The raw material used for manufacturing the specimens is in the form of cylindrical tube as shown in Fig.3. According to the requirements of the international standard ASTM D638, the specimen should be in the shape of dog-bone and the thickness should be between 1.0 mm and 14.0 mm. In order to ensure that the specimen is not curved in both the length and width directions, the thickness of the specimen should not be too large. The thickness of the specimen is 2.5 mm. The geometry of the specimen is shown in Fig.4.

Both ASTM D638 and ASTM D2991 state that the tensile and relaxation properties of polymer materials are related to conditions such as tensile rate, temperature and other environment conditions during testing. Therefore, in this paper, the MTS universal testing machine in Shanghai Jiao Tong University Engineering Mechanics Test Center is used for the test, which has been verified by the relevant international standards. The tensile machine has a stroke of ±25 mm, a static loading capacity of ±2000 N, a dynamic loading capacity of ±3000 N and 1 high temperature reactor controlled by two groups of thermocouples. The basic structure of this stretching machine is shown in Fig.5, and the specimen mounted on the machine is shown in Fig.6. For this test, an extensometer with 25 mm pitch is used to measure the tensile displacement of the pitch section to calculate its strain.

In this paper, four kinds of polymer materials provided by the producer: PA11, PVDF, PEX-B, and HDPE are firstly subjected to tension at ambient temperature. According to the test requirements of ASTM D638, the tensile test procedure is as follows: Firstly, the specimen is installed on the testing machine as shown in Fig.6; Secondly, the three-quarter force gauge measuring the force load is zeroed; Then, the specimen is stretched at a speed of 4 mm/min and the specimen status is recorded with the force-displacement graph on the monitor. When the force load decreases rapidly, it indicates that the specimen has partially failed and the stretching can be stopped. The maximum displacement that can be measured is 12.5 mm, which corresponds to 50% of the nominal strain. For most polymer materials, the specimen will partially fail before reaching this nominal strain value, so the measuring range of extensometer is enough.

After the tensile testing, PA11 and PVDF materials for stress relaxation tests under high temperature conditions are selected. The reason for choosing these two materials is that in the flexible pipes, they are usually used for the inner sheath, and this structure is often required to directly contact the high-temperature fluid transported inside the flexible pipe, where the stress relaxation phenomenon is more serious. The 18 loading cases for stress relaxation are shown in Tab.1.

The testing procedure for stress relaxation is: (1) The specimen is mounted on the tensile machine where the high-temperature reactor is arranged. Considering that a certain gap will exist between the temperature of thermocouple and the specimen, it is necessary to utilize adhesive tape to attach the temperature sensor to the specimen; (2) The high-temperature reactor should be installed; (3) The thermocouple should be controlled to raise the temperature at a uniform rate so that the temperature of specimen reaches a preset temperature and maintains for 3 hours; (4) The specimen should be stretched to the pre-strain position at a speed of 2.5 mm/min and the indications of the triptych start to be recorded, every 10 s until after 24 h.

The curve used to describe the relaxation behavior is called stress relaxation curve, and its horizontal axis is time, while the vertical axis is the axial load, the stress in material, or the relaxation modulus of material. The classical form of the relaxation curve is shown in Fig.7.

The relaxation modulus is defined as

where σ (t) is the value of the stress in the material at any point in time; ε is the fixed strain when stress relaxation occurs in a material.

The Prony parameter, commonly used to describe stress relaxation phenomena, has two forms

where E₀ is the relaxation modulus of the material at the beginning of relaxation, i.e., at t=0; g_i, τ_i are Prony series describing the relaxation, which have j orders; E_∞ is the relaxation modulus of a material at its relaxation limit. In order to facilitate the discussion of the mechanical characterization of stress relaxation using the normalization method later on, Method 1 is chosen to characterize the stress relaxation curves.

In this paper, the axial force load-time curve F(t)−t can be obtained from the stress relaxation test. Then, according to the relationship between the axial force load F and the stress σ, the testing data is processed to obtain the nominal stress-time curve σ(t)−t. Finally, by Eq.(1), the relaxation modulus-time curve E(t)−t is obtained.

The program of its own is writtern in this paper to identify the Prony parameters in Method 1 shown in Eq.(2) from the relaxation modulus-time data points using the L-M method of nonlinear regression, and the program architecture is shown in Fig.8.

In this paper, tensile tests of four polymer materials at ambient temperature have been carried out, the four materials are: PA11, PVDF, PEX-B and HDPE. Two specimens of each material have been tested, and the results are shown in Fig.9.

From Fig.9, it can be seen that the stress-strain curves of each polymer material have a good coincidence, which indicates that the experimental procedure designed in this paper is valid and reliable. Meanwhile, it can be found from Fig.9 that all the stress-strain curves of the four polymer materials show significant nonlinear characteristics, with no completely linear phase in the curve, which indicates that no purely elastic phase exists in the stretching process of the polymer materials. This means that elastic deformation and plastic deformation always exist simultaneously in the specimen during its deformation. That is:

It can be assumed that when the stress increases with the increase of strain, elastic deformation dominates the interior of the material. When the stress no longer increases or even decreases with strain, the internal plastic deformation increases. Fig.9 shows a rough estimate of the critical strain values for these four materials where elastic or plastic deformation is the dominant component within the material. These data are listed in Tab.2.

The data of the stress-strain curves are processed. Firstly, the data at the initial stage of the stress-strain curve which can be considered as a linear stage are selected for linear regression to obtain the so-called "linear stage modulus of elasticity". Then the tangent moduli at different strains are calculated, and the yield strength is obtained. These data are shown in Tab.3.

From the data in Tab.3, it can be seen that the difference in the "Linear Stage Modulus of Elasticity" of all the materials is very large. The reason is that the selection of the linear stage is too long. However, in practical engineering applications, strain of the material often exceeds this range, and the value of elastic modulus estimated by using these linear stages will lead to a large difference in results of mechanical analysis. Therefore, in practical engineering applications, it is recommended to estimate the deformation of the material beforehand, and then use the stress-strain curve tangent modulus under the corresponding strain as the basis for design and calibration.

The high-temperature stress relaxation testing conditions in this paper are shown in Tab.1, and the test results are expressed in the form of relaxation curves with time as the horizontal coordinate and stress as the vertical coordinate. The stress-time curves are given in Fig.10. The 3^rd-order Prony parameters obtained by the identification method in Section 1.3 are presented in Tab.4.

It is worth noting that the 3^rd-order Prony parameters obtained in Tab.4 are not unique with respect to the initial input forecasting matrix Prony and the target difference shown in the block diagram of the program in Fig.8.

The viscoelastic materials module in the nonlinear finite element software ABAQUS describes the stress relaxation behavior of the material using Method 1 in Eq.(2). Therefore, for the purpose of verifying the Prony parameters in Tab.4, ABAQUS software is used to establish the finite element model of the specimen part within the extensometer. The model is a cuboid with a section of 6 mm×2.5 mm and a length of 25 mm. In the calculation, the 6 degrees of freedom of one boundary of the model are all fixed, and the other boundary is controlled by displacement, so that the model reaches the predetermined strain. When the specimen generates the predetermined strain, the analysing step is set as the viscoelastic analysis step, and two boundaries of the model are fixed at the current positions, and the calculation time is 90 000 s.

However, the model in ABAQUS is three-dimensional, which will produce "shape effect" and "boundary effect", as shown in Fig.11, so the stress of each node on the cross-section will be different due to the location and distance between itself and the boundary, and thus it is difficult to find a node whose stresses can represent the average stress of the cross-section. To solve this problem, a reference point is inserted on the free boundary section and the reference point is coupled with the free boundary section, as shown in Fig.12.

After the calculation, the reaction force (RF3) of the reference point is extracted, and the value of reaction force is divided by the cross-sectional area to obtain the average stress on the cross section. The Prony parameters in Tab.4 is then used to fit the stress curve for comparison with the finite element results and test data. Due to the large amount of cases, Fig.13 only samples the comparison curves for Cases 1, 7, 12, and 18, and the comparison curves for the remaining cases can be found in Appendix A.

As can be seen from Fig.13 and Appendix A, the three curves for each load case have good consistence, and the final relaxation limits do not differ by more than 1 MPa, which indicates that the stress relaxation constitutive model in the form of Prony parameters chosen in this paper can accurately describe the stress relaxation behavior, and the parameter identification program established in this paper can identify the Prony parameters in the constitutive model with the experimental data.

The reason for the difference between finite element result and testing data in relaxation limit for each case is that, in the test, the cross-section of the specimen's pitch section does not have deformation constraints in both the width and thickness directions (x, y-axis directions in Fig.11 and Fig.12), but the cross section coupled with reference point in the finite element model makes constraints in these two directions. When the test data are processed, the stress relaxation is considered to occur only in the tensile direction of the specimen, and the stress σ is obtained by dividing the force F at each moment by the cross-sectional area of the specimen A (6 mm×2.5 mm), and the relaxation modulus E is obtained by dividing the stress σ by a fixed strain ε. These data are applicable only to the dimension of the tensile direction (z-axis direction in Fig.11 and Fig.12), and changes in both the width and thickness directions of the cross-section are ignored. In the simulation calculation of ABAQUS software, the model of the specimen is three-dimensional, in addition to the stress relaxation phenomenon in the tensile direction of the specimen, the stress relaxation phenomenon also exists in the other two directions of the specimen. In the simulation process, when stress relaxation occurs in the other two directions of the specimen (x, y-axis directions in Fig.11 and Fig.12), there is a tendency of shortening the specimen in the axial direction (z direction) and reducing the cross-sectional area under the influence of the shape effect. In the stress relaxation process simulated by finite element, in order to ensure that the elongation of the specimen can always meet the requirements of the material to generate pre-strain, the cross-section coupled with reference point in Fig.12 is fixed throughout the entire process at the position after the displacement occurs, and no point on the cross-section is allowed to generate any displacement in the degrees of freedom, which leads the reference point to bear an additional support reaction force (RF3) to resist the tendency of axial shortening and reduction of cross-sectional area. Therefore, the stress results obtained by dividing the branch reaction force data by the cross-sectional area in the extracted finite element simulation results in this paper will be slightly larger than the test results. This is the reason that the relaxation limit of "Finite Element Result" is slightly larger than those of the other two curves in the three curves of 18 load cases.

In order to further discuss the mechanical properties of the material during stress relaxation, the relaxation modulus during relaxation is normalized using the fitted 3^rd-order Prony parameter

Fig.14 gives the normalized relaxation modulus curves for PVDF and PA11 materials at the same temperature and different pre-strain, respectively.

From Fig.14(a)−(c), it can be seen that the decrease in relaxation modulus of PVDF specimens during stress relaxation is in the following order from the smallest to the largest under the same temperature conditions: 1% pre-strain < 2.5% pre-strain < 15% pre-strain < 10% pre-strain. Reviewing the data given in Tab.2, at 10% nominal strain and above, plastic deformation has slowly begun to possess a greater share of the overall deformation of the specimen. It is generally accepted that the stress relaxation phenomenon in material is the conversion of a portion of the elastic strain to plastic strain on the premise that the overall strain in material remains unchanged. When the overall deformation of the material is dominated by elastic deformation, the material has a sufficiently large base of elastic deformation that can be converted to plastic deformation. As the proportion of plastic deformation increases in the overall deformation of the material before relaxation, the base of elastic deformation that can be converted to plastic deformation in the material becomes smaller, and thus the decrease in modulus during relaxation becomes smaller. For the above reason, number of relaxation modulus decreases for the "15% pre-strain" curve < that for the "10% pre-strain" curve.

Observing the nominal stress curve of the PVDF specimens in Fig.9(b), it is not difficult to find that the material can take up a larger stress at 15% strain than at 1% strain and 2.5% strain. Consequently, when polymer materials are used in high temperature environments, if the method for reducing the effects caused by stress relaxation is considered solely, the strain of the polymer materials can be appropriately increased to a larger proportion of their internal plastic deformation.

However, the PA11 materials show a different pattern at 75 ℃: the decrease of relaxation modulus has: 15% pre-strain > 30% pre-strain, which may indicate that the relaxation phenomenon of PA11 materials is not significant at 75 ℃. Fig.15 gives the normalized relaxation modulus curves of these two materials at different external temperatures under the same pre-strain conditions.

As can be seen in Fig.15(a)−Fig.15(d), the relaxation of PVDF materials at external temperatures of 75 ℃, 85 ℃ and 95 ℃ are similar. Observing Fig.15(f), the two curves at 85 ℃ and 95 ℃ show a significant increase in relaxation amplitude compared to the 75 ℃ curve, whereas this phenomenon is not obvious in Fig.15(e). This may indicate that the stress relaxation properties of PA11 are less sensitive to temperature at 15% pre-strain, while the temperature sensitivity of the stress relaxation properties of PA11 is stronger when the value of strain reaches 30%, which makes the relaxation amplitude of the material at 75 ℃ much smaller than those at 85 ℃ and 95 ℃. From the data in Tab.2, it can be seen that no matter 15% strain or 30% strain, the elastic strain inside PA11 is dominant, and at temperatures of 85 ℃ and 95 ℃, which are with larger relaxation sensitivity, it still conforms to the law that the larger the percentage of elastic deformation inside the specimen is, the larger the relaxation amplitude will be.