2.1. Creep Equation of a Spring at Room Temperature
In room-temperature creep, the creep deformation increases logarithmically with time, which is consistent with the first stage of typical creep. Strain hardening and fatigue models are usually used in creep theory at room temperature. The fatigue model is more accurate in fast loading, while the strain hardening model can be used in room-temperature creep under arbitrary loading [
23].
According to the microscopic situation of room-temperature creep, Schoeck [
24] proposed the constitutive equation of room-temperature creep:
where:
—room-temperature creep rate;
—dislocation density;
A—the area of dislocations swept after passing an obstacle;
V—activation volume;
—vibration frequency of the dislocation line;
—thermal activation energy required to pass obstacles;
—Boltzmann constant;
T—experimental temperature.
can be expressed as the product of the thermal activation energy,
, minus the effective stress,
, acting on the dislocation line and the activation volume,
, namely:
In the strain hardening model, the external stress,
, is constant, but due to the hardening effect, the effective stress,
, decreases with the increase of creep value,
, namely:
where:
—hardening coefficient at room-temperature creep;
—creep value.
The relationship between room-temperature creep and creep time can be obtained by introducing Equations (2) and (3) into Equation (1), namely:
where:
where:
B—Burgers vector;
—strain hardening coefficient.
Derived from Equation (4), the relationship between creep rate,
, and creep time,
, at room temperature can be obtained as follows:
After the end of loading, creep just appears. At this time,
and
, which can be substituted into Equation (6) to obtain the following relationship:
It can be seen from Equation (7) that the main influencing factors of creep rate are and , which can be directly obtained by experiment. By fitting the experimental data with Equation (7), the creep rate equation can be obtained.
According to Equation (6), the room-temperature creep rate,
, can be obtained only after obtaining the influencing factors
and
. However, these two factors are a measure of micro performance, which are difficult to obtain and not suitable for the situation of large individual differences. Therefore, Xiao [
24] adopted a method to calculate room-temperature creep only with macro parameters, and the relevant parameters can be obtained through routine experiments, which is a more simple and convenient method in engineering applications.
The initial creep rate,
, at the beginning of creep can be combined with Equation (4) to obtain Equation (8):
Therefore, the parameter τ is transformed into the initial creep rate, . There is no difference between creep loading at room temperature and tensile-test loading. Therefore, the strain rate at the moment when the room-temperature creep loading is completed is equal to the rate when the creep is just carried out, and the creep stress is equal to the stress at the end of the loading.
The Ramberg–Osgood model [
25] is usually used to describe the stress–strain curve of steel. This model was put forward in 1943. The main idea is that the strain of a material is composed of elastic deformation and plastic deformation. The nominal flow limit,
, of a material is selected by the classical method, and the corresponding deformation
= 0.002, then the equation for the Ramberg–Osgood model is:
where:
—nominal flow limit;
n—strain hardening coefficient.
The strain hardening coefficient can be selected by the classical method. If
is used [
26], the strain hardening coefficient is:
The above equation is accurate when the stress is less than the nominal flow limit, , but when the stress exceeds the nominal flow limit the calculated result of this model is larger than the actual result.
On the basis of the Ramberg–Osgood model [
25], Kim J. R. Rasmussen [
27] put forward the method of subsection fitting through experimental research. The boundary point is the nominal flow limit,
. When the stress is less than the nominal flow limit, the Ramberg–Osgood model is used. After
is exceeded, the Ramberg–Osgood model is calculated in the translation coordinate system. Through a large number of experimental calculations and statistical analysis, an improved Ramberg–Osgood model is obtained:
- (1)
When , n is the strain hardening coefficient, which can be calculated by Equation (10).
- (2)
When , is the initial Young’s modulus at this stage, that is, the tangent modulus at 0.2% yield strength. Its value can be calculated by Equation (12):
where:
e—parameter,
;
—total strain at final fracture;
—stress at final fracture, i.e., tensile strength;
m—index, m = 1 + 3.5
;
—
corresponding total engineering strain,
.
The strain rate at the loading stage can be obtained by deriving the time
t from both sides of Equation (11) at the same time.
It is known that the state at the end of loading is the initial state at the beginning of creep, that is,
, and
. By substituting Equation (8), Equation (14) can be obtained:
where
is the constant stress in the creep stage of the material and its value is equal to the material stress at the completion of loading. Therefore,
. By using Equations (12) and (13), the creep value increases with the increase of creep time.
2.3. Calculation of Temperature Creep in Spring Chamber
According to article [
28], the maximum stress of a cylindrical helical torsional spring when it only withstands external torque T is:
and
where:
—torsion spring mounting Angle;
—flexural section coefficient;
—spring index;
—torsion section coefficient;
—helical angle.
According to Mohr’s strength theory, the equivalent stress at the danger point of the spring is:
and
where:
—tensile yield point;
—compressive yield point.
When the spring is compressed by 30
, the moment of a single spring is 300 N
mm. By substituting the relevant values in
Table 2 into Equations (15)–(17),
= 16.869 MPa,
= 539.33 MPa, and
= 20.99 MPa can be obtained. By substituting these three values into Equation (18) (where m = 0.9231), the equivalent stress,
= 525.38 MPa, of the spring danger point can be obtained.
Since the equivalent stress
= 525.38 MPa is
at the spring danger point, the strain rate at the spring danger point under constant external load can be obtained by substituting the parameter into Equation (13).
When the equivalent stress
, in the loading stage, the relationship between stress and strain rate is given by Equation (20). Assuming that the loading rate is 1, the relationship between them is shown in
Figure 1.
Upon substitution of the equivalent stress of the spring danger point into Equation (20), and applying the result to Equation (14), the calculation formula for creep strain under external load T = 300 N∙mm is:
In Equation (21),
and
can be obtained by fitting the data obtained from tensile and creep tests, where
n = 1.113 and
is:
According to the relevant experiment experience of loading rate, in the tensile test, the value of loading rate is generally set at 5~40 MPa/min. According to relevant literature [
29], the magnitude of strain in the first stage of the material obtained from loading rates within the range of 5~40 MPa/min remains basically unchanged, with slight differences in subsequent stages, but the difference is not significant. Therefore, to simplify the calculation, the loading rate is selected as 20 MPa/min, that is,
0.333 MPa/s.
By integrating Equations (14), (20) and (22), and taking the loading rate as 0.333 MPa/s, the expression formula of creep stress can be obtained as follows:
It can be seen from the above formula that creep strain is mainly related to time and stress, and the relationship between the three is shown in
Figure 2.
As can be seen in
Figure 2, the room-temperature creep of a cylindrical helical torsion spring shows a typical creep curve trend when the stress is determined. In the case of low stress, the creep of the torsion spring enters the second stage of stable creep in a short time. In the stable creep stage, the creep strain rate is small, and there is little increase in creep strain with time. In the condition of high stress, the first stage of creep of a torsion spring ends after a longer time, and the creep of the torsion spring enters the second stage of stable creep after a longer time. It can be seen that, in the same case, the greater the stress produced by a torsion spring, the longer the time it experiences in the first creep stage. In the stable creep stage, compared with the lower stress condition, the creep strain rate is larger, and the increase of creep strain is larger for a long time.
After derivation of time t on both sides of Equation (23), the relationship between creep rate of a cylindrical helical torsional spring, stress, and time can be obtained, as shown in
Figure 3. As can be seen from
Figure 3, the creep rate decreases significantly with the increase of time. Compared with a state of low stress, the time of torsional spring creep in the first stage of creep increases obviously in a state of high stress, and the creep rate in the second stage of creep also increases obviously.
Since the stress values in each part of the spring are not equal and cannot be calculated in detail, it is not practical to calculate the creep strain. In order to simplify the calculation and improve the safety margin, the creep strain values at the spring danger points are chosen to replace the creep strain values at each part of the spring. It can be seen from the above that the equivalent stress of the spring danger point is
= 525.38 MPa, and the relationship between creep strain value and time can be obtained by substituting it into Equation (23), as shown in
Figure 4.
It can be seen from
Figure 4 that the creep strain curve after the torsional spring loading is in line with the first and second stages of the theoretical creep curve. With the increase of time, the creep strain continues to increase and the rate decreases to a fixed value. The creep strain of the torsional spring will not enter the third stage because it is at room temperature and the loading stress is not large.
In the elastic deformation stage, according to data [
30], the stress–strain relationship in pure bending can be written as:
where:
y is the distance between the linear strain on the section and the neutral axis, and assuming that each fiber is only subject to axial tension and compression, it can be obtained according to Hooke’s law:
and
Therefore, the stress–strain relationship at the lower boundary of the section in the elastic stage of a torsion spring is:
According to the relationship between creep strain and elastic strain of a torsion spring, the change of rotation angle during creep of a torsion spring can be obtained, as shown in
Figure 5.
In this paper, the creep process of a torsion spring is calculated theoretically, and the creep strain of a torsion spring at room temperature, and the relationship between creep angle and time, are obtained. The following uses the finite-element method to simulate the creep process of a torsion spring.
The simulation was carried out by using Abaqus. One end of the torsion spring was fixed, and a torque of 300 N∙mm was applied to the other end. In the first analysis step, a torque of 300 N∙mm was applied to make the spring undergo elastic deformation, and the time was 1 s. The second analysis step was creep analysis, which lasted for 54,000 s. The displacement results obtained are shown in
Figure 6.
The shadow in
Figure 6 is the image before the torsion spring is deformed. It can be seen from the figure that the displacement at the elastic deformation stage is 8.761 mm from the farthest point of the central axis of the spring. After 54,000 s, its deformation increases to 8.847 mm. Compared with the elastic stage, the creep deformation is 0.98% of the elastic deformation and the creep strain is
. In order to display the creep curve more clearly, the curve at the elastic deformation stage is ignored and only the curve within a period of time at the beginning of creep is truncated. The point is selected as the lower endpoint of the torsional spring applying force, and the creep curve is shown in
Figure 7.
It can be seen from
Figure 7 that the creep simulation curve of a torsion spring is similar to the theoretical calculation curve, and a comparison between the simulation curve and the theoretical curve is shown in
Figure 8.
It can be seen from
Figure 8 that the theoretical calculation value of torsion spring creep is slightly less than the simulation value in the early stage, and the theoretical calculation value is slightly greater than the simulation result as time goes on. In the later period, the theoretical calculation is smaller than the simulation result. At 54,000 s, the theoretical calculation value of torsion spring creep is 0.2935
, while the simulation result is 0.3105°, which is 5.79% larger than the theoretical calculation, proving that the theoretical calculation formula is more accurate. Regarding the error between the theoretical calculation and simulation results: on the one hand, it may be because the software adopts traditional age-hardening creep theory in the finite-element simulation process, without considering the influence of some material properties, such as interaction and microstructure. In addition, when using this theory, parameters such as creep strain rate, creep activation energy, and initial hardness of materials need to be determined. If the actual material parameters are different from those used in the theoretical calculation, the calculation results will be biased. On the other hand, it may be because some small quantities are omitted in the derivation of theoretical formulas, which leads to the change of calculation accuracy. However, on the whole, the error between the theoretical calculation results and the finite-element simulation results is within an acceptable error range, which shows that the theoretical calculation results are more accurate and can accurately predict and estimate the performance and life of a spring in use, which is of great significance for designing high-performance and reliable spring components.
2.4. Experimental Method of Spring Chamber Temperature Creep
The structure of the cylindrical helical torsional spring loading table is shown in
Figure 9, and is mainly composed of two identical unilateral loading mechanisms, a bottom plate, and a steering gear fixed seat. The unilateral loading mechanisms are fixed onto the bottom plate through support legs, and the rudder wing is locked by a bolt onto the gripper. The steering gear is installed on the steering gear seat and is locked by bolts. The deflection of the rudder wing is driven by the clamping claw to rotate the rotating shaft of the two unilateral loading mechanisms, and the torque is provided by the torsion springs in the unilateral loading mechanisms.
The internal structure of a unilateral loading mechanism is shown in
Figure 10. The torsional spring loading platform mainly realizes the change of loading torque by replacing the unilateral loading mechanism. Different torsional springs correspond to different torque.
Each unilateral loading mechanism contains two torsion springs that are arranged in a coaxial reverse direction, and the rotating ends of the two torsion springs wind in opposite directions. The installation diagram is shown in
Figure 11. Both torsion springs are pre-compressed by 30°. When the axial rotation is on one side, the force of one torsion spring increases and the compression Angle increases, while the compression Angle of the other torsion spring decreases.
The rotation of the steering gear is transmitted to the torsion springs through the clamping claw, which provides the corresponding torque. The change of the output voltage of the rotary potentiometer is measured by a multimeter, and the rotation angle can be obtained by a certain conversion formula.
The specific experimental process is shown in
Figure 12.
After loading the weight, the output voltage value of the Angle sensor can be measured by keeping the weight unchanged. After converting the voltage value into Angle, the relationship of the spring shape variable with time and the spring creep curve can be obtained.