickers hardness of the matrix that does not contain micro-

Based on the central limit theorem, the relationship between the sample mean *μHVM*<sup>1</sup> and the population mean *μHVM*<sup>0</sup> is *μHVM*<sup>1</sup> ¼ *μHVM*0. Further, the relationship

> *χ*2 2 � �*<sup>n</sup>* <sup>2</sup>�1

The microcrack non-propagation limit *σ<sup>w</sup>*<sup>0</sup> is determined by the average characteristics of the material properties around the microcrack. *σ<sup>w</sup>*<sup>0</sup> can be empirically

� �<sup>2</sup> 2*s* 2 *HVM*1

*HVM*<sup>1</sup> is the sample variance.

2

*:* (13)

*HVM*<sup>0</sup> is described by

(14)

(15)

( )

#### **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.

*σwd* is divided into two crack non-propagation limits, namely, the nonpropagation limit *σwdI* of the small crack originating from the interior defect and the non-propagation limit *σwdS* 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 *R* is 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 *R* as follows:

$$
\sqrt{area\_R} = \sqrt{\pi}R.\tag{22}
$$

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

$$K\_{IR} = 0.5 \, F\_R \, \gamma\_m \, \sigma\_{1,j} \sqrt{\pi \sqrt{area\_R}} \, \tag{23}$$

$$F\_R = \frac{4}{\pi^{5/4}} \left\{ 1 - \left( \frac{2}{\sqrt{\pi}} - \frac{4}{3\pi} \right) \frac{\sqrt{area\_R}}{\rho} \right\}. \tag{24}$$

Moreover, the threshold stress intensity factor range *ΔKw* of the small surface defect of size ffiffiffiffiffiffiffiffiffiffiffi *areaP* p in the metal with Vickers hardness *HVM* is given by the following equation [33, 34]:

$$
\left.\Delta K\_w\right|\_{\sigma\_m=0} = \frac{2a\beta\sqrt{area\_P}^{1/3}}{\ln\left(2\beta/H\_{VM} + 1\right)},\tag{25}
$$

*<sup>α</sup>* <sup>¼</sup> <sup>3</sup>*:*<sup>3</sup> � <sup>10</sup>�3and *<sup>β</sup>* <sup>¼</sup> 120.

(*ΔKw* is in MPa ffiffiffiffi <sup>m</sup><sup>p</sup> , *HVM* is in kgf/mm2 , and ffiffiffiffiffiffiffiffiffiffiffi *areaP* p is in μm).

The limit hardness that determines whether the fatigue crack originating from the interior microstructural crack is arrested, *gR σ*1*,j;* ffiffiffiffiffiffiffiffiffiffiffi *areaR* � � p , is given based on the relationship ffiffiffiffiffiffiffiffiffiffiffi *areaR* <sup>p</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>7</sup> ffiffiffiffiffiffiffiffiffiffiffi *areaP* p by the following equation [25, 35]:

$$\log\_R = 240 / \left\{ \exp \left( \frac{1.56 \times 240}{F\_R \gamma\_m \sigma\_{1,j} \sqrt{area\_R}^{1/6}} \right) - 1 \right\} \tag{26}$$

(*σ*1*,j* is in MPa, *gR* is in kgf/mm<sup>2</sup> , and ffiffiffiffiffiffiffiffiffiffiffi *areaR* p is in μm.)

The relationship between *μHVMR* and *μHVM*<sup>0</sup> can be expressed as follows [25, 35]:

$$
\mu\_{H\_{\rm VM}R} = \mu\_{H\_{\rm VM}0} = \mu\_{H\_{\rm VM}1}.\tag{27}
$$

*S* ð Þ *m <sup>σ</sup>wdS,j* ¼ ð<sup>∞</sup> 0

*f <sup>χ</sup>*<sup>2</sup> exp �

*DOI: http://dx.doi.org/10.5772/intechopen.88413*

The fatigue survival rate *S*ð Þ *<sup>m</sup>*

surface elements as follows [25, 35]:

The fatigue survival rate *S*ð Þ *<sup>m</sup>*

obtained by multiplying *S*ð Þ *<sup>m</sup>*

ð ffiffiffiffiffiffiffiffi *areaP* <sup>p</sup> *<sup>c</sup>*

ð*gP* 0

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense…*

*S*ð Þ *<sup>m</sup> <sup>σ</sup>wdS* <sup>¼</sup> <sup>Y</sup>*nS*

*3.3.3 Fatigue survival rate of element with microstructural defects*

*<sup>σ</sup>wdI* and *<sup>S</sup>*ð Þ *<sup>m</sup>*

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

*σ<sup>w</sup>*2j

and its fatigue limit reliability *F<sup>σ</sup><sup>w</sup>* is obtained as follows [35]:

�

**4.1 Material, shape of specimen, and experimental procedure**

survival rate S*<sup>σ</sup>wd* is given by the following equation:

constant regardless of the crack length. *σ<sup>w</sup>*2j

**3.5 Prediction of fatigue limit reliability**

**4. Fatigue experiment**

ical properties.

**49**

*S*ð Þ *<sup>m</sup> <sup>σ</sup>wd* <sup>¼</sup> *<sup>S</sup>*ð Þ *<sup>m</sup>*

*p*ð Þ *<sup>m</sup> dS*

structural cracks is obtained by multiplying the fatigue survival rates of all the

*j*¼1 *S* ð Þ *m σwdS,j*

*<sup>σ</sup>wdS* as follows [25, 35]:

*<sup>σ</sup>wdI* � *<sup>S</sup>*ð Þ *<sup>m</sup>*

*S*ð Þ *<sup>m</sup>*

Because the material contains *Md* types of inhomogeneous particles, the fatigue

*m*¼1

*<sup>σ</sup>m*¼<sup>0</sup> <sup>¼</sup> *<sup>Δ</sup>KwUL*j*<sup>σ</sup>m*¼<sup>0</sup> 2*F* ffiffiffiffi

The probability that fatigue fracture is caused by microcracks or microstructural defects is obtained by the complementary event defined by the product of *S<sup>σ</sup>w*<sup>1</sup> and *S<sup>σ</sup>wd* . Because the fatigue limit of a notched specimen, *σ<sup>w</sup>* cannot be lower than *σw*2,

*<sup>F</sup><sup>σ</sup><sup>w</sup>* <sup>¼</sup> <sup>0</sup> ð Þ *<sup>σ</sup><sup>w</sup>* <sup>≤</sup> *<sup>σ</sup><sup>w</sup>*<sup>2</sup>

The material used for the experiment was Al-Si-Cu alloy (JIS AC4B). The agehardened aluminum cast alloy is identified as AC4B-T6. **Table 2** shows its mechan-

1 � *S<sup>σ</sup>w*<sup>1</sup> � *S<sup>σ</sup>wd* ð Þ *σw*>*σ<sup>w</sup>*<sup>2</sup>

<sup>S</sup>*<sup>σ</sup>wd* <sup>¼</sup> <sup>Y</sup> *Md*

*σ<sup>w</sup>*<sup>2</sup> of the notched specimen with *ρ*≤ *ρ*<sup>0</sup> is equal to the fatigue limit of the cracked specimen obtained by *ρ* ! 0. *ΔKwUL* is the upper limit of *ΔKw* and is

*dM*ð Þ *<sup>m</sup> S,j d* ffiffiffiffiffiffiffiffiffiffiffi *areaP* <sup>p</sup> *<sup>f</sup> HVMPdhvm* !*<sup>d</sup>* ffiffiffiffiffiffiffiffiffiffiffi

( )

*<sup>σ</sup>wdS* of a surface element containing surface micro-

*<sup>σ</sup>wd* of an element containing microstructural defects is

*areaP* p

*:* (35)

*<sup>σ</sup>wdS :* (36)

*<sup>σ</sup>wd :* (37)

*<sup>σ</sup>m*¼<sup>0</sup> can be obtained as follows [35]:

*<sup>π</sup><sup>t</sup>* <sup>p</sup> *:* (38)

(39)

*dχ*<sup>2</sup> (34)

∞

Moreover, the relationship between *s* 2 *HVMR* and *s* 2 *HVM*<sup>0</sup> can be expressed as follows [25, 35]:

$$\mathcal{A}\_{\text{npc}} \cdot \mathfrak{s}\_{H\_{\text{VM}} \text{R}}^2 = \mathcal{A}\_{H\_{\text{VM}} \text{0}} \cdot \mathfrak{s}\_{H\_{\text{VM}} \text{0}}^2. \tag{28}$$

Here, *AHVM*<sup>0</sup> ¼ *P=μHVM*0, *Anpc R* ¼ *PR=gR*, and *PR* are the loads used to create the indentation for obtaining the Vickers hardness *gR* and the indentation area *Anpc R*.

The fatigue survival rate of *j*-th solid element containing interior defects, *S* ð Þ *m σwdI,j* , is given by the following equation [25, 35]:

$$\mathcal{S}^{(m)}\_{\sigma\_{\text{add}},j} = \int\_0^\infty f\_{\chi^2} \exp\left\{-\int\_\infty^{\mathbb{R}\_c} \left(\int\_0^{\mathbb{R}\_R} p\_{dI}^{(m)} \frac{d\mathcal{M}\_{V,j}^{(m)}}{dR} f\_{H\_{V\mathcal{M}}R} dh\_{vm}\right) dR\right\} d\chi^2. \tag{29}$$

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 *S*ð Þ *<sup>m</sup> <sup>σ</sup>wdI* 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]:

$$\mathcal{S}\_{\sigma\_{\text{wall}}}^{(m)} = \prod\_{j=1}^{n\_V} \mathcal{S}\_{\sigma\_{\text{wall}},j}^{(m)}. \tag{30}$$

Here, *nV* is the number of solid elements.

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

The stress intensity factor *KIP* of a small surface crack of size ffiffiffiffiffiffiffiffiffiffiffi *areaP* p is given by the following equation [20, 35]:

$$K\_{IP} = 0.65 \, F\_R \, \text{y}\_m \, \sigma\_{1,j} \sqrt{\pi \sqrt{area\_P}} \tag{31}$$

$$F\_P = 0.968 - 1.021 \frac{\sqrt{area \rho}}{\rho}. \tag{32}$$

Further, the limit hardness *gP σ*1*,j;* ffiffiffiffiffiffiffiffiffiffiffi *areaP* � � p that determines whether the small surface crack is arrested is given by the following equation [25, 35]:

$$\mathbf{g}\_P = 240 / \left\{ \exp \left( \frac{\mathbf{1}.43 \times 240}{F\_R \ \boldsymbol{\gamma}\_m \ \sigma\_{1,j} \sqrt{\boldsymbol{\sigma} \boldsymbol{\sigma} \boldsymbol{\sigma} \boldsymbol{\Phi}\_P}} \right) - \mathbf{1} \right\}. \tag{33}$$

(*σ*1*,j* is in MPa, *gP* is in kgf/mm<sup>2</sup> , and ffiffiffiffiffiffiffiffiffiffiffi *areaP* p is in μm.)

The fatigue survival rate of *j*-th surface element containing surface microstructural cracks, *S* ð Þ *m σwdS,j* , is given by

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*

$$\mathcal{S}^{(m)}\_{\sigma\_{\text{endS},j}} = \int\_0^\infty f\_{\chi^2} \exp\left\{-\int\_\infty^{\sqrt{\text{area}\_p}} \left(\int\_0^{\mathbf{g}\_P} p\_{\text{dS}}^{(m)} \frac{d\mathcal{M}^{(m)}\_{\mathbf{S},j}}{d\sqrt{\text{area}\_P}} f\_{\text{Hvm},p} dh\_{\text{vm}}\right) d\sqrt{area\_P}\right\} d\chi^2 \tag{34}$$

The fatigue survival rate *S*ð Þ *<sup>m</sup> <sup>σ</sup>wdS* of a surface element containing surface microstructural cracks is obtained by multiplying the fatigue survival rates of all the surface elements as follows [25, 35]:

$$\mathcal{S}^{(m)}\_{\sigma\_{\text{subS}}} = \prod\_{j=1}^{n\_{\text{S}}} \mathcal{S}^{(m)}\_{\sigma\_{\text{subS}},j}. \tag{35}$$

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

The fatigue survival rate *S*ð Þ *<sup>m</sup> <sup>σ</sup>wd* of an element containing microstructural defects is obtained by multiplying *S*ð Þ *<sup>m</sup> <sup>σ</sup>wdI* and *<sup>S</sup>*ð Þ *<sup>m</sup> <sup>σ</sup>wdS* as follows [25, 35]:

$$\mathbf{S}^{(m)}\_{\sigma\_{\text{wd}}} = \mathbf{S}^{(m)}\_{\sigma\_{\text{wall}}} \times \mathbf{S}^{(m)}\_{\sigma\_{\text{wd}}}.\tag{36}$$

Because the material contains *Md* types of inhomogeneous particles, the fatigue survival rate S*<sup>σ</sup>wd* is given by the following equation:

$$\mathbf{S}\_{\sigma\_{\text{adv}}} = \prod\_{m=1}^{M\_d} \mathbf{S}\_{\sigma\_{\text{adv}}}^{(m)}.\tag{37}$$

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

*σ<sup>w</sup>*<sup>2</sup> of the notched specimen with *ρ*≤ *ρ*<sup>0</sup> is equal to the fatigue limit of the cracked specimen obtained by *ρ* ! 0. *ΔKwUL* is the upper limit of *ΔKw* and is constant regardless of the crack length. *σ<sup>w</sup>*2j *<sup>σ</sup>m*¼<sup>0</sup> can be obtained as follows [35]:

$$
\left.\sigma\_{w2}\right|\_{\sigma\_m=0} = \frac{\Delta K\_{wUL}|\_{\sigma\_m=0}}{2F\sqrt{\pi t}}.\tag{38}
$$

#### **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 *S<sup>σ</sup>w*<sup>1</sup> and *S<sup>σ</sup>wd* . Because the fatigue limit of a notched specimen, *σ<sup>w</sup>* cannot be lower than *σw*2, and its fatigue limit reliability *F<sup>σ</sup><sup>w</sup>* is obtained as follows [35]:

$$F\_{\sigma\_w} = \begin{cases} 0 & (\sigma\_w \le \sigma\_{w2}) \\ \mathbf{1} - \mathbf{S}\_{\sigma\_{w1}} \times \mathbf{S}\_{\sigma\_{w2}} & (\sigma\_w \succ \sigma\_{w2}) \end{cases} \tag{39}$$

## **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 agehardened aluminum cast alloy is identified as AC4B-T6. **Table 2** shows its mechanical properties.

*gR* ¼ 240*=* exp

*Anpc R* � *s* 2

> ð*Rc* ∞

ð*gR* 0

*p*ð Þ *<sup>m</sup> dI*

multiplying the fatigue survival rates of all the solid elements as follows [25, 35]:

*S*ð Þ *<sup>m</sup> <sup>σ</sup>wdI* <sup>¼</sup> <sup>Y</sup>*nV*

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

> *j*¼1 *S* ð Þ *m σwdI,j*

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

The stress intensity factor *KIP* of a small surface crack of size ffiffiffiffiffiffiffiffiffiffiffi

*KIP* ¼ 0*:*65 *FR γ<sup>m</sup> σ*1*,j*

surface crack is arrested is given by the following equation [25, 35]:

*FP* ¼ 0*:*968 � 1*:*021

*areaP*

*FR γ<sup>m</sup> σ*1*,j*

, and ffiffiffiffiffiffiffiffiffiffiffi *areaP* p is in μm.)

The fatigue survival rate of *j*-th surface element containing surface microstruc-

1*:*43 � 240

!

( )

ffiffiffiffiffiffiffiffiffiffiffi *areaP* p <sup>1</sup>*=*<sup>6</sup>

*<sup>σ</sup>wdI* of a solid element containing interior microstructural defects is obtained by

*dM*ð Þ *<sup>m</sup> V,j*

( )

!

(*σ*1*,j* is in MPa, *gR* is in kgf/mm<sup>2</sup>

*Fracture Mechanics Applications*

Moreover, the relationship between *s*

is given by the following equation [25, 35]:

*f <sup>χ</sup>*<sup>2</sup> exp �

Here, *nV* is the number of solid elements.

Further, the limit hardness *gP σ*1*,j;* ffiffiffiffiffiffiffiffiffiffiffi

, is given by

(*σ*1*,j* is in MPa, *gP* is in kgf/mm<sup>2</sup>

ð Þ *m σwdS,j*

tural cracks, *S*

**48**

*gP* ¼ 240*=* exp

ð<sup>∞</sup> 0

the following equation [20, 35]:

[25, 35]:

*S* ð Þ *m <sup>σ</sup>wdI,j* ¼

rate *S*ð Þ *<sup>m</sup>*

1*:*56 � 240

!

( )

ffiffiffiffiffiffiffiffiffiffiffi *areaR* p 1*=*6

2

2

*dR <sup>f</sup> HVMRdhvm*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *π* ffiffiffiffiffiffiffiffiffiffiffi *areaP*

ffiffiffiffiffiffiffiffiffiffiffi *areaP* p

� � p that determines whether the small

� 1

p q

*μHVMR* ¼ *μHVM*<sup>0</sup> ¼ *μHVM*1*:* (27)

� 1

*HVM*<sup>0</sup> can be expressed as follows

*HVM*0*:* (28)

*dR*

*dχ*<sup>2</sup>

*:* (30)

*areaP*

*,* (31)

*<sup>ρ</sup> :* (32)

p is given by

*:* (33)

(26)

ð Þ *m σwdI,j* ,

*:* (29)

*FRγ<sup>m</sup> σ*1*,j*

, and ffiffiffiffiffiffiffiffiffiffiffi *areaR* p is in μm.)

2

The relationship between *μHVMR* and *μHVM*<sup>0</sup> can be expressed as follows [25, 35]:

*HVMR* and *s*

*HVMR* ¼ *AHVM*<sup>0</sup> � *s*

Here, *AHVM*<sup>0</sup> ¼ *P=μHVM*0, *Anpc R* ¼ *PR=gR*, and *PR* are the loads used to create the indentation for obtaining the Vickers hardness *gR* and the indentation area *Anpc R*. The fatigue survival rate of *j*-th solid element containing interior defects, *S*


**Table 2.**

*Mechanical properties.*

**Figure 6.** *Specimen configuration.*

**Figure 6** shows the configurations of the specimens. The notch depth *t* and opening angle *θ* were set at 0.5 mm and 60°, respectively. The notch root radii *ρ* 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 [27–31] under stress amplitude *σ<sup>a</sup>* ¼ 60–110 MPa and frequency *f* ¼ 50 Hz. An Ono-type rotating-bending fatigue machine of a capacity 15 Nm was used for the tests. The nominal stress *σ<sup>n</sup>* used for the analyses of the experimental results was the stress at the minimum cross section, where the diameter *d* was 5 mm. The fatigue life *Nf* was defined as the total number of stress cycles to failure.

limit when *ρ* ¼ 1 and 2 mm, it can be said that the microcrack non-propagating limit *σ<sup>w</sup>*<sup>1</sup> or the small macrocrack non-propagating limit *σwd* appears as the fatigue limit. When *ρ* = 0.1 mm for *σ<sup>w</sup>* = 90 MPa and *ρ* = 0.3 mm for *σ<sup>w</sup>* = 95 MPa, the nonpropagating macrocracks were observed along the notch root. Therefore, the long

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense…*

*DOI: http://dx.doi.org/10.5772/intechopen.88413*

**5.1 Notch sensitivity to crack initiation limit in age-hardened aluminum alloy**

**Figure 9** shows the relationship between *Ktσ<sup>w</sup>*1*=σ<sup>w</sup>*<sup>0</sup> and *ρ* using early fatigue data of previous studies [28–30]. *ε*<sup>0</sup> values of the curves are shown in **Figure 10**. When *ε*<sup>0</sup> values were approximated with the lines by the least squares method, the

macrocrack non-propagating limit *σ<sup>w</sup>*<sup>2</sup> = 90 MPa.

*Optical micrograph of non-propagating crack under notch root.*

following equation was obtained:

**Figure 7.** *S* � *N curve.*

**Figure 8.**

**51**

**5. Examination of validity of prediction method**

#### **4.2 Experimental results**

**Figure 7** shows *S*-*N* curves obtained from the results of the tests. **Figure 8** shows optical micrographs of a specimen when *ρ* = 0.1 mm for *σ<sup>w</sup>* = 90 MPa. Fatigue limit *σ<sup>w</sup>* ¼ 105 when *ρ* ¼ 2 mm. *σ<sup>w</sup>* = 95 when *ρ* ¼ 1 and 0*:*3 mm. *σ<sup>w</sup>* = 90 when *ρ* = 0.1 mm. Since the non-propagating macrocrack was not observed at the fatigue

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*

**Figure 8.** *Optical micrograph of non-propagating crack under notch root.*

limit when *ρ* ¼ 1 and 2 mm, it can be said that the microcrack non-propagating limit *σ<sup>w</sup>*<sup>1</sup> or the small macrocrack non-propagating limit *σwd* appears as the fatigue limit. When *ρ* = 0.1 mm for *σ<sup>w</sup>* = 90 MPa and *ρ* = 0.3 mm for *σ<sup>w</sup>* = 95 MPa, the nonpropagating macrocracks were observed along the notch root. Therefore, the long macrocrack non-propagating limit *σ<sup>w</sup>*<sup>2</sup> = 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 *Ktσ<sup>w</sup>*1*=σ<sup>w</sup>*<sup>0</sup> and *ρ* using early fatigue data of previous studies [28–30]. *ε*<sup>0</sup> values of the curves are shown in **Figure 10**. When *ε*<sup>0</sup> values were approximated with the lines by the least squares method, the following equation was obtained:

**Figure 6** shows the configurations of the specimens. The notch depth *t* and opening angle *θ* were set at 0.5 mm and 60°, respectively. The notch root radii *ρ* 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 [27–31] under stress amplitude *σ<sup>a</sup>* ¼ 60–110 MPa and frequency *f* ¼ 50 Hz. An Ono-type rotating-bending fatigue machine of a capacity 15 Nm was used for the tests. The nominal stress *σ<sup>n</sup>* used for the analyses of the experimental results was the stress at the minimum cross section, where the diameter *d* was 5 mm. The fatigue life *Nf* was defined as the total number of stress

*E σ***<sup>0</sup>***:***<sup>2</sup>** *σ<sup>B</sup> δ HV HVM*

74 292 349 1.5 152 92 *E*: Young's modulus (GPa) *σ*0*:*2: 0.2% proof stress (MPa)

σB: ultimate tensile strength (MPa) *δ*: elongation (%) *HV*: Vickers hardness of matrix with inhomogeneous particles (kgf/mm<sup>2</sup>

*HVM*: Vickers hardness of matrix without inhomogeneous particles (kgf/mm<sup>2</sup>

**9.8 N, 30 sec 29.4 mN, 30 sec**

)

)

**Figure 7** shows *S*-*N* curves obtained from the results of the tests. **Figure 8** shows optical micrographs of a specimen when *ρ* = 0.1 mm for *σ<sup>w</sup>* = 90 MPa. Fatigue limit

*ρ* = 0.1 mm. Since the non-propagating macrocrack was not observed at the fatigue

*σ<sup>w</sup>* ¼ 105 when *ρ* ¼ 2 mm. *σ<sup>w</sup>* = 95 when *ρ* ¼ 1 and 0*:*3 mm. *σ<sup>w</sup>* = 90 when

cycles to failure.

**50**

**Figure 6.**

*Specimen configuration.*

**Table 2.**

*Mechanical properties.*

*Fracture Mechanics Applications*

**4.2 Experimental results**

$$\left.\varepsilon\_{0}\right|\_{\sigma\_{n}=0} = \mathbf{5.0} \times \mathbf{10}^{-4} \ H\_{B} - \mathbf{0.0164},\tag{40}$$
 
$$\rho \ge \mathbf{0.5, 97} \le H\_{B} \le 207.$$

**5.2** *ΔKwUL* **of age-hardened aluminum alloy**

*DOI: http://dx.doi.org/10.5772/intechopen.88413*

*ΔKwUL*j

used to derive the following equation:

(*ΔKwUL* and *ΔKwLL* are in MPa ffiffiffiffi

by the following equation [12, 37]:

alloys.

**Figure 11.**

**53**

*Relation between ΔKwUL and HB.*

**Figure 11** shows the values of *ΔKwUL* obtained from the early fatigue data of *σw*<sup>2</sup>

40 ≤ *HB* ≤100*:*

<sup>m</sup><sup>p</sup> , and *HB*is in kgf/mm<sup>2</sup>

*<sup>σ</sup>m*¼<sup>0</sup> <sup>¼</sup> *<sup>Δ</sup>KwLL* <sup>þ</sup> <sup>0</sup>*:*<sup>03</sup> *HB,* (41)

.)

*<sup>π</sup>* <sup>p</sup> *:* (42)

mp for the Al-Si-X

for different Al-Si-X alloys, where X is a transition element [28, 29, 31]. An approximation of *ΔKwUL* obtained from the lines by the least squares method was

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense…*

Here, *ΔKwLL* is the lower limit of *ΔKw* and *ΔKwLL* = 0.5 MPa ffiffiffiffi

**5.3 Evaluation of statistical characteristics of inhomogeneous particles**

*r* ¼

porosity. The relationship between *MV*<sup>0</sup> and *MA*<sup>0</sup> is as follows [12, 37]:

ffiffiffiffiffiffiffiffiffiffiffiffi *areaA* p

ffiffiffi

**Figure 13** shows the measured *MA*<sup>0</sup> values of eutectic Si and Fe compounds and

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 *r* are obtained from *areaA*

(*ε*<sup>0</sup> and *ρ* are in mm, and *HB* is in kgf/mm<sup>2</sup> .) Once *σw*<sup>1</sup> has been predicted, *HVM* can be used instead of *HB*.

**Figure 9.** *Relation between Ktσ<sup>w</sup>*1*=σ<sup>w</sup>*<sup>0</sup> *and* 1*=ρ.*

**Figure 10.** *Relation between ε*<sup>0</sup> *and HB.*

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*

## **5.2** *ΔKwUL* **of age-hardened aluminum alloy**

*<sup>ε</sup>*0j*<sup>σ</sup>m*¼<sup>0</sup> <sup>¼</sup> <sup>5</sup>*:*<sup>0</sup> � <sup>10</sup>�<sup>4</sup> *HB* � <sup>0</sup>*:*0164*,* (40)

*ρ* ≥0*:*5*,* 97 ≤ *HB* ≤ 207*:*

Once *σw*<sup>1</sup> has been predicted, *HVM* can be used instead of *HB*.

.)

(*ε*<sup>0</sup> and *ρ* are in mm, and *HB* is in kgf/mm<sup>2</sup>

*Fracture Mechanics Applications*

**Figure 9.**

**Figure 10.**

**52**

*Relation between ε*<sup>0</sup> *and HB.*

*Relation between Ktσ<sup>w</sup>*1*=σ<sup>w</sup>*<sup>0</sup> *and* 1*=ρ.*

**Figure 11** shows the values of *ΔKwUL* obtained from the early fatigue data of *σw*<sup>2</sup> for different Al-Si-X alloys, where X is a transition element [28, 29, 31]. An approximation of *ΔKwUL* obtained from the lines by the least squares method was used to derive the following equation:

$$
\Delta K\_{wUL}|\_{\sigma\_w = 0} = \Delta K\_{wLL} + \text{0.03 } H\_{B\text{s}}.\tag{41}
$$

$$
4\mathbf{O} \le H\_B \le \mathbf{100}.
$$

(*ΔKwUL* and *ΔKwLL* are in MPa ffiffiffiffi <sup>m</sup><sup>p</sup> , and *HB*is in kgf/mm<sup>2</sup> .)

Here, *ΔKwLL* is the lower limit of *ΔKw* and *ΔKwLL* = 0.5 MPa ffiffiffiffi mp for 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 *r* are obtained from *areaA* by the following equation [12, 37]:

$$r = \frac{\sqrt{area\_A}}{\sqrt{\pi}}.\tag{42}$$

**Figure 13** shows the measured *MA*<sup>0</sup> values of eutectic Si and Fe compounds and porosity. The relationship between *MV*<sup>0</sup> and *MA*<sup>0</sup> is as follows [12, 37]:

**Figure 11.** *Relation between ΔKwUL and HB.*

*Fracture Mechanics Applications*

$$M\_{A0}(r\_0) = 2\int\_{\infty}^{r\_0} \sqrt{R^2 - r^2} \, \frac{d\mathcal{M}\_{V0}}{dR} dR. \tag{43}$$

Here, *MA*0ð Þ *r*<sup>0</sup> is the number of cross-sectional particles in a unit area for which *r*≥*r*<sup>0</sup> on a unit area. Considering the assumption that *MV*0ð Þ *R*<sup>0</sup> is given by Eqs. (1) and (2), the asymptotic characteristics of Eq. (42) are expressed by the following equation [12, 37]:

$$M\_{A0}(r\_0) \cong \sqrt{\frac{2\pi}{\nu}} \lambda \left(\frac{r\_0}{\lambda}\right)^{1-\frac{\nu}{2}} \overline{N}\_{V0} \, \exp\left\{-\left(\frac{r\_0}{\lambda}\right)^{\nu}\right\}.\tag{44}$$

The line of Eq. (44) is drawn to best fit the *MA*<sup>0</sup> values obtained by Eq. (43) to determine the values of *NV*0, *ν*, and *λ*.

**Figure 14** shows the values of *MV*<sup>0</sup> for porosity, Fe compounds, and eutectic Si. **Figure 15** shows the values of *MS*<sup>0</sup> for the porosity, Fe compounds, and eutectic Si.

**Figure 12.** *Definition of areaA.*

**5.4 Evaluation of statistical characteristics of Vickers hardness of matrix**

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense…*

*DOI: http://dx.doi.org/10.5772/intechopen.88413*

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* ¼ 29*:*4 mN in consideration with the stress distribution at

**without inhomogeneous particles**

*areaP* p *.*

**Figure 14.**

**Figure 15.**

**55**

*Relation between MS*<sup>0</sup> *and* ffiffiffiffiffiffiffiffiffiffiffi

*Relation between MV*<sup>0</sup> *and R.*

**Figure 13.** *MA*<sup>0</sup> *of porosity and eutectic Si and Fe compounds.*

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*

**Figure 14.** *Relation between MV*<sup>0</sup> *and R.*

*MA*0ð Þ¼ *r*<sup>0</sup> 2

ffiffiffiffiffi 2*π ν* r

*<sup>λ</sup> <sup>r</sup>*<sup>0</sup> *λ* � �1�*<sup>ν</sup>* 2

Si. **Figure 15** shows the values of *MS*<sup>0</sup> for the porosity, Fe compounds, and

*MA*0ð Þffi *r*<sup>0</sup>

determine the values of *NV*0, *ν*, and *λ*.

equation [12, 37]:

*Fracture Mechanics Applications*

eutectic Si.

**Figure 12.** *Definition of areaA.*

**Figure 13.**

**54**

*MA*<sup>0</sup> *of porosity and eutectic Si and Fe compounds.*

ð*r*<sup>0</sup> ∞

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>R</sup>*<sup>2</sup> � *<sup>r</sup>*<sup>2</sup> p *dMV*<sup>0</sup>

*NV*<sup>0</sup> exp � *<sup>r</sup>*<sup>0</sup>

*λ* n o � �*<sup>ν</sup>*

Here, *MA*0ð Þ *r*<sup>0</sup> is the number of cross-sectional particles in a unit area for which *r*≥*r*<sup>0</sup> on a unit area. Considering the assumption that *MV*0ð Þ *R*<sup>0</sup> is given by Eqs. (1) and (2), the asymptotic characteristics of Eq. (42) are expressed by the following

The line of Eq. (44) is drawn to best fit the *MA*<sup>0</sup> values obtained by Eq. (43) to

**Figure 14** shows the values of *MV*<sup>0</sup> for porosity, Fe compounds, and eutectic

*dR dR:* (43)

*:* (44)

**Figure 15.** *Relation between MS*<sup>0</sup> *and* ffiffiffiffiffiffiffiffiffiffiffi *areaP* p *.*

## **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* ¼ 29*:*4 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 *μHVM*<sup>1</sup> and sample variance *s* 2 *HVM*<sup>1</sup> were 91.8 kgf/mm<sup>2</sup> and 486.0 (kgf/mm2 ) 2 , respectively.

## **5.5 Evaluation of** *γ<sup>m</sup>* **value**

Because the values of *Rm=pm* for the eutectic Si were much greater than those of the Fe compounds and porosity, as shown in **Table 3**, *γ<sup>m</sup>* was 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 *ν<sup>M</sup>* = *ν<sup>I</sup>* = 0.3 [35]. Using *Rm=pm* = 0.192, *γ<sup>m</sup>* was determined to be 1.055.

## **5.6 Evaluation of** *Anpc S***,** *Anpc R***, and** *Anpc P* **values**

## *5.6.1 Evaluation of Anpc S value*

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

*Anpc S* 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 *ρ* and it is sufficiently small, its non-propagation limit *σwdS* is given by the following equation:

> > *<sup>σ</sup>wdS* <sup>¼</sup> <sup>1</sup>*:*43ð Þ *HVM* <sup>þ</sup> <sup>120</sup> *FP*

> > > *,* ffiffiffiffiffiffiffiffiffiffiffi *areaP*

Conversely, when the macrocrack is sufficiently large, *ΔKw* is greater than *ΔKwUL*. The macrocrack is thus treated as being large, and its non-propagation limit

> *<sup>σ</sup><sup>w</sup>*<sup>2</sup> <sup>¼</sup> <sup>13</sup>*:*0ð Þ *HVM* <sup>þ</sup> <sup>16</sup>*:*<sup>7</sup> *FP*

ffiffiffiffiffiffiffiffiffiffiffi *areaP*

, and ffiffiffiffiffiffiffiffiffiffiffi *areaP* p : in μm.)

*FP* is approximated to be 1. Because the average Vickers hardness of the matrix of

*Anpc R* is a function of *PR* and *gR*; *Anpc P* is a function of *PP* and *gP*. Because *gR* and *gP* can be calculated using Eqs. (26) and (33), respectively, only *PR* and *PP* need to

*σ<sup>w</sup>*<sup>0</sup> was 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 *σ<sup>n</sup>* = 120 MPa. From these experimental results, the *lnpc* <sup>0</sup> of the present study was assumed to be 70 μm. *lnpc* was set to achieve *b=l* = 0.4. Using *c* = 2.5 mm, *lnpc* <sup>0</sup> = 70 μm is equivalent to

*area* <sup>p</sup> *npc* <sup>0</sup> of 39.2 <sup>μ</sup>m. **Figure 18** shows the relationship between *lnpc* and 1*=ρ*.

*FP* <sup>¼</sup> <sup>0</sup>*:*<sup>968</sup> � <sup>1</sup>*:*<sup>021</sup> � <sup>10</sup>�<sup>3</sup>

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense…*

*DOI: http://dx.doi.org/10.5772/intechopen.88413*

is categorized as *σw*2, which is given by the following equation:

(*σwdS* is in MPa, *HVM* is in kgf/mm<sup>2</sup>

(*σ<sup>w</sup>*<sup>2</sup> is in MPa, *HVM* is in kgf/mm<sup>2</sup>

ffiffiffiffiffiffiffiffiffi

**57**

**Figure 17.**

*Schematic illustration of the microcrack.*

the present AC4B-T6 is about 91.8 kgf/mm2

*5.6.2 Evaluation of Anpc R and Anpc P values*

ffiffiffiffiffiffiffiffiffiffiffi *areaP*

<sup>p</sup> <sup>1</sup>*=*<sup>6</sup> *,* (45)

<sup>p</sup> <sup>1</sup>*=*<sup>2</sup> *:* (47)

, the microcrack non-propagation limit

*<sup>ρ</sup> :* (46)

ffiffiffiffiffiffiffiffiffiffiffi *areaP* p

p is in μm, and *ρ* is in mm.)

**Figure 16.** *Evaluation of Vickers hardness of matrix from normal probability paper.*


#### **Table 3.**

*Parameters of particle size distribution.*

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*

**Figure 17.** *Schematic illustration of the microcrack.*

the notch root. **Figure 16** shows the results plotted on a normal probability paper.

Because the values of *Rm=pm* for the eutectic Si were much greater than those of the Fe compounds and porosity, as shown in **Table 3**, *γ<sup>m</sup>* was 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 *ν<sup>M</sup>* = *ν<sup>I</sup>* = 0.3

Because a microcrack often grows radially, it is approximated by the

*Anpc S* in Eq. (20) can also be roughly evaluated. The approximation of the microcrack by the semielliptical macrocrack is such that the crack non-propagation

2

*HVM*<sup>1</sup> were 91.8 kgf/mm<sup>2</sup> and

**]** *ν λ* **[**μ**m]** *Rm* **[**μ**m]** *pm* **[**μ**m]**

The sample mean *μHVM*<sup>1</sup> and sample variance *s*

, respectively.

[35]. Using *Rm=pm* = 0.192, *γ<sup>m</sup>* was determined to be 1.055.

**5.6 Evaluation of** *Anpc S***,** *Anpc R***, and** *Anpc P* **values**

*Evaluation of Vickers hardness of matrix from normal probability paper.*

Eutectic Si 8.73 <sup>10</sup><sup>6</sup> 1.6 1.04 0.932 4.86 Fe compound 2.20 <sup>10</sup><sup>7</sup> 0.5 0.10 0.200 3.57 Porosity (*R*≥105*μ*m) 1.20 102 0.3 0.180 0.167 21.5

(*<sup>R</sup>* < 105*μ*m) 1.00 <sup>10</sup><sup>5</sup> 0.3 0.0180

**Inhomogeneous particle** *NV***<sup>0</sup> [1/mm<sup>3</sup>**

*Parameters of particle size distribution.*

) 2

**5.5 Evaluation of** *γ<sup>m</sup>* **value**

*Fracture Mechanics Applications*

*5.6.1 Evaluation of Anpc S value*

**Figure 16.**

**Table 3.**

**56**

semielliptical crack shown in **Figure 17**.

486.0 (kgf/mm2

limits are equal. If the macrocrack is located near the notch root of radius *ρ* and it is sufficiently small, its non-propagation limit *σwdS* is given by the following equation:

$$
\sigma\_{\text{wdS}} = \frac{1.43(H\_{\text{VM}} + 120)}{F\_P \sqrt{area\_P}^{1/6}},
\tag{45}
$$

$$F\_P = 0.968 - 1.021 \times 10^{-3} \frac{\sqrt{area \rho}}{\rho}. \tag{46}$$

(*σwdS* is in MPa, *HVM* is in kgf/mm<sup>2</sup> *,* ffiffiffiffiffiffiffiffiffiffiffi *areaP* p is in μm, and *ρ* is in mm.)

Conversely, when the macrocrack is sufficiently large, *ΔKw* is greater than *ΔKwUL*. The macrocrack is thus treated as being large, and its non-propagation limit is categorized as *σw*2, which is given by the following equation:

$$
\sigma\_{w2} = \frac{\mathbf{13.0} (H\_{VM} + \mathbf{16.7})}{F\_P \sqrt{area\_P}^{1/2}}.\tag{47}
$$

(*σ<sup>w</sup>*<sup>2</sup> is in MPa, *HVM* is in kgf/mm<sup>2</sup> , and ffiffiffiffiffiffiffiffiffiffiffi *areaP* p : in μm.)

*FP* 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 *σ<sup>w</sup>*<sup>0</sup> was 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 *σ<sup>n</sup>* = 120 MPa. From these experimental results, the *lnpc* <sup>0</sup> of the present study was assumed to be 70 μm.

*lnpc* was set to achieve *b=l* = 0.4. Using *c* = 2.5 mm, *lnpc* <sup>0</sup> = 70 μm is equivalent to ffiffiffiffiffiffiffiffiffi *area* <sup>p</sup> *npc* <sup>0</sup> of 39.2 <sup>μ</sup>m. **Figure 18** shows the relationship between *lnpc* and 1*=ρ*.

## *5.6.2 Evaluation of Anpc R and Anpc P values*

*Anpc R* is a function of *PR* and *gR*; *Anpc P* is a function of *PP* and *gP*. Because *gR* and *gP* can be calculated using Eqs. (26) and (33), respectively, only *PR* and *PP* need to

**Figure 18.** *Relation between lnpc and* 1*=ρ.*

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* ¼ *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 *σ* <sup>∗</sup> <sup>1</sup> ¼ *σ*1*=σ*max = [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/mm<sup>2</sup> is used, the *ΔKwUL* = 5.06 MPa ffiffiffiffi mp is predicted from Eq. (41). Because the value of *ξ* for the present specimen is 0.167 (i.e., using *d* = 5 mm and *t* = 0.5 mm, as in Section 4.1), *F* is 0.754 [39]. In this case, the predicted value of *σ<sup>w</sup>*<sup>2</sup> is 84.7 MPa. Considering that the experimentally determined value of *σ<sup>w</sup>*<sup>2</sup> is 90 MPa, the prediction is confirmed to be good.

**Figure 19** shows the fatigue limit reliability *F<sup>σ</sup><sup>w</sup>* . 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 *F<sup>σ</sup><sup>w</sup>* for *ρ* = 0.3 mm suddenly changes from 0 to 1 when *σ<sup>w</sup>* = 84.7 MPa, which is due to *σ<sup>w</sup>*<sup>1</sup> and *σwd* being cut off by *σw*2. 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 *σ<sup>w</sup>* and 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 *ΔKwUL* of the present AC4B-T6 was unknown and

the corresponding value for the of Al-Si-X alloy was used for predicting *σw*2. It is expected that an even better prediction accuracy would be achieved by using the true *ΔKwUL:* Nevertheless, *σ<sup>w</sup>* was well predicted, which validated the proposed

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense…*

*DOI: http://dx.doi.org/10.5772/intechopen.88413*

method for notched AC4B-T6 specimens.

**Figure 19.**

**Figure 20.**

**59**

*Relation between σ<sup>w</sup> and* 1*=ρ.*

*Fatigue limit reliability F<sup>σ</sup><sup>w</sup> .*

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*

**Figure 19.** *Fatigue limit reliability F<sup>σ</sup><sup>w</sup> .*

be further examined. In this study, it is assumed that *PP* = 0.3 kgf. Considering the

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

**Figure 19** shows the fatigue limit reliability *F<sup>σ</sup><sup>w</sup>* . 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 *F<sup>σ</sup><sup>w</sup>* for *ρ* = 0.3 mm suddenly changes from 0 to 1 when *σ<sup>w</sup>* = 84.7 MPa, which is due to *σ<sup>w</sup>*<sup>1</sup> and *σwd* being cut off by *σw*2. In other words, the inhomogeneous particles have almost no effect on the fatigue limit reliability in terms of initiating a

**Figure 20** shows the relationship between *σ<sup>w</sup>* and 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 *ΔKwUL* of the present AC4B-T6 was unknown and

<sup>1</sup> ¼ *σ*1*=σ*max = [0.95, 1] at the center of the specimen was

mp is predicted from

**5.7 Comparison and examination of predicted and experimental results**

Eq. (41). Because the value of *ξ* for the present specimen is 0.167 (i.e., using *d* = 5 mm and *t* = 0.5 mm, as in Section 4.1), *F* is 0.754 [39]. In this case, the predicted value of *σ<sup>w</sup>*<sup>2</sup> is 84.7 MPa. Considering that the experimentally determined

adopted as the control volume. In this case, the region was ring-like. When *HB* = 152 kgf/mm<sup>2</sup> is used, the *ΔKwUL* = 5.06 MPa ffiffiffiffi

value of *σ<sup>w</sup>*<sup>2</sup> is 90 MPa, the prediction is confirmed to be good.

fatigue crack. Instead, the eutectic Si actually strengthens the matrix.

difficulty in evaluating *PR*, it is also assumed that *PR* ¼ *PP*.

within the range of *σ* <sup>∗</sup>

*Relation between lnpc and* 1*=ρ.*

*Fracture Mechanics Applications*

**Figure 18.**

**58**

**Figure 20.** *Relation between σ<sup>w</sup> and* 1*=ρ.*

the corresponding value for the of Al-Si-X alloy was used for predicting *σw*2. It is expected that an even better prediction accuracy would be achieved by using the true *ΔKwUL:* Nevertheless, *σ<sup>w</sup>* was 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. Rotatingbending 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 *σw*<sup>2</sup> appears as the fatigue limit when *ρ* = 0.1 and 0.3 mm. On the other hand, since the non-propagating macrocrack was not observed when *ρ* ¼ 1 and 2 mm, it can be said that the microcrack non-propagating limit *σw*<sup>1</sup> or the small macrocrack non-propagating limit *σwd* appears as the fatigue limit. The fatigue limits predicted by the present method were in good agreement with the experimental ones.

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[9] Lea V-D, Morela F, Belletta D, Saintierb N, Osmondc P. Multiaxial high

Journal of Materials Processing Technology. 2010;**210**:1249-1259

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.

## **Author details**

Tatsujiro Miyazaki<sup>1</sup> , Shigeru Hamada<sup>2</sup> and Hiroshi Noguchi<sup>2</sup> \*

1 Energy and Environment Program, School of Engineering, University of the Ryukyus, Okinawa, Japan

2 Department of Mechanical Engineering, Kyushu University, Fukuoka, Japan

\*Address all correspondence to: nogu@mech.kyushu-u.ac.jp

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*

## **References**

**6. Conclusions**

*Fracture Mechanics Applications*

**Author details**

**60**

Tatsujiro Miyazaki<sup>1</sup>

Ryukyus, Okinawa, Japan

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. Rotatingbending 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 *σw*<sup>2</sup> appears as the fatigue limit when *ρ* = 0.1 and 0.3 mm. On the other hand, since the non-propagating macrocrack was not observed when *ρ* ¼ 1 and 2 mm, it can be said that the microcrack non-propagating limit *σw*<sup>1</sup> or the small macrocrack non-propagating limit *σwd* appears as the fatigue limit. The fatigue limits predicted by the present

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

, Shigeru Hamada<sup>2</sup> and Hiroshi Noguchi<sup>2</sup>

1 Energy and Environment Program, School of Engineering, University of the

2 Department of Mechanical Engineering, Kyushu University, Fukuoka, Japan

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: nogu@mech.kyushu-u.ac.jp

provided the original work is properly cited.

\*

method were in good agreement with the experimental ones.

and can be applied to the economic development of materials.

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[8] Wang QG, Apelian D, Lados DA. Fatigue behavior of A356/357 aluminum cast alloys. Part II—Effect of microstructural constituents. Journal of Light Metals. 2001;**1**(1):85-97

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[10] Laz PJ, Hillberry BM. Fatigue life prediction from inclusion initiated cracks. International Journal of Fatigue. 1998;**20**(4):263-270

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[12] Miyazaki T, Kang H, Noguchi H, Ogi K. Prediction of high-cycle fatigue life reliability of aluminum cast alloy from statistical characteristics of defects at meso-scale. International Journal of Mechanical Sciences. 2008;**50**(2): 152-162

[13] Wang QG, Apelian D, Lados DA. Fatigue behavior of A356-T6 aluminum cast allys—Part I. Effect of casting defects. Journal of Light Metals. 2001; **1**(1):73-84

[14] Yia JZ, Leea PD, Lindleya TC, Fukuib T. Statistical modeling of microstructure and defect population effects on the fatigue performance of cast A356-T6 automotive components. Materials Science and Engineering A. 2006;**432**(1):59-68

[15] Gao YX, Yi JZ, Lee PD, Lindley TC. The effect of porosity on the fatigue life of cast aluminum-silicon alloys. Fatigue & Fracture of Engineering Materials & Structures. 2004;**27**:559-570

[16] JSME. JSME S 002-1994, Standard Method of Statistical Fatigue Testing. Tokyo, Japan: Japanese Society of

Mechanical Engineering; 1994. (in Japanese)

[17] ISO. ISO12107: Metallic Materials-Fatigue Testing-Statistical Planning and Analysis of Data. Geneva, Switzerland: International Organization for Standardization; 2012

[18] Murakami Y, Endo M. Effects of defects, inclusions and inhomogeneities on fatigue strength. International Journal of Fatigue. 1994;**16**:163-182

[19] Murakami Y, Tazunoki Y, Endo T. Existence of the coaxing effect and effects of small artificial holes on fatigue strength of an aluminum alloy and 70-30 brass. Metallurgical and Materials Transactions A. 1984;**15**(11):2029-2038

[20] Murakami Y. Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions. Oxford: Elsevier Science Ltd.; 2002

[21] Kobayashi M, Mutsui T. Prediction of fatigue strength of aluminum casting alloys by the √area parameter model. Transactions of the Japan Society of Mechanical Engineers, Series A. 1996; **62**(594):341-346. (in Japanese)

[22] Ceschini L, Morri A, Morri A. Estimation of local fatigue behaviour in A356-T6 gravity die cast engine head based on solidification defects content. International Journal of Cast Metals Research. 2014;**27**(1):56-64

[23] Tajiria A, Nozakib T, Uematsub Y, Kakiuchib T, Nakajimac M, Nakamurac Y, et al. Fatigue limit prediction of large scale cast aluminum alloy A356. Procedia Materials Science. 2014;**3**: 924-929

[24] Roya MJ, Nadotb Y, Nadot-Martinb C, Bardinb P-G, Maijera DM. Multiaxial Kitagawa analysis of A356-T6. International Journal of Fatigue. 2011; **33**(6):823-832

[25] Miyazaki T, Noguchi H, Kage M, Imai R. Estimation for fatigue limit reliability of a metal with inhomogeneities under stress ratio *R* ¼ �1. International Journal of Mechanical Sciences. 2005;**47**(2): 230-250

limit of a metal with an arbitrary crack under a stress controlled condition (stress ratio *R* ¼ �1). International Journal of Fracture. 2004;**129**:21-38

*DOI: http://dx.doi.org/10.5772/intechopen.88413*

*Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense…*

[34] Miyazaki T, Noguchi H, Kage M. Fatigue limit of steel with an arbitrary crack under a stress controlled constant

International Journal of Fracture. 2005;

[35] Miyazaki T, Noguchi H, Kage M. Prediction of fatigue limit reliability of high strength steel with deep notch under mean stress σ<sup>m</sup> = 0. International Journal of Fracture. 2011;**168**(1):73-91

[36] Miyazaki T, Noguchi H, Ogi K, Aono Y. Examination of fatigue characteristics of Aluminum cast alloy from meso-level consideration (2nd report, prediction for the fatigue limit reliability of plain specimen of metal

[37] Hashimoto A, Miyazaki T, Noguchi H, Ogi K. Estimation of particle size distribution in materials in the case of spheroidal particles using quantitative microscopy. Journal of Testing and Evaluation. 2000;**28**(5):367-377

[38] Nisitani H. Method of approximate calculation of interference of notch effects and its application. Bulletin of the JSME. 1968;**11**(47):725-738

[39] Benthem JP, Koiter WT. Asymptotic approximations to crack problems. In: Method of Analysis and Solutions of Crack Problems. Leyden: Noordhoff International; 1973. pp. 131-178

**63**

containing different sorts of inhomogeneities under *R* = �1). Transactions of the Japan Society of Mechanical Engineers, Series A. 2005; **71**(712):1699-1707. (in Japanese)

with a positive mean stress.

**134**:109-126

[26] Isibasi T. Prevention of Fatigue and Fracture of Metals. Tokyo: Yokendo Ltd.; 1967. (in Japanese)

[27] Nisitani H. Effect of size on the fatigue limit and the branch point in rotary bending tests of carbon steel specimen. Bulletin of the JSME. 1968; **11**(47):725-738

[28] Nisitani H, Goto T. Fatigue notch sensitivity of an aluminum alloy. Transaction of the Japan Society of Mechanical Engineers. 1976;**42**(361): 2666-2672. (in Japanese)

[29] Nisitani H, Kawagoishi N. Comparison of notch sensitivities in three age-hardened aluminum alloys. Transaction of the Japan Society of Mechanical Engineers, Series A. 1985; **51**(462):530-533. (in Japanese)

[30] Takao K, Nisitani H, Sakaguchi H. Relation between crack initiation process and notch sensitivity in rotating bending fatigue. Journal of the Society of Materials Science, Japan. 1980; **29**(325):982-987. (in Japanese)

[31] Kawagoishi N, Nisitani H, Tsuno T. Notch sensitivity of squeeze cast aluminum alloy in rotating bending fatigue. Transaction of the Japan Society of Mechanical Engineers, Series A. 1990; **56**(521):10-14. (in Japanese)

[32] Dowling NE. Notched member fatigue life predictions combining crack initiation and propagation. Fatigue & Fracture of Engineering Materials & Structures. 1979;**2**(2):129-138

[33] Miyazaki T, Noguchi H, Ogi K. Quantitative evaluation of the fatigue *Fatigue Limit Reliability Analysis for Notched Material with Some Kinds of Dense… DOI: http://dx.doi.org/10.5772/intechopen.88413*

limit of a metal with an arbitrary crack under a stress controlled condition (stress ratio *R* ¼ �1). International Journal of Fracture. 2004;**129**:21-38

Mechanical Engineering; 1994. (in

*Fracture Mechanics Applications*

International Organization for

Standardization; 2012

[17] ISO. ISO12107: Metallic Materials-Fatigue Testing-Statistical Planning and Analysis of Data. Geneva, Switzerland:

[25] Miyazaki T, Noguchi H, Kage M, Imai R. Estimation for fatigue limit

inhomogeneities under stress ratio *R* ¼ �1. International Journal of Mechanical Sciences. 2005;**47**(2):

[26] Isibasi T. Prevention of Fatigue and Fracture of Metals. Tokyo: Yokendo

[27] Nisitani H. Effect of size on the fatigue limit and the branch point in rotary bending tests of carbon steel specimen. Bulletin of the JSME. 1968;

[28] Nisitani H, Goto T. Fatigue notch sensitivity of an aluminum alloy. Transaction of the Japan Society of Mechanical Engineers. 1976;**42**(361):

[30] Takao K, Nisitani H, Sakaguchi H. Relation between crack initiation process and notch sensitivity in rotating bending fatigue. Journal of the Society of Materials Science, Japan. 1980; **29**(325):982-987. (in Japanese)

[31] Kawagoishi N, Nisitani H, Tsuno T. Notch sensitivity of squeeze cast aluminum alloy in rotating bending fatigue. Transaction of the Japan Society of Mechanical Engineers, Series A. 1990;

**56**(521):10-14. (in Japanese)

[32] Dowling NE. Notched member fatigue life predictions combining crack initiation and propagation. Fatigue & Fracture of Engineering Materials & Structures. 1979;**2**(2):129-138

[33] Miyazaki T, Noguchi H, Ogi K. Quantitative evaluation of the fatigue

reliability of a metal with

Ltd.; 1967. (in Japanese)

2666-2672. (in Japanese)

[29] Nisitani H, Kawagoishi N. Comparison of notch sensitivities in three age-hardened aluminum alloys. Transaction of the Japan Society of Mechanical Engineers, Series A. 1985; **51**(462):530-533. (in Japanese)

**11**(47):725-738

230-250

[18] Murakami Y, Endo M. Effects of defects, inclusions and inhomogeneities on fatigue strength. International Journal of Fatigue. 1994;**16**:163-182

[19] Murakami Y, Tazunoki Y, Endo T. Existence of the coaxing effect and effects of small artificial holes on fatigue strength of an aluminum alloy and 70-30 brass. Metallurgical and Materials Transactions A. 1984;**15**(11):2029-2038

[20] Murakami Y. Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions. Oxford: Elsevier Science

[21] Kobayashi M, Mutsui T. Prediction of fatigue strength of aluminum casting alloys by the √area parameter model. Transactions of the Japan Society of Mechanical Engineers, Series A. 1996; **62**(594):341-346. (in Japanese)

[22] Ceschini L, Morri A, Morri A. Estimation of local fatigue behaviour in A356-T6 gravity die cast engine head based on solidification defects content. International Journal of Cast Metals

[23] Tajiria A, Nozakib T, Uematsub Y, Kakiuchib T, Nakajimac M, Nakamurac Y, et al. Fatigue limit prediction of large

[24] Roya MJ, Nadotb Y, Nadot-Martinb C, Bardinb P-G, Maijera DM. Multiaxial

International Journal of Fatigue. 2011;

Research. 2014;**27**(1):56-64

scale cast aluminum alloy A356. Procedia Materials Science. 2014;**3**:

Kitagawa analysis of A356-T6.

Japanese)

Ltd.; 2002

924-929

**33**(6):823-832

**62**

[34] Miyazaki T, Noguchi H, Kage M. Fatigue limit of steel with an arbitrary crack under a stress controlled constant with a positive mean stress. International Journal of Fracture. 2005; **134**:109-126

[35] Miyazaki T, Noguchi H, Kage M. Prediction of fatigue limit reliability of high strength steel with deep notch under mean stress σ<sup>m</sup> = 0. International Journal of Fracture. 2011;**168**(1):73-91

[36] Miyazaki T, Noguchi H, Ogi K, Aono Y. Examination of fatigue characteristics of Aluminum cast alloy from meso-level consideration (2nd report, prediction for the fatigue limit reliability of plain specimen of metal containing different sorts of inhomogeneities under *R* = �1). Transactions of the Japan Society of Mechanical Engineers, Series A. 2005; **71**(712):1699-1707. (in Japanese)

[37] Hashimoto A, Miyazaki T, Noguchi H, Ogi K. Estimation of particle size distribution in materials in the case of spheroidal particles using quantitative microscopy. Journal of Testing and Evaluation. 2000;**28**(5):367-377

[38] Nisitani H. Method of approximate calculation of interference of notch effects and its application. Bulletin of the JSME. 1968;**11**(47):725-738

[39] Benthem JP, Koiter WT. Asymptotic approximations to crack problems. In: Method of Analysis and Solutions of Crack Problems. Leyden: Noordhoff International; 1973. pp. 131-178

**Chapter 4**

**Abstract**

(0o

Composites

), mixed-mode I/II (30o

**1. Introduction**

**65**

, 45o

the narrowest pattern width for higher resistance to fracture.

, and 60o

the J-integral method was used to determine the fracture toughness. Experimental and numerical results were found to be consistent. When the results obtained from pure and hybrid fabrics are compared, it is seen that hybridization had positive effects on the fracture strength of composite material compared to pure glass/epoxy material. Additionally, as the width of the pattern decreased, the fracture strength of the hybrid composites increased. In this respect, the hybridization processing should be done in

**Keywords:** fracture toughness, strain energy release rate, Arcan fracture test, pure and hybrid laminated composite, knitted fabric, J-integral method

High strength to low weight ratio is a sought-after feature in the materials used in the structural elements of today's world. With the technological advances in recent years, composite materials are used in many industries, where durability and lightness are at the forefront, especially from the aerospace to automotive sectors. Polymeric composites have been used in many engineering applications due to their high strength in proportion to their weight, high stability, rigidity, superior corrosion, and fatigue resistance [1–3]. Woven or knitted fabrics of durable synthetic fibers such as glass, carbon, or aramid are used to reinforce polymer matrix

The Fracture Behavior of Pure

and Hybrid Intraply Knitted

Fabric-Reinforced Polymer

*Huseyin Ersen Balcioglu and Hayri Baytan Ozmen*

Due to the high synergistic effects of the components, hybrid composite materials are more advantageous than nonhybrid composite materials for advanced engineering applications. Additionally, knitted fabrics may have a different behavior than woven ones. Although the nonhybrid composites have only one reinforcing fiber type, the hybrid composites have multiple reinforcing fibers. In this chapter, fracture characterizations of laminated composites reinforced with intraply pure and hybrid knitted fabrics are experimentally and numerically investigated under different loading conditions. For this purpose, pure (100%) and hybrid fabrics (50–50%), which have 1 1 rib-knitted structure, were knitted by using glass and carbon fibers. Also, hybrid fabrics were knitted in three different widths in order to investigate the effect of knitting pattern width on the fracture toughness. Fracture toughness and energy strain release rates of pure and hybrid Arcan test specimens were determined under mode I

), and mode II (90o

) loading conditions. Also,

## **Chapter 4**
