Prediction Modeling

To characterize the reliability model as a function of dwelling, the relationship between characteristic life and stress amplitudes for joints cycled with various dwellings are plotted as shown in Figure 13, where data are fitted to power equations for each dwelling period. A decreasing trend for the power value (the material ductility exponent) is identified when the dwell time is increased. However, the constant C was observed to decrease with dwelling. The R-square values are above 99% for all fitting lines.

**Figure 13.** Characteristic life as a function of stress amplitude for SAC305 solder joints at various dwelling times.

To predict the reliability as a function of dwell time and stress amplitude, the correlation between *n* and *C* values as a function of dwell time (*td*) must be identified. The correlations to predict the power values of *n* and *C* value as a function of dwell time are shown in Figures 14 and 15. From Equation (2), the characteristic life is predicted as a function of stress amplitude and dwell time as shown in Equation (3).

$$N\_{63} = 5 \times 10^8 \times e^{(-0.03t\_d)} \times P^{-(-0.0075t\_d + 4.5188)} \tag{3}$$

where *N*63 is the characteristic life, *P* is the stress amplitude, and *td* is the dwell time. To find out the general reliability model of solder joints as a function of dwell time, parameters in Equation (1) must be determined. In our case, there is no observed trend for the shape

parameter of the Weibull plot at different dwell times and stress amplitudes. The shape parameter values were found between 3.5 and 14 with an average of 6.66. For the scale parameter (*θ*); the finding of characteristic life from Equation (3) is substituted in Equation (1). As a result, the general reliability model as a function of dwell time is established as shown in Equation (4).

*t*

 )6.66

(4)

<sup>5</sup>×108×*e*(−0.03×*td*)×*<sup>P</sup>*(0.0075×*td*+4.5188)

 = *e* −(

**Figure 14.** Power value (*n*) as a function of dwell time.

**Figure 15.** Constant (*C*) as a function of dwell time.

## *3.2. Stress–Strain Analysis*

Hysteresis loops or stress–strain loops are essential to determine the damage parameters for each cycle represented by the inelastic work and plastic strain per cycle. These parameters are directly related to the accumulated damage. The loop area represents the inelastic (damage) work per cycle, where its width along the *x*-axis represents plastic strain. Figure 16 shows the average hysteresis loops for joints cycled with various stress levels at no dwelling. Both inelastic work (area inside the loop) and the plastic strain are obviously increased with higher stress amplitude. In Figure 17, average hysteresis loops are generated for 10 s of dwelling periods. At a specific dwelling, the same trend is observed; the hysteresis loop is enlarged drastically at higher stress levels. Moreover, more stress–relaxation is noticed with higher stress magnitudes. The enlargement of hysteresis loops is due to a massive increase in acculturated damaged work due to the creep effect in addition to fatigue damage. Consequently, evolution in hysteresis loops is generated for various dwelling times at certain stress levels of 24 MPa, as shown in Figure 18. Similar behavior is noticed for the plastic strain parameter. The higher stress or more prolonged dwelling would cause more plastic strain to be accumulated at a particular dwelling or stress levels, respectively.

**Figure 16.** The hysteresis loops for SAC305 joints cycled with different stress amplitudes at no dwelling.

**Figure 17.** The hysteresis loops for SAC305 joints cycled with different stress amplitudes at 10 s dwelling.

**Figure 18.** The hysteresis loops for SAC305 joints cycled at various dwellings with 24 MPa stress level.

The evolution in the hysteresis loop is directly related to the damage parameters, inelastic work, and plastic strain per cycle. Therefore, it is essential to examine the progression of such parameters during the joint lifetime to explain the evolvement of stress–strain loops. Figure 19 represents the typical development of inelastic work during the joint's life. This would provide an understanding of such behavior. The evolution is divided into three

regions; the first region is the strain hardening, and it lasts for a few cycles. The second region includes the constant or steady-state region, which represents most of the joint's life. The last phase is the crack initiation and propagation. The same behavior is noticed for all combinations of dwellings and stress levels, as shown in Figures 20 and 21.

**Figure 19.** Inelastic work vs. the number of cycles for SAC305 solder joints cycled at 16 MPa stress amplitude until failure.

**Figure 20.** Inelastic work vs. the number of cycles for SAC305 solder joints cycled at various stress amplitudes at no dwelling until failure.

**Figure 21.** Inelastic work vs. the number of cycles for SAC305 solder joints cycled at 20 MPa stress amplitudes at various dwellings until failure.

## *3.3. Creep Effect*

In this study, the creep effect is demonstrated by evaluating the damaged work per cycle and comparing it for both conditions of no and 10 s of dwelling. At the no dwelling conditions, the damaged work is related to cyclic fatigue only, since the only effect is mechanical fatigue. The creep effect is considered negligible due to the fast ramp rate employed. This would not allow a considerable creep effect to take place between ramps. However, the damage due to creep is dominant during dwelling, and the damage due to fatigue is acquired between ramps in the case of a combined creep–fatigue test. According to the above, it is assumed that inelastic (damaged) work is caused by fatigue only in the cyclic fatigue test with no dwelling, and inelastic (damaged) work due to creep– fatigue is generated due to the combined effects of creep–fatigue at dwelling conditions. Consequently, the damaged work due to creep is determined approximately by subtracting damage due to fatigue only (at no dwelling) from the one generated in the dwelling experiment. Figure 22, for example, shows the hysteresis loops for both conditions of no dwelling (green-colored) and with 10 s of dwelling (yellow-colored) in addition to a bar chart summarizing the damaged work generated for both cases. For more clarification, the damaged work due to fatigue is colored as green in the bar graph, and the damage due to creep is added accordingly for each case with more extended dwelling periods. Results show that work due to creep increases with longer dwelling times, where the related damage on a life reduction basis is reflected obviously. Despite the amount of inelastic work due to creep being almost similar to fatigue in the case of 10 s dwelling, the reflection on life reduction is massive by reducing life by a factor of 3. The same trends for other stress levels with higher damaged work are noticed due to more damaged being created due to higher stress.

Figure 23 illustrates the average accumulated work until complete failure for SAC305 joints as a function of the dwell times time for different stress levels. For a certain stress level, it is noticed that increasing dwelling periods generates lower accumulated work until complete failure. This can be explained by the smaller amount of cycles observed until complete failure for extended dwellings which are accompanied with less accumulated work. Accumulated work includes other types than damaged or plastic work, which might be work due to friction or generated heat accompanied with each cycle. The trend is different during the dwellings. At no dwelling, 10 s, and 60 s of dwellings, the lower stress level generates more accumulated work than higher stress ones due to the significant additional cycles (accompanied with more accumulated work) observed until complete failure. On the other hand, the non-significant difference in the number of cycles until complete failure at 16 MPa stress level compared with the other stress levels of 20 and 24 MPa causes less accumulated work at 180 s of dwelling. Moreover, the number of cycles until complete failure for 20 and 24 MPa are close together, but more accumulated work is generated in the case of 24 MPa due to the higher average damage work during the 180 s dwelling. That is why it shows more accumulated damage work in the case of 24 MPa.

**Figure 23.** Accumulated work until complete failure vs. dwell time at different stress amplitudes.

Figures 24 and 25 illustrate the average inelastic work per cycle and plastic strain as a function of dwell time, respectively. Both the average work per cycle and the plastic strain were observed to increase with higher stress levels at specific dwellings drastically. This can be explained by more damage accumulating with higher stress levels at fixed dwelling and generating more inelastic work and plastic strain. Moreover, the longer dwelling would produce more inelastic work and plastic strain at a specific stress level. This is due to creep damage accumulated with extended dwellings. However, the creep effect might be substantial on both quantities of inelastic work and plastic strain compared with stress level. In many cases, inelastic work and plastic strain are higher at lower stress levels with longer dwellings than higher stress amplitude with shorter dwell times.

**Figure 24.** Average work per cycle as a function of dwelling times at various stress levels.

#### *3.4. The Coffin–Manson and Morrow Energy Models* Coffin–MansonModel

Figure 26 describes the plastic strain as a function of dwell time. It is clearly shown that there is a trend of plastic strain as a function of dwell time at various stress amplitudes. With a high *R-square* for all curves, data are fitted to a power equation, so the plastic strains can be predicted as a function of dwell time according to Equation (5).

$$PS = D \times t\_d^{0.119} \tag{5}$$

where *PS* is the plastic strain, *D* is constant, and *td* is the dwell time. The *D*-value could also be formulated as a function of stress amplitude according to Figure 27. Thus, Equation (6) can be expressed as shown in Equation (6). The dwell time's impact on the value of the plastic strain is depicted by the *D* coefficient in terms of its magnitude.

$$PS = 0.0003 \text{ } e^{0.1248} \text{ } ^\circ \text{ } t\_d^{0.119} \text{ } \tag{6}$$

**Figure 26.** Plastic strain per cycle as a function of dwelling time curves at various stress levels.

**Figure 27.** D-value as a function of stress amplitude.

Coffin–Manson is one of the most common models employed for fatigue life prediction as a function of plastic strain. The correlation between fatigue life and plastic strain is illustrated in Equation (7).

$$N\_{63} = PS^{\frac{1}{\pi}} Z^{\frac{-1}{\pi}} \tag{7}$$

where *N*63 is the characteristic fatigue life, *Z* is the fatigue ductility coefficient, *PS* is the plastic strain, and *R* is the fatigue exponent. Based on the Coffin–Manson equation, the general reliability model as a function of plastic strain could be developed under certain conditions. If there is no clear trend for the coefficient of fatigue ductility (*Z*) and the fatigue exponent (*R*) at various dwellings, this implies that dwelling does not affect the Coffin–Manson equation. Moreover, data points for all conditions should have a similar trend (slope) to be fitted to a global Coffin–Manson equation no matter what the dwelling time is. To establish such a model, the characteristic life as a function of plastic strain (Equation (7)) must be obtained. Then, the new equation is substituted in Equation (1) to obtain the reliability model. To examine the model applicability in our case, the abovestated conditions must be checked. Figure 28 demonstrates the characteristic life as a function of plastic strain for various dwelling periods. Data points have the same trend (slope) and could be fitted to a global Coffin–Manson equation. The values for Coffin– Manson equation constants at various dwellings are generated accordingly, as shown in Table 2. It is obviously shown that there is no clear trend in these constants regardless of the dwelling time. This means dwelling has no effect on the Coffin–Manson model, and a general model could be developed. The values of the coefficient of fatigue ductility (*Z*) and the fatigue exponent (*R*) for the global equation are 0.19 and 0.646, respectively. Moreover, the global Coffin–Manson model is illustrated in Figure 29.

**Figure 28.** Characteristic life vs. plastic strain for SAC305 joints at various dwelling times on a log–log scale.


**Table 2.** Fatigue ductility coefficient and fatigue exponent of the Coffin–Manson model.

**Figure 29.** Characteristic life (global Coffin–Manson equation) vs. plastic strain for SAC305 joints at various dwelling times on a log–log scale.

Finally, the characteristic life equation as a function of plastic strain (Equation (8)) must be obtained and substituted in the reliability equation (Equation (1)) to develop the reliability model as a function of plastic strain. The characteristic life as a function of plastic strain is obtained from Figure 29 as shown in Equation (8).

$$N\_{63} = 0.0771 \times PS^{-1.546} \tag{8}$$

Substituting Equation (8) in the reliability equation (Equation (1)) and considering the average shape parameter for all combinations as the defined shape parameter, the general reliability model is developed as shown in Equation (9).

$$R(t) = e^{-\left(\frac{t}{0.071 \times 10^{-1.456}}\right)^{6.66}}\tag{9}$$

where *PS* is the plastic strain, and *t* is the number of cycles.

#### *3.5. Morrow Energy Model*

Inelastic work is another parameter considered to measure damage, where the Morrow Energy equation is the common related model. Figure 30 illustrates the inelastic work as a function of dwell time for various stress levels. It shows a trend in inelastic work as a function of dwell time at various stress amplitudes. With high R-square for all curves, data are fitted to a power equation, so the inelastic work is predicted as a function of dwell time according to Equation (10)

$$\mathcal{W} = H \times t\_d^{0.12} \tag{10}$$

where *W* is the inelastic work, *H* is constant, and *td* is the dwell time. On the other hand, the *H*-value could also be formulated as a function of stress amplitude according to Figure 31. Thus, Equation (10) can be expressed as shown in Equation (11). The H coefficient illustrates the magnitude of the dwell time impact on the value of the inelastic work per cycle.

$$W = \left(-5.0 \times 10^{-6} \times P - 7 \times 10^{-5}\right) t\_d^{0.12} \tag{11}$$

**Figure 30.** Inelastic work per cycle as a function of dwelling time curves at various stress levels.

**Figure 31.** *H*-value as a function of stress amplitude.

Morrow Energy is a common model used to predict life as a function of inelastic work as shown in Equation (12).

$$N\_{63} = G^{\frac{1}{m}} \,\,\, \mathcal{W}^{\frac{-1}{m}} \tag{12}$$

where *N*63 is the characteristic fatigue life, *G* is the fatigue ductility coefficient, *W* is the inelastic work, and *m* is the fatigue exponent. In the same way as the Coffin–Manson model, the reliability model as a function of inelastic work based on the Morrow Energy model could be established under similar circumstances defined above. Our data show that fatigue ductility and fatigue exponent have no clear trend at various dwellings, as shown in Figure 32. Furthermore, the data points on a log–log scale demonstrate having a similar trend (slope) and could be fitted to the global Morrow Energy equation. The fatigue ductility and the fatigue exponent constants for all dwellings are specified accordingly, as shown in Table 3. It is obviously shown that there is no clear trend in these constants regardless of the dwelling time. This means dwelling has no effect on the Morrow Energy model, and the global model could be developed. Figure 33 shows the global model for Morrow Energy equation considering that the global constants for fatigue ductility coefficient and fatigue exponent are 0.0025 and 0.737, respectively.

**Figure 32.** Characteristic life vs. inelastic work for SAC305 joints at various dwelling times on a log–log scale.

**Table 3.** Fatigue ductility coefficient and fatigue exponent of the Morrow Energy model.


**Figure 33.** Characteristic life (global Morrow Energy equation) vs. plastic work for SAC305 joints at various dwelling times on a log–log scale.

To generate the reliability model (Equation (1)) as a function of inelastic work, the characteristic life equation as a function of plastic work (Equation (12)) must be attained. Therefore, the characteristic life as a function of plastic work is developed from Figure 33 as shown in Equation (13).

$$N\_{63} = 0.0003 \text{ W}^{-1.596} \tag{13}$$

Finally, substituting Equation (13) in the reliability equation (Equation (1)) and considering the average shape parameter for all combinations as the defined shape parameter, the general reliability model is developed as shown in Equation (14).

$$R(t) = e^{-\left(\frac{t}{0.0009 \times W^{-1.356}}\right)^{\rho \text{-} \rho \text{-} \rho}} \tag{14}$$

where *W* is the plastic work, and *t* is the number of cycles.

## *3.6. Microstructure Analysis*

The failure mode for samples at room temperature was determined by studying SEM images for tested joints after various dwelling times. This would provide an idea about the effect of dwelling on material evolution under various dwelling periods. Results show that failure is located within the bulk region among all cases of fatigue and creep–fatigue tests as shown in Figure 34.

**Figure 34.** SEM images for tested joints under various dwelling periods compared with no dwelling condition (most left).
