Modeling of AP-HTPB Solid Propellant Viscoelastic Behavior Using Modified Maxwell Model

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F. Nour Eldin, Walid. M. Adel, Ahmed El sabbagh This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4545250/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 3 You are reading this latest preprint version Abstract In this research, an initial exploration of the viscous effects present in heterogeneous solid rocket propellant is conducted through experimental analysis. To achieve this, uniaxial tensile and relaxation experiments were conducted on the Joint Army-Navy-NASA-Air Force Propulsion Committee (JANNAF) standard[ 1 ], specimens using the Zwick Z050 universal testing machine. The work involved conducting destructive tensile tests at different strain rates and relaxation tests at different strain levels and temperatures, with three values being considered. The resulting data from the experiments are illustrated in appropriate diagrams, and depending on these data, the viscoelastic behavior of this material was confirmed. Additionally, mathematical modeling of this studied phenomenon using the generalized Maxwell model, identified on the basis of experimental data, is presented. The material parameters of the constitutive model are determined numerically using MATLAB software based on the Prony series procedure. The efficiency of the model and the identification approach are discussed. A high agreement between the calculation and the experimental results was found based on the Prony coefficients of relaxation modulus, with a maximum error of about 2.35%. Finally, numerical modeling of the relaxation tests was conducted to simulate the stress relaxation behavior of the AP-HTPB composite solid propellant using the ANSYS program. The Prony coefficients were added to the nonlinear viscoelastic components to define the material model in ANSYS, and the maximum difference between the numerical and experimental results is 3.98%. Viscoelastic material relaxation modulus hydroxyl-terminated polybutadiene (HTPB) propellant Prony series generalized Maxwell model Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 1. Introduction Composite solid propellants are crucial in strategic and tactical missiles due to their simplicity, ease of operation, safety, reliability, high specific impulse, extended shelf life, and low cost [ 1 – 2 ]. Composite solid propellant is a heterogeneous system in which solid particles of oxidizer and solid metal fuel are distributed in a polymeric matrix [ 3 – 4 ]. Hydroxyl terminated poly-butadiene (HTPB) based composite solid propellants are the most common category of composite propellants due to their improved performance and conventional processing technology. Polymeric binder is responsible for holding the solid ingredients into the matrix and providing the composite propellant with its viscoelastic nature. Studying the behaviors of solid propellant under different load conditions is crucial to ensuring the structural reliability and propellant grain integrity of SRM [ 5 – 6 ]. Viscoelasticity is the characteristic of a substance exhibiting both elastic and viscous behavior, the application of stress causes temporary deformation if the stress is quickly removed but permanent deformation if it is maintained. The essence of viscoelasticity is that the motion of a polymer molecule has relaxation characteristics. The change in the polymer's mechanical property with time is collectively referred to as mechanical relaxation. The mechanical relaxation of high molecule materials under fixed stress [ 7 – 8 – 9 ]. Common mechanical properties are investigated during tensile tests, such as peak tensile strength and strain at peak strength [ 10 – 11 – 12 ]. The goal of this work is to propose a novel approach that builds upon the accumulated knowledge with respect to the mechanical behavior of solid propellants. It does this by utilizing experimental data from uniaxial tensile tests to analyze the behavior of the material and how the solid propellant responds to different strain rates, relaxation tests, modeling, and simulation results. In order to do a structural analysis of the solid propellant grain, the following mechanical characteristics are essential, Young's modulus, ultimate strength, strain at ultimate strength, relaxation modulus, and shift factor. The earlier investigators carried out a number of studies. Park and Schapery [ 8 ] presented a mechanical model founded on thermodynamics that explains how temperature and time depend on a particulate composite with a changing microstructure and growing damage. Park and Schapery[ 14 ] implementing the so-called pseudo strain theory, the time-temperature superposition principle (TTSP), and the rate-type evolution equation of two internal damage variables, the model was extended to a thermo-viscoelastic model that could simulate the effects of temperature, confining pressure, and axial strain rate on Hydroxyl-Terminated Polybutadiene (HTPB) propellant. Schapery [ 15 ] devised a constitutive model to describe the solid propellant's nonlinear elastic deformation characteristics under axial tension and confining pressure. Canga et al[ 16 ] modified the model to allow an efficient numerical implementation and presented the comparisons between finite element analysis results and test data. Joseph[ 17 ] stress relaxation tests were used to determine the time dependence of the composite solid propellant. To determine the CSP's time dependence, Joseph applied stress relaxation tests. Sanal Kumar[ 18 ] suggested that temperature variations have a greater impact on the relaxation modulus of HTPB-based propellant than does time. Swanson and Christensen[ 19 ] developed a constitutive model with a strain softening function to describe the viscoelastic response of a high elongation solid propellant, by fitting experimental data, the model revealed the influence of pressure on the strain-softening function. based on a micromechanical model and the work potential theory. Kadiresh P. N and Sridhar B. T. N [ 20 ] applied the Ansys software to compute the stress relaxation behavior of the AP-HTPB solid propellant using a 3D finite element model, however, the model geometry deviated from the standard specimen used in their experiments. Additionally, the comparison between the simulated results and experimental data was limited to the summation of reaction forces for the initial 30 seconds, omitting the influence of diverse loading conditions and temperature variations. Hui Li and Jin‑sheng Xu [ 21 ] used the Prony series called by the generalized Maxwell model to compare the model results of relaxation modulus with the experimental results of the CSP at room temperature, the next step is to compare the experimental data with Ansys's calculation of the reaction forces for the first 30 seconds, without changing the temperatures or loading conditions. A relaxation test can be used to determine the relaxation modulus, which is important for SRM design because propellants with higher relaxation moduli will cause surface cracks during actual firing, while propellants with lower relaxation moduli will cause grain deformation during storage [ 9 – 18 ]. It must therefore be essential to conduct more realistic experiments and make more accurate predictions about the solid propellant's relaxation modulus. In actuality, a lot of studies have used relaxation test results to analyze grain under various loading conditions, but there aren't many that simulate stress relaxation tests, particularly when using standard specimens. Therefore, in order to verify the efficacy of the relaxation model obtained, a 3D finite element model must be established in order to reproduce the solid propellant specimen's relaxation behavior under actual test conditions [9- 15 – 19 ] .Composite propellants are subjected to temperature and mechanical loads during the production, storage, and operation processes. Using a Zwick universal test machine, tensile tests were performed with different operating temperatures (− 40, 25, and 75°C) and strain rates (0.000656 1/s, 0.0328 1/s). This work performs complete uniaxial tension and stress relaxation tests to describe how solid propellant acts under different loadings, as mentioned in Section 2 . The constitutive equations that create the nonlinear viscoelastic model are used in Section 3 . A 3D finite element model of the solid propellant specimen was constructed to simulate stress relaxation under different conditions using nonlinear viscoelastic material properties. 2. Experimental Work 2.1. Preparation of Propellant Formulations and Specimens. The material under investigation in the present study is a nutritional solid propellant. The primary compounds of the solid propellant comprise differing percentages of 68% ammonium perchlorate and 18% aluminum powder that are distributed in a polymeric binder matrix consisting of hydroxyl-terminated poly-butadiene and the residual percentage consists of other substances. These compounds were mixed, and the resulting propellant slurry was cast vibrating under vacuum in specialized molds that have internal sizes of 200 mm × 150 mm × 150 mm. Consequently, the molds were cased in a large curing oven with a maintained minimum temperature of 60 ℃ for a total period of 240 hours. After curing, the molds were cut into sheets with uniform thickness using a special cut press according to the Joint Army-Navy-NASA-Air Force Propulsion Committee (JANNAF) standard, as shown in Fig. 1[ 1 ]. Figure 1. Main dimensions of the JANNAF specimen . Figure 2. Flowchart of methodology work. 2.2. Uniaxial Tensile Test All tensile tests were conducted using a computer-controlled universal test machine (Zwick/Roell) Z050. The machine was equipped with a thermal chamber to set and maintain the desired temperature constant during the test [ 25 ]. Prior to testing, the thickness and width of each specimen were accurately measured in its gauge length region. The effective gauge length (L O ) was determined to be 74.4 mm, and the crosshead speed of the machine was maintained at 50.8 mm/min. Each experimental test was conducted using five samples, with the mean measurement value being utilized for the analysis in simulations. Additionally, prior to each test, the specimens underwent pre-conditioning for two hours in an external conditioning chamber. The specimens were tested at three different temperatures (+ 25, + 75, and − 40 o C), with a constant crosshead speed of 50 mm/min, corresponding to a strain rate ℇ o =0.016404 s − 1. Furthermore, each experiment were carried out under constant temperature conditions maintained by a digital control temperature chamber, ensuring an accuracy of ± 0.1 o C in relation to the designated set point throughout the duration of the testing phase. The stress-strain curves shown in Fig. 3 were generated using the load-displacement data. The curves depicted demonstrate the highest levels of stress and strain that occur at the point of maximum stress. The temperature has a significant impact on the deformation behavior of the AP-HTPB solid propellant. The maximum stress and maximum strain increased significantly by 155.6% and 49.23%, respectively, while the break stress and break strain increased by 178% and 49.9%. Moreover, Young's modulus increased significantly by 2.21% when the temperature decreased from + 75°C to − 40°C at a strain rate of ℇ o = 0.016404 s − 1 . The stress and strain values at the yield point, maximum point, break point, and Young's modulus were experimentally determined and are presented in Table 1 . Table 1 The values of the uniaxial tensile test result. Temperature Yield Stress (MPa) Yield Strain (%) Young’s Modulus (MPa) Max Stress (MPa) Max Strain (%) Break Stress (MPa) Break Strain (%) T = − 40 o C T = + 25 o C T = + 75 o C 0.832 0.524 0.626 14.80 20.39 35.77 5.62 2.55 1.75 1.40 0.735 0.626 53.70 39.50 35.77 1.25 0.64 0.51 56.22 41.53 39.50 Figure 4 presents the results of tensile testing conducted at different strain rates while maintaining a constant temperature of 25°C. The significant impact of strain rate or loading rate on the material's mechanical behavior highlights the viscoelastic properties of the data presented in the text. This is in contrast to completely elastic materials, where the rate at which deformation occurs is not critical. Furthermore, at lower strain rates, the material also shows a linear viscoelastic response, solid particles noticeably improve the binder’s properties. Simultaneously, at high strain rates, the interfacial cohesion of the oxidizer with the binder decreases, and the work of the reinforcement reduces, which causes nonlinearity of the curve. 2.3. Relaxation Test The viscoelastic properties of a propellant can be demonstrated by a relaxation test, where a sample is subjected to a constant strain and the change in stress σ (t) over time is evaluated [ 26 ], as shown in Fig. 5 . This test was conducted using the same test sample that was utilized in the tensile test. This figure shows that at a constant strain level, the stress generated in the propellant specimen relaxes with time. Additionally, the experiment results of the stress test indicate that the stress applied to the specimen will gradually decrease as it is subjected to a constant level of strain for a period of time. This demonstrates the impact of strain level on stress. Before conducting the experimental procedures, the samples endured a three-hour conditioning process in an external environment chamber to establish thermal equilibrium. In addition, the relaxation test and uniaxial tensile test were performed in an environment-controlled chamber with an error in temperature of ± 0.2 o C. The sampling frequency for recording load change against time was 1 Hz. the modulus curve of relaxation is shown at 5% 10% 15% 20% and 25% strain level in Fig. 5 . The curves remain constant for 45 minutes at a constant a temperature of 25 o C. From comparing the stress results obtained at different strain levels in Fig. 5 , it is clear that stress relaxation tests at higher strain levels show greater relaxation than those at lower strain levels. This indicates that a more rapid reduction in stress occurs as strain levels diminish from 25–5% with the passage of time, particularly when the initial deformation is higher (20% or 25%) as opposed to being lower (5% or 10%).Essentially, there is a nonlinear decrease in stress with decreasing strain levels, indicating that higher strain levels disrupt the material's internal structure more significantly and lead to faster relaxation and stress reduction. Figure 6 shows relaxation tests performed on the propellant under a constant strain level of 10% at various temperatures. It can be observed that stress reduction increases with temperature rise (− 40 ºC to + 75 ºC) and also with longer periods of time for nonlinear stress relaxation. This observation aligns with the general principle of increased molecular mobility at higher temperatures. When molecules move more readily, they can easily rearrange and recover from imposed deformation, resulting in quicker stress reduction. The relaxation modulus E r (t) is expressed by: $$\text{E}\text{r} \left(\text{t}\right)=\frac{{\sigma } \left(\text{t}\right)}{\text{ℇ}}$$ 1 The relaxation modulus E R (t) is expressed by Eq. ( 1 ) [ 27 ], where it is influenced by both strain level and temperature Fig. 7 demonstrate affects strain level on relaxation modulus, as strain level increases within the same period 45 min from 5–25% at + 25°C, relaxation modulus decreases nonlinearly. Similarly, Fig. 8 , shows that temperature affects relaxation modulus, as temperature rises from − 40 ºC to + 75 ºC for the same period (45 min) at a constant strain level of 10%, relaxation modulus also decreases nonlinearly. Higher temperatures increase internal energy and molecular mobility, facilitating faster rearrangement and reducing stress relaxation. Understanding these interactions between strain level, temperature, and material behavior is crucial for predicting material response under different loading conditions and designing applications that require optimal stress response. Figure 7. Relaxation modulus at different strain levels and a constant temperature of 25 o C. At lower temperatures, the material is in a glassy state and has a high modulus, meaning it is stiff and rigid. With increasing t to the glass transition temperature T g, the material becomes more rubbery, and the modulus decreases [ 1 ]. As the temperature continues to rise beyond (Tg), the decline in modulus becomes slower. This is because the material is undergoing a transition from a glassy state to a rubbery state, which involves molecular rearrangements and increased mobility of polymer chains. These changes result in a decrease in stiffness but at a slower rate compared to the initial decrease at lower temperatures. The behavior of the nonlinear relaxation modulus is adapted with temperature variations. As the temperature rises from − 40°C to + 75°C, there is a decline in the nonlinear relaxation modulus over time. This suggests a decrease in the material's capacity to endure deformation resulting from applied stress or strain over time. Furthermore, the values of the equilibrium relaxation modulus diminish as temperature rises. The equilibrium relaxation modulus characterizes the prolonged reaction of the material once it has attained a state of stability. As temperature escalates, the material's capability to regain its original form post-deformation diminishes, resulting in decreased values of the equilibrium relaxation modulus. As the temperature rises towards and exceeds the glass transition temperature (Tg), significant changes occur in the material's physical properties. These alterations encompass a decreased rate of decline in modulus, along with decreases in both nonlinear relaxation modulus and equilibrium relaxation modulus values. 2.4. Mathematical Modeling The Prony series is a mathematical technique that excels at representing the time-dependent relaxation behavior of viscoelastic materials, including solid propellants [ 24 ]. For a typical viscoelastic material, the relaxation modulus can be represented by a power law, which is in the form of a Prony series[ 28 ]. The modulus obtained based on the Prony series has not only high accuracy, but is also more convenient in data processing, it can closely model the behavior of many viscoelastic materials. In this paper have adopted a particular Prony series formulation in Eq. (2) to model the relaxation modulus of the generalized Maxwell model of the solid propellant they are studying [ 26 ]. (2) Where, Er - relaxation modulus, MPa; Ƞi - Viscosity coefficient of the i-th Maxwell element, MPa.s. Ei - Elastic modulus of the i-th Maxwell element, MPa. E \(\infty\) - Equilibrium relaxation modulus coefficient Identification of linear viscoelastic parameters [ 29 ]. For the one-dimensional condition, the linear viscoelastic model with relaxation modulus described by the Prony series, namely the generalized Maxwell model, is composed of multiple spring-dashpot elements connected in parallel, each with its own relaxation time, see Fig. 9. can be presented [ 30 ]. Figure 9. Schematic diagram of the generalized Maxwell model. Combining the above formulas and relaxation test data, each coefficient of Eq. (2) can be obtained by the nonlinear Least Squares Method, which is implemented by the fit function of MATLAB software [ 24 ]. The MATLAB program was built based on the nonlinear least squares method to determine the Prony coefficients [ 11 ]. By choosing the number of Prony components and performing curve fitting for the relaxation test data, we can get the unknown coefficients. In the following calculations, the Prony series is taken as 5 terms. It’s clear from comparing the experimental results obtained by testing the relaxation modulus temperature at 25 o C with the mathematical equation implemented by the appropriate function of MATLAB shown in Eq. (2) that various strain levels with a constant temperature of 25 o C influence the nonlinear relaxation modulus over time. Figure 10 shows that the curves of E r and curve fitting at strain levels (5%, 10%, 15%, 20%, and 25) are almost identical to each other and, the maximum differences between calculation and experimental, analysis results are 2.35% respectively. It investigates the prediction of outcomes of propellant behavior as a viscoelastic generalization Maxwell model. The results presented in Table 2 , the relevant Prony coefficients according to the relaxation test data for solid propellant at a temperature of 25 o C and strain levels (5%, 10%, 15%, 20%, and 25%,) obtain from this data the influence of various of strain levels on equilibrium relaxation modulus. Table 2 Prony series coefficients of HTPB propellant’s relaxation modulus of various strain levels at a temperature of 25 o C. Parameters t i 5% 10% 15% 20% 25% E \(\varvec{\infty }\) - 6.266 − 18.11 0.7176 -12.16 − 12.58 E 1 0.1 0.1519 0.4054 0.413 0.4349 0.3559 E 2 1 0.4467 0.3038 0.2891 0.2973 0.3037 E 3 10 0.1329 0.3933 0.3787 0.3268 0.3494 E 4 100 1.972 − 2.069 0.3167 -1.409 − 1.62 E 5 1000 − 6.49 21.3 0.1045 14.6 15.12 Model evaluation. In order to assess the predictive capacity of the proposed model, the root mean square error (RMSE) is introduced and calculated as follows. [ 24 ](3). where E r experiment denotes the experimental relaxation modulus of CSP, E r model is the model prediction corresponding to the same strain levels, and n is the number of data points. Table 3 shows the calculated RMSE values for various combinations of relaxation and strain levels. The highest RMSE value is 0.006127 MPa. This indicates the largest discrepancy between the model's prediction and the experimental data for a specific combination of relaxation and strain levels. In each cases, the RMSE values are below 0.01 MPa. This suggests a good agreement between the model's predictions and the experimental data for the majority of tested conditions. Table 3. RMSE values of experimental and predicted results under different strain levels. Strain level 5% 10% 15% 20% 25% RMSE 0.006127 0.004058 0.00253 0.002733 0.002296 Figure 11. Relaxation modulus and curve fitting vs time at various temperatures at a strain level of 10%. Figure 11 shows that comparing experimental data and the data obtained from the calculation of the Prony Series for curves of Er and curve fitting for a specific strain level of 10% at three different temperatures (− 40 o C, 25 o C, and 75 o C) are identical to each other, generalization Maxwell model components are found to lead to a good fit and high accuracy. From investigating the results of the relaxation test in Table 4 , it becomes clear that influence various temperature on the equilibrium relaxation modulus values and the constant strain level in each component of the series. It investigates the prediction of outcomes of propellant behavior as a viscoelastic generalization Maxwell model. The results presented in Table 4 , the relevant Prony coefficients according to the relaxation test data for solid propellant at strain level 10% and temperature of 25 o C, − 40 o C, and 75 o C are listed in Table 4 . Table 4 Prony series coefficients of HTPB propellant’s relaxation modulus of various temperature at strain level 10% Parameters t i − 40 o C 25 o C 75 o C E \(\varvec{\infty }\) - -56.99 − 18.11 39.09 E 1 0.1 2.449 0.4054 0.4024 E 2 1 1.259 0.3038 0.03085 E 3 10 1.111 0.3933 0.2201 E 4 100 -8.514 − 2.069 6.789 E 5 1000 66.35 21.3 − 44.63 Table 5 . shows the values of RMSE under relaxation and different temperatures at a strain level of 10%. As can be observed, the majority of examples had an RMSE below 0.015 MPa, with a highest value of 0.01202 MPa. Table 5 RMSE values of experimental and predicted results under different strain levels. Temperature − 40 o C 25 o C 75 o C RMSE 0.008261 0.004058 0.01202 The shift factors a T describe the temperature dependence of the relaxation time and usually follow the empirical Williams-Landel-Ferry (WLF) equation, which determines how the viscoelastic property of propellant is expressed as follows: Log (a T ) = \(\frac{\text{C}1 ( \text{T}-\text{T}\text{r}\text{e}\text{f})}{\text{C}2+\text{T}-\text{T}\text{r}\text{e}\text{f}}\) (4) Where C 1 and C 2 are the material constants C 1 = − 5.35, C 2 = 779.2, according to the WLF method based on reference temperature T ref = 25 o C .The time-temperature superposition principle (TTSP) can be used to create the relaxation modulus master curve, which is shown in Fig. 11, using the previously described data. As shown in Fig. 13 . The time – temperature shift is one of the most important values in the structural analysis of viscoelastic materials because it allows the temperature impact to be converted to the time influence, which makes the analysis much easier. 2.5 Simulation, Experimental and Model Verification A finite element analysis was performed using the ANSYS software in order to examine the stress relaxation properties of the AP-HTPB composite solid propellant [ 10 ]. The simulation was experimented with to precisely replicate the conditions of uniaxial tensile loading by discretizing the same specimen utilized during actual tests, thereby ensuring accurate dimensions. The mathematical data acquired from the relaxation model at ε = 10% and T = 298K were initially tested. Within the nonlinear viscoelastic parts of the generalized Maxwell model branches, the Prony coefficient of the relaxation modulus was incorporated to characterize the material model in ANSYS software. Figure 14. 3D Finite element model of a solid propellant specimen. The mechanical boundary conditions constrained all degrees of freedom at one end and subjected the other end's nodes to X-direction shifts at strain values of 5%, 10%, 15%, 20%, and 25% with an initial tension rate of 10 mm/min. The tension was maintained for 45 minutes, and the variation of the maximum stress over that time was plotted against time with a one-second interval. In the thermal boundary conditions, the initial temperature was the strain-free temperature, and the final temperature was the experimental test temperature. In order to validate the numerical results obtained from finite element analysis, two different loading conditions were chosen based on the experimental study: relaxation stress at various strain levels and constant temperature and relaxation stress at various temperature and constant strain levels. The Prony coefficient of relaxation modulus according to Tables 2 and 4 was added to the generalized Maxwell model branches under the nonlinear viscoelastic components to define the material model in ANSYS software. To get the relaxation modulus from the results of ANSYS software, it will be compensated in Eq. (2). The other material properties of solid propellant were taken from Tables 6 and 7 . Table 6 The change of viscoelastic solid propellant properties of various strain levels at temperature 25 o C. Number of Term Shear modulus (G) (MPa) Bulk modulus (K) (MPa) 5% 10% 15% 20% 25% 5% 10% 15% 20% 25% 1 0.051 0.135 0.138 0.145 0.119 12.66 33.78 34.42 36.24 29.66 2 0.149 0.101 0.096 0.099 0.101 37.23 25.32 24.1 24.78 25.31 3 0.044 0.131 0.126 0.109 0.117 11.08 32.77 31.56 27.23 29.11 4 0.658 -0.691 0.105 -0.47 -0.541 164.33 -172.42 26.39 -117.42 -135 5 -2.166 7.109 0.035 4.87 5.047 -540.83 1775 8.71 1216. 7 1260 Table 7 The Change of viscoelastic solid propellant properties of various temperature at strain level of 10% Number of Term Shear Modulus (G) Bulk Modulus (K) − 40 o C 25 o C 75 o C − 40 o C 25 o C 75 o C 1 0.817 0.135 0.134 204.08 33.78 33.53 2 0.4202 0.1014 0.0103 104.92 25.32 2.571 3 0.3709 0.1313 0.074 92.58 32.78 18.34 4 -2.8402 -0.691 2.266 -709.5 -172.42 565.75 5 22.147 7.11 -14.89 5529.2 1775 -3719.17 In the following calculation, for the studied of composite solid propellant is treated as a nonlinear viscoelastic material. The thermal and mechanical parameters needed for this calculation are listed in Table 8 . Table 8 Parameters of composite solid propellant. Material Parameter The Value Modulus E (MPa) E (t, t i ) Expansion coefficient α(1/k) 9×10 − 5 Poisson ratio 0.498 Density ρ (kg/m3) 1750 Thermal conductivity k (w/m · k) 0.51 Specific Heat (J/kg · k) 1247.6 A comparison of the relaxation modulus between the finite element results and the experimental data at three distinct strain levels (5%, 15%, and 25%) at a constant temperature of 25°C is presented in Fig. 15 . Additionally, Fig. 16 compares the results of the finite element results with the experimental data for the relaxation modulus at three different temperatures (-40°C, 25°C and 75°C) with a constant strain level of 10%. . Figure 16. Relaxation modulus at different temperature and constant strain level of 10%. The comparison between the simulation or numerical results and the experimental data is clearly demonstrated by the data depicted in Figs. 15 and 16. The numerical results show that the material nonlinear viscoelastic effect is very close to the experimental data from the change of strain levels and temperature effect, which must be considered in the analysis of a composite solid propellant, and the maximum differences between the numerical and experimental, analysis results are 2.76%, and 3.98% respectively. This indicates that the proposed method demonstrates a significant level of accuracy in replicating the relaxation behavior of the solid propellant under different loading conditions and maintains stability throughout the whole testing period. 3. Conclusions In this work, a nonlinear viscoelastic model was proposed based on experimental data to define the response of this material under different loading conditions. In addition, a 3D finite element model has been established to stimulated solid propellant relaxation behavior under realistic test conditions. From the results, the main conclusions can be summarized as follows: 1. Young`s modulus, maximum stress, and strain at maximum stress all increase proportionally as the strain rate increase. 2. At constant strain levels the stresses generated in the propellant relax with time. The relaxation modulus decreases when the temperature increases or when the strain level decreases. 3. Increasing the applied load in the propellant leads to an increase in deformation of the composite because more stress transferred to the matrix interface, which results in the de-bonding of fibers and the failure of the composite structure happen. Also, increasing the temperature results in higher macromolecular mobility. 4. For all strain levels, relaxation modulus –time curves are generated using Maxwell model. The model was able to capture all facts of material response and close matching with actual test curves has been observed. This indicates the suitability of using the Maxwell model to describe the relaxation behavior of solid propellant. 5. To increase the level of the simulation of solid propellant behavior, the material must be well defined as a nonlinear viscoelastic material, and the Prony series coefficients must be calculated at different strain levels and temperatures. 6. The nonlinear viscoelastic model demonstrates excellent predictive performance when compared to the experimental relaxation modulus. The largest RMSE value at various strain levels is 0.006127 MPa, and most cases are lower than 0.01 MPa. The largest RMSE value at various temperatures with a strain level of 10% is 0.008261 MPa, and most cases are lower than 0.015 MPa. 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Park, S. & W S. “Development of a Nonlinear Thermo-Viscoelastic Constitutive Equation for Particulate Composites with Growing Damage.pdf.” Schapery,R. A. 1987 . Nonlinear constitutive equations for solid propellant based on a work potential and micromechanical model. no. March 1987, pp. 0–10, doi: 10.13140/RG.2.2.34351.79527. Canga,M. E. Becker, E. B. & Upek,Oz. .Constitutive modeling of viscoelastic materials with damage computational aspects.”. Available: www.elsevier.com/locate/cma J. H. Stoker, 1964. Use of stress relaxation tests to characterize time dependencies of a composite solid propellant. AIAA J., vol. 2, no. 10, pp. 1816–1818, doi: 10.2514/3.2671. Sabarinath,K. R. K. Sandeep. 2013 . Influence of Viscoelastic properties of Solid Propellants on Starting Transient of Solid Rocket Motors. 5th Eur. Conf. Aerosp. Sci., no. July, pp. 2–5, doi: 10.13140/2.1.2140.5122. Swanson, S. R. & Christensen,L. W.1983. A constitutive formulation for high-elongation propellants. J. Spacecr. Rockets, vol. 20, no. 6, pp. 559–566, doi: 10.2514/3.8587. Kadiresh, P. N. & Sridhar,B. T. N.2008. Experimental evaluation and simulation on aging characteristics of aluminised AP-HTPB composite solid propellant. Mater. Sci. Technol . , vol. 24, no. 4, pp. 406–412, doi: 10.1179/174328408X278420. Li, H. J. Xu,sheng. Chen, X. J. fa Zhang, & Li, J. 2023 . A nonlinear viscoelastic constitutive model with damage and experimental validation for composite solid propellant. Sci. Rep., vol. 13, no. 1, pp. 1–20, doi: 10.1038/s41598-023-29214-7. Adel, W. M. and Guo-zhu, L. 2017. Analysis of Mechanical Properties for AP / HTPB Solid Propellant under Different Loading Conditions. in International Journal of Aerospace and Mechanical Engineering, vol. 11, no. 12, pp. 1915–1919. Deng,B. Xie,Y. & Tang, G. J. 2014. Three-dimensional structural analysis approach for aging composite solid propellant grains. Propellants, Explos. Pyrotech . , vol. 39, no. 1, pp. 117–124, doi: 10.1002/prep.201300120. Chen,S. Wang,C. Zhang, K. Lu, X. & Li, Q. 2022. A Nonlinear Viscoelastic Constitutive Model for Solid Propellant with Rate-Dependent Cumulative Damage. Materials (Basel) . , vol. 15, no. 17, doi: 10.3390/ma15175834. Noureldin, A. F. Adel, W. M. Attai,Y. A. & Ismail, M. A. 2020 . An experimental study on mechanical and ballistic characteristics of different HTPB composite propellant formulations. IOP Conf. Ser. Mater. Sci. Eng. , vol. 973, no. 1, 2020, doi: 10.1088/1757-899X/973/1/012030. Tong,X. Xu,J. Doghri,I. El Ghezal,M. I. Krairi, A. & Chen, X.2020. A nonlinear viscoelastic constitutive model for cyclically loaded solid composite propellant. Int. J. Solids Struct., vol. 198, pp. 126–135, Aug. 2020, doi: 10.1016/j.ijsolstr.2020.04.011. Chyuan,S.2004. The Journal of Strain Analysis for Engineering Design Computational studies of variations in Poisson ’ s ratio for thermoviscoelastic solid propellant grains. doi: 10.1177/030932470403900109. Ji,Y. Cao, L. Li, Z. Chen, G. Cao,P. & Liu,T. 2023. Numerical Conversion Method for the Dynamic Storage Modulus and Relaxation Modulus of Hydroxy-Terminated Polybutadiene (HTPB) Propellants. Polymers (Basel)., vol. 15, no. 1, Jan. 2023, doi: 10.3390/polym15010003. Jung,G.-D. Youn, S. K. & Kim ,B. K. 2000. A three-dimensional nonlinear viscoelastic constitutive model of solid propellant, vol. 37. doi: 10.1016/S0020-7683(99)00180-8. Jrad,H. Dion, J. Renaud, L. F. Tawfiq, I. & Haddar, M. 2012. Non-linear generalized maxwell model for dynamic characterization of viscoelastic components and parametric identification techniques. Proc. ASME Des. Eng. Tech. Conf., vol. 1, no. PARTS A and B, pp. 291–300, 2012, doi: 10.1115/DETC2012-70264. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editor assigned by journal 13 Jun, 2024 Submission checks completed at journal 10 Jun, 2024 First submitted to journal 07 Jun, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4545250","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":313834363,"identity":"c0dc28e8-1954-47cb-9dc6-cddff207d4fe","order_by":0,"name":"A. F. Nour Eldin","email":"","orcid":"","institution":"Ain Shams University","correspondingAuthor":false,"prefix":"","firstName":"A.","middleName":"F. Nour","lastName":"Eldin","suffix":""},{"id":313834364,"identity":"ae5dabeb-31a0-4af9-a3d8-d8fb71fd846b","order_by":1,"name":"Walid. M. Adel","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Walid.","middleName":"M.","lastName":"Adel","suffix":""},{"id":313834365,"identity":"1cdafa56-b33d-4f5d-9315-f3c1e8d0d612","order_by":2,"name":"Ahmed El sabbagh","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQElEQVRIie2QMUvDQBTHXzhIlgtdI4f1K5wU6hBIvkpKhlsu0C5SUPCmy9Lq2oAfQhetW+TALhkcAy6thU4dBEGKRvG6SLAp4iaY33C8x/G7/7sHUFPzB/FjAGQIAApg6h41S5dBpULRN6X1e6Ujflas2XNv7HnXgBaPed9lZyS+h1WhoGFxCk+vm39BuEWSLAxvhHnQ4hmLkvOsawylgp3BkhrJaUUKBmJLFNIU2iSSKrrIeQC2UEBzTpE9qBps/mbLE61YLyT6UMzXilHowfy18l6l6MdtqTyaYp0iVEAdliJs6hRHK7CqGqztJtkkoAofEn7H9kc5B7UrGXayRfd2KDYUP47nD73xkU8n8RXhx+5eY8Rms2XhNhtxeDldFZWLXtMR6KvGehX6XJepIbcqfqm2pqVme0pNTU3Nv+ETQjVs0n1TXZEAAAAASUVORK5CYII=","orcid":"","institution":"Ain Shams University","correspondingAuthor":true,"prefix":"","firstName":"Ahmed","middleName":"El","lastName":"sabbagh","suffix":""}],"badges":[],"createdAt":"2024-06-07 09:46:31","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4545250/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4545250/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":58951997,"identity":"1d4c6d62-da9a-45aa-8ab5-8834c06d8773","added_by":"auto","created_at":"2024-06-24 14:08:54","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":9971,"visible":true,"origin":"","legend":"\u003cp\u003eMain dimensions of the JANNAF specimen\u003c/p\u003e","description":"","filename":"fig1.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/527d62b4104b48ca2d8ab748.png"},{"id":58952573,"identity":"2fd24b50-9a12-4157-a0f4-668cb4c2bead","added_by":"auto","created_at":"2024-06-24 14:16:54","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":49149,"visible":true,"origin":"","legend":"\u003cp\u003eFlowchart of methodology work.\u003c/p\u003e","description":"","filename":"fig2.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/4e113adcbabe46869b59d5a0.png"},{"id":58952569,"identity":"7b445a11-3042-4e8c-ac70-f9bfb4ca17dd","added_by":"auto","created_at":"2024-06-24 14:16:54","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":26037,"visible":true,"origin":"","legend":"\u003cp\u003eEffect of temperature stress-strain curves.\u003c/p\u003e","description":"","filename":"fig3.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/c2d2c9c1f1d94bfe0a649cd4.png"},{"id":58952570,"identity":"ce064d23-3bd0-4302-9b32-726d11f5f00b","added_by":"auto","created_at":"2024-06-24 14:16:54","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":35085,"visible":true,"origin":"","legend":"\u003cp\u003eEffect of strain rate on stress-strain curves.\u003c/p\u003e","description":"","filename":"fig4.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/93c73fe57e2c714b528e53b7.png"},{"id":58952007,"identity":"e194f490-1be3-4d67-a947-258845aab197","added_by":"auto","created_at":"2024-06-24 14:08:54","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":34416,"visible":true,"origin":"","legend":"\u003cp\u003eStress relaxation at different strain levels at temperature of 25ºC.\u003c/p\u003e","description":"","filename":"fig5.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/4fa96b63b509d0e464ee96a1.png"},{"id":58953902,"identity":"32ce7507-a5cf-498e-958b-61d0ffadb7ee","added_by":"auto","created_at":"2024-06-24 14:32:54","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":30612,"visible":true,"origin":"","legend":"\u003cp\u003eEffect of temperature on stress relaxation at strain level 10%.\u003c/p\u003e","description":"","filename":"fig6.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/bdbad215ca5a518d99964a71.png"},{"id":58953131,"identity":"29bf027e-71fc-4233-a303-81ce10e03888","added_by":"auto","created_at":"2024-06-24 14:24:54","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":35863,"visible":true,"origin":"","legend":"\u003cp\u003eRelaxation modulus at different strain levels and a constant temperature of 25 \u003csup\u003eo\u003c/sup\u003eC.\u003c/p\u003e","description":"","filename":"fig7.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/a7a034edfe1d523d4d897c31.png"},{"id":58952003,"identity":"a3291074-e920-471f-ac82-9b5c24468222","added_by":"auto","created_at":"2024-06-24 14:08:54","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":27645,"visible":true,"origin":"","legend":"\u003cp\u003eRelaxation modulus at different temperatures and constant strain levels ε =10%.\u003c/p\u003e","description":"","filename":"fig8.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/d3349266fd44b96658f3eaec.png"},{"id":58951999,"identity":"b9d13adc-0643-4bcc-9733-a81c62c88ed6","added_by":"auto","created_at":"2024-06-24 14:08:54","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":18209,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of the generalized Maxwell model.\u003c/p\u003e","description":"","filename":"fig9.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/849737ae9ed6491f8326109f.png"},{"id":58952574,"identity":"781d23a1-2b5f-4e16-aff5-ed3b013c5b3b","added_by":"auto","created_at":"2024-06-24 14:16:54","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":119888,"visible":true,"origin":"","legend":"\u003cp\u003eRelaxation modulus and curve fitting vs time at various strain levels at temperature of 25 \u003csup\u003eo\u003c/sup\u003eC.\u003c/p\u003e","description":"","filename":"fig10.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/7c40a10bdf660a203bc725eb.png"},{"id":58952575,"identity":"549dda7f-751e-4145-85c0-91f13b3aa3a4","added_by":"auto","created_at":"2024-06-24 14:16:55","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":101636,"visible":true,"origin":"","legend":"\u003cp\u003eRelaxation modulus and curve fitting vs time at various temperatures at a strain level of 10%.\u003c/p\u003e","description":"","filename":"fig11.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/0d06661dc7777055357618ab.png"},{"id":58952008,"identity":"6caab9f2-1c62-4695-ba22-e41e670f6850","added_by":"auto","created_at":"2024-06-24 14:08:55","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":24078,"visible":true,"origin":"","legend":"\u003cp\u003eDetermination of log (a\u003csub\u003eT\u003c/sub\u003e)\u003c/p\u003e","description":"","filename":"fig12.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/8f715648c5009cfb6bc9b116.png"},{"id":58952013,"identity":"52c34ff8-15b8-4447-8702-d27890acc1b9","added_by":"auto","created_at":"2024-06-24 14:08:56","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":14338,"visible":true,"origin":"","legend":"\u003cp\u003eTime-temperature equivalence.\u003c/p\u003e","description":"","filename":"fig13.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/3f6cbac43cf059e2f37a38f0.png"},{"id":58952012,"identity":"a8605f17-fd4d-4898-887e-10cd512035a1","added_by":"auto","created_at":"2024-06-24 14:08:55","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":53101,"visible":true,"origin":"","legend":"\u003cp\u003e3D Finite element model of a solid propellant specimen.\u003c/p\u003e","description":"","filename":"fig14.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/02dbfafff8d4a86fed86f319.png"},{"id":58952005,"identity":"7cb33f7c-fb7b-46e4-9769-bf6d936a06b1","added_by":"auto","created_at":"2024-06-24 14:08:54","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":86943,"visible":true,"origin":"","legend":"\u003cp\u003eRelaxation modulus at different strain levels and a constant temperature of 25 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e","description":"","filename":"fig15.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/11ae332441363c71ab22f161.png"},{"id":58952010,"identity":"2f63b8e7-8a52-43f7-bee5-5803f8f73646","added_by":"auto","created_at":"2024-06-24 14:08:55","extension":"png","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":118385,"visible":true,"origin":"","legend":"\u003cp\u003eRelaxation modulus at different temperature and constant strain level of 10%.\u003c/p\u003e","description":"","filename":"fig16.png","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/0c0f2f812bf3c451b4286b84.png"},{"id":58954695,"identity":"c2abb717-e856-4a6b-9360-2e294735c586","added_by":"auto","created_at":"2024-06-24 14:40:55","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1403390,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4545250/v1/ca66ccd1-e0c2-4a38-a43d-4498968ee94b.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Modeling of AP-HTPB Solid Propellant Viscoelastic Behavior Using Modified Maxwell Model","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eComposite solid propellants are crucial in strategic and tactical missiles due to their simplicity, ease of operation, safety, reliability, high specific impulse, extended shelf life, and low cost [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Composite solid propellant is a heterogeneous system in which solid particles of oxidizer and solid metal fuel are distributed in a polymeric matrix [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eHydroxyl terminated poly-butadiene (HTPB) based composite solid propellants are the most common category of composite propellants due to their improved performance and conventional processing technology. Polymeric binder is responsible for holding the solid ingredients into the matrix and providing the composite propellant with its viscoelastic nature. Studying the behaviors of solid propellant under different load conditions is crucial to ensuring the structural reliability and propellant grain integrity of SRM [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. Viscoelasticity is the characteristic of a substance exhibiting both elastic and viscous behavior, the application of stress causes temporary deformation if the stress is quickly removed but permanent deformation if it is maintained. The essence of viscoelasticity is that the motion of a polymer molecule has relaxation characteristics. The change in the polymer's mechanical property with time is collectively referred to as mechanical relaxation. The mechanical relaxation of high molecule materials under fixed stress [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. Common mechanical properties are investigated during tensile tests, such as peak tensile strength and strain at peak strength [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. The goal of this work is to propose a novel approach that builds upon the accumulated knowledge with respect to the mechanical behavior of solid propellants. It does this by utilizing experimental data from uniaxial tensile tests to analyze the behavior of the material and how the solid propellant responds to different strain rates, relaxation tests, modeling, and simulation results. In order to do a structural analysis of the solid propellant grain, the following mechanical characteristics are essential, Young's modulus, ultimate strength, strain at ultimate strength, relaxation modulus, and shift factor. The earlier investigators carried out a number of studies. Park and Schapery [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] presented a mechanical model founded on thermodynamics that explains how temperature and time depend on a particulate composite with a changing microstructure and growing damage. Park and Schapery[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] implementing the so-called pseudo strain theory, the time-temperature superposition principle (TTSP), and the rate-type evolution equation of two internal damage variables, the model was extended to a thermo-viscoelastic model that could simulate the effects of temperature, confining pressure, and axial strain rate on Hydroxyl-Terminated Polybutadiene (HTPB) propellant. Schapery [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] devised a constitutive model to describe the solid propellant's nonlinear elastic deformation characteristics under axial tension and confining pressure. Canga et al[\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] modified the model to allow an efficient numerical implementation and presented the comparisons between finite element analysis results and test data.\u003c/p\u003e \u003cp\u003eJoseph[\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] stress relaxation tests were used to determine the time dependence of the composite solid propellant.\u003c/p\u003e \u003cp\u003eTo determine the CSP's time dependence, Joseph applied stress relaxation tests. Sanal Kumar[\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] suggested that temperature variations have a greater impact on the relaxation modulus of HTPB-based propellant than does time. Swanson and Christensen[\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] developed a constitutive model with a strain softening function to describe the viscoelastic response of a high elongation solid propellant, by fitting experimental data, the model revealed the influence of pressure on the strain-softening function. based on a micromechanical model and the work potential theory. Kadiresh P. N and Sridhar B. T. N [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] applied the Ansys software to compute the stress relaxation behavior of the AP-HTPB solid propellant using a 3D finite element model, however, the model geometry deviated from the standard specimen used in their experiments. Additionally, the comparison between the simulated results and experimental data was limited to the summation of reaction forces for the initial 30 seconds, omitting the influence of diverse loading conditions and temperature variations. Hui Li and Jin‑sheng Xu [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e] used the Prony series called by the generalized Maxwell model to compare the model results of relaxation modulus with the experimental results of the CSP at room temperature, the next step is to compare the experimental data with Ansys's calculation of the reaction forces for the first 30 seconds, without changing the temperatures or loading conditions. A relaxation test can be used to determine the relaxation modulus, which is important for SRM design because propellants with higher relaxation moduli will cause surface cracks during actual firing, while propellants with lower relaxation moduli will cause grain deformation during storage [\u003cspan additionalcitationids=\"CR10 CR11 CR12 CR13 CR14 CR15 CR16 CR17\" citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. It must therefore be essential to conduct more realistic experiments and make more accurate predictions about the solid propellant's relaxation modulus. In actuality, a lot of studies have used relaxation test results to analyze grain under various loading conditions, but there aren't many that simulate stress relaxation tests, particularly when using standard specimens. Therefore, in order to verify the efficacy of the relaxation model obtained, a 3D finite element model must be established in order to reproduce the solid propellant specimen's relaxation behavior under actual test conditions [9- \u003cspan additionalcitationids=\"CR16 CR17 CR18\" citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] .Composite propellants are subjected to temperature and mechanical loads during the production, storage, and operation processes. Using a Zwick universal test machine, tensile tests were performed with different operating temperatures (\u0026minus;\u0026thinsp;40, 25, and 75\u0026deg;C) and strain rates (0.000656 1/s, 0.0328 1/s).\u003c/p\u003e \u003cp\u003eThis work performs complete uniaxial tension and stress relaxation tests to describe how solid propellant acts under different loadings, as mentioned in Section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The constitutive equations that create the nonlinear viscoelastic model are used in Section \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e3\u003c/span\u003e. A 3D finite element model of the solid propellant specimen was constructed to simulate stress relaxation under different conditions using nonlinear viscoelastic material properties.\u003c/p\u003e"},{"header":"2. Experimental Work","content":"\u003cdiv id=\"Sec3\"\u003e\n \u003ch2\u003e2.1. Preparation of Propellant Formulations and Specimens.\u003c/h2\u003e\n \u003cp\u003eThe material under investigation in the present study is a nutritional solid propellant. The primary compounds of the solid propellant comprise differing percentages of 68% ammonium perchlorate and 18% aluminum powder that are distributed in a polymeric binder matrix consisting of hydroxyl-terminated poly-butadiene and the residual percentage consists of other substances. These compounds were mixed, and the resulting propellant slurry was cast vibrating under vacuum in specialized molds that have internal sizes of 200 mm \u0026times; 150 mm \u0026times; 150 mm. Consequently, the molds were cased in a large curing oven with a maintained minimum temperature of 60 ℃ for a total period of 240 hours. After curing, the molds were cut into sheets with uniform thickness using a special cut press according to the Joint Army-Navy-NASA-Air Force Propulsion Committee (JANNAF) standard, as shown in Fig.\u0026nbsp;1[\u003cspan\u003e1\u003c/span\u003e].\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eFigure 1.\u003c/strong\u003e Main dimensions of the JANNAF specimen\u003c/p\u003e\n \u003cp\u003e.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eFigure 2.\u003c/strong\u003e Flowchart of methodology work.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec4\"\u003e\n \u003ch2\u003e2.2. Uniaxial Tensile Test\u003c/h2\u003e\n \u003cp\u003eAll tensile tests were conducted using a computer-controlled universal test machine (Zwick/Roell) Z050. The machine was equipped with a thermal chamber to set and maintain the desired temperature constant during the test [\u003cspan\u003e25\u003c/span\u003e]. Prior to testing, the thickness and width of each specimen were accurately measured in its gauge length region. The effective gauge length (L\u003csub\u003eO\u003c/sub\u003e) was determined to be 74.4 mm, and the crosshead speed of the machine was maintained at 50.8 mm/min. Each experimental test was conducted using five samples, with the mean measurement value being utilized for the analysis in simulations. Additionally, prior to each test, the specimens underwent pre-conditioning for two hours in an external conditioning chamber. The specimens were tested at three different temperatures (+\u0026thinsp;25, +\u0026thinsp;75, and \u0026minus;\u0026thinsp;40 \u003csup\u003eo\u003c/sup\u003eC), with a constant crosshead speed of 50 mm/min, corresponding to a strain rate ℇ\u003csub\u003eo\u003c/sub\u003e =0.016404 s\u003csup\u003e\u0026minus;\u003c/sup\u003e1. Furthermore, each experiment were carried out under constant temperature conditions maintained by a digital control temperature chamber, ensuring an accuracy of \u0026plusmn;\u0026thinsp;0.1\u003csup\u003eo\u003c/sup\u003eC in relation to the designated set point throughout the duration of the testing phase.\u003c/p\u003e\n \u003cp\u003eThe stress-strain curves shown in Fig. \u003cspan\u003e3\u003c/span\u003e were generated using the load-displacement data. The curves depicted demonstrate the highest levels of stress and strain that occur at the point of maximum stress. The temperature has a significant impact on the deformation behavior of the AP-HTPB solid propellant. The maximum stress and maximum strain increased significantly by 155.6% and 49.23%, respectively, while the break stress and break strain increased by 178% and 49.9%. Moreover, Young\u0026apos;s modulus increased significantly by 2.21% when the temperature decreased from +\u0026thinsp;75\u0026deg;C to \u0026minus;\u0026thinsp;40\u0026deg;C at a strain rate of ℇ\u003csub\u003eo\u003c/sub\u003e = 0.016404 s\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e. The stress and strain values at the yield point, maximum point, break point, and Young\u0026apos;s modulus were experimentally determined and are presented in Table \u003cspan\u003e1\u003c/span\u003e.\u003c/p\u003e\n \u003cdiv\u003e\n \u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 1\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eThe values of the uniaxial tensile test result.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"8\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eTemperature\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eYield Stress\u003c/p\u003e\n \u003cp\u003e(MPa)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eYield Strain\u003c/p\u003e\n \u003cp\u003e(%)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eYoung\u0026rsquo;s Modulus\u003c/p\u003e\n \u003cp\u003e(MPa)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMax Stress\u003c/p\u003e\n \u003cp\u003e(MPa)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMax Strain\u003c/p\u003e\n \u003cp\u003e(%)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eBreak Stress\u003c/p\u003e\n \u003cp\u003e(MPa)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eBreak Strain\u003c/p\u003e\n \u003cp\u003e(%)\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eT\u0026thinsp;=\u0026thinsp;\u0026minus;\u0026thinsp;40 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003cp\u003eT\u0026thinsp;=\u0026thinsp;+\u0026thinsp;25 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003cp\u003eT\u0026thinsp;=\u0026thinsp;+\u0026thinsp;75 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.832\u003c/p\u003e\n \u003cp\u003e0.524\u003c/p\u003e\n \u003cp\u003e0.626\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e14.80\u003c/p\u003e\n \u003cp\u003e20.39\u003c/p\u003e\n \u003cp\u003e35.77\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.62\u003c/p\u003e\n \u003cp\u003e2.55\u003c/p\u003e\n \u003cp\u003e1.75\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.40\u003c/p\u003e\n \u003cp\u003e0.735\u003c/p\u003e\n \u003cp\u003e0.626\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e53.70\u003c/p\u003e\n \u003cp\u003e39.50\u003c/p\u003e\n \u003cp\u003e35.77\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.25\u003c/p\u003e\n \u003cp\u003e0.64\u003c/p\u003e\n \u003cp\u003e0.51\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e56.22\u003c/p\u003e\n \u003cp\u003e41.53\u003c/p\u003e\n \u003cp\u003e39.50\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eFigure \u003cspan\u003e4\u003c/span\u003e presents the results of tensile testing conducted at different strain rates while maintaining a constant temperature of 25\u0026deg;C. The significant impact of strain rate or loading rate on the material\u0026apos;s mechanical behavior highlights the viscoelastic properties of the data presented in the text. This is in contrast to completely elastic materials, where the rate at which deformation occurs is not critical. Furthermore, at lower strain rates, the material also shows a linear viscoelastic response, solid particles noticeably improve the binder\u0026rsquo;s properties. Simultaneously, at high strain rates, the interfacial cohesion of the oxidizer with the binder decreases, and the work of the reinforcement reduces, which causes nonlinearity of the curve.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec5\"\u003e\n \u003ch2\u003e2.3. Relaxation Test\u003c/h2\u003e\n \u003cp\u003eThe viscoelastic properties of a propellant can be demonstrated by a relaxation test, where a sample is subjected to a constant strain and the change in stress \u0026sigma; (t) over time is evaluated [\u003cspan\u003e26\u003c/span\u003e], as shown in Fig. \u003cspan\u003e5\u003c/span\u003e. This test was conducted using the same test sample that was utilized in the tensile test. This figure shows that at a constant strain level, the stress generated in the propellant specimen relaxes with time. Additionally, the experiment results of the stress test indicate that the stress applied to the specimen will gradually decrease as it is subjected to a constant level of strain for a period of time. This demonstrates the impact of strain level on stress. Before conducting the experimental procedures, the samples endured a three-hour conditioning process in an external environment chamber to establish thermal equilibrium. In addition, the relaxation test and uniaxial tensile test were performed in an environment-controlled chamber with an error in temperature of \u0026plusmn;\u0026thinsp;0.2\u003csup\u003eo\u003c/sup\u003eC. The sampling frequency for recording load change against time was 1 Hz. the modulus curve of relaxation is shown at 5% 10% 15% 20% and 25% strain level in Fig. \u003cspan\u003e5\u003c/span\u003e. The curves remain constant for 45 minutes at a constant a temperature of 25 \u003csup\u003eo\u003c/sup\u003eC.\u003c/p\u003e\n \u003cp\u003eFrom comparing the stress results obtained at different strain levels in Fig. \u003cspan\u003e5\u003c/span\u003e, it is clear that stress relaxation tests at higher strain levels show greater relaxation than those at lower strain levels. This indicates that a more rapid reduction in stress occurs as strain levels diminish from 25\u0026ndash;5% with the passage of time, particularly when the initial deformation is higher (20% or 25%) as opposed to being lower (5% or 10%).Essentially, there is a nonlinear decrease in stress with decreasing strain levels, indicating that higher strain levels disrupt the material\u0026apos;s internal structure more significantly and lead to faster relaxation and stress reduction.\u003c/p\u003e\n \u003cp\u003eFigure \u003cspan\u003e6\u003c/span\u003e shows relaxation tests performed on the propellant under a constant strain level of 10% at various temperatures. It can be observed that stress reduction increases with temperature rise (\u0026minus;\u0026thinsp;40 \u0026ordm;C to +\u0026thinsp;75 \u0026ordm;C) and also with longer periods of time for nonlinear stress relaxation. This observation aligns with the general principle of increased molecular mobility at higher temperatures. When molecules move more readily, they can easily rearrange and recover from imposed deformation, resulting in quicker stress reduction.\u003c/p\u003e\n \u003cp\u003eThe relaxation modulus E\u003csub\u003er\u003c/sub\u003e (t) is expressed by:\u003c/p\u003e\n \u003cdiv id=\"Equ1\"\u003e\n \u003cdiv id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$$\\text{E}\\text{r} \\left(\\text{t}\\right)=\\frac{{\\sigma } \\left(\\text{t}\\right)}{\\text{ℇ}}$$\u003c/div\u003e\n \u003cdiv\u003e1\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eThe relaxation modulus E\u003csub\u003eR\u003c/sub\u003e(t) is expressed by Eq.\u0026nbsp;(\u003cspan\u003e1\u003c/span\u003e) [\u003cspan\u003e27\u003c/span\u003e], where it is influenced by both strain level and temperature Fig. 7 demonstrate affects strain level on relaxation modulus, as strain level increases within the same period 45 min from 5\u0026ndash;25% at +\u0026thinsp;25\u0026deg;C, relaxation modulus decreases nonlinearly. Similarly, Fig. \u003cspan\u003e8\u003c/span\u003e, shows that temperature affects relaxation modulus, as temperature rises from \u0026minus;\u0026thinsp;40 \u0026ordm;C to +\u0026thinsp;75 \u0026ordm;C for the same period (45 min) at a constant strain level of 10%, relaxation modulus also decreases nonlinearly. Higher temperatures increase internal energy and molecular mobility, facilitating faster rearrangement and reducing stress relaxation.\u003c/p\u003e\n \u003cp\u003eUnderstanding these interactions between strain level, temperature, and material behavior is crucial for predicting material response under different loading conditions and designing applications that require optimal stress response.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eFigure 7.\u003c/strong\u003e Relaxation modulus at different strain levels and a constant temperature of 25 \u003csup\u003eo\u003c/sup\u003eC.\u003c/p\u003e\n \u003cp\u003eAt lower temperatures, the material is in a glassy state and has a high modulus, meaning it is stiff and rigid. With increasing t to the glass transition temperature T g, the material becomes more rubbery, and the modulus decreases [\u003cspan\u003e1\u003c/span\u003e]. As the temperature continues to rise beyond (Tg), the decline in modulus becomes slower. This is because the material is undergoing a transition from a glassy state to a rubbery state, which involves molecular rearrangements and increased mobility of polymer chains. These changes result in a decrease in stiffness but at a slower rate compared to the initial decrease at lower temperatures.\u003c/p\u003e\n \u003cp\u003eThe behavior of the nonlinear relaxation modulus is adapted with temperature variations. As the temperature rises from \u0026minus;\u0026thinsp;40\u0026deg;C to +\u0026thinsp;75\u0026deg;C, there is a decline in the nonlinear relaxation modulus over time. This suggests a decrease in the material\u0026apos;s capacity to endure deformation resulting from applied stress or strain over time. Furthermore, the values of the equilibrium relaxation modulus diminish as temperature rises. The equilibrium relaxation modulus characterizes the prolonged reaction of the material once it has attained a state of stability. As temperature escalates, the material\u0026apos;s capability to regain its original form post-deformation diminishes, resulting in decreased values of the equilibrium relaxation modulus.\u003c/p\u003e\n \u003cp\u003eAs the temperature rises towards and exceeds the glass transition temperature (Tg), significant changes occur in the material\u0026apos;s physical properties. These alterations encompass a decreased rate of decline in modulus, along with decreases in both nonlinear relaxation modulus and equilibrium relaxation modulus values.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec6\"\u003e\n \u003ch2\u003e2.4. Mathematical Modeling\u003c/h2\u003e\n \u003cp\u003eThe Prony series is a mathematical technique that excels at representing the time-dependent relaxation behavior of viscoelastic materials, including solid propellants [\u003cspan\u003e24\u003c/span\u003e]. For a typical viscoelastic material, the relaxation modulus can be represented by a power law, which is in the form of a Prony series[\u003cspan\u003e28\u003c/span\u003e]. The modulus obtained based on the Prony series has not only high accuracy, but is also more convenient in data processing, it can closely model the behavior of many viscoelastic materials. In this paper have adopted a particular Prony series formulation in Eq.\u0026nbsp;(2) to model the relaxation modulus of the generalized Maxwell model of the solid propellant they are studying [\u003cspan\u003e26\u003c/span\u003e].\u003c/p\u003e\n \u003cp\u003e\u003cimg 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\"\u003e(2)\u003c/p\u003e\n \u003cp\u003eWhere, Er - relaxation modulus, MPa;\u003c/p\u003e\n \u003cp\u003eȠi - Viscosity coefficient of the i-th Maxwell element, MPa.s.\u003c/p\u003e\n \u003cp\u003eEi - Elastic modulus of the i-th Maxwell element, MPa.\u003c/p\u003e\n \u003cp\u003eE\u003cspan\u003e\u003cspan\u003e\\(\\infty\\)\u003c/span\u003e\u003c/span\u003e - Equilibrium relaxation modulus coefficient\u003c/p\u003e\n \u003cp\u003eIdentification of linear viscoelastic parameters [\u003cspan\u003e29\u003c/span\u003e]. For the one-dimensional condition, the linear viscoelastic model with relaxation modulus described by the Prony series, namely the generalized Maxwell model, is composed of multiple spring-dashpot elements connected in parallel, each with its own relaxation time, see Fig.\u0026nbsp;9. can be presented [\u003cspan\u003e30\u003c/span\u003e].\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eFigure 9.\u003c/strong\u003e Schematic diagram of the generalized Maxwell model.\u003c/p\u003e\n \u003cp\u003eCombining the above formulas and relaxation test data, each coefficient of Eq.\u0026nbsp;(2) can be obtained by the nonlinear Least Squares Method, which is implemented by the fit function of MATLAB software [\u003cspan\u003e24\u003c/span\u003e]. The MATLAB program was built based on the nonlinear least squares method to determine the Prony coefficients [\u003cspan\u003e11\u003c/span\u003e]. By choosing the number of Prony components and performing curve fitting for the relaxation test data, we can get the unknown coefficients. In the following calculations, the Prony series is taken as 5 terms.\u003c/p\u003e\n \u003cp\u003eIt\u0026rsquo;s clear from comparing the experimental results obtained by testing the relaxation modulus temperature at 25 \u003csup\u003eo\u003c/sup\u003eC with the mathematical equation implemented by the appropriate function of MATLAB shown in Eq. (2) that various strain levels with a constant temperature of 25 \u003csup\u003eo\u003c/sup\u003eC influence the nonlinear relaxation modulus over time. Figure \u003cspan\u003e10\u003c/span\u003e shows that the curves of E\u003csub\u003er\u003c/sub\u003e and curve fitting at strain levels (5%, 10%, 15%, 20%, and 25) are almost identical to each other and, the maximum differences between calculation and experimental, analysis results are 2.35% respectively. It investigates the prediction of outcomes of propellant behavior as a viscoelastic generalization Maxwell model.\u003c/p\u003e\n \u003cp\u003eThe results presented in Table \u003cspan\u003e2\u003c/span\u003e, the relevant Prony coefficients according to the relaxation test data for solid propellant at a temperature of 25 \u003csup\u003eo\u003c/sup\u003eC and strain levels (5%, 10%, 15%, 20%, and 25%,) obtain from this data the influence of various of strain levels on equilibrium relaxation modulus.\u003c/p\u003e\n \u003cdiv\u003e\n \u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 2\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eProny series coefficients of HTPB propellant\u0026rsquo;s relaxation modulus of various strain levels at a temperature of 25 \u003csup\u003eo\u003c/sup\u003eC.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eParameters\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003et\u003csub\u003ei\u003c/sub\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e5%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e10%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e15%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e20%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e25%\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE\u003cspan\u003e\u003cspan\u003e\\(\\varvec{\\infty }\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e6.266\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e\u0026minus;\u0026thinsp;18.11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.7176\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-12.16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e\u0026minus;\u0026thinsp;12.58\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.1519\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.4054\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.413\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.4349\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.3559\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.4467\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.3038\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.2891\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.2973\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.3037\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.1329\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.3933\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.3787\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.3268\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.3494\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE\u003csub\u003e4\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e100\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1.972\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e\u0026minus;\u0026thinsp;2.069\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.3167\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-1.409\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e\u0026minus;\u0026thinsp;1.62\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE\u003csub\u003e5\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e\u0026minus;\u0026thinsp;6.49\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e21.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.1045\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e14.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e15.12\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cstrong\u003eModel evaluation.\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003eIn order to assess the predictive capacity of the proposed model, the root mean square error (RMSE) is introduced and calculated as follows.\u003c/p\u003e\n \u003cp\u003e\u003cimg src=\"https://myfiles.space/user_files/122228_c8a1650c59388082/122228_custom_files/img171923777495.png\"\u003e[\u003cspan\u003e24\u003c/span\u003e](3).\u003c/p\u003e\n \u003cp\u003ewhere E\u003csub\u003er\u003c/sub\u003e experiment denotes the experimental relaxation modulus of CSP, E\u003csub\u003er\u003c/sub\u003e model is the model prediction corresponding to the same strain levels, and n is the number of data points.\u003c/p\u003e\n \u003cdiv\u003e\n \u003cdiv align=\"left\"\u003eTable 3 shows the calculated RMSE values for various combinations of relaxation and strain levels. The highest RMSE value is 0.006127 MPa. This indicates the largest discrepancy between the model\u0026apos;s prediction and the experimental data for a specific combination of relaxation and strain levels. In each cases, the RMSE values are below 0.01 MPa. This suggests a good agreement between the model\u0026apos;s predictions and the experimental data for the majority of tested conditions. \u0026nbsp;\u003c/div\u003e\n \u003ctable id=\"Tab3\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003e\n \u003cp\u003e\u003cstrong\u003eTable 3.\u003c/strong\u003e RMSE values of experimental and predicted results under different strain levels.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"6\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eStrain level\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e5%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e10%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e15%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e20%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e25%\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eRMSE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.006127\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.004058\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.00253\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.002733\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.002296\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cstrong\u003eFigure 11.\u003c/strong\u003e Relaxation modulus and curve fitting vs time at various temperatures at a strain level of 10%.\u003c/p\u003e\n \u003cp\u003eFigure 11 shows that comparing experimental data and the data obtained from the calculation of the Prony Series for curves of Er and curve fitting for a specific strain level of 10% at three different temperatures (\u0026minus;\u0026thinsp;40 \u003csup\u003eo\u003c/sup\u003eC, 25 \u003csup\u003eo\u003c/sup\u003eC, and 75 \u003csup\u003eo\u003c/sup\u003eC) are identical to each other, generalization Maxwell model components are found to lead to a good fit and high accuracy. From investigating the results of the relaxation test in Table \u003cspan\u003e4\u003c/span\u003e, it becomes clear that influence various temperature on the equilibrium relaxation modulus values and the constant strain level in each component of the series. It investigates the prediction of outcomes of propellant behavior as a viscoelastic generalization Maxwell model.\u003c/p\u003e\n \u003cp\u003eThe results presented in Table \u003cspan\u003e4\u003c/span\u003e, the relevant Prony coefficients according to the relaxation test data for solid propellant at strain level 10% and temperature of 25 \u003csup\u003eo\u003c/sup\u003eC, \u0026minus;\u0026thinsp;40 \u003csup\u003eo\u003c/sup\u003eC, and 75 \u003csup\u003eo\u003c/sup\u003eC are listed in Table \u003cspan\u003e4\u003c/span\u003e.\u003c/p\u003e\n \u003cdiv\u003e\n \u003ctable id=\"Tab4\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 4\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eProny series coefficients of HTPB propellant\u0026rsquo;s relaxation modulus of various temperature at strain level 10%\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"5\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eParameters\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003et\u003csub\u003ei\u003c/sub\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u0026minus;\u0026thinsp;40 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e25 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e75 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE\u003cspan\u003e\u003cspan\u003e\\(\\varvec{\\infty }\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-56.99\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e\u0026minus;\u0026thinsp;18.11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e39.09\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2.449\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.4054\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.4024\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1.259\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.3038\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.03085\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1.111\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.3933\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.2201\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE\u003csub\u003e4\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e100\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-8.514\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e\u0026minus;\u0026thinsp;2.069\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e6.789\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE\u003csub\u003e5\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1000\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e66.35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e21.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e\u0026minus;\u0026thinsp;44.63\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eTable \u003cspan\u003e5\u003c/span\u003e. shows the values of RMSE under relaxation and different temperatures at a strain level of 10%. As can be observed, the majority of examples had an RMSE below 0.015 MPa, with a highest value of 0.01202 MPa.\u003c/p\u003e\n \u003cdiv\u003e\n \u003ctable id=\"Tab5\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 5\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eRMSE values of experimental and predicted results under different strain levels.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"4\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eTemperature\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u0026minus;\u0026thinsp;40 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e25 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e75 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eRMSE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.008261\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.004058\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.01202\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eThe shift factors a\u003csub\u003eT\u003c/sub\u003e describe the temperature dependence of the relaxation time and usually follow the empirical Williams-Landel-Ferry (WLF) equation, which determines how the viscoelastic property of propellant is expressed as follows:\u003c/p\u003e\n \u003cp\u003eLog (a\u003csub\u003eT\u003c/sub\u003e) = \u003cspan\u003e\u003cspan\u003e\\(\\frac{\\text{C}1 ( \\text{T}-\\text{T}\\text{r}\\text{e}\\text{f})}{\\text{C}2+\\text{T}-\\text{T}\\text{r}\\text{e}\\text{f}}\\)\u003c/span\u003e\u003c/span\u003e (4)\u003c/p\u003e\n \u003cp\u003eWhere C\u003csub\u003e1\u003c/sub\u003e and C\u003csub\u003e2\u003c/sub\u003e are the material constants C\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u0026minus;\u0026thinsp;5.35, C\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;779.2, according to the WLF method based on reference temperature T\u003csub\u003eref\u003c/sub\u003e = 25 \u003csup\u003eo\u003c/sup\u003eC .The time-temperature superposition principle (TTSP) can be used to create the relaxation modulus master curve, which is shown in Fig.\u0026nbsp;11, using the previously described data.\u003c/p\u003e\n \u003cp\u003eAs shown in Fig. \u003cspan\u003e13\u003c/span\u003e. The time \u0026ndash; temperature shift is one of the most important values in the structural analysis of viscoelastic materials because it allows the temperature impact to be converted to the time influence, which makes the analysis much easier.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec7\"\u003e\n \u003ch2\u003e2.5 Simulation, Experimental and Model Verification\u003c/h2\u003e\n \u003cp\u003eA finite element analysis was performed using the ANSYS software in order to examine the stress relaxation properties of the AP-HTPB composite solid propellant [\u003cspan\u003e10\u003c/span\u003e]. The simulation was experimented with to precisely replicate the conditions of uniaxial tensile loading by discretizing the same specimen utilized during actual tests, thereby ensuring accurate dimensions. The mathematical data acquired from the relaxation model at \u0026epsilon;\u0026thinsp;=\u0026thinsp;10% and T\u0026thinsp;=\u0026thinsp;298K were initially tested. Within the nonlinear viscoelastic parts of the generalized Maxwell model branches, the Prony coefficient of the relaxation modulus was incorporated to characterize the material model in ANSYS software.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eFigure 14.\u003c/strong\u003e 3D Finite element model of a solid propellant specimen.\u003c/p\u003e\n \u003cp\u003eThe mechanical boundary conditions constrained all degrees of freedom at one end and subjected the other end\u0026apos;s nodes to X-direction shifts at strain values of 5%, 10%, 15%, 20%, and 25% with an initial tension rate of 10 mm/min. The tension was maintained for 45 minutes, and the variation of the maximum stress over that time was plotted against time with a one-second interval. In the thermal boundary conditions, the initial temperature was the strain-free temperature, and the final temperature was the experimental test temperature. In order to validate the numerical results obtained from finite element analysis, two different loading conditions were chosen based on the experimental study: relaxation stress at various strain levels and constant temperature and relaxation stress at various temperature and constant strain levels. The Prony coefficient of relaxation modulus according to Tables \u003cspan\u003e2\u003c/span\u003e and \u003cspan\u003e4\u003c/span\u003e was added to the generalized Maxwell model branches under the nonlinear viscoelastic components to define the material model in ANSYS software. To get the relaxation modulus from the results of ANSYS software, it will be compensated in Eq. (2). The other material properties of solid propellant were taken from Tables \u003cspan\u003e6\u003c/span\u003e and \u003cspan\u003e7\u003c/span\u003e.\u003c/p\u003e\n \u003cdiv\u003e\n \u003ctable id=\"Tab6\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 6\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eThe change of viscoelastic solid propellant properties of various strain levels at temperature 25 \u003csup\u003eo\u003c/sup\u003eC.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"11\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eNumber\u003c/p\u003e\n \u003cp\u003eof\u003c/p\u003e\n \u003cp\u003eTerm\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"5\"\u003e\n \u003cp\u003eShear modulus (G)\u003c/p\u003e\n \u003cp\u003e(MPa)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"5\"\u003e\n \u003cp\u003eBulk modulus (K)\u003c/p\u003e\n \u003cp\u003e(MPa)\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e5%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e10%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e15%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e20%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e25%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e5%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e10%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e15%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e20%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e25%\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.051\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.135\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.138\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.145\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.119\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e12.66\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e33.78\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e34.42\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e36.24\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e29.66\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.149\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.101\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.096\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.099\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.101\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e37.23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e24.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e24.78\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25.31\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.044\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.131\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.126\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.109\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.117\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e11.08\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e32.77\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e31.56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e27.23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e29.11\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.658\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.691\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.105\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.47\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.541\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e164.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-172.42\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e26.39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-117.42\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-135\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-2.166\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e7.109\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.035\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4.87\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e5.047\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-540.83\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1775\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e8.71\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1216. 7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1260\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cdiv\u003e\n \u003ctable id=\"Tab7\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 7\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eThe Change of viscoelastic solid propellant properties of various temperature at strain level of 10%\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eNumber\u003c/p\u003e\n \u003cp\u003eof\u003c/p\u003e\n \u003cp\u003eTerm\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"3\"\u003e\n \u003cp\u003eShear Modulus\u003c/p\u003e\n \u003cp\u003e(G)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"3\"\u003e\n \u003cp\u003eBulk Modulus\u003c/p\u003e\n \u003cp\u003e(K)\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u0026minus;\u0026thinsp;40 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e25 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e75 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u0026minus;\u0026thinsp;40 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e25 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e75 \u003csup\u003eo\u003c/sup\u003eC\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.817\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.135\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.134\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e204.08\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e33.78\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e33.53\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.4202\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.1014\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.0103\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e104.92\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2.571\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.3709\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.1313\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.074\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e92.58\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e32.78\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e18.34\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-2.8402\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-0.691\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2.266\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-709.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-172.42\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e565.75\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e22.147\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e7.11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-14.89\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e5529.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1775\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e-3719.17\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eIn the following calculation, for the studied of composite solid propellant is treated as a nonlinear viscoelastic material. The thermal and mechanical parameters needed for this calculation are listed in Table \u003cspan\u003e8\u003c/span\u003e.\u003c/p\u003e\n \u003cdiv\u003e\n \u003ctable id=\"Tab8\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv\u003eTable 8\u003c/div\u003e\n \u003cdiv\u003e\n \u003cp\u003eParameters of composite solid propellant.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"2\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMaterial Parameter\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eThe Value\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eModulus E (MPa)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eE (t, t\u003csub\u003ei\u003c/sub\u003e )\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eExpansion coefficient \u0026alpha;(1/k)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePoisson ratio\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.498\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDensity \u0026rho; (kg/m3)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1750\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eThermal conductivity k\u003c/p\u003e\n \u003cp\u003e(w/m \u0026middot; k)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.51\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSpecific Heat (J/kg \u0026middot; k)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1247.6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eA comparison of the relaxation modulus between the finite element results and the experimental data at three distinct strain levels (5%, 15%, and 25%) at a constant temperature of 25\u0026deg;C is presented in Fig. \u003cspan\u003e15\u003c/span\u003e. Additionally, Fig.\u0026nbsp;16 compares the results of the finite element results with the experimental data for the relaxation modulus at three different temperatures (-40\u0026deg;C, 25\u0026deg;C and 75\u0026deg;C) with a constant strain level of 10%.\u003c/p\u003e\n \u003cp\u003e.\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eFigure 16.\u003c/strong\u003e Relaxation modulus at different temperature and constant strain level of 10%.\u003c/p\u003e\n \u003cp\u003eThe comparison between the simulation or numerical results and the experimental data is clearly demonstrated by the data depicted in Figs. \u003cspan\u003e15\u003c/span\u003e and 16. The numerical results show that the material nonlinear viscoelastic effect is very close to the experimental data from the change of strain levels and temperature effect, which must be considered in the analysis of a composite solid propellant, and the maximum differences between the numerical and experimental, analysis results are 2.76%, and 3.98% respectively.\u003c/p\u003e\n \u003cp\u003eThis indicates that the proposed method demonstrates a significant level of accuracy in replicating the relaxation behavior of the solid propellant under different loading conditions and maintains stability throughout the whole testing period.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"3. Conclusions","content":"\u003cp\u003eIn this work, a nonlinear viscoelastic model was proposed based on experimental data to define the response of this material under different loading conditions. In addition, a 3D finite element model has been established to stimulated solid propellant relaxation behavior under realistic test conditions. From the results, the main conclusions can be summarized as follows:\u003c/p\u003e\n\u003cp\u003e\u003cspan\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e1. Young`s modulus, maximum stress, and strain at maximum stress all increase proportionally as the strain rate increase.\u003c/p\u003e\u003cspan\u003e\n \u003cp\u003e2. At constant strain levels the stresses generated in the propellant relax with time. The relaxation modulus decreases when the temperature increases or when the strain level decreases.\u003c/p\u003e\n\u003c/span\u003e\u003cspan\u003e\n \u003cp\u003e3. Increasing the applied load in the propellant leads to an increase in deformation of the composite because more stress transferred to the matrix interface, which results in the de-bonding of fibers and the failure of the composite structure happen. Also, increasing the temperature results in higher macromolecular mobility.\u003c/p\u003e\n\u003c/span\u003e\u003cspan\u003e\n \u003cp\u003e4. For all strain levels, relaxation modulus \u0026ndash;time curves are generated using Maxwell model. The model was able to capture all facts of material response and close matching with actual test curves has been observed. This indicates the suitability of using the Maxwell model to describe the relaxation behavior of solid propellant.\u003c/p\u003e\n\u003c/span\u003e\u003cspan\u003e\n \u003cp\u003e5. To increase the level of the simulation of solid propellant behavior, the material must be well defined as a nonlinear viscoelastic material, and the Prony series coefficients must be calculated at different strain levels and temperatures.\u003c/p\u003e\n\u003c/span\u003e\u003cspan\u003e\n \u003cp\u003e6. The nonlinear viscoelastic model demonstrates excellent predictive performance when compared to the experimental relaxation modulus. The largest RMSE value at various strain levels is 0.006127 MPa, and most cases are lower than 0.01 MPa. The largest RMSE value at various temperatures with a strain level of 10% is 0.008261 MPa, and most cases are lower than 0.015 MPa.\u003c/p\u003e\n\u003c/span\u003e\n"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eAhmed F., Walid, and Ahmed ElSabbagh wrote the main part of the manuscript and prepared itFigures 1 -16 and Tables 1 -8. All authors reviewed the manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eNevi\u0026egrave;re, R. 2006 . An extension of the time-temperature superposition principle to non-linear viscoelastic solids. Int. J. Solids Struct., vol. 43, no. 17, pp. 5295\u0026ndash;5306, doi: 10.1016/j.ijsolstr..09.009.\u003c/li\u003e\n\u003cli\u003eWang, Z., H. Qiang, T. Wang, and G. Wang. 2019. Tensile behaviors of thermal aged HTPB propellant at low temperatures under dynamic loading. Mech Time-Depend Mate\u003cem\u003er\u003c/em\u003e, doi: 10.1007/s11043-019-09413-4.\u003c/li\u003e\n\u003cli\u003eFleeman, E. L. 2010 . Aerodynamics , Propulsion , and Heat Transfer of Missiles. Aerosp. Eng. 2010 by John Wiley Sons, Ltd, doi: 10.1002/9780470686652.eae375.\u003c/li\u003e\n\u003cli\u003eIshitha, K., and Ramakrishna .P. A. 2014. Studies on the role of iron oxide and copper chromite in solid propellant combustion. Combust. Flame, doi: 10.1016/j.combustflame. 03.015.\u003c/li\u003e\n\u003cli\u003eElhedery ,T. M., and Guozhu ,L., \u0026ldquo;Experimental Determination of Mechanical Properties of a Thermoplastic Composite Solid Propellant.\u0026rdquo;\u003c/li\u003e\n\u003cli\u003eWang, Z., Qiang, H., Wang, G., and Huang, Q. 2015. Tensile mechanical properties and constitutive model for HTPB propellant at low temperature and high strain rate. J. Appl. Polym. Sci\u003cem\u003e.\u003c/em\u003e, vol. 132, no. 24, pp. 1\u0026ndash;9, doi: 10.1002/app.42104.\u003c/li\u003e\n\u003cli\u003eChyuan, S. W. 2002. Nonlinear thermoviscoelastic analysis of solid propellant grains subjected to temperature loading. Finite Elem. Anal.\u003cem\u003e Des.\u003c/em\u003e, vol. 38, no. 7, pp. 613\u0026ndash;630, doi: 10.1016/S0168-874X(01)00095-6.\u003c/li\u003e\n\u003cli\u003ePark, S. W., \u0026amp; Schapery, R. A. 1997. A viscoelastic constitutive model for particulate composites with growing damage. Int. J. Solids Struct., vol. 34, no. 8, pp. 931\u0026ndash;947, doi: 10.1016/S0020-7683(96)00066-2.\u003c/li\u003e\n\u003cli\u003eDrozdov, A. D. 1997. A constitutive model for nonlinear viscoelastic media. Int. J. Solids Struct\u003cem\u003e.\u003c/em\u003e, vol. 34, no. 21, pp. 2685\u0026ndash;2707, doi: 10.1016/S0020-7683(96)00178-3.\u003c/li\u003e\n\u003cli\u003eAdel, W. M. \u0026amp; Liang, G. Z. 2017. Developing a viscoelastic relaxation model for AP-HTPB composite solid propellant based on experimental data. in 21st AIAA International Space Planes and Hypersonics Technologies Conference, Hypersonics American Institute of Aeronautics and Astronautics Inc, AIAA. doi: 10.2514/6.2017-2377.\u003c/li\u003e\n\u003cli\u003eAdel, W. M. \u0026amp; Liang,G. Z. 2017. Study of cooldown thermal loading effect on the bore deformation of viscoelastic solid propellant grain. in 53rd AIAA/SAE/ASEE Joint Propulsion Conference, American Institute of Aeronautics and Astronautics Inc, AIAA, 2017. doi: 10.2514/6.2017-4692.\u003c/li\u003e\n\u003cli\u003eGligorijević, N. et al. 2014. Mechanical Properties of HTPB Composite Propellants in the Initial Period of Service Life. \u003c/li\u003e\n\u003cli\u003eGligorijević , N. I. et al\u003cem\u003e. \u003c/em\u003e 2016. Mechanical characterization of composite solid rocket propellant based on hydroxy-terminated polybutadiene. Hem. ind.70, vol. 70, no. 5, pp. 581\u0026ndash;594.\u003c/li\u003e\n\u003cli\u003ePark, S. \u0026amp; W S. \u0026ldquo;Development of a Nonlinear Thermo-Viscoelastic Constitutive Equation for Particulate Composites with Growing Damage.pdf.\u0026rdquo; \u003c/li\u003e\n\u003cli\u003eSchapery,R. A. 1987 . Nonlinear constitutive equations for solid propellant based on a work potential and micromechanical model. no. March 1987, pp. 0\u0026ndash;10, doi: 10.13140/RG.2.2.34351.79527.\u003c/li\u003e\n\u003cli\u003eCanga,M. E. Becker, E. B. \u0026amp; Upek,Oz. .Constitutive modeling of viscoelastic materials with damage computational aspects.\u0026rdquo;. Available: www.elsevier.com/locate/cma\u003c/li\u003e\n\u003cli\u003eJ. H. Stoker, 1964. Use of stress relaxation tests to characterize time dependencies of a composite solid propellant. AIAA J., vol. 2, no. 10, pp. 1816\u0026ndash;1818, doi: 10.2514/3.2671.\u003c/li\u003e\n\u003cli\u003eSabarinath,K. R. K. Sandeep. 2013 . Influence of Viscoelastic properties of Solid Propellants on Starting Transient of Solid Rocket Motors. 5th Eur. Conf. Aerosp. Sci., no. July, pp. 2\u0026ndash;5, doi: 10.13140/2.1.2140.5122.\u003c/li\u003e\n\u003cli\u003eSwanson, S. R. \u0026amp; Christensen,L. W.1983. A constitutive formulation for high-elongation propellants. J. Spacecr. Rockets, vol. 20, no. 6, pp. 559\u0026ndash;566, doi: 10.2514/3.8587.\u003c/li\u003e\n\u003cli\u003eKadiresh, P. N. \u0026amp; Sridhar,B. T. N.2008. Experimental evaluation and simulation on aging characteristics of aluminised AP-HTPB composite solid propellant. Mater. Sci. Technol\u003cem\u003e.\u003c/em\u003e, vol. 24, no. 4, pp. 406\u0026ndash;412, doi: 10.1179/174328408X278420.\u003c/li\u003e\n\u003cli\u003eLi, H. J. Xu,sheng. Chen, X. J. fa Zhang, \u0026amp; Li, J. 2023 . A nonlinear viscoelastic constitutive model with damage and experimental validation for composite solid propellant. Sci. Rep., vol. 13, no. 1, pp. 1\u0026ndash;20, doi: 10.1038/s41598-023-29214-7.\u003c/li\u003e\n\u003cli\u003eAdel, W. M. and Guo-zhu, L. 2017. Analysis of Mechanical Properties for AP / HTPB Solid Propellant under Different Loading Conditions. in International Journal of Aerospace and Mechanical Engineering, vol. 11, no. 12, pp. 1915\u0026ndash;1919.\u003c/li\u003e\n\u003cli\u003eDeng,B. Xie,Y. \u0026amp; Tang, G. J. 2014. Three-dimensional structural analysis approach for aging composite solid propellant grains. Propellants, Explos. Pyrotech\u003cem\u003e.\u003c/em\u003e, vol. 39, no. 1, pp. 117\u0026ndash;124, doi: 10.1002/prep.201300120.\u003c/li\u003e\n\u003cli\u003eChen,S. Wang,C. Zhang, K. Lu, X. \u0026amp; Li, Q. 2022. A Nonlinear Viscoelastic Constitutive Model for Solid Propellant with Rate-Dependent Cumulative Damage. Materials (Basel)\u003cem\u003e.\u003c/em\u003e, vol. 15, no. 17, doi: 10.3390/ma15175834.\u003c/li\u003e\n\u003cli\u003eNoureldin, A. F. Adel, W. M. Attai,Y. A. \u0026amp; Ismail, M. A. 2020 . An experimental study on mechanical and ballistic characteristics of different HTPB composite propellant formulations. \u003cem\u003eIOP Conf. Ser. Mater. Sci. Eng.\u003c/em\u003e, vol. 973, no. 1, 2020, doi: 10.1088/1757-899X/973/1/012030.\u003c/li\u003e\n\u003cli\u003eTong,X. Xu,J. Doghri,I. El Ghezal,M. I. Krairi, A. \u0026amp; Chen, X.2020. A nonlinear viscoelastic constitutive model for cyclically loaded solid composite propellant. Int. J. Solids Struct., vol. 198, pp. 126\u0026ndash;135, Aug. 2020, doi: 10.1016/j.ijsolstr.2020.04.011.\u003c/li\u003e\n\u003cli\u003eChyuan,S.2004. The Journal of Strain Analysis for Engineering Design Computational studies of variations in Poisson \u0026rsquo; s ratio for thermoviscoelastic solid propellant grains. doi: 10.1177/030932470403900109.\u003c/li\u003e\n\u003cli\u003eJi,Y. Cao, L. Li, Z. Chen, G. Cao,P. \u0026amp; Liu,T. 2023. Numerical Conversion Method for the Dynamic Storage Modulus and Relaxation Modulus of Hydroxy-Terminated Polybutadiene (HTPB) Propellants. Polymers (Basel)., vol. 15, no. 1, Jan. 2023, doi: 10.3390/polym15010003.\u003c/li\u003e\n\u003cli\u003eJung,G.-D. Youn, S. K. \u0026amp; Kim ,B. K. 2000. \u003cem\u003eA\u003c/em\u003e three-dimensional nonlinear viscoelastic constitutive model of solid propellant, vol. 37. doi: 10.1016/S0020-7683(99)00180-8.\u003c/li\u003e\n\u003cli\u003eJrad,H. Dion, J. Renaud, L. F. Tawfiq, I. \u0026amp; Haddar, M. 2012. Non-linear generalized maxwell model for dynamic characterization of viscoelastic components and parametric identification techniques. Proc. ASME Des. Eng. Tech. Conf., vol. 1, no. PARTS A and B, pp. 291\u0026ndash;300, 2012, doi: 10.1115/DETC2012-70264.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"mechanics-of-time-dependent-materials","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"mtdm","sideBox":"Learn more about [Mechanics of Time-Dependent Materials](http://link.springer.com/journal/11043)","snPcode":"11043","submissionUrl":"https://submission.nature.com/new-submission/11043/3","title":"Mechanics of Time-Dependent Materials","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Viscoelastic material, relaxation modulus, hydroxyl-terminated polybutadiene (HTPB) propellant, Prony series, generalized Maxwell model","lastPublishedDoi":"10.21203/rs.3.rs-4545250/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4545250/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this research, an initial exploration of the viscous effects present in heterogeneous solid rocket propellant is conducted through experimental analysis. To achieve this, uniaxial tensile and relaxation experiments were conducted on the Joint Army-Navy-NASA-Air Force Propulsion Committee (JANNAF) standard[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e], specimens using the Zwick Z050 universal testing machine. The work involved conducting destructive tensile tests at different strain rates and relaxation tests at different strain levels and temperatures, with three values being considered. The resulting data from the experiments are illustrated in appropriate diagrams, and depending on these data, the viscoelastic behavior of this material was confirmed.\u003c/p\u003e \u003cp\u003eAdditionally, mathematical modeling of this studied phenomenon using the generalized Maxwell model, identified on the basis of experimental data, is presented. The material parameters of the constitutive model are determined numerically using MATLAB software based on the Prony series procedure. The efficiency of the model and the identification approach are discussed. A high agreement between the calculation and the experimental results was found based on the Prony coefficients of relaxation modulus, with a maximum error of about 2.35%.\u003c/p\u003e \u003cp\u003eFinally, numerical modeling of the relaxation tests was conducted to simulate the stress relaxation behavior of the AP-HTPB composite solid propellant using the ANSYS program. The Prony coefficients were added to the nonlinear viscoelastic components to define the material model in ANSYS, and the maximum difference between the numerical and experimental results is 3.98%.\u003c/p\u003e","manuscriptTitle":"Modeling of AP-HTPB Solid Propellant Viscoelastic Behavior Using Modified Maxwell Model","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-06-24 14:08:48","doi":"10.21203/rs.3.rs-4545250/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"editorAssigned","content":"","date":"2024-06-13T04:08:29+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-06-10T05:01:49+00:00","index":"","fulltext":""},{"type":"submitted","content":"Mechanics of Time-Dependent Materials","date":"2024-06-07T09:45:12+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"mechanics-of-time-dependent-materials","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"mtdm","sideBox":"Learn more about [Mechanics of Time-Dependent Materials](http://link.springer.com/journal/11043)","snPcode":"11043","submissionUrl":"https://submission.nature.com/new-submission/11043/3","title":"Mechanics of Time-Dependent Materials","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"ace696be-4054-444f-b8e1-b56207f77a7c","owner":[],"postedDate":"June 24th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2024-06-24T14:08:49+00:00","versionOfRecord":[],"versionCreatedAt":"2024-06-24 14:08:48","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4545250","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4545250","identity":"rs-4545250","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}