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## Mathematical Methods and Modeling in Machine Fault Diagnosis

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Research Article | Open Access

Volume 2014 |Article ID 713678 | https://doi.org/10.1155/2014/713678

Shan Jiang, Wei Zhang, Xiaoyang Li, Fuqiang Sun, "An Analytical Model for Fatigue Crack Propagation Prediction with Overload Effect", Mathematical Problems in Engineering, vol. 2014, Article ID 713678, 9 pages, 2014. https://doi.org/10.1155/2014/713678

# An Analytical Model for Fatigue Crack Propagation Prediction with Overload Effect

Revised22 Jun 2014
Accepted23 Jun 2014
Published23 Jul 2014

#### Abstract

In this paper a theoretical model was developed to predict the fatigue crack growth behavior under the constant amplitude loading with single overload. In the proposed model, crack growth retardation was accounted for by using crack closure and plastic zone. The virtual crack annealing model modified by Bauschinger effect was used to calculate the crack closure level in the outside of retardation effect region. And the Dugdale plastic zone model was employed to estimate the size of retardation effect region. A sophisticated equation was developed to calculate the crack closure variation during the retardation area. Model validation was performed in D16 aluminum alloy and 350WT steel specimens subjected to constant amplitude load with single or multiple overloads. The predictions of the proposed model were contrasted with experimental data, and fairly good agreements were observed.

#### 1. Introduction

The paper is organized as follows. First, the fatigue crack growth model and the modified virtual crack annealing crack closure model are briefly reviewed. Second, our proposed model for crack growth retardation behavior estimation is discussed in detail. The model is derived based on plastic zone and crack closure variation. And then model validation is performed using the experimental data in D16 aluminum alloy and 350WT steel from the literature. Finally, some conclusions and future work are given based on the current study.

#### 2. Methodology Development

A general fatigue crack growth model can be expressed as (1) by Wolf [4]. The fatigue crack growth rate is determined by the effective stress intensity factor range where is the crack growth rate, is the effective stress intensity factor, is the stress intensity factor of the peak load, and is the crack closure level; , are calibration parameters.

In this paper the virtual crack annealing model is employed to avoid considering the complex contact of crack closure [7]. This analytical closure model is based on plasticity ahead of crack tip, but it does not consider the Bauschinger effect. So a brief derivation of the modified closure model is discussed in the following part.

Since the crack overlapping length is very small compared with the true crack length (i.e., ), its effect on the stress intensity factor calculation can be ignored. Equation (2) can be rewritten as where is equal to . Equation (3) is the general solution using the virtual crack annealing model for the crack opening stress calculation under constant amplitude loadings. Equation (3) can be further simplified as

There are two possible solutions for the opening stress level under the proposed virtual crack annealing model. One solution is , which indicates that there is no crack closure and the overlapped length is zero. The other solution is . This observation indicates that either there is no crack closure or there is a unique crack closure level under constant amplitude loadings. In this paper, the modified virtual crack annealing model is used to make predictions of the fatigue crack growth behavior in the outside of retardation effect region.

The crack growth behavior is shown in Figure 3. After the overload, the crack growth rate increases transitorily and then decreases sharply. Once the value reaches to minimum, crack growth rate increases gradually and finally recovers to the equilibrium constant amplitude loading growth rate.

The crack tip plastic zone associated with the application of a single overload is shown in Figure 4(a). A new approach to estimation of crack opening stress under the influence of single overload is proposed, as shown in Figure 4(b). When the crack grows to point , a single overload stress is applied and a large plastic zone appears ahead of the crack tip. As the crack penetrates into this plastic zone and grows to point , the retardation effect gradually recedes until it vanishes.

A function of and the crack length is established to calculate the crack closure variation during the retardation region, which is shown in Figure 4(b). The points and in this figure, respectively, represent the situation of crack growing to points and in Figure 4(a). Equation (5) can be obtained based on the modified virtual crack annealing model above: where is Bauschinger effect factor, is the load ratio, and is the overload ratio. The left side of (5) is the ordinate of point . Similarly, the ordinate of point can be illustrated by the following:

The horizontal ordinate of point is where is the horizontal ordinate of point , which is the crack length when single overload is applied. is the diameter of plastic zone created by that single overload. If point is considered to be the ordinate origin, the equation of the curve between points and is where and are, respectively, the ordinate of points and and is the horizontal ordinate of point . When the value of can be 2, 1, and 0.5, the curve is, respectively, shown as (A), (B), and (C) in Figure 4. Based on the current study, is a good approximation for aluminum alloy. If the is considered to be the ordinate origin, (7) can be rewritten as

Based on the discussion above, the new crack closure model is established as (10), which is used to describe fatigue crack propagation behavior under single overload interspersed in a constant amplitude loading spectrum. Consider

The cycle by cycle algorithm is used to implement the above model. The flow chart for fatigue crack growth prediction based on the modified crack closure model is shown in Figure 5.

#### 3. Experimental Validation

The experimental data for model validation are from the literature [27, 28]. The material used in both papers was D16 aluminum alloy. The standard chemical composition is shown in Table 1. And the mechanical properties of this material in longitudinal (LT) orientation were as follows:  MPa,  MPam0.5, and percentage elongation = 12%.

 Chemical composition of D16 aluminum alloy Element Cu Mg Mn Si Fe Zn Weight % 3.8–4.9 1.2–1.8 0.3–0.9 0.5 0.5 0.3

Before any predictions can be made, there are several unknown parameters (see (1) and (10)) that need to be calibrated. Three sets of testing data are used to evaluate these parameters (Figure 6), which are assumed to depend on the material only [27]. The calibrated parameters are and , and Bauschinger effect factor is 0.97.

Then additional two sets of - data are used to validate the proposed model. The specimen configuration for these tests is [27] width = 100 mm, length = 500 mm, and thickness = 4 mm. Initial half crack length is  mm. The specimen was subjected to a constant amplitude load sequence with  MPa, . And the overload ratio is 2.

The model predictions are compared to the experimental data in Figure 7. The -axis is the number of cycles, and the -axis is crack length. The red circlets represent the fatigue crack growth under the constant amplitude loading; and the small blue squares are the counterpart under the constant amplitude load with single overload. It is observed that the predictions by proposed model match the testing data well.

More detailed information can be obtained in the curve in Figure 8. Right after the single overload, the fatigue crack growth rate increases in a short time and then decreases sharply. After a certain crack length, the retardation phenomenon vanishes and the crack growth rate turns back to the original trend. Note that very good agreement is observed between the model prediction and experimental data.

In this section, the model validation will extend to 350WT steel. The experimental data are given by Taheri et al. [29]. The specimen dimensions are as follows: length = 300 mm, width = 100 mm, and thickness = 5 mm. Initial center crack length is  mm. Recording test data was started at crack length of 11.17 mm. In addition, the mechanical properties of 350WT in longitudinal (LT) orientation are as follows:  MPa,  MPa, Modulus of elasticity = 205 GPa, and Poisson’s ration = 0.30 [30].

Similarly, the parameters are calibrated by the data in the literature [19], as shown in Figure 10. The fitting coefficients are and , and the Bauschinger effect factor is 1.05.

Figure 11 shows the comparison between the predicted - curve and the experimental data in 350WT steel specimens. The single overloads are applied at the crack length of 15 and 25 mm. It can be seen that there are two obvious retarded regions right after the overload in the experimental data. The prediction follows almost the same trend as the experimental observation. Predicted total fatigue lives are slightly lower than the experimental values.

In Figure 12, crack growth rate variation caused by overloads is also compared with our model prediction. It is clear that the prediction matches the experimental data very well, especially in the retardation region. Hence, it is proven that the method proposed in this paper is applicable to predict the fatigue crack growth behavior under the constant loading with single overloads in 350WT steel.

#### Nomenclature

 : Crack length : Crack growth in one cycle : Infinitesimal crack increment : Fatigue crack growth rate per cycle , : Minimum and maximum stress in one loading cycle : Stress level at which the crack begins to grow : Stress level of single overload : Stress ratio : Overload ratio , : Maximum/minimum stress intensity factor : Stress intensity factor range : Stress intensity factor at which the crack begins to grow : Effective stress intensity factor range : Monotonic plastic zone size : Forward plastic zone size , : Reverse plastic zone size : Material yield strength : Bauschinger effect factor in loading process : Bauschinger effect factor in unloading process : Bauschinger effect factor.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The research is financially supported by the specialized research fund for the doctoral program of higher education funding under the contract no. 20131102120047.

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