## 2.1 Anand constitutive equation and parameter fitting

The Anand model is a viscoplastic model of metal thermal work proposed by Anand and Brown [4]. Suitable for describing large viscoplasticity and small elastic deformation. The relationship between the saturation stress and the strain rate of the viscoplastic Anand model is as follows:

The relationship between stress and strain at different temperature and strain rates:

The above equation contains the nine parameters of the Anand constitutive equation:

According to the formula derivation, they are constant, activation energy / gas constant, stress multiplier, saturation value coefficient of deformation impedance, strain rate sensitivity, strain rate sensitivity index, strain hardening parameter, strengthening coefficient, and initial value of deformation impedance. The parameters of the constitutive equation model of nano-silver are calculated through formula deduction, which are listed in Table 1 [5].

## 2.2 ANSYS finite element model

In order to facilitate the study of thermal fatigue effects of solder joints, the model needs to be simplified. Due to the symmetry, a quarter of the chip can be used as the research object, which contains 12 solder joints. The model is shown in Figure 2. The material parameters of each component are listed in Table 2 and Table 3 [6].

## 2.3. Meshing

Because the size of the mesh division will have a greater impact on the calculation time and the final analysis result. In order to make the effect of the mesh division on the basis of satisfying the calculation accuracy and save the calculation time, there are differences in the meshing of different parts. This time, the mesh of the model is divided into 0.005 using the size controller for the solder joints and copper pillars, and 0.02 is used for the chip controller. All of them are mapped grids, which are divided into 1334400 grid cells Figure 3 shows the meshing situation.

## 2.4. Loading loads and boundary conditions

The thermal cycling load is determined according to various conditions that may be encountered during service. The temperature range is −50 ° C to 150 ° C, the temperature rise and fall rate is 25 ° C / min, and it is maintained for 10 minutes when it rises to the highest temperature and 10 minutes when it is lowered to the lowest temperature. The cycle period is 36min. According to existing research, the stress and strain of solder joints show periodic changes during thermal cycling. After loading for several cycles, the results will stabilize. This model calculates 5 cycles [7], as shown in Figure 4.

The base plate is mounted on a tooling fixture. The bottom surface of the lower chip can be imposed with zero displacement constraints in all directions, and the two symmetrical surfaces of the upper chip can be imposed with unidirectional zero displacement constraints.

## 2.5. Stress and strain analysis

The simulation lasted for 20400 seconds. After the simulation, check the stress and strain diagrams of the entire chip using the POST_26 post-processing program. It can be seen that there is basically no obvious stress and strain on the chip part, as shown in Figure 5.

Enlarge the stress-strain diagram at the solder joint. As shown in Figure 6, it can be seen that the outer part of the solder joint where contact with the upper and lower copper pillars generates a large stress, while the interior of the solder joint generates a large strain. The strain is gradually extended from the center of the solder joint to the periphery with the load of the temperature cycle. When it reaches the contact surface with the upper and lower copper pillars, stress deformation occurs due to the different material properties of the two. It can be speculated that if the simulation is continued, fatigue cracking will occur at the interface between the solder joint and the upper and lower copper pillars, which will cause the chip to fail.

In order to more clearly analyze the stress and strain of the solder joint under the thermal cycling load, a node in the middle of the solder joint is now selected as the research object, and the stress and strain curve is drawn for analysis. As shown in Figure 7, the curve of the equivalent shear stress of the joint at the middle of the solder joint over time. In the figure, the horizontal axis represents time, and the vertical axis represents the numerical value of the stress change of the node. It can be seen that the equivalent shear stress of the node changes periodically with the progress of the thermal cycle.

It can be seen from Fig. 8 that the viscoplastic strain energy density of the solder joint gradually increases with time, and the increase in each cycle is basically the same.

The stress-strain hysteresis curve of the internal node of the solder joint is shown in Figure 9, where the horizontal axis represents the equivalent shear strain of the internal node of the solder joint and the vertical axis represents the equivalent shear stress of the node. It can be seen from the figure that as the temperature cyclic load is loaded, the stress-strain hysteresis curve moves to the right, and the interior of the solder joint is undergoing cumulative plastic deformation.

## 2.6. Fatigue life prediction

In the case of loading thermal cycle load, the stress and strain behavior of solder joints is very complicated, and factors such as time, temperature, and solder all affect the performance of solder joints. In general, plastic strain is the root cause of cracking of solder joints until failure. Therefore, we use the strain fatigue model Manson-Coffin to predict solder joint life.

Fifties of last century, the relationship equation proposed in the famous paper published by Coffin and Manson [8, 9]. It relates the fatigue life to the magnitude of the inelastic strain applied. The index β is found to be very common in pure metallic materials and very close to 2, it has nothing to do with their microstructure. When the plastic strain amplitude is large, this law applies to the definition of low cycle fatigue state (LCF) [10]. In the low cycle thermal cycle, the fatigue life () and the nonlinear strain amplitude () have the following relationship regardless of the residence time:

In the formula, is the fatigue life, C and β are the constant numbers of the sample, and is the inelastic strain amplitude.

Above we introduced a method for predicting thermal fatigue life—C-M empirical equation based on strain range. However, since this equation only deals with the influence of the magnitude of the strain amplitude on the thermal fatigue life, it does not take into account the influence of temperature on it, thus reducing the credibility of its predicted life, so there are limitations. In order to solve this problem and make the results of life prediction more accurate, Engel-Maier's C-M correction equation becomes a more accurate one of many derivative equations:

Among them, represents the equivalent plastic shear strain range , and represents the fatigue toughness coefficient (take 0.325) [11]. C stands for the fatigue toughness index, which can be determined by the following formula.

In the formula, Tm is the arithmetic mean of high and low temperatures, and f is the number of loads per unit time. The Engel-Maier correction formula not only reduces the number of model parameters, but also simplifies its structure. Because of the influence of temperature, the result of this formula is relatively accurate. It is widely used in calculating the thermal fatigue life of solder joints.

Through the comparison and calculation of the strain range of each node in the central part of the solder joint by the ANSYS program, the node with the maximum strain range can be obtained. The equivalent plastic strain range of the node = 0.0082, and the plastic shear strain range = 1.42×10-2. The average temperature of the temperature cyclic loading load is Tm = (150-50)/2=50 °C, and each thermal cycle is 36 min. The temperature cycle frequency f=1.67cyc/h can be calculated and substituted into Equation (5), and it can be calculated C=-0.455, substituting the above data into Equation (4), the fatigue life of the silver tin solder joint can be calculated = 2166 weeks.