# Investigation on Aging Model of Solid Propellant Using the Degree of Crosslinking

Authors: Junqi Wang; Huiru Cui; Weili Ma
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

Solid propellant is the main part of solid rocket motor (SRM). To further investigate the service life of SRM, aging of solid propellant should be concerned. The aim of this study is to propose a new aging constitutive model of solid propellant using the degree of crosslinking. The aging development equation of viscoelastic Poisson’s ratio (VPR) and relaxation modulus are considered in establishing the model. A series accelerated aging experiments accompanied with some tensile tests are designed to verify the new model. Aging experiments show that degree of crosslinking (DCL) can be used to act as a link to facilitate the communication between chemical aging and aged mechanical parameters well.


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## Body

## 1. Introduction

Aging of the solid propellant will seriously affect the mechanical property of the solid propellant, especially the VPR and relaxation modulus [1–6]. It has been proved that a microchange of Poisson’s ratio strongly affects structural integrity of the solid rocket motor (SRM) [7–9]. A solid propellant grain is the power origin and main structural part of SRM. The life of the solid propellant grain plays a decisive role in the service length of SRM. It is very essential to establish a new aging model of the solid propellant.The start point of life prediction research of solid propellant grains is the study on the aging of the solid propellant. There are some methods, like mechanical property method [10–12], activation energy method [13–15], dynamic mechanical property method [15, 16], and DCL method [17, 18], used in the aging research. The mechanical property method always based on the aging phenomenon during natural aging or accelerated aging. The mechanical property is watched and fitted by some mathematical method. Like mechanical property method, dynamic mechanical property method does not research based on the aging mechanism; the results always come from apparent measurement. The aging development equation of mechanical property is obtained based on the aging kinetic equation of polymer in the activation energy method. However, much more research results were funded by the DCL method. Chemical property and the relationship with mechanical property are studied in the DCL method. Zhou [19] proposed an aging development equation of relaxation modulus based on this method and a constitutive model of solid propellant considering aging effect. Furthermore, experimental data shows that the increase of the DCL will company with the decrease of Poisson’s ratio during aging process [20]. Therefore, it is feasible to use the DCL in establishing the aging development equation of VPR.The paper proposed a new aging constitutive model of solid propellant based on the aging development equation of VPR and relaxation modulus. To determine and verify the parameters of the constitutive model, accelerate aging tests and uniaxial tensile tests are designed. Figure1 shows the research idea of the whole paper.Figure 1
Research idea of the whole paper.


## 2. Aging Model

During the age process of propellant, the crosslinking network will change for the unstable of polymer binder matrix. The DCL is always used to represent the degree of aging. Furthermore, the DCL can be expressed as chain members of branching point, numbers of branching point, and average mean molecular weight. In this paper, the molarity of crosslinking chain segment is used to represent the DCL. Zhou [19] had given the development equation of the DCL.
(1)μta=μm−Be−βcta,T1+αcBe−βcta,T,whereta is the aging time, E0c represents the activation energy in aging reaction, αc is the undetermined parameter, μm is the maximum DCL, and μ0 is the initial DCL. Coefficients B and βc have the form
(2)B=μm−μ01+αcμ0,βc=β0c∫0taTexp−E0ckBTdτ,where
(3)β0c=AkB1+αcμmR,where kB is the Boltzmann constant and A is the speed constant of aging reaction.For the aging test in constant temperature,T0, βc can be degraded into the next form
(4)βc=β0cT0exp−E0ckBT0ta.For the homogeneous isotropic material, the aging slack variable can be expressed fully by relaxation modulusEt,ta, and shear modulus Gt,ta with aging parameters [19].
(5)Et,ta=E0t+μta−μ0‘E1t,(6)Gt,ta=G0t+μta−μ0‘G1t,whereE0t is the unaged relaxation modulus, μta−μ0‘E1t represents the aged relaxation modulus, and G0t and μta−μ0‘G1t represent the unaged and aged shear modulus, respectively. However, the change of DCL, μta−μ0‘, is used rather than relative change of DCL, μ¯=μta−μ0‘/μ0. In order to keep the uniform dimension, the paper selects the relative change of DCL in the modified aging development equation.
(7)Et,ta=E0t+μ¯taE1t,Gt,ta=G0t+μ¯taG1t.In Equations (5) and (6), the VPR never changes during the aging process which did not correspond to the phenomenon above. Zhou [19] had proved the validity of Equation (5). But Equation (6) had not been verified. It seems hard to find a suitable model for aged VPR from the two equations.Based on the elastic physics of rubbery, for the ideal crosslinking network, the shear modulus can be expressed as(8)G=μRT,whereμ represents DCL, R is the Planck constant, and T represents the temperature.But, for the real crosslinking polymer film, the crosslinking network is always incomplete. Flory [21] gives the modified form of Equation (8):
(9)G=RTμ−2ρM,whereρ is the density and M is the unaged initial molecular weight.To express simply, Equation (9) can be rewritten as
(10)G=RTμ−k.For the homogeneous isotropic elastomers, shear modulus, tensile modulus, and Poisson’s ratio have the relationship(11)Gt=Et21+ν.Considering two time status as shown in Figure2, T0 and T1, where T0 is the initial time and E0 represents initial tensile modulus. The DCL at T0 and T1 are μ0 and μ separately.Figure 2
DCL and its change at different times of solid propellant.The corresponding shear modulus atT0 and T1 can be expressed based on Equation (11):
(12)μ0−kRT=E021+ν0,μ−kRT=E0+μ¯E121+ν1,whereE0 represents the unaged tensile modulus, E1 represents the aged tensile modulus, and ν0 and ν1 represent unaged and aged Poisson’s ratio.Combining the above equations, the unaged and aged Poisson’s ratio can be expressed as(13)21+ν1−21+ν0=−RTμ0−kE0+μ0−kμ¯E1−μ−kE0RTμ0−kRTμ−k.Simplified as(14)ν1−ν0=Δμμ−kμ0−kμ0E1E0−11+ν0.For the crosslinking polymer films, considering the time of aging, the model of aged VPR can be expressed as(15)νt,ta=ν0t+Δμtaμta−kμ0−kμ0E1tE0t−11+ν0t,where t represents loading time and ta represents aging time.The above equation has only one undetermined parameter, the equation may be hard to fit the experimental data, and the modified equation is used finally without changing the basic shape of curve.(16)νt,ta=ν0t+δ1Δμtaμta−kμ0−kμ0E1tE0t−11+ν0t+δ2,where δ1 and δ2 are undermined coefficients.Using the correspondence principle and considering the aging development equation of VPR and relaxation modulus, for isotropic materials, the aging constitutive model of the solid propellant using the degree of crosslinking can be written as(17)σijt=Sijt+13δijσkkt,with
(18)Sijt+∫−∞tνθ−θ′,ta∂Sijτ∂τdτ=∫−∞tEξ−ξ′,ta∂eijτ∂τdτ,σkkt−2∫−∞tνθ−θ′,ta∂σkkτ∂τdτ=∫−∞tEξ−ξ′,ta∂ε^kkτ∂τdτ,where
(19)ε^kk=εkk−3αTΩ.In which,θ and ξ are reduced time, αklT is the expansion coefficient, and Ω is the change of temperature.


## 3. Experimental Procedures

To determine the parameters in the development equation of the DCL, the next tests are designed. Some results related to the DCL and relaxation modulus can be found in the previous work [22]. According to their research, DCL can be used to describe the aged relaxation modulus well.The DCL of the propellant is determined at the designed aging time in Table1 at the constant temperature of 75°C. The determined data is used in fitting the curve overusing aging time. The fitted curve will establish the foundation of the next texts.Table 1
Tests for determination of aging development equations of relaxation modulus and VPR.
TemperatureAging time (days)Number of testsStretching velocity (mm·min-1)75°C0, 3, 5, 7, 9, 11, 16, 21, 26, 31, and 413−−20°C0, 3, 5, and 7−−−−20°C5, 11, 26, and 31−−1.68, 3.36, and 5.04The designed tests below are to determine the parameters in the aging development equations of the relaxation modulus and VPR. Furthermore, the fitting results are used to directly verify the reasonability of the two equations.The relaxation modulus and VPR are determined at the designed aging time by using the test specimen as shown in Figure3(a). The determination of VPR will use the digital image correction method (DICM) which is proposed by Cui et al. [1] as shown in Figure 3(b). The fitted parameters will be used in the calculation of theory value in the next tests.Figure 3
Test specimen and image of testing scenarios.
(a)(b)The determined aging development equations of the relaxation modulus and VPR should be verified by some tests. The stretch tests are very effective tests.The longitudinal stress measured by the force sensor is used to verify the aging development equations of relaxation modulus while the transverse strain measured by DICM is used to verify the aging development equations of VPR.The longitudinal stress has the next form during stretch tests(20)σxt=∫0+tEt∂εx∂tdt.Introducing the aged relaxation modulus in Prony series (see Equation (40)), the longitudinal stress has the form(21)σxt=E∞0εx+μ¯t′E∞1εx+∫0+t∑n=1NEEn0+μ¯t′En1e−t/τnE∂εx∂tdt.Simplifying the above equation(22)σxt=E∞0RVt+μ¯t′E∞1RVt+RV∑n=1NEτnE1−e−t/τnEEn0+μ¯t′En1,whereRV is stretching velocity, εx=RVt.For the linear elastic materials, the volume strain tensorεkk in stretch tests has the form
(23)εkk=εx+εy+εz=1−2νεx,whereν is Poisson’s ratio, εy and εz represent two transverse strains, and εx represents the longitudinal strain.The corresponding expression in complex domain based on the elastic-viscoelastic corresponding principles can be expressed as(24)ε¯kks=ε¯xs−2sε¯xsν¯s,wheres is the Laplace transform variable.Considering the Laplace inversion of the above equation, the volume strain tensorεkk has the form
(25)εkkt=εxt−2εx0νt+∫0tνt−τ∂εxτ∂τdτ.The stretching velocity in the stretch tests is a constant. The above equation can be simplified as(26)εkkt=RVt−2RV∫0+tνtdt.Considering the symmetry of the problem and combining Equations (28) and (35), transverse strain can be expressed as
(27)εyt=−RV∫0+tνtdt.


## 4. Results and Discussion

As illustrated in Figure4, the initial DCL of propellant is nonzero for the crosslinking reaction during the natural storage before accelerate aging test. The DCL increases sharply during the initial phase (1-10 days) and fluctuates wildly for the sudden change of storage temperature. As aging time goes by, the propellant begins to adapt to the high temperature. The increase of the DCL becomes slow. But the DCL still increases with time. At the end of aging period, the DCL increases slowly and even comes to a standstill, but the data still appears fluctuation. Equation (28) is the fitted results of DCL. The fitted curve is obtained by the least square method.
(28)μta=4.519−3.0390e−0.0657ta1−0.4284e−0.0657ta.Figure 4
Measurement of DCL during the aging period.The parameters of the aging development equations of relaxation modulus are fitted by the DCL curve and relaxation modulus measured. The parameters of Prony series of the aging development equations of relaxation modulus (see Equation (29)) are illustrated in Table 2, while the number of the terms in the Prony series is 5. Figure 5 gives the relaxation modulus under different aging times and its fitting curve. As illustrated, the relaxation modulus increases with the aging time which means the propellant hardens during the aging process.
(29)Et,ta=E∞0+μ¯taE∞1+∑n=1NEEn0+μ¯taEn1e−t/τnE,Table 2
Parameters of Prony series of the aging development equations of relaxation modulus.
n∞12345En01.08861.0001.0001.85890.77730.5924τnE—0.01070.21344.268085.36051707.2109En11.25901.0000.10000.73700.5604-0.0995Figure 5
Relaxation modulus under different aging times (A0, A3, A5, and A7 represent the specimen aged for 0, 3, 5, and 7 days).whereE∞0 and E∞1 are the equilibrium relaxation modulus, NE is the number of terms in the Prony series, t represents time, and En0, En1, and τnE are undetermined coefficients.Based on the results of DCL, aging development equations of the relaxation modulus, and VPR measured at different aging times, the parameters in the aging development equations of VPR are fitted. Figure6(a) illustrates the original fitted curve of the aging development equations of VPR. The original curve seems cannot fit the experimental data well, but the experimental data reflects some regulations of the aging development of VPR. VPR decreases with aging time. The curves of VPR at different aging times have the similar shape. Figure 6(b) illustrates the modified curve of the aging development equations of VPR. The modified curve fits the experimental data pretty good. Furthermore, the data witnessed that the VPR decreases while the increase of DCL in literature [20]. Equation (30) gives the modified aging development equation of VPR. The fitting curve shows that the aging development equations of VPR can predict the VPR of the propellant during the aging process directly.
(30)νt,t′=ν0t+0.0803Δμt′μt′−0.4427μ0−0.4427μ0E1tE0t−11+ν0t+0.0002.Figure 6
VPR under different aging times and its fitting curves.
(a)
Original aging development equations of VPR(b)
Modified aging development equations of VPRFigure7 gives the curves of longitudinal stress vary with time under different stretching velocity. The longitudinal strain is 5% at the end of stretch tests to avoid introducing damage effect. The experimental data fits the longitudinal stress predicted very well. Furthermore, longitudinal stress increases with stretching velocity which is in accord with theory.Figure 7
The curves of longitudinal stress vary with time under different stretching velocity.
(a)
Agingtime=5 d(b)
Agingtime=26 dFigure8 illustrates the curves of longitudinal stress vary with longitudinal strain under different aging times. The longitudinal stress increases with aging time. The fitting results prove that the modified aging development equation of relaxation modulus is reasonable during the aging process indirectly.Figure 8
The curves of longitudinal stress vary with longitudinal strain under different aging times (stretchingvelocity=5.04 mm·min-1).Figure9 illustrates the curves of transverse strain vary with time under different stretching velocity. The measured data fits the predicted well, while the data is small. Furthermore, the stretching velocity has no directly effect on transverse strain. Figure 10 gives the curves of transverse strain vary with longitudinal strain under different aging times. Furthermore, the transverse strain decreases with aging time which is arisen from the decrease of VPR during aging process. The fitting results prove that the modified aging development equation of VPR is reasonable during the aging process indirectly.Figure 9
The curves of transverse strain vary with time under different stretching velocity.
(a)
Agingtime=11 d(b)
Agingtime=31 dFigure 10
The curves of transverse strain vary with longitudinal strain under different aging times (stretchingvelocity=5.04 mm·min-1).


## 5. Conclusion

To accurately describe the aging properties of the solid propellant, chemically related changes DCL is used to act as a link to facilitate the communication between chemical aging and aged mechanical parameters. With the aged mechanical parameters, a new aging constitutive model of solid propellant is proposed from the aging phenomenon of propellant.The DCL curve is generated first through accelerate aging tests. In addition, the parameters of the aging development equations of relaxation modulus are fitted by the DCL curve and relaxation modulus measured. Stretch tests reveals that the modified aging development equation of VPR and aging development equation of the relaxation modulus are reasonable in different aging periods. Compared with the traditional statistical methods, the DCL method as well as the validation examples is more convincing. The method in this research provides a new method to further investigate the aging properties of the solid propellant.



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