*4.1. The Weibull Statistical Analysis on the Failure Probability of Al–Cu Joints*

The failure of an engineering component, as defined by reliability engineering, can be recognized as "the event, or inoperable state, in which any item or part of an item does not, or would not, perform as previously specified" [42]. The advantage of Weibull statistical analysis for reliability engineering is to effectively evaluate failure probability and provide reasonable failure predictions for engineering components from the stages of design and manufacturing processes with extremely small amount of testing samples. In this study, the cumulative failure probability function (Equation (1)) determined by the three-parameter Weibull model is applied to simulate the data variability of the measured tensile strength of FSW Al–Cu dissimilar joints. The cumulative failure probability *F*(*σi*) of the testing samples is estimated using Bernard's median rank [43], and the reliability function *R*(*σi*) with a relation of *R*(*σi*)=1 − *F*(*σi*) is determined as the survival probability. The cumulative failure probability is controlled by parameters *m*, *σc*, and *σ*0. The Weibull modulus (*m*) is a measure of data variability. The characteristic life (*σc*) corresponds to the tensile strength at which the cumulative failure probability is equal to 63.2%. The minimum strength (*σ*0) is so-called the failure-free strength, which means that the failure probability of the FSW Al–Cu dissimilar joints below this tensile strength is zero.

$$F(\sigma\_i) = \int\_{\sigma=0}^{\sigma=\sigma\_i} f(\sigma)d\sigma = 1 - \exp\left[-\left(\frac{\sigma\_i - \sigma\_0}{\sigma\_c}\right)^m\right] \tag{1}$$

Fitting the measured tensile strength data into the cumulative failure probability function of Equation (1), and subsequently the failure probability density function (*f*(*σi*)) of FSW A1/C1, A6/C1, and A1/C2 dissimilar joints are calculated and plotted in Figure 7a. Figure 7b illustrates the Weibull plot, which is a natural logarithmic (ln) graph for the cumulative failure probability at each corresponding *σ<sup>i</sup>* of various FSW Al–Cu dissimilar joints. The horizontal scale of the Weibull plot is a measure of tensile strength, and the vertical scale is the cumulative percentage failed. The Weibull modulus, which is particularly significant and provides a clue to the physics of the failure, is graphically evaluated from the slope of Weibull plots by the least squares fitting method at a maximum coefficient of determination (*R*2). According to the definition of the Weibull model, the critical coefficient of determination (CCD) for 20 failures (*n* = 20) should be higher than 0.95, and then the Weibull plot

can be considered a good fit for the three-parameter Weibull cumulative distribution function [44]. Therefore, it is recognized that a good linear relationship is obtained for the experimental data in Figure 7b, and the Weibull model is valid to describe the failure behaviors of the FSW Al–Cu joints. The Weibull statistical analysis results are listed in Table 3.

**Figure 7.** (**a**) The failure probability density function *f*(*σi*) curves, and (**b**) the Weibull distribution plots of various FSW Al–Cu dissimilar joints. *F*(*σi*) is the cumulative failure probability for a corresponding tensile strength (*σi*), and the slope represents the Weibull modulus (*m*) calculated by the least squares fitting method at a maximum coefficient of determination (*R*2).

**Table 3.** Statistical analysis results \* of the Weibull model for the tensile strength of various FSW A1–Cu dissimilar joints.


\* Data were calculated from Weibull plots (Figure 7b).
