Abstract
The purpose of this work is to study the effects of different loading rate ratios and loading speeds on the biaxial tension of hydroxyl-terminated polybutadiene (HTPB) solid propellant. A proper kind of biaxial tensile specimen with which the stresses in its central part can be obtained with the loads acted on each loading direction is designed and used in the study, and the strains in its central parts are obtained with the digital image correlation (DIC) method. The stress and strain relationship at each direction can be obtained by experiments. The uniaxial stress vs. strain curves and the biaxial stress vs. strain curves were obtained, and it was found that the loading speed remarkably influenced the biaxial tensile behaviors of HTPB propellant. The Mises equivalent stress and strain could be used to describe the biaxial tension stress and strain state, and the exponential constitutive model obtained in the study could be used to predict the stress vs. strain curve under different test conditions.
1. Introduction
In the course of manufacturing, storage, transportation, ignition, and flight, solid rocket motor (SRM) grain was exposed to stresses such as heat, impact, vibration, acceleration, and ignition pressure [1–3]. If the grain’s stress and strain levels exceeded acceptable levels, damages like grain fracture, deformation, and other things could happen. This would seriously impair the SRM’s working performance and could even result in the engine’s total destruction [4, 5]. It is necessary to determine the materials’ permissible limits for stress and strain in order to evaluate the SRM’s structural integrity. The structural integrity study of grain is based on research on the mechanical characteristics and constitutive model of solid propellant [6–9]. External temperature, loading strain rate, and loading mode all had a clear impact on the mechanical characteristics of solid propellants [10–12]. To inform the structural design of grain and guarantee the appropriate operation of SRM, research on the mechanical characteristics and constitutive model of solid propellants under various loading circumstances was of major value [13]. The mechanical characteristics of propellant and its deterioration under various loading circumstances have received a lot of attention [14, 15]. Temperature, stress and strain condition, etc., all have a significant impact on the mechanical characteristics and fracture mechanisms of solid propellants [16].
In order to understand the behavior of composite propellants during engine ignition, the mechanical and ultimate performance of propellants filled with water-terminated polybutadiene were studied under imposed hydrostatic pressure [17]. The mechanical response of the propellant was obtained by uniaxial tensile and simple shear tests at several temperatures, strain rates, and superimposed pressures from atmospheric pressure to 15 MPa. The difficulty of directly measuring stress in solid propellants was noted and discussed. In order to evaluate the stress in solid propellants, the piezoresistive phenomenon was explored and demonstrated [18]. The uniaxial compressive mechanical response of two high-energy polymer propellants, M30 and JA2, was studied using statistical factor design techniques. It was found that the aspect ratio and end lubrication variables can affect the overall mechanical behavior of these materials [19]. The stress vs. strain curve and tensile fracture surface of HTPB propellant were studied using uniaxial and biaxial tensile tests over a wide range of temperature and strain rates, respectively. The results indicate that the strength of the material increases with the decrease of environmental temperature and the increase of loading strain rate [20, 21]. Previous studies have shown that uniaxial testing could not determine the accurate mechanical properties of high particle-filled elastomers subjected to complex loads. Therefore, conducting uniaxial tests of solid propellants alone could not meet the research requirements for the mechanical properties of solid propellants.
In fact, the loading of solid propellants during storage and transport is very complex. In recent decades, attempts have been made to investigate the mechanical properties of materials under biaxial loading using a variety of specimens and loading devices. The study of viscoelastic properties as well as the effect of softening and the occurrence of induced anisotropy in filled elastomers was completed by biaxial loading tests using a four-vector test rig, the Zwick/Roell test machine [22]. The effect of vertical tensile load and bidirectional tensile load ratio on the mechanical properties of single-ply nylon rope and rubber composites under bidirectional tensile conditions was investigated [23]. A new test fixture that can be used for biaxial compression tests on a uniaxial testing machine was designed, and the biaxial compression behavior of polymer foam was investigated [24]. To design and improve the protective properties of Kevlar 49 fabrics, the stress-strain response, the apparent Poisson’s ratio, and the in-plane shear response of Kevlar 49 fabrics in the warp and weft directions were investigated by biaxial tensile tests. The experimental results showed that it has the same Young’s modulus in the warp and filling directions, but different curl strains, tensile strengths, and ultimate strains [25]. A newly designed aluminum apparatus and uniaxial Instron tester were used [26]. Then, quasibiaxial tensile stress responses were obtained for HTPB propellants at room temperature and at different high strain rates. Based on the above studies, it can be seen that the biaxial tensile test type has been developed through the biaxial tensile test of film convex expansion, biaxial tensile test of internally pressed thin-walled cylinders or spherical shells, and biaxial tensile test of plane cross-shaped specimens [27–30]. Comparison of the results of their research can be seen, cross-shaped specimen biaxial tensile results are closer to the mechanical properties of materials subjected to complex loads, and cross-shaped biaxial tensile test to promote the development of biaxial tensile testing machine, especially the plane cross-shaped specimen for biaxial tensile test, has the advantages of frictionless, measurement of noncontact, and so on, and at the same time, changing the two-axis loading displacement ratio can be obtained by the different strain paths of linear or nonlinear. At the same time, by changing the loading displacement ratio of the two axes, the linear or nonlinear equivalent force strain curves of different strain paths can be obtained, which is very popular. However, there are few studies on the biaxial tensile experiments on cross specimens of solid propellant HTTP. So far, the main conclusions of the biaxial mechanical property studies on solid propellant HTPB are based on quasibiaxial mechanical experiments [10, 20]. Since the quasibiaxial experiments cannot control the loading rate ratio in both directions very accurately, it is doubtful whether the results of quasibiaxial experiments can accurately reflect the biaxial mechanical properties. In reality, the biaxial loading to which the material is subjected is complex and varied, in which the nonequivalent biaxial loading is dominant, and the nonequivalent loading situation cannot be skipped in the study of the biaxial mechanical properties. Due to experimental difficulties, it is difficult to effectively obtain the biaxial tensile behavior of solid propellant materials. Quasibiaxial stretching can only obtain biaxial tensile properties of the material for which the equivalent load ratios are approximated. Biaxial tests on solid propellant butyl hydroxyl propellants with different load ratios have not been successfully carried out and do not provide relevant mechanical properties. Therefore, in order to gain a comprehensive understanding of solid propellants, this work will investigate the biaxial tension of solid propellants at different load ratios.
At present, quasistatic biaxial tensile mechanical property tests of solid propellants can be divided into two categories according to the type of experimental setup. One type is the uniaxial tensile testing machine with strip specimens [31]. The other type is the biaxial tensile testing machine test method with crossed specimens [32]. Crossed specimens are more suitable for biaxial tensile tests due to the biaxial tensile limitations of strip specimens [33–36]. Due to the low modulus and low strength physical properties of solid propellants, as well as the high level of risk and the complex carving process of solid propellant cross-shaped specimens, measures such as center thinning or slotting in the arm alone are not well suited to improve the reliability of biaxial tensile mechanical property testing of cross-shaped specimens [37].
In this paper, the biaxial stretching experiment of hydroxyl-terminated polybutadiene (HTPB) propellant at two different loading speeds, four kinds of loading rate ratios, and at room temperature is going to be performed. A newly designed cruciform sample with a reduced center section and slots is used. The relationship between load and stress in the central portion of the specimen in each corresponding direction is obtained by means of the linear viscoelastic intrinsic structure that comes with ABAQUS. The tensile behaviors of HTPB propellant under different loading speed and different loading rate ratios are discussed, and a constitutive equation is established.
2. Experimental
2.1. Material
The test material was HTPB solid rocket propellant, and the matrix was a viscoelastic material and contains metal particles. The solid propellant consisted of AP (ammonium perchlorate, 69.50 wt. %) as an oxidizer, Al (aluminum powder, 18.50 wt. %) powders as a metal fuel, DOS (di-2-ethylhexyl sebacate, 3.40 wt. %) as a plasticizer, MAPO (tri-(2-methylaziridinyl) phosphine oxide, 0.05-0.10 wt. %) as a bonding agent, TDI (2,4-toluene diisocyanate, 1.0-2.0 wt. %) as a curing agent, HTPB (hydroxyl-terminated polybutadiene, 0.6-0.7 wt. %) as a binder, and additives (liquid, 0.50-1.0 wt. %). Moreover, the sizes of larger ammonium perchlorate (AP) particles were within the range of 100–400 μm in diameter. The sizes of smaller AP particles were less than 35 μm in diameter. The mean size of aluminum particles was 22 μm. These components eventually form an over reticulated HTPB matrix.
2.2. Sample Preparation
Biaxial tensile test is the extension and expansion of uniaxial tensile test, so in the study of biaxial tensile properties, it is indispensable to compare the results of biaxial tensile test with the results of uniaxial tensile test. According to the Chinese aerospace industry standard, GJB 770B-2005, the uniaxial tensile sample was designed as a type B dumbbell shape, as shown in Figure 1(a). But there were no standard available for the dimensions of biaxial tensile sample. Based on the previous research, a new sample was designed which was slotted on the arm and widened the radius of the arc segment. A schematic diagram of the cruciform sample is shown in Figure 1(b).
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(b)
The following are the benefits and enhancements of the biaxial tensile specimens developed in this paper as opposed to earlier specimens, as shown by the biaxial specimen diagram in Figure 1(b): (1)Compared to previous biaxial tensile specimens [38, 39], the center of the specimen in this paper has been reduced to a very thin thickness of 5 mm, which can achieve significant deformation at the center(2)In the past, the shear stress in the middle area was removed from the wall using just one groove to simplify the operation, but the results were not satisfactory. Due to this, two grooves with a 3 mm radius were carved into the specimen’s wall. As a result, the shear stress in the central region will be significantly reduced or even eliminated, causing the central deformation zone’s stress primary axis to coincide with the tensile direction. The major tension in the central region may be measured and calculated more easily using this groove(3)As opposed to earlier biaxial examples, each corner of the specimen only had one chamfered corner. Large shear stresses at the individual chamfered corners can cause unequal positive stresses in the central zone and deviate from the direction of tension, which leads to mistakes in the calculation of stresses in the central zone. Shear stresses can also extend towards the central zone as a result. Each corner of the biaxial specimens in this paper has two 8 mm radius chamfered corners. This will significantly lower the shear stress at the corners and hence lower the stress in the central zone calculation error
In order to verify the validity of the newly designed biaxial tensile specimen, this paper uses numerical simulation calculations and test pictures. The deformation of 1/8 finite element model of cross biaxial tensile cruciform specimen with different loading rate ratios is calculated and analyzed by using ABAQUS software. Figure 2 shows the Mises stress diagram of the specimen at 20% strain under isometric biaxial tensile loading. In the calculations, the propellant is modelled by the “Prony series” viscoelastic structure model with Poisson’s ratio of 0.495 and a mesh of C3D8RH mesh elements. From Figure 2, it can be seen that the maximum stress occurs in the central region, and the stress value in this region is generally larger than that in the noncentral region, which is sufficient to ensure that the specimen is damaged from the central region. In addition, the stress concentration in the central region is not obvious.
By photographing the tensile damage process of specimen B18 under 1 : 1 axial loading, as shown in Figure 3, the cracks expanded towards the corners of the specimen limbs until they ran through the whole specimen. The tensile load on the specimen limbs forms a resultant force in the central region of the specimen, i.e., the tensile loads at the two adjacent ends are combined to form a larger load pointing in the direction of the chamfered corners of the limbs, which causes the specimen to fracture along the chamfered direction of the limbs. At the same time, the central region of the specimen is the thinnest part of the whole specimen, while the joints of the specimen limbs are relatively thick, so the crack initiation point of the specimen occurs in the relative central region of the specimen. The fracture behavior is in accordance with the expected results of the isometric biaxial tensile test. Combined with Figures 2 and 3, it can be seen that the newly designed biaxial tensile specimen is effective and can meet the test requirements.
(a) Prefracture
(b) Fracturing
(c) Complete fracture
2.3. Experiment
The newly designed biaxial tensile specimens processed in this paper were produced using a casting production method with the highest degree of fault tolerance. Firstly, the HTPB propellant formulation shown in section 2.1 was mixed proportionally at 58°C. The release agent was applied to the surface of the custom mould coated with the release agent, and then, the mixed propellant slurry was poured into the mould. Finally, the mould is placed at a constant temperature of 20°C, and the propellant sample is removed after the slurry has completely solidified. After processing, the specimen is placed in a drying oven for drying. Due to the most fault-tolerant casting production method, initial damage to the specimen during processing is almost negligible and does not affect the accuracy of subsequent test results.
The uniaxial tensile test was carried out on a universal test machine Zwick Z005 at a room temperature of approximately 23°C, as shown in Figure 4(a). An extensometer and load cell were used to record the deformation and the load acted on the uniaxial tensile specimen simultaneously. The biaxial tensile test was carried out on the MTS biaxial tensile testing machine, as shown in Figure 4(b), the load cells in each direction record the load acted on each arm of the cruciform specimen. The deformation of the center part of the cruciform specimen was obtained with camera, and the strains vs. time in each direction were calculated with the digital image correlation (DIC) method.
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(b)
The experiments were performed with displacement control. The uniaxial tension experiment was carried out at three displacement loading speeds of 1 mm/min, 2 mm/min, and 20 mm/min, respectively. The biaxial experiment was carried out with four different loading rate ratios and two different loading speeds in the direction. The specific implementation method was to apply 2 mm/min and 100 mm/min displacement loading speed in the direction. For the displacement loading speed of 100 mm/min in the direction, by changing the displacement loading speed in the direction, such as applying 100 mm/min, 50 mm/min, 33.33 mm/min, and 25 mm/min in the direction, then the loading rate ratios of : of 1 : 1, 1 : 2, 1 : 3, and 1 : 4 could be gotten. For the displacement loading speed of 2 mm/min in the direction, the displacement loading speeds in the direction were applied with 2 mm/min, 1 mm/min, 0.666 mm/min, and 0.5 mm/min, respectively, to get the loading rate ratios of : of 1 : 1, 1 : 2, 1 : 3, and 1 : 4 loading scheme. The test conditions have been listed in Table 1, which includes the test number specimen and the loading method, as follows.
A Nikon D800 camera was used to take pictures. For the test with a loading rate of 100 mm/min in thedirection, the photo interval was one second for one picture. About the test with a loading rate of 2 mm/min in the direction, the photo interval was five seconds a picture. To give a better characterization of the specimen surfaces, random speckle pattern with two different colors (black and red) was sprayed onto the specimen surface in the center part.
2.4. DIC Method
The digital image correlation (DIC) method was adopted to measure the displacement field of the center part of the cruciform specimen during the biaxial tensile test. The principle of the DIC method was given in reference [40]. The principal features and procedures of the displacement measurement system were developed in this paper as shown in Figure 5.
2.5. Linear Viscoelasticity
In order to obtain the relationship between the load applied to the specimen arms and the stress in the central portion of the specimen, this paper is carried out by using the linear viscoelastic material [41] model that comes with the finite element software ABAQUS. By simulating the loading conditions with different displacement loading rates, the relationship between loads and stresses in all directions of HTPB composite solid propellant can be obtained.
The constitutive model chosen for this paper is the time-based integral constitutive model as follows: where is the stress, is the strain, is the time, and is the relaxation modulus equation. The relaxation modulus in this paper is expressed in terms of a Prony series of the third order, as follows:
It is sufficient to input each relaxation modulus and its corresponding relaxation time constant in Eq. (2) into the model built in ABAQUS to give the model linear viscoelastic material properties.
3. Results and Discussion
3.1. Biaxial Tensile Displacement Measurement
The images of the center part of the cruciform specimen recorded with camera during experiment were processed with the DIC program, and the displacement field vs. time could be obtained.
Figure 6 shows three of the images of solid propellant HTPB biaxial tensile No. 5 specimen (abbreviated as B5) which were the images of the center part of the cruciform specimen. It was shown that the area of the center part of the specimen increases with the time of the biaxial tension test.
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(b)
(c)
With the image shown in Figure 6(a) as the initial image at , the displacement fields in direction and in direction for the picture shown in Figure 6(a), respectively, were obtained with DIC method, and the results are given in Figures 7(a) and 7(b). The unit of the displacements , , , and was pixel. It was shown from Figure 7 that the displacement fields in the and direction in the central part of the specimen were approximately plane; therefore, the strain field in the central region of the specimen was approximately uniform. So, the design of the specimen shown in Figure 1(b) was suitable for the biaxial tensile experiment. The average strain in this region could be used to represent the strain in the central part of the specimen.
(a)
(b)
3.2. Biaxial Tensile Strain Calculation
The main purpose of this paper was to obtain the stress and strain relationship of HTPB solid propellant. It could be seen from Figure 7(a) that the displacement field in direction was nearly related only with the coordinate of and nearly had no effect with the coordinate of . By linear fitting to the coordinates and , respectively, the results were shown as follows:
It was proved from Eqs. (3) and (4) that the displacement field in direction had a strong linear relationship with the coordinate but had almost no effect with the position. The slope in Eq. (3) was the engineering strain in the direction, and its value was . Similarly, the engineering strain in the direction could also be obtained by linear fitting the displacement field in direction with the coordinate , the engineering strain in the direction could be obtained, and its value was . The tensile true strains in the two lateral directions could be calculated as
By calculating all the images taken during the biaxial tension process, the displacement fields and the corresponding strains in the center part of the all specimens at different times could be obtained. So that, the strain (true)-time curves for all the specimens could be gotten.
Parts of the strain (true) vs. time curves with four loading rate ratios for the specimens tested with the displacement loading speed of 100 mm/min are shown in Figure 8. From the strain (true) vs. time curves shown in Figure 8, it could be seen that the ratio of true strain in the two directions at different times had a little different with the displacement loading rate. For example, in Figure 8(a), the ideal strain (true) vs. time curves in and directions should be the same; although the tendency of the curves was the same, the value of the strains in and directions at the same time had a little difference. This difference was caused by that the loading in two directions could not be exactly synchronous.
(a)
(b)
(c)
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3.3. Biaxial Tensile Stress Calculation
During the experiment, the values of the loads and in both directions could be recorded directly on the MTS experimental machine. In order to obtain the stresses in the center part of the specimens tested, a cross-sample model shown in Figure 1(b) corresponding to the 1 : 1 size of the experimental sample is established. Due to its symmetry, in order to simplify the calculation, a 1/8 model was established, as shown in Figure 9.
The simulations of the biaxial loading with different displacement loading rate ratios were done. The relationship of the load acted on the arms of the specimen and the average stresses in the corresponding directions for four different loading rate ratios are obtained and shown in Figure 10.
From the stress vs. load calculation results in Figure 10, it could be found that the stresses in the central part were only related to the load acted on the arms of the specimen and nearly not related with the loading rate ratios. It was a very useful result that we could get the stresses in the center part with the load acted on the arms of the specimen, regardless of the deviation of the loading rate ratio in a real experiment (such as that in Figure 8). The relationship between the average stress in the central region and the load acted on the arms of the specimen in the corresponding directions could be obtained by fitting the data shown in Figure 10 and was given as Eq. (6).
The unit of stress is MPa, and the unit of load is kN.
With the load acted on each arm of the specimen, the stresses in the center part of the specimen in and direction, respectively, could be obtained with Eq. (7), so that the stress vs. time curves could be drawn. Figure 11 shows parts of the stress vs. time curves that correspond to the strain vs. time curves shown in Figure 8.
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(b)
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With true strain vs. time curves in Figure 8 and the true stress vs. time curves in Figure 11, the true stress vs. strain curves in and direction, respectively, for all the specimens tested in the study could be obtained. Parts of the true stress vs. strain curves with four loading rate ratios for the specimens tested under the displacement loading speed of 100 mm/min are given in Figure 12.
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3.4. The Equivalent Stress vs. Strain Curve
The uniaxial tensile stress and strain was obtained by uniaxial tensile test with test machine Zwick Z005. The true stress vs. true strain curves are shown in Figure 13.
The Mises equivalent stress vs. strain was used to analyze the tensile behavior of HTPB propellant. In this paper, the Mises stress refers neither to nominal nor true stress, but rather to an equivalent stress to the experimentally measured true stress. The Mises stress and strain formulas were given as
The unit of stress is MPa.
With the true stress vs. true strain curves in and directions obtained for the specimens in the biaxial test as shown in Figure 12, the biaxial tensile Mises equivalent stress vs. strain curves of HTPB propellant under different test conditions could be obtained with Eqs. (8) and (9). The Mises equivalent stress vs. strain curves for specimens tested with two different loading speeds of 2 mm/min and 100 mm/min in the direction and different load ratios are shown in Figure 14. The biaxial equivalent stress vs. strain curves for various loading rate ratios exhibit a steady trend, as shown in Figure 14; however, the values vary slightly. The substance HTPB used in this study is a randomly filled composite that contains energy particles; therefore, there will obviously be some variability in the material itself. The mechanical behavior of the material will vary due to this inherent unpredictability, but this variability is acceptable for the purposes of this study.
(a)
(b)
With the test results shown in Figures 13 and 14, the following mechanical parameters were obtained according to the Chinese aerospace industry standard of P.R.C, GJB 770B-2005, and the American JANNAF standard. The uniaxial tensile elastic modulus and biaxial equivalent modulus were defined in the conventional manner as the slope on the initial linear portion of the stress vs. strain curves in Figures 13 and 14. The maximum equivalent tensile stress and the corresponding equivalent strain were also obtained. Tables 2–4 list the mechanical parameters and the average equivalent strain rates under different loading condition, respectively.
It is shown from Figures 13 and 14 that the propellant displays nonlinear material behavior under all the test conditions. From Tables 2–4, it could be seen that the equivalent elastic modulus , the biaxial maximum tensile equivalent stress , and the corresponding equivalent strain increase with the increasing of the average strain rate. It could also be seen that under the same loading speed, the equivalent elastic modulus , the biaxial maximum tensile equivalent stress , and the corresponding equivalent strain did not relate with the loading rate ratios for biaxial tensile test results. It means that the biaxial tensile behaviors of HTPB propellant were remarkably influenced by loading rates but nearly not related with the loading rate ratios for biaxial tension experiment.
3.5. Discussion
The biaxial Mises equivalent stress vs. strain curves shown in Figure 14(a) and the uniaxial stress vs. strain curves tested under a loading speed of 2 mm/min are compared in Figure 15. It was shown that the uniaxial stress vs. strain curve and the Mises equivalent biaxial stress vs. strain curves could be approximately considered as the same. This meant that the Mises equivalent stress and strain could be approximately used to describe the biaxial tensile behaviors of the HTPB propellant material.
Eq. (8) shows an exponential function. where the , , and were the corresponding fitting parameters.
All the stress vs. strain curves excluding the uniaxial tensile tests under loading speed of 20 mm/min would be fitted with Eq. (10). Figure 16 shows parts of the fitting curves, as follows.
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(b)
(c)
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It was found that the exponential function (Eq. (10)) could fit the stress vs. strain curve well in Figure 16. It was found that the fitting parameters value of could be fixed as -1.4. The values of the fitting parameters are shown in Table 5.
It was found that the value of the fitting parameter was linearly related to the strain rate , and the value of the fitting parameter was exponentially related to the strain rate . Their fitting formulas were as follows:
The unit of strain rate is s-1.
Combined with the Eqs. (10), (11), (12), and (13), Eq. (14) could be obtained.
Figure 17(a) shows a comparison of the uniaxial stress vs. strain curve tested at a loading speed of 20 mm/min with the predict curve of Eq. (14). It was shown that although the Eq. (14) was obtained by the fitting results of all the stress vs. strain curves excluding the uniaxial tensile tests under loading speed of 20 mm/min, the predict curve with Eq. (14) fitted well with the experiment curves. The predict curves with Eq. (14) of other strain rates which correspond to the uniaxial tension of 1 mm/min, biaxial tension of 2 mm/min, and biaxial tension of 100 mm/min are also shown in Figures 17(a)–17(c), respectively.
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(b)
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It is shown from Figure 17 that the predict stress vs. strain curve obtained with Eq. (14) coincided well with the tested stress vs. strain curves of uniaxial tensile speed 1 mm/min and 20 mm/min, biaxial tension 2 mm/min, and biaxial tension 100 mm/min. It meant that Eq. (14) could be used to predict the stress vs. strain curve under different test conditions. It was also proved that Mises equivalent stress and strain could be used to describe the biaxial tension stress and strain state.
4. Conclusions
(a)The uniaxial tensile and biaxial tensile experiment for HTPB propellant was performed. It was shown that with the newly designed biaxial specimen, the stresses in the central part of the specimen were only related to the load acted on the arms of the specimen(b)There were still some shortcomings in this article because the biaxial tensile stress values are obtained based on the linear constitutive model. Further calculations based on the constitutive model with damage were needed to more accurately reveal the changes in biaxial tensile mechanical behavior of solid propellants(c)The elastic modulus , , and increased with the increasing of the loading speed(d)The Mises equivalent stress and strain could be used to describe the biaxial tension stress and strain state(e)The constitutive model obtained in the study (Eq. (14)) could be used to predict the stress vs. strain curve under different test conditions
Data Availability
The data that support the findings of this study are available upon reasonable request from the authors.
Conflicts of Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Authors’ Contributions
Li Jin conceptualized the paper, conducted the methodology, wrote the original draft, and reviewed and edited the paper. Qin Zhi Fang wrote the original draft and reviewed and edited the paper. XingWei Yan conducted the investigation. Qin Wei Hu conducted the investigation.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (11021202 and 11272244).