### Material property for part and die

The material of the ball stud parts was 34CrMo4 (Table 1) with a diameter of 22 mm, and spheroidizing heat treatment was applied to enhance hardenability. To obtain simulation properties, tensile and compression specimens were processed, as shown in Fig. 2, in accordance with ASTM E8 (sub size) standard^{26}. A tensile test was performed at a speed of 10 mm/min, and a compression test was performed at a speed of 2 mm/min up to a compression rate of 80%. As a result of the tensile test, the mechanical properties were obtained, as shown in Table 2. The engineering stress and strain obtained from the tensile and compression tests were converted to true stress and strain by the following equations.

$$\epsilon _true=\int _l_0^l\fracdll=\mathrmln(1+\epsilon _eng)$$

(1)

$$\sigma _true=\sigma _eng(1+\epsilon _eng)$$

(2)

The tensile and compression true stress–true strain curves of the 34CrMo4 material were derived, as shown in Fig. 3. 34CrMo4 is an exclusive material for cold heading, and as a result of the tensile test, the uniform elongation section is very small, so it cannot sufficiently simulate work hardening in compression deformation.

On the other hand, in the compression test, the true stress–strain curve of a fairly wide section can be obtained because the material does not break to a compression ratio of 80%. Therefore, a compression curve was used for the simulation properties of the mulit-stage cold forging process.

The die used in the cold forging process of the ball stud parts is generally composed of a core, reinforcing ring, and case, and the materials used are different. WC–Co alloy is used for the core, where breakage of die occurs owing to the concentration of stress in the forging process. WC exhibits high hardness and abrasion resistance, and Co is related to toughness^{27}. In general, the mechanical properties of the WC–Co alloy are determined by the Co content, and it is manufactured via a sintering process of press-molding while being heated to an appropriate temperature. The material of the core die used in the manufacturing process of the ball stud parts was a WC–Co alloy with a Co content of 20%, and its mechanical properties are shown in Table 3. WC–Co alloy has high compressive strength but is vulnerable to tensile strength, so the concentration of tensile stress is suppressed by the reinforcing ring. However, when the cyclic tensile stress applied to the die material by high-speed cyclic loading exceeds a certain strength, fatigue failure occurs. Therefore, to define the limiting life of the cold forging die, it is necessary to acquire the fatigue properties of the mold material. Fatigue test specimens were manufactured through sintering, grinding, and polishing processes, as shown in Fig. 4, in accordance with the ASTM E 466 standard^{28}.

A radius of curvature of 3 mm was reflected to prevent stress concentration in the part in contact with the jig of the testing machine. Further, the radius of curvature of the area corresponding to the gage length was 12.7 mm, which was designed so that stress concentration could occur effectively. Using Instron 8801 equipment, the S–N diagram of the die material was derived, as shown in Fig. 5, for a stress ratio of 0.1 and a frequency of 10 Hz. Starting with the load condition corresponding to low life, the life curve progressed to the level at which the fatigue limit was secured until the flattened section.

### Simulation result of ball stud manufacturing process with die structure

The manufacturing process of the ball stud parts comprised a total of six stages with forming apparatus, as shown in Fig. 6. Different molds for each of the 6 processes are placed in one die block. After one stroke, the material automatically transfer to the next process. Accurate prediction of the tensile stress in the weak point of the core die at which tensile stress is repeatedly applied should be preceded. For this, a finite element simulation was performed on the multi-stage cold forging process using FORGE, a finite element analysis program. As shown in Fig. 7, all die structures at each stage were modeled, and a fully coupled method was applied to improve the accuracy of die stress prediction. Figure 8 shows the detailed die modeling for ball stud forming procedure. WC–Co, SKD-61, and SKD-51/SKD-11 were used for the core die (WC), reinforcement ring (H13), and case (D2/M2) material of each stage, respectively. The physical property values provided by the analysis program were used as shown in Table 4. For the analysis properties of 34CrMo4, the compression diagram shown in Fig. 3 was used. The amount of shrink fit of the reinforcing ring was applied differently at each stage within the range of 0.1–0.14%. In addition, a friction coefficient of 0.08 between the material and core die was applied, and a coefficient of friction of 0.12 was applied to the rest of the contact regions. The movement speed of the punch was the same at 150 mm/s for all stages. The maximum principal stress acting on the die due to the pressurization of the material was confirmed through the fully coupled analysis. Figure 9 shows the point where the maximum principal stress in each stage acts. This analysis process is then used to derive the history of the maximum principal stress value according to the forming load in each process. The maximum principal stress value shows a constant trend according to the change of the forming load. The time it takes to confirm the results of a single analysis case is 24 h. Since it is not possible to follow the production cycle at the manufacturing site, it is simplified to the model for calculating the maximum principal stress based on the trend.

### Calculation of limit die life

The maximum principal stress acting on the core die mainly occurs at the edge of the die, and this value cannot be directly substituted on the *y*-axis in Fig. 5. This is because the result value of the finite element analysis corresponds to a stress concentration dependent on the element and shape functions. Both the finite element analysis result and stress corresponding to the *y*-axis in Fig. 5 should be converted into nominal stress values. The stress concentration factor (*k*_{t}) value cab be calculated based on the shape factor (corner curvature radius and depth) of the corner where breakage is expected to occur^{29}. The stress concentration coefficient is a numerical value indicating the degree of stress concentration distributed in notches, holes, and grooves. By applying the stress concentration factor to the finite element analysis result value, it is possible to convert the maximum principal stress to nominal stress.

$$\sigma _analysis=Max.\; Principal\; stress/k_t$$

(3)

Similarly, the fatigue stress concentration factor (*k*_{f}) is applied to the *y*-axis stress value in Fig. 5 to convert it into nominal stress. As shown in Fig. 4, since there is a notch in the center of the specimen, the stress values are not nominal stresses. The fatigue stress concentration factor is a numerical value indicating the degree of stress concentration due to the notch in the fatigue load state.

A fatigue test specimen without a notch was additionally prepared. Under the same fatigue test conditions, *k*_{f} is calculated as the ratio of fatigue strength without notch and fatigue strength with notch.

$$k_f=\sigma _without\_notch/\sigma _with\_notch$$

(4)

Then, by dividing the stress amplitude in Fig. 5 by *k*_{f}, it is converted to the nominal fatigue stress.

$$\sigma _fatigue=stress \; amplitude/k_f$$

(5)

It is converted into nominal stress (*σ*_{analysis}) by substituting the maximum principal stress, which is the analysis result, into Eq. (3). Substituting this into the nominal fatigue stress (*σ*_{fatigue}) of Eq. (5), it becomes the fatigue strength that can be substituted into the S–N curve.

$$stress \; amplitude\_FEM=k_f*\sigma _analysis$$

(6)

Quantitative evaluation of the die life was performed by predicting the life corresponding to the fatigue stress. The equation was derived by fitting the S–N diagram in Origin, a commercial S/W. By substituting the value of Eq. (6) into the fitted equation, it is possible to derive the cycle corresponding to the lifespan. The results are shown together with the actual die stress in Table 5. Comparison of the predicted data with the actual die life in the field reveals an error range of ± 20%, which is attributed to the fact that working environment variables are not taken into account in the die life prediction process. In the actual working environment, the forming load changes flexibly owing to die alignment, dispersion of material properties, and changes in friction conditions, which means that the maximum principal stress acting on the die changes according to the working environment. However, in the process of quantitatively predicting the die life, the maximum principal stress acting on the die is assumed to be an ideally fixed value, so this error is indicated. Another problem is that it inhibits applications to the field because it is difficult for non-experts to use it as the simulation of the forming process must be performed to predict the die life.