Area of isostress near the notch root.

## Abstract

This study proposes a quantitative method for predicting fatigue limit reliability of a notched metal containing inhomogeneities. Since the fatigue fracture origin of the notched metal cannot be determined in advance because of stress nonuniformity, randomly distributed particles, and scatter of a matrix, it is difficult to predict the fatigue limit. The present method utilizes a stress-strength model incorporating the “statistical hardness characteristics of a matrix under small indentation loads” and the “statistical hardness characteristics required for non-propagation of fatigue cracks from microstructural defects”. The notch root is subdivided into small elements to eliminate the stress nonuniformity. The fatigue limit reliability is predicted by unifying the survival rates of the elements obtained by the stress-strength model according to the weakest link model. The method is applied to notched specimens of aluminum cast alloy JIS AC4B-T6 containing eutectic Si, Fe compounds and porosity. The fatigue strength reliability at 107 cycles, which corresponds to the fatigue limit reliability, is predicted. The fatigue limits of notch root radius ρ = 2, 1, 0.3, and 0.1 mm are obtained by rotating-bending fatigue tests. It is shown that the fatigue limits predicted by the present method are in good agreement with the experimental ones.

### Keywords

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

## 1. Introduction

Aluminum cast alloys are widely applied, for example, in motor vehicles, ships, aircraft, machines, and structures, owing to the high cast ability and high specific strength [1, 2, 3]. They can be improved so as to meet specific mechanical properties by tuning the casting method, the alloying elements, and the cooling and heat treatment conditions [4, 5, 6]. Generally, precipitation hardening, also called age-hardening, is used to strengthen the aluminum cast alloys, which brings the dense precipitate of particles such as eutectic Si. The precipitations form fine microstructures such as dendrites, which significantly improve the mechanical properties. However, the resultant stress concentrations by the precipitations further to fatigue fracture unfortunately [7, 8, 9]. Moreover, the possibility of the fatigue fracture increases more and more if microstructural flaws such as porosity are created in the casting process [10, 11, 12, 13, 14, 15]. Because the precipitate particles and the microstructural defects are unique, the fatigue strength of the aluminum cast alloys is obliged to treat statistically.

Statistical fatigue test methods [16, 17] are standardized to determine the reliability of the fatigue strength. However, because they require many fatigue tests, it is time-consuming to determine the fatigue strength reliability at 10^{7} 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^{7} stress cycles, which are equivalent to the statistically determined fatigue limit of aluminum cast alloys, have been proposed [18, 19, 20, 21, 22, 23, 24]. Through a series of stress analyses and fatigue experiments, Murakami et al. [18, 19, 20] clarified the non-propagation limit of a fatigue crack initiated by a microstructural defect and proposed a simple formula for predicting the fatigue limit of a plain specimen containing defects [18, 19, 20]. The non-propagation limit of a fatigue crack initiated by microstructural defect is determined by the defect size and mechanical characteristics of the matrix near the defect. The maximum defect, which is often estimated by extreme statistics, is therefore assumed to be the origin of the fatigue fracture. Most of the methods are based on the assumption that fatigue fracture begins at the maximum defects, and they often do not consider the interference effects of inhomogeneities and the scatter of the hardness of the matrix [25]. Because aluminum cast alloys have much higher densities of inhomogeneities, it is presumed that the interference effect is not negligible and the maximum inhomogeneity is not in the severest mechanical state necessarily. Additionally, in the case of a notched specimen, the stress varies significantly. The most severe mechanical defect should be used for prediction, even if it is not maximal. Generally, the fatigue limit of a notched specimen of a homogeneous metal in which microstructural defect is not the origin of the fatigue fracture consists of the microcrack and macrocrack non-propagation limits [26, 27, 28, 29, 30, 31, 32]. This fact is widely used in predicting fatigue limit. However, since microstructural defects act as crack initiation sites, the fatigue limit of an inhomogeneous metal also cannot be predicted by these two types of crack non-propagation limits.

In this study, a quantitative method for predicting the fatigue limit reliability of a notched metal containing inhomogeneous particles is proposed. The present method is also based on the stress-strength model and is applied to notched specimens of an Al-Si-Cu alloy (JIS AC4B). The inhomogeneous particle in the alloy comprises eutectic Si and Fe compounds and porosity in the matrix. Rotating-bending fatigue tests are performed on the notched specimens of AC4B-T6 by changing notch root radius variously. The validity of the present method is examined by comparing its numerical prediction with experimental results.

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

notch depth | |

notch root radius | |

branch point | |

limit notch root radius | |

fatigue limit of notched specimen | |

fatigue limit of plain specimen | |

microcrack non-propagation limit | |

long macrocrack non-propagation limit | |

small macrocrack non-propagation limit |

Generally, when fatigue tests are performed on a notched specimen by changing the notch root radius

If

In the case of metals containing microstructural defects, the non-propagation limit of the fatigue crack that originates from the microstructural defect may be the fatigue limit. Because the defect is categorized as a macrocrack, the low macrocrack non-propagation limit is differentiated from

This is the non-propagation limit of a microcrack that is initiated by repeated irreversible plastic strains in a homogeneous notch stress field without microstructural and structural stress concentrations | |

This is the non-propagation limit of a three-dimensional fatigue crack that originates from microstructural defects such as nonmetallic inclusions and pits in a homogeneous notch stress field without other microstructural and structural stress concentrations | |

This is the non-propagation limit of structural long macrocracks such as deep notches with |

Figure 2 is a schematic illustration of the relationships between

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,

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

Nomenclature | |

size of | |

size of | |

region required for the non-propagation of fatigue crack | |

size of surface defect | |

lower limit size of small surface crack | |

size of internal defect | |

geometric correction factor | |

fatigue limit reliability of notched specimen | |

limit hardness | |

stress relaxation effect | |

Vickers hardness | |

stress intensity factor | |

stress concentration factor | |

threshold stress intensity factor range | |

lower limit value of | |

upper limit value of | |

the number of surface cracks with | |

the number of particles with | |

types of inhomogeneous particles | |

the number of particles in a unit volume | |

the number of surface elements | |

the number of solid elements | |

indentation load | |

existence probability of particles with | |

limit size of small interior crack | |

survival rate of notched specimen | |

survival rate of surface element with microcracks | |

survival rate of solid element with microstructural defects | |

population of | |

size of | |

material constant | |

mean of | |

first principal stress | |

relative first principal stress | |

stress produced by the spherical particle in the infinite body under | |

mean stress | |

stress amplitude | |

stress gradient of first principal stress |

This section presents a method for predicting the fatigue limit reliability of a notched specimen with stress concentration factor

### 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]:

Here,

The total number of particles in a unit volume is denoted by

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

Here,

#### 3.1.2 Average radius and distance

The average particle radius is evaluated by the following equation:

If

#### 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,

Here,

The stress equilibrium condition in the

Here,

#### 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

1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|

1–0.95 | 0.95–0.9 | 0.9–0.8 | 0.8–0.7 | 0.7–0.6 | 0.6–0.5 | |

0.463 | 0.216 | 0.361 | 0.358 | 0.402 | 0.496 | |

0.0083 | 0.0181 | 0.0703 | 0.144 | 0.296 | 0.657 |

To predict the fatigue limit reliability, the control volume is set at the notch root and divided into surface and solid elements. The sizes of the solid and surface elements are denoted by

### 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

If

Here,

Based on the central limit theorem, the relationship between the sample mean

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

The microcrack non-propagation limit

(^{2}, and

If the stress relaxation effect

Here,

It is supposed that

Here,

The fatigue survival rate

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,

Here,

### 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.

#### 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

The stress intensity factor

Moreover, the threshold stress intensity factor range

(^{2}, and

The limit hardness that determines whether the fatigue crack originating from the interior microstructural crack is arrested,

(^{2}, and

The relationship between

Moreover, the relationship between

Here,

The fatigue survival rate of

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

Here,

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

The stress intensity factor

Further, the limit hardness

(^{2}, and

The fatigue survival rate of

The fatigue survival rate

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

The fatigue survival rate

Because the material contains

### 3.4 Prediction of long macrocrack non-propagation limit σ w 2

### 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

## 4. Fatigue experiment

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

The material used for the experiment was Al-Si-Cu alloy (JIS AC4B). The age-hardened aluminum cast alloy is identified as AC4B-T6. Table 2 shows its mechanical properties.

9.8 N, 30 sec | 29.4 mN, 30 sec | ||||
---|---|---|---|---|---|

74 | 292 | 349 | 1.5 | 152 | 92 |

^{2}) | |||||

^{2}) |

Figure 6 shows the configurations of the specimens. The notch depth

### 4.2 Experimental results

Figure 7 shows

## 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

(^{2}.)

Once

### 5.2 ΔK wUL of age-hardened aluminum alloy

Figure 11 shows the values of

(^{2}.)

Here,

### 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

Figure 13 shows the measured

Here,

The line of Eq. (44) is drawn to best fit the

Figure 14 shows the values of

### 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 ^{2} and 486.0 (kgf/mm^{2})^{2}, respectively.

### 5.5 Evaluation of γ m value

Because the values of

Inhomogeneous particle | ^{3}] | |||||
---|---|---|---|---|---|---|

Eutectic Si | 8.73 × 10^{6} | 1.6 | 1.04 | 0.932 | 4.86 | |

Fe compound | 2.20 × 10^{7} | 0.5 | 0.10 | 0.200 | 3.57 | |

Porosity | ( | 1.20 × 10^{2} | 0.3 | 0.180 | 0.167 | 21.5 |

( | 1.00 × 10^{5} | 0.3 | 0.0180 |

### 5.6 Evaluation of A npc S , A npc R , and A npc P values

#### 5.6.1 Evaluation of A npc S value

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

(^{2}

Conversely, when the macrocrack is sufficiently large,

(^{2}, and

^{2}, the microcrack non-propagation limit

#### 5.6.2 Evaluation of A npc R and A npc P values

### 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

When ^{2} is used, the

Figure 19 shows the fatigue limit reliability

Figure 20 shows the relationship between

## 6. Conclusions

This study proposed a nondestructive method for predicting the fatigue limit reliability of notched specimens of a metal containing inhomogeneous particles densely. The method was applied to aluminum cast alloy JIS-AC4B-T6. Rotating-bending fatigue tests were performed on the notched specimens of AC4B-T6 with notch root radius

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.