*3.1. Structural Micro-Division*

From the structure of the unloading valve, it can be known that the initial state of contact between the valve ball and the valve seat is a line contact on the circumference. In order to facilitate the analysis of the contact force, the structure is divided into n(n→∞) equal divisions as shown in Figure 4. The force of each micro-element is shown in Figure 5.

**Figure 4.** The micro-division of the structure.

**Figure 5.** The force of each micro-element.

Under uniform oil pressure and initial spring force, the maximum force that the ball is subjected to can be expressed as:

$$F = F\_t + (p\_{1\text{max}} - p\_2) \cdot 2\pi r \cos\theta. \tag{1}$$

The force that each micro-element is subjected to can be expressed as:

$$
\Delta F = \frac{F}{n}.\tag{2}
$$

It can be obtained from the force model of the micro-element that:

$$
\Delta F = \Delta N \cdot \sin \theta + \Delta f \cdot \cos \theta. \tag{3}
$$

Δ*N* and Δ*f* are the positive pressure and friction of the micro-element. From the Equation (3), the relationship between Δ*N* and Δ*f* is not known; it cannot be solved directly, so it is necessary to obtain the equivalent friction coefficient between the two.

#### *3.2. Equivalent Friction Coe*ffi*cient Calculation*

We can suppose a situation that a block on a horizontal plane is subjected to horizontal thrust *F*<sup>1</sup> and forward pressure *N*1. When the horizontal thrust *F*<sup>1</sup> is within the range of 0 ≤ *F*<sup>1</sup> ≤ *fmax* = μ1*N*<sup>1</sup> (μ1—maximum static friction between the two materials), the slider has no displacement. However, there is a slight slip Δ*x* between the slider and the horizontal plane. It can be considered that the micro slip range in which the slider does not move is 0 ≤ Δ*x* ≤ Δ*x*max. It can be also considered that the equivalent friction coefficient *fx* satisfies *fx*∝Δ*x* [15].

The meaning of μ*<sup>x</sup>* is the ratio of the value of non-critical equivalent friction to the positive pressure:

$$
\mu\_x = \frac{f\_x}{N}.\tag{4}
$$

The relationship between the equivalent friction coefficient μ*<sup>x</sup>* and the micro-slip Δ*x* is experimentally verified and obtained by Dr. Liu [16]. μ*<sup>x</sup>* and Δ*x* are proportional to each other.

$$
\mu\_{\mathbf{x}} = k\_f \Delta \mathbf{x} \tag{5}
$$

The valve ball and valve seat structure are steel materials, *kf* = 0.25 μm−<sup>1</sup> [16].

The micro-element structure of the ball valve is analyzed, and its force deformation is shown in Figure 6. Δ*X* is the slight slippage of the valve ball under the action of the spring force that can be calculated from the Hertz contact theory in Section 3.3. Here, *a* is the half width of the contact between the valve ball and the valve seat under the action of positive pressure Δ*N*. The initial radius of the ball is *r*. After the force is deformed, the distance between the center point of the contact and the center of the sphere is *r*1, which can be expressed as:

$$r\_1 = \sqrt{r^2 - a^2}.\tag{6}$$

**Figure 6.** The force model of the ball valve.

The micro-slip distance Δ*X* can be expressed as:

$$\frac{r - r\_1}{\Delta X} = \tan \theta.\tag{7}$$

The relationship between Δ*N* and Δ*f* in the force model of micro-element can be described as:

$$
\Delta f = k\_f \Delta X \Delta N. \tag{8}
$$

Therefore, the relationship between the equivalent friction and the positive pressure can be linked by the Hertz contact theory.

#### *3.3. Structural Contact Stress Analysis*

The contact part of the micro-element structure of the spherical unloading valve can be regarded as a model of contact between the cylinder and the plane. According to the Hertz contact theory [17], the contact half width *a* satisfies:

$$a = \sqrt{\frac{4 \cdot r \cdot \Delta N}{\pi \cdot E^\* \cdot \Delta l}}.\tag{9}$$

The maximum contact stress in the contact portion appears at the contact center, and the maximum value is:

$$p\_{\text{max}} = \frac{2N}{\pi a L'} \tag{10}$$

where Δ*l* is the length of the micro-element of the contact between the valve seat and the valve ball line, which satisfies:

$$
\Delta l = \frac{2\pi r \cos \theta}{n}.\tag{11}
$$

*E\** is the equivalent elastic modulus, which satisfies:

$$\frac{1}{E^\*} = \frac{\left(1 - \upsilon\_1^2\right)}{E\_1} + \frac{\left(1 - \upsilon\_2^2\right)}{E\_2}.\tag{12}$$

*E*<sup>1</sup> and *E*<sup>2</sup> are the elastic modulus values of the valve ball and the valve seat material, respectively. υ<sup>1</sup> and υ<sup>2</sup> are the Poisson s ratios of the valve ball and the valve seat material, respectively. *E*<sup>1</sup> = 200 GPa, *E*<sup>2</sup> = 213 GPa. υ<sup>1</sup> = 0.3, υ<sup>1</sup> = 0.29.

It can be obtained by solving Equations (1)–(3) and (6)–(12) that: Δ*N* = 145.4/*n*, Δ*f* = 12.9/*n*. The equivalent friction coefficient is:

$$
\mu = \frac{\Delta f}{\Delta N} \approx 0.0887.\tag{13}
$$

At the same time, the micro-slip distance Δ*x* = 0.355 μm can be obtained; the contact half width *a* = 17.15 μm.

The maximum contact stress value of the contact between the valve ball and the valve seat is expressed as follows:

$$p\_{\text{max}} = 1220 \,\text{MPa}.\tag{14}$$

#### **4. Simulation Analysis**

The valve ball seat structure shown in Figure 7 is added to the ANSYS model. In order to facilitate the addition of force, a plane is selected on the left side of the valve ball. The elastic modulus and Poisson's ratio parameters of the ball and seat are added to the material properties.

**Figure 7.** The simulation structure of the pressure relief valve.

The s-N curve of the structural material needs to be known when calculating the fatigue safety factor of the structure. The material used for the valve ball and the valve seat is bearing steel 9Cr18, and its p -s-N curve is used to set the material fatigue property parameters [18]. The meaning of the p -s-N curve expression is that the life distribution of the material is a normal distribution form under the same stress level [19]. The lifetime of most tested materials is distributed at *p* = 0.5, the middle of the normal distribution curve. In this simulation, in order to meet the requirement of one-millionth of the destruction probability of the structure in engineering applications, it is necessary to fit the s-N curve data in the case of the probability *p* = 0.000001 according to the *p*<sup>1</sup> = 0.01, *p*<sup>2</sup> = 0.05, *p*<sup>3</sup> = 0.1 and *p*<sup>4</sup> = 0.5 data given by the material according to the normal distribution.

The fatigue life curve formula is as shown in Equation (15):

$$\mathbf{N}' = \mathbb{C} \cdot \mathbf{s}^{-m} \,, \tag{15}$$

where *N* is the number of cycles; *s* (Pa) is the stress size; the values of *C* and *m* refer to Table 2.


**Table 2.** Parameter values under different damage probabilities.

The s-N curve data table with a probability of destruction of one part per million is calculated as shown in Table 3. The meshing parameter is set for the structure, and the local mesh encryption is performed on the key contact portion of the calculation, as shown in Figure 8. The overall mesh size is 0.3 mm and the contact portion mesh is refined to 0.015 mm.


**Table 3.** The data sheet of fatigue life curve.

**Figure 8.** Meshing and simulation boundary conditions.

The structural stress boundary conditions, i.e., the pressure acting surface and the fixed surface, are set. The contact portion was set to frictional contact and the coefficient of friction was defined as 0.0887 as calculated in Section 3.3. The pressure is calculated according to the maximum force under working conditions, that is 50 MPa oil pressure, equivalent to 78.58 N.

The fatigue simulation force is in the form of pulsating circulation. The force application is still the same as the external force in the static simulation. The alternating equivalent oil pressure is 5–50 MPa, and the minimum oil pressure σ*min* = 0.1 × 50 MPa = 5 MPa. The pulsating form is shown in Figure 9.

**Figure 9.** Constant amplitude load ratio.

The contact stress distribution obtained by simulation is shown in Figure 10. The maximum contact stress value calculated by simulation is *pmax* = 1107.8 MPa, and the theoretical calculation result is *pmax* = 1220 MPa. The error between the two is about 9.2%, less than 10%, which verifies the correctness of the theoretical model. The overall equivalent stress distribution is obtained as shown in Figure 11. The maximum equivalent stress is 725.63 MPa. The yield strength of the ball and seat material is σ*<sup>t</sup>* = 1900 MPa, and the tensile yield strength and compressive yield strength of the elastoplastic material are generally considered to be the same, i.e., σ*<sup>t</sup>* = σ*<sup>c</sup>* (σ*<sup>c</sup>* is the compressive yield strength). The von Mises equivalent stress theoretical shear yield strength satisfies τ*<sup>s</sup>* = σ*t*/ <sup>√</sup>3, and τ*<sup>s</sup>* ≈ 1096 MPa can be calculated [20]. The check is performed with the minimum shear yield strength, which is much larger than the equivalent stress of 725.63 MPa that is theoretically calculated. It can be judged that the ball valve is safe under static conditions.

**Figure 10.** The simulation results of contact stress.

**Figure 11.** The simulation results of equivalent stress.

The fatigue safety factor distribution of the whole structure under the stress condition of 1.728 <sup>×</sup> <sup>10</sup><sup>9</sup> times is shown in Figure 12; the maximum is 3.6558, which indicates that the structural ball will not suffer structural fatigue damage under the action of fatigue alone.

**Figure 12.** The distribution of fatigue safety factor.

#### **5. Discussion**

In this paper, the calculation of the stress of valve ball and valve seat is solved by phenomenological mathematics, obtained by the experiment, and the result is verified by simulation. Through the static and fatigue check, the damage mechanism of the valve ball and the valve seat under working conditions is not static and fatigue damage.

For a further study of the damage mechanism, a high frequency test rig as shown in Figure 13 was designed. The loading force can reach more than 100 N and the operating frequency can reach 500 Hz. A 50-h experiment has been done between the valve ball and the valve seat at 500 Hz and 8–80 N. The damage mechanism needs further research to explain the experimental phenomena or to illustrate the problems in combination with the experiments.

**Figure 13.** Test bench, (1) piezoelectric actuator; (2) fixed disc; (3) cylinder 1; (4) force sensor; (5) cylinder 2; (6) testing object; (7–9) base.

The ball and seat wear band under the 50-h test is shown in Figure 14. The diameter of the ball is small. It is only a little more than a millimeter. Therefore, the amount of damage cannot be assessed by quality. At present, the best evaluation method is to take the damage surface morphology by microscopy and estimate the damage by width and depth.

**Figure 14.** Damage zone (8–80 N, 500 Hz, 50 h).

Although the amount of damage can be estimated by microscopy, this method of evaluation is not particularly accurate. It is also impossible to monitor the damage during the experiment. The test bench can be improved to add better calculation methods. For example, a distance sensor can be added to monitor the depth of the damage zone to determine the depth of damage during the experiment.

#### **6. Conclusions**

The injection pressure of the gasoline direct injection vehicle is currently developing from low pressure to high pressure. The structural damage problems brought by the increase of the injection pressure should be solved urgently. Therefore, based on theoretical analysis and numerical analysis, the paper first determines whether there is traditional structural damage.

The theoretical stress calculation and static and fatigue simulation analysis of the unloading valve structure were carried out around the ball damage problem of the high pressure pump unloading valve in a gasoline direct injection vehicle. The following conclusions were obtained:

(1) According to the maximum static force of the valve ball, the theoretical calculation is carried out, the equivalent friction coefficient is obtained by solving the statically indeterminate problem, and the maximum contact stress value of 1220 MPa is obtained by the Hertz contact theory.

(2) Through simulation, the maximum contact stress is 1107.8 MPa and the maximum equivalent stress is 725.63 MPa under maximum static force. The simulated contact stress values are compared with the theoretical calculations and the difference between the two is less than 10%, which verifies the correctness of the theoretical model. At the same time, the equivalent stress is used for static checking, and it is judged that the unloading valve structure will not be damaged under the action of static force.

(3) By the simulation analysis, the fatigue safety factor of the unloading valve is 3.6558 under the condition of one-millionth of the failure probability and 1.728 <sup>×</sup> 10<sup>9</sup> cycles. It is verified that the traditional structural fatigue is not the cause of the ball failure of the unloading valve.

It can be seen from the above verification analysis that the damage of the ball valve structure is not caused by static force and fatigue damage; further analysis of the structural damage mechanism is needed. In the theoretical calculation, the tangential displacement (0.355 μm) and radial displacement (17.15 μm) of the valve ball are all in the micron range. The motion state belongs to the fretting category and the surface damage morphology of the valve ball is similar to the fretting damage. It can be preliminarily speculated that the surface damage of the valve ball is a fretting damage. The mechanism still needs further research.

**Author Contributions:** L.L. (Liang Lu) and Q.X. provided theoretical research methods; L.L. (Liang Lu) and Q.X. conceived and designed the experiments; Q.X. performed the experiments; M.Z. and L.L. (Liangliang Liu) analyzed the data; M.Z., L.L. (Liangliang Liu) and Z.W. contributed to the project supervision and management; Z.W. contributed to the problem research and provided research object and resources. Q.X. provided original draft preparation. L.L. (Liang Lu) provided review and editing of writing.

**Funding:** The authors are grateful to the National Natural Science Foundation of China (No. 51605333) and the Fundamental Research Funds for the Central Universities (kx0138020173443) for financial support.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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