Measurement of Relaxation Modulus of Viscoelastic Materials and Design of Testing Device

Abstract

Viscoelastic parameters of viscoelastic materials should be measured to ensure their effective use. Unfortunately, devices for effective and accurate measurement of the relaxation modulus of materials are not currently available. In this study, we designed a mechanical device that can control the spherical indenter to press on the material at a uniform speed to allow measurement of the relaxation modulus of a viscoelastic material. The results showed that the device is accurate and easy to operate. The load expressions of ramp loading (the motion of the indenter is a uniform velocity) and ramp-constant loading (the motion of the indenter is stationary after uniform velocity) were derived for the spherical indenter. The relaxation modulus function of the material was calculated by measuring the displacement and load of the indenter and fitting it using the non-negative least square method. The relaxation modulus expression of viscoelastic materials calculated using the ramp loading experiment was substituted into the derived load expression of ramp-constant loading. The values were compared with the load values obtained using the ramp-constant loading experiment and the error was evaluated. The error was less than 2.5%, indicating high feasibility and practicability of the experimental device designed in this study.

1. Introduction

Coal is one of the major sources of energy in China [1]. In 2021, approximately 6.1 billion tons of coal were consumed in China, accounting for 68% of the country’s energy consumption. Coal materials obtained from mines are mainly transported using mine conveyance equipment. A conveyor belt is the most commonly used material for conveying equipment. The conveyor belt has several advantages, including large throughput and high reliability and is thus widely used [2]. The conveyor belt is currently being developed to achieve high power, long distance and high throughput owing to the growth of the economy and advances in science and technology. Notably, the power of the conveyor belt is closely related to its operation resistance [3].

The running resistance of the conveyor belt comprises the main resistance, the inclination resistance of the material and the specific resistance of the conveyor belt under a stable state [4]. The inclination resistance of the material is positively correlated with the installation angle of the conveyor. Therefore, it can be measured accurately under certain conditions [5]. The specific resistance is dependent on the type of conveyor belt [6]. It can be accurately calculated based on the specific structural form of the conveyor belt [7]. The additional resistance mainly comprises the bending deformation resistance when the moving conveyor belt contacts the drum, the friction resistance between the material and the guide chute, and the resistance generated during the cleaning of the conveyor belt. There are several types of running resistance components of the conveyor belt, but the most important types are the collapse resistance of the conveyor belt and the rotation resistance of the idler [8]. Previous findings indicate that, in the running resistance composition of a 1000-m-long heavy conveyor with a conveyor belt applying the specified preload, the rotation resistance accounts for about 9% and the collapse resistance accounts for 61% [9]. In summary, the theoretical analysis of collapse resistance and the systematic research on the relationship between common working conditions and collapse resistance are important for energy-saving and green design of conveyor belts because collapse resistance accounts for the highest proportion of running resistance [10].

The upper covering layers and lower covering layers of conveyor belts are mainly made of rubber. The upper covering layer comes in contact with the material to be transported to achieve transportation of the material, while the lower covering layer comes in contact with the idler to transfer the load to the idler. Collapse resistance occurs when the lower covering layer made of rubber comes into contact with the idler. The mechanical properties of the idler material are predominantly rigid, whereas the mechanical properties of the covering material are mainly viscoelastic [11,12]. When the conveyor belt passes through the rigid idler at a constant speed, the contact between the two materials causes deformation of the lower covering layer of the conveyor belt and contact stress. The deformed contact area gradually recovers with the continuous forward movement of the conveyor belt, and the non-deformed area exhibits contact deformation. Notably, deformation recovery and deformation production cannot occur synchronously due to the viscoelastic characteristics of the lower covering layer of the conveyor belt [13,14]. Therefore, the actual contact area between the conveyor belt and the idler is asymmetric with respect to the center of the idler, resulting in asymmetric distribution of the contact stress [15,16,17]. As a result, resistance that hinders the movement of the conveyor belt, which is denoted by the collapse rolling resistance, is generated [18].

Therefore, it is imperative to explore the viscoelastic properties of materials to understand the collapse rolling resistance of rubber. Evaluating the viscoelastic properties of the material is essential in industries to ensure effective functioning of conveyor belts. The aim of this study was to test the linear viscoelastic relaxation modulus of a 75 HA conveyor belt. The conveyor belt exhibits nonlinear viscoelasticity because when strain is applied to the viscoelastic body, the resulting effect is nonlinear. This is a property often observed in large deformations. Analysis of nonlinear viscoelasticity is more complex than the analysis of linear viscoelasticity. Linear viscoelasticity occurs when the strain applied to a viscoelastic material can be expressed linearly. The Maxwell model is usually used to represent this attribute. The nonlinear viscoelasticity of a viscoelastic material can be approximated as linear viscoelasticity if its deformation is negligible [19].

Several methods have been developed to determine the linear viscoelastic properties of materials, including the time domain and frequency domain methods [20]. The most commonly used method in the frequency domain is the dynamic mechanical analysis (DMA) method [21]. The DMA method can determine the change in the function of relaxation modulus with frequency, but it is challenging to control the experimental conditions for this method. Several time domain methods have been reported, including stretching, twisting and pressing. The time-based methods can measure the change in the function of the relaxation modulus with time. In this study, the linear viscoelastic relaxation modulus in the time domain of the rubber sheet was determined by applying a small strain to the rubber sheet. The experimental device designed in this study causes the sliding table to move downwards by controlling the rotation of the stepping motor. Subsequently, it presses the spherical indenter into the rubber sheet using the lever principle. The displacement and load of the spherical indenter are then measured using the displacement sensor and the force sensor, respectively. The relaxation modulus parameters of the rubber sheet are obtained by fitting the experimental results.

2. Related Studies

In the time domain indentation measurement methods, Lu [22] proposed a variable rate loading method using the spherical indenter. In this method, the most important thing is to ensure that the displacement of the indenter changes according to the requirements of the variable speed rate. This method requires extremely high-speed control accuracy of the indentation instrument. The speed requirement at the initial moment is extremely high, and the rate variation range during the testing process should be wide so that long-term load-displacement data can be measured. Due to the limitations of the indentation device, the maximum speed achievable in the initial stage is very limited. Therefore, in general experiments, the relaxation modulus function of the material will not be obtained over a long time range.

Du et al. [23] studied the flat punch loading method and derived a set of methods for measuring material relaxation modulus using the ramp constant loading test method. In the study, the ramp constant loading test was divided and the theoretical calculation formulas for measuring the relaxation modulus of the ramp and constant parts were obtained. Du et al. derived the theoretical formula for measuring the relaxation modulus using the ramp constant loading method under a flat punch. Since the theoretical formulas for the ramp and constant parts have the same unknown variables, the two formulas can be used to fit the overall measured data and obtain the required variable values when calculating the relaxation modulus. By substituting the generalized Maxwell model of linear viscoelastic materials, the relaxation modulus function of the required solution can be obtained.

The experimental data obtained using the ramp constant loading test method needs to be measured through a force sensor. Therefore, the force change curve data for the ramp loading section will be affected by the accuracy of the force sensor, thereby affecting the minimum value of the data measured in the time domain. Constant data can be measured for an infinite time under ideal conditions. Therefore, the calculation of the relaxation modulus function based on the obtained experimental data is only affected by the accuracy of the force sensor and is not limited by the measurement method.

Using the above analysis, the ramp constant loading test method can be used to calculate the relaxation modulus function values in the entire time domain during the experimental measurement process compared with the variable rate loading method. However, the rigid flat punch indenter used by Du et al. was prone to material damage during the experiment due to its sharp edges. Second, when using a flat punch for testing, it is difficult to ensure that the flat punch moves vertically into the material plane, which seriously affects the accuracy of the measurements.

Therefore, in order to comprehensively determine the viscoelastic properties of materials, we need a suitable measurement method. Analyzing various relaxation modulus testing methods under the existing instrument indentation method showed that the ramp constant loading test method is more suitable for measuring the relaxation modulus of materials compared with other testing methods. However, due to the shortcomings of the flat punch, we propose the use of a spherical indenter to measure the relaxation modulus through ramp constant loading.

3. Design of Relaxation Modulus Device for Viscoelastic Materials

To test the relaxation modulus, it is necessary to press the spherical indenter into viscoelastic materials at a lower speed during the experiment. The stepping motor responsible for power output is unstable at lower speeds, thus the stepping motor was set to rotate at a higher appropriate speed to ensure the accuracy of the indenter while in motion. Deceleration was then conducted using the deceleration lever connected to the sliding table so that the spherical indenter can obey the predetermined law of motion. We divided the mechanical system of the experimental device into the power part and the loading part.

A representation of the front view of the power part of the device is shown in Figure 1. The power part of the experimental device comprises a stepping motor, ball screw and sliding table. These parts were fixed on the channel steel of the bottom plate through an L-shaped support base. The stepping motor was fixed on the ball screw slide module and the shaft of the motor was connected to the ball screw through coupling. When the motor rotates, it drives the ball screw to rotate and forces the sliding table to move up and down along the guide rail to achieve power output. The sliding table is connected to the nonfixed end of the lever through a shaft pin. When the sliding table moves up and down, it drives the shaft pin to move back and forth in the groove of the lever, thus ensuring the rising and pressing of the lever.

We chose the KL42BYGH405 stepping motor for the dynamic part of the viscoelastic material relaxation modulus test device. This type of motor has several advantages, including high motion accuracy, small step angle error, wide working temperature range, large output torque and high-temperature resistance, and thus meets the requirements of the experiment. We adopted the stepping motor linear ball screw slide assembly with the CBX1204-100 model for the stepping motor ball screw slide component. The module is characterized by high positioning accuracy, large maximum operating load and long effective working distance and can be used to transmit high-speed motion. The L-shaped support plate is a Q235 steel sheet with a thickness of 5 mm, which was folded into a 90° angle by a bending machine. This support plate has high stiffness and facilitates processing. The stepping motor was fixed to the ball screw slide assembly using screws, and the ball screw slide assembly was rigidly connected to the L-shaped support plate using screws. Finally, the L-shaped support seat was fixed to the channel steel at the bottom with four bolts. This connection method ensures a rigid connection between parts, prevents movement between parts when the stepping motor moves and reduces the vibration of the power region. The specific technical parameters for the assembly of the CBX1204-100 stepping motor linear ball screw slide are shown in

Table 1. Parameters of the CBX1204-100 sliding table component.

Parameter	Value
Material	Aluminum alloy + stainless steel + iron
Wire rod	1204
Maximum velocity	100 mm/s
Ball screw pitch	4 mm
Maximum load	80 kg
Positioning accuracy	0.001 mm
Effective range	400 mm
Screw installed	M4
Supporting motor	42/57 stepper motor shaft (8 mm)
Recommended effective length	within 1500 mm

.

The lever principle should be used for the test device to decelerate the linear motion of the slide table. We designed a lever with a reduction ratio of 20:1 to achieve a 20-fold reduction in the speed of the slide table through the lever principle and to ensure that the speed of the spherical indenter transmitted by the sliding table through the lever and the loading rod (Figure 2) met the predetermined speed requirements. This was achieved by pressing the spherical indenter onto the viscoelastic material and predetermining the stable operation speed of the stepping motor. The speed of the sliding table can be obtained and then the corresponding speed of the stepping motor can be deduced based on the reduction ratio of the lever. The design of the loading part is described in detail in the next paragraph.

The screw was connected to the bottom channel steel and connector 1 using nuts, and the lever and connector 1 were connected using pin 1 to fix one end of the lever (Figure 2). Pin 2 was used to connect the lever and connector 2. The pin 2 hole is oval to allow the left and right movement of pin 2. Both ends of pin 1 and pin 2 have threads to make sure the two pins are fixed to the two connectors without moving through the threaded nut connection. Connector 2 is rigidly connected to the loading rod using the threaded nut. A spring base was placed at the lower end of connector 2. The spring was squeezed by the spring base and the bearing to provide an upward force to connector 2, thus ensuring that pin 2 and the lever are always in contact. The bearing was vertically fixed on horizontal plate 1 with a thickness of 5 mm using hexagon socket bolts and nuts. The loading rod moves up and down in the bearing, hence the direction of movement of the loading rod is maintained in the vertical direction. The lower end of the loading rod is connected to the force sensor through threading and the force sensor is connected to the spherical indenter using a short screw. As a result, the downward moving distance of the loading rod was the depth of the spherical indenter pressing into the viscoelastic material and the pressure data measured by the force sensor was the stress response of the viscoelastic material. The pressing depth can be obtained by measuring the moving distance between the displacement sensor and the sensing plate (plate 3) fixed using the loading rod. The viscoelastic material to be measured was fixed on the bottom channel steel using plate 2 and bolts. A hole is present at the leftmost end of the lever for hanging heavy objects. We hang a sufficient weight through the hole when measuring the relaxation modulus of viscoelastic materials to reduce the vibration of the device during movement and improve movement accuracy.

In summary, the measuring device can convert the high-speed rotation of the stepping motor into linear motion of the sliding table. Subsequently, it can convert the motion of the sliding table into the linear motion of the low-speed vertical direction of the loading rod through the lever. The spherical indenter rigidly connected to the loading rod presses the viscoelastic material vertically. We can make the rigid spherical indenter press into the viscoelastic material by controlling the speed of the stepping motor, and a specific strain loading of the viscoelastic material can be achieved. The force sensor fixed on the loading rod can determine the stress response of viscoelastic material to a specific strain loading. The displacement sensor can indirectly monitor whether the movement law of the indenter meets the experimental requirements by measuring the law of movement of plate 3.

4. Least Square Method

Previous scholars used a nano-indentation instrument to determine the creep compliance of viscoelastic materials, then used the conversion relationship between creep compliance and relaxation modulus to calculate the relaxation modulus. However, the use of a conversion relationship to calculate the relaxation modulus is not reliable due to the conversion error. Therefore, accurate relaxation modulus can be obtained by directly measuring it rather than indirectly obtaining the value through conversion. In this study, the relaxation modulus was measured using a direct method.

The load-displacement relationship of a viscoelastic material can be derived directly from the load-displacement relationship of elastic material under the condition that the contact area between the indenter and the test material does not change. As a result, the relaxation modulus of a viscoelastic material can be obtained. The indenter is gradually pressed down during the test process, that is, the contact area between the indenter and viscoelastic material increases with time. Therefore, the load-displacement relationship of viscoelastic materials cannot be derived directly from the load-displacement relationship of elastic materials. Lee and Radok [24] solved this problem by introducing a generic integral operator.

In this study, a series of expressions for linear viscoelasticity of materials were derived based on the Hertz contact theory of linear elastic objects. The load-displacement relationship of a linear elastic material pressed by a spherical indenter can be expressed as follows:

$$P=\frac{4\sqrt{R}}{3(1-v^2)}E{h}^{\frac{3}{2}}$$

(1)

where $P$ is the load; $R$ is the radius of a small ball; $h$ is the displacement; $\nu$ is Poisson’s ratio of elastic materials; $E$ is the elastic modulus of elastic materials.

After introducing the generic integral operator, the load-displacement relationship between linear viscoelastic materials and time $t$ can be expressed as shown below:

$$P(t)=\frac{4\sqrt{R}}{3(1-v^2)}{\displaystyle {\mathsf{\int }}_0^{t}E(t-\xi )\frac{d{h}^{\frac{3}{2}}(\xi )}{d\xi }}d\xi$$

(2)

The speed of pressing the small ball into the viscoelastic material is controlled as a uniform linear motion with velocity $V_0$ as follows:

$$h(t)=V_{0}t$$

(3)

Equation (3) was substituted in Equation (2) to yield:

$$P(t)=\frac{2V_0{}^{\frac{3}{2}}\sqrt{R}}{1-v^2}{\displaystyle {\mathsf{\int }}_0^{t}E(\xi )\sqrt{t-\xi }}d\xi$$

(4)

The relaxation modulus equation of viscoelastic materials under the generalized Maxwell model [25] is shown below:

$$E(t)=E_0+{\displaystyle \sum _{i=1}^N{E}_i}{e}^{-\frac{t}{{\tau }_i}}$$

(5)

where $N$ is the number of components; $E_0,E_i(i=1,2,\dots ,N)$ represents various relaxation modulus parameters; ${\tau }_i$ represents each relaxation time.

Equation (5) was substituted in Equation (4) to yield:

$$P(t)=\frac{2V_0{}^{\frac{3}{2}}\sqrt{R}}{(1-{\nu }^2)}(\frac{2}{3}{E}_0{t}^{\frac{3}{2}}+{\displaystyle \sum _{i=1}^N{\displaystyle {\mathsf{\int }}_0^t{E}_i}}{e}^{-\frac{\xi }{{\tau }_i}}\sqrt{t-\xi }d\xi )$$

(6)

The long expressions can be simplified as follows:

$${\displaystyle {\mathsf{\int }}_0^{t_k}{e}^{-\frac{\xi }{{\tau }_i}}}\sqrt{t_k-\xi }d\xi =I_{ik}$$

(7)

$$\frac{V_0{}^{\frac{3}{2}}\sqrt{R}}{1-{\nu }^2}=V$$

(8)

where $t_k$ represents each time point $(k=1,2,3,\dots M)$. ${\tau }_i(i=1,2,3,\dots ,N-1)$ was defined as equally spaced and can be expressed as shown below:

$$\lg {\tau }_{i+1}-\lg {\tau }_i=a(cons\tan t)$$

(9)

The least square method was used in this study. We obtained the load data through experiments. Subsequently, we obtained the relaxation modulus function of the material by minimizing the sum of the squares of the errors between the fitting data and the experimental data.

Let the right side of Equation (6) be equal to $f(t)$ and each time point be $t_k(k=1,2,3,\dots ,M)$. The sum of squared errors of $P(t)$ can be denoted as shown below:

$$S={{\displaystyle \sum _{k=1}^M\left[P(t_k)-f(t_k)\right]}}^2$$

(10)

If $S$ is minimized then:

$$\frac{\partial S}{\partial E_0}=0$$

(11)

$$\mathrm{i}.\mathrm{e}.,-\frac{8}{3}V{\displaystyle \sum _{k=1}^M\left[P(t_k)-f(t_k)\right]}{t}_k{}^{\frac{3}{2}}=0$$

(12)

$$\mathrm{i}.\mathrm{e}., {\displaystyle \sum _{k=1}^M\left[P(t_k)-f(t_k)\right]}{t}_k{}^{\frac{3}{2}}=0$$

(13)

$$\mathrm{i}.\mathrm{e}., E_0{\displaystyle \sum _{k=1}^M\frac{4}{3}}V{t}_k{}^3+2V{\displaystyle \sum _{i=1}^N{E}_i{\displaystyle \sum _{k=1}^M{I}_{ik}{t}_k{}^{\frac{3}{2}}}}={\displaystyle \sum _{k=1}^{M}P(t_k})t_k{}^{\frac{3}{2}}$$

(14)

For the jth $E$, i.e., $E_j(j=1,2,3,\dots ,N)$, take its derivative and make it zero to get the following expression:

$$\frac{\partial S}{\partial E_j}=0$$

(15)

$$\mathrm{i}.\mathrm{e}.,-4V{\displaystyle \sum _{k=1}^M{I}_{jk}\left[P(t_k)-f(t_k)\right]}=0$$

(16)

$$\mathrm{i}.\mathrm{e}., {\displaystyle \sum _{k=1}^M{I}_{jk}\left[P(t_k)-f(t_k)\right]}=0$$

(17)

$$\mathrm{i}.\mathrm{e}., E_0{\displaystyle \sum _{k=1}^M\frac{4}{3}}V{t}_k{}^{\frac{3}{2}}{I}_{jk}+2V{\displaystyle \sum _{i=1}^N{E}_i}{\displaystyle \sum _{k=1}^M{I}_{ik}}{I}_{jk}={\displaystyle \sum _{k=1}^M{I}_{jk}}P(t_k)$$

(18)

The form of the matrix is as follows:

$$AE=B$$

(19)

$$A=\left[\begin{array}{ccccccc}\frac{4}{3}V{\displaystyle \sum _{k=1}^M{t}_k{}^3}& 2V{\displaystyle \sum _{k=1}^M{I}_{1k}{t}_k{}^{\frac{3}{2}}}& 2V{\displaystyle \sum _{k=1}^M{I}_{2k}{t}_k{}^{\frac{3}{2}}}& \dots & 2V{\displaystyle \sum _{k=1}^M{I}_{ik}{t}_k{}^{\frac{3}{2}}}& \dots & 2V{\displaystyle \sum _{k=1}^M{I}_{Nk}}{t}_k{}^{\frac{3}{2}}\\ \frac{4}{3}V{\displaystyle \sum _{k=1}^M{t}_k{}^{\frac{3}{2}}{I}_{1k}}& 2V{\displaystyle \sum _{k=1}^M{I}_{1k}{}^2}& 2V{\displaystyle \sum _{k=1}^M{I}_{2k}{I}_{1k}}& \dots & 2V{\displaystyle \sum _{k=1}^M{I}_{ik}{I}_{1k}}& \dots & 2V{\displaystyle \sum _{k=1}^M{I}_{Nk}{I}_{1k}}\\ \frac{4}{3}V{\displaystyle \sum _{k=1}^M{t}_k{}^{\frac{3}{2}}{I}_{2k}}& 2V{\displaystyle \sum _{k=1}^M{I}_{1k}{I}_{2k}}& 2V{\displaystyle \sum _{k=1}^M{I}_{2k}{}^2}& \dots & 2V{\displaystyle \sum _{k=1}^M{I}_{ik}{I}_{2k}}& \dots & 2V{\displaystyle \sum _{k=1}^M{I}_{Nk}{I}_{2k}}\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \frac{4}{3}V{\displaystyle \sum _{k=1}^M{t}_k{}^{\frac{3}{2}}{I}_{jk}}& 2V{\displaystyle \sum _{k=1}^M{I}_{1k}{I}_{jk}}& 2V{\displaystyle \sum _{k=1}^M{I}_{2k}{I}_{jk}}& \dots & 2V{\displaystyle \sum _{k=1}^M{I}_{ik}{I}_{jk}}& \dots & 2V{\displaystyle \sum _{k=1}^M{I}_{Nk}^{}{I}_{jk}}\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \frac{4}{3}V{\displaystyle \sum _{k=1}^M{t}_k{}^{\frac{3}{2}}{I}_{Nk}}& 2V{\displaystyle \sum _{k=1}^M{I}_{1k}{I}_{Nk}}& 2V{\displaystyle \sum _{k=1}^M{I}_{2k}{I}_{Nk}}& \dots & 2V{\displaystyle \sum _{k=1}^M{I}_{ik}{I}_{Nk}}& \dots & 2V{{\displaystyle \sum _{k=1}^M{I}_{Nk}{}^2}}^{}\end{array}\right]$$

(20)

$$E^T=\left[\begin{array}{ccc}{E}_0& E_1& \begin{array}{cccc}\begin{array}{cc}{E}_2& \dots \end{array}& E_i& \dots & E_N\end{array}\end{array}\right]$$

(21)

$$B^T=\left[\begin{array}{ccc}\begin{array}{cccc}\begin{array}{cc}{\displaystyle \sum _{k=1}^{M}P(t_k)t_k{}^{\frac{3}{2}}}& {\displaystyle \sum _{k=1}^{M}P(t_k)I_{1k}}\end{array}& {\displaystyle \sum _{k=1}^{M}P(t_k)I_{2k}}& \dots & {\displaystyle \sum _{k=1}^{M}P(t_k)I_{jk}}\end{array}& \dots & {\displaystyle \sum _{k=1}^{M}P(t_k)I_{Nk}}\end{array}\right]$$

(22)

$E_0,E_i(i=1,2,3,\dots ,N)$ can then be solved using the least square method.

Knauss and Zhao [26] proposed that the ramp-constant strain is equivalent to the sum of two ramp strains. The ramp-constant strain can be divided into the sum of two ramp strains and the constant section is equal to the sum of the ramp behavior and its inverse behavior (Figure 3). Ping et al. used this relationship to derive the load formula of the ramp constant strain of the flat punch indenter. We can obtain the load formula of the ramp constant strain of the spherical indenter by referring to the derivation process. The load formula of the constant part can be obtained as follows:

$$P(t)=\frac{2\sqrt{R}{V}_0{}^{\frac{3}{2}}}{1-{\nu }^2}{\displaystyle {\mathsf{\int }}_0^{t}E(\xi )\sqrt{t-\xi }}d\xi -\frac{2\sqrt{R}{V}_0{}^{\frac{3}{2}}}{1-{\nu }^2}{\displaystyle {\mathsf{\int }}_0^{t-t_0}E(\xi )\sqrt{t-\xi }}d\xi$$

(23)

Equation (5) was substituted to Equation (23) to yield:

$$P(t)=\frac{2\sqrt{R}{V}_0{}^{\frac{3}{2}}}{1-{\nu }^2}⟮\frac{2}{3}{E}_0{t}_0{}^{\frac{3}{2}}+{\displaystyle \sum _{i=1}^N{\displaystyle {\mathsf{\int }}_{t-t_0}^t{E}_i{e}^{-\frac{\xi }{{\tau }_i}}\sqrt{t-\xi }d\xi }}⟯$$

(24)

The expressions for calculating the ramp-constant strain are shown below:

Ramp loading part:

$$P(t)=\frac{2V_0{}^{\frac{3}{2}}\sqrt{R}}{(1-{\nu }^2)}(\frac{2}{3}{E}_0{t}^{\frac{3}{2}}+{\displaystyle \sum _{i=1}^N{\displaystyle {\mathsf{\int }}_0^t{E}_i}}{e}^{-\frac{\xi }{{\tau }_i}}\sqrt{t-\xi }d\xi )(t<t_0)$$

(25)

Constant part:

$$P(t)=\frac{2\sqrt{R}{V}_0{}^{\frac{3}{2}}}{1-{\nu }^2}⟮\frac{2}{3}{E}_0{t}_0{}^{\frac{3}{2}}+{\displaystyle \sum _{i=1}^N{\displaystyle {\mathsf{\int }}_{t-t_0}^t{E}_i{e}^{-\frac{\xi }{{\tau }_i}}\sqrt{t-\xi }d\xi }}⟯(t\ge t_0)$$

(26)

5. Results and Analysis

5.1. Process and Experiment Results

The purpose of this experiment was to use the experimental device to measure the viscoelastic function over a long period within the small deformation (linear viscoelasticity) range of the rubber material. Therefore, it was necessary to ensure that the spherical indenter pressed onto the rubber material at low speeds. The minimum speed that the experimental device could control was about 0.15 μm/s, so the experimental device was used to control the pressing of the indenter on the 75 HA rubber sheet during ramp loading at a uniform speed of 0.18 μm/s, 0.30 μm/s and 0.45 μm/s. Notably, multiple speeds were used for the test to obtain reliable test results. The displacement scatter plot of the spherical indenter measured by the displacement sensor is shown in Figure 4. We only read the data for 900 microns to ensure small deformation of the rubber sheet. Repeated experiments showed that the linear boundary of the rubber material was about 1000 microns. The linear boundary of rubber material will not be described in detail since it is not the main focus of this study.

The velocity of the spherical indenter calculated fluctuated due to vibration, but the velocity of the spherical indenter was approximately expressed as 0.18 μm/s, 0.30 μm/s and 0.45 μm/s under uniform linear motion (Figure 4).

$E_0=3,550,288.64Pa$, $E_1=23,140,015,764.1Pa$, $E_2,E_3,\dots ,E_{13},E_{14}=0Pa$ can be obtained using the nonnegative least square method in MATLAB if we make ${\tau }_1$ equal to 0.01, $a$ equal to 0.5 and $N$ equal to 14 in Equation (9). The lsqnonneg MATLAB function was used for this calculation.

When $t$ is within the range of 0 s–5000 s, the relaxation modulus of the material can be expressed as:

$$E(t)=3,550,288.64+23,140,015,764.1e^{-100t}$$

(27)

The relaxation modulus of the rubber sheet can be characterized using the three-element Maxwell model according to Equation (27).

We carried out ramp-constant loading and controlled the small ball to press the material at a constant speed of 0.5 μm/s to verify the accuracy of the measured material relaxation modulus. The small ball was stopped after it had pressed the material for 2000 s, and the time was then increased to 5000 s. A displacement scatter plot of the ramp-constant load is shown in Figure 5. The scatter plot in the static phase cannot be completely horizontal due to the effect of the mechanical vibration of the device, but it can be approximated to the static state within the precision range of the micrometer level.

The loads measured and calculated with the material relaxation modulus fitted parameters using Equations (25) and (26) are shown in Figure 6.

5.2. Error Analysis

A small relative error of less than 2.5% (in the initial stage and the transition stage of uniform motion and standstill, the error was slightly larger, ranging from 2 to 2.5%) was observed between the experimentally measured load and the calculated load (Figure 7). This error can be attributed to the vibration of the experimental device and the calculation error. The small value of the error indicates the correctness and practicability of Equations (25) and (26).

6. Conclusions

The experimental device designed in this study can use the spherical indenter to measure the load and displacement. The load expressions of ramp loading and ramp constant loading were derived. The parameter value of the relaxation modulus of linear viscoelastic materials can be obtained directly and accurately using the experimental data and the non-negative least square method, and the error can be maintained within a certain range. The findings indicate that the experimental device and calculation method have relatively high feasibility and practicality and can be used to test the viscoelastic parameters of any conveyor belt.

However, the study has some limitations that should be addressed. For example, the minimum speed of the pressing of the indenter controlled by this device was 0.15 μm/s, and a speed less than 0.15 μm/s could not be achieved. Therefore, the relaxation modulus function was not tested over a longer time range. In addition, the fitting error was greater than 2% at the initial stage and the transition stage of the constant speed and static state during ramp-constant loading due to the mechanical vibration of the device and the calculation error. Therefore, the experimental device should be further improved to reduce the fitting error.