Strength criterion of composite solid propellants under dynamic loading

Zhe-jun Wang,Hong-fu Qiang,Guang Wang,Biao Geng

206Staff Room,Xi'an Hi-Tech Institute,Xi'an,710025,China

Keywords:Strength criterion Unified strength theory Composite solid propellant Dynamic loading Biaxial tension

ABSTRACT Based on the dynamic loading(1-100 s-1)experiments under different temperatures(223-298 K)and stress states,uniaxial and biaxial strength criterion of a Hydroxyl-terminated polybutadiene(HTPB)based composite solid propellant were further investigated.These experiments were conducted through the use of a new uniaxial INSTRON testing machine,different new designed gripping apparatus and samples with different configurations.According to the test results,dynamic uniaxial tensile strength criterion of the propellant was directly constructed with the master curve of the uniaxial maximum tensile stress.Whereas,a new method was proposed to determine the dynamic uniaxial compressive strength of the propellant in this study.Then uniaxial compressive strength criterion of the propellant was constructed based on the related master curve.Moreover,it found that the uniaxial tensile compressive strength ratio of the propellant is more sensitive to loading temperature under the test conditions.The value of this parameter is about 0.4 at room temperature,and it reduces to 0.2-0.3 at low temperatures.Finally,the theoretical biaxial strength criterion of HTPB propellant under dynamic loading was constructed with the unified strength theory,the uniaxial strength and the typical biaxial tensile strength.In addition,the theoretical limit lines of the principal stress plane for the propellant under dynamic loading at different temperatures were further plotted,and the scope of the limit line increases with decreasing temperature.

1.Introduction

Up to now,solid rocket motor(SRM)has found extensively applications in current military and space technologies[1].In addition,the propellant grain serves as the most prime component and the energysource of this motor.In-service SRMs are often stored for long time,and transported from one place to another before operation,thus they are exposed to various environmental conditions,in which the loading rate and temperature play an important role[2-4].When the maximum stress or strain capacities of solid propellants was exceeded under those different conditions,cracks may arise and propagate in the propellant grain,which can further cause the motor to explode and prevent missiles from fulfilling their mission[5,6].Therefore,it is very necessary and important to construct the suitable failure criterion of solid propellant to assess the structural integrity of propellant grain.In general,the failure criterion of solid propellant includes strength criterion and fracture criterion[7],and the later one will be only studied in this investigation.

With the measured data in laboratory,the strength criterion of solid propellant is usually employed to provide a criterion for evaluating its damage under other loading conditions.Over the last few decades,a considerable amount of experiments have been done in studying the uniaxial properties of solid propellants under various loading conditions,such as the constant strain-rate loading tests,constant stress-rate loading tests,constant strain loading tests,constant stress loading tests and dynamic mechanical analysis(DMA)tests[8-12].These test results indicate that the viscoelastic properties of solid propellant are very complex and significantly dependent on the loading factors(for example strain rate,temperature,aging,and stress state).Moreover,according to these test results and the time-temperature superposition principle(TTSP),the uniaxial strength criterion of solid propellant was constructed,and the analysis of the structural integrity for propellant grain during the lifetime of SRM has been widely performed[13-15].However,for the port pressurization condition during ignition of SRM,the propellant grain experiences a biaxial stress field[16].Hence,the biaxial strength criterion of solid propellant under dynamic loading are more useful for analyzing the structural integrity of propellant grain.Otherwise,using uniaxial test results can result in gross inaccuracies.In recent years,there are strict requirement for the structural integrity of propellant grain during ignition of SRM at low temperatures,with the increasing demand for the multiple military tasks and the continuous development of high performance tactical missiles[17,18].The propellant withstands the coupled effects of low temperature and dynamic loading(1-100 s-1)under this specific engineering condition[19].Therefore,the need to construct the biaxial strength criterion of solid propellant at strain rates(1-100 s-1)and low temperatures becomes more urgent and necessary.However,up to now,the related researches are very inadequate.Wherefore,new test methods and theories should be applied for the further investigation.

In the presented paper,uniaxial strength criterion of a composite solid propellant under dynamic loading(1-100 s-1)was constructed firstly based on the related test results from our previous works.Afterwards,the effects of loading strain rate and temperature on the uniaxial tensile-compressive strength ratio were discussed.Finally,the theoretical biaxial strength criterion of the propellant under dynamic loading(1-100 s-1)was further constructed with new strength theory.

2.Experiments

Hydroxyl-terminated polybutadiene(HTPB,binder fuel)based composite solid propellant has been widely used in current SRM worldwide,thus it was selected for this investigation.Its components are as follows:6.0-7.0 mass-%HTPB,60.0 mass-%larger ammonium perchlorate(AP,oxidizer),9.5 mass-%smaller AP,18.5 mass-%Aluminium(Al,metal fuel)powder,0.05-0.10 mass-%tris 1(2 methylazirindinyl)phosphine oxide(MAPO,bonding agent),1.0-2.0 mass-%toluene diisocyanate(TDI,curative),3.4 mass-%dicapryl sebacate(DOS,plasticizer)and 0.5-1.0 mass-%other liquid additives.

In general,the uniaxial mechanical properties of materials were investigated based on the uniaxial tensile tests and uniaxial compressive tests.The biaxial tensile tests,biaxial compressive tests and biaxial tensile-compressive tests should be all conducted to study the biaxial mechanical properties of materials.However,up to now,it is very difficult to conduct the dynamic biaxial tests on materials due to the scarcity of suitable testing machine.Therefore,the typical dynamic biaxial test on HTPB propellant was investigated here.

For dynamic testing,it is very important to design the suitable configuration and dimensions of the sample.According to the Chinese national standard of P.R.C,GJB 770B-2005,the American JANNAF(Joint Army-Navy-NASA-Air Force)standard[20]and the previous researches[21,22],the uniaxial tensile test samples,uniaxial compressive test samples and biaxial tensile test samples were designed as shown in Fig.1.As stated in the introduction section,it is more important to assess the structural integrity of propellant grain during ignition of SRM at low temperatures.Therefore,most tests in this study were conducted to investigate the mechanical properties of solid propellant under dynamic loading at low temperatures.The sample test matrix is shown in Table 1.Furthermore,all tests were performed using the new uniaxial testing machine INSTRON VHS 160/100-20 and different new designed gripping jaws.More detailed information about the test setup and the test processing had been all stated in our previous works[23-25],please refer to them.The results obtained from these tests were further analyzed in this investigation.

3.Uniaxial strength criterion

3.1.Uniaxial tensile strength criterion

According to the Chinese national standard of P.R.C,GJB 770B-2005 and the American JANNAF standard[20],the maximum tensile stress obtained directly from the uniaxial tensile stress-strain curve is defined as the uniaxial tensile strengthσumtof solid propellant.

Table 1 Sample test matrix.

In general,the mechanical properties of highly particle- filled elastomers such as solid propellant are sensitive to the loading strain rate and temperature.Thus the effects of strain rate and temperature must be considered when constructing the strength criterion of these materials.Based on lots of test results and some theories,Williams et al.stated that the viscoelastic behavior of materials at one temperature T2can be related to that at another temperature T1by a change in the frequency or time scale only[26].In other words,the viscoelastic behavior of materials at high temperature and fast loading rate can be equivalent to that at low temperature and slow loading rate.This relationship is called the TTSP and can be described with Equation(1).According to this principle,the curve of the typical mechanical parameter versus logarithmic loading frequency or logarithmic time at each temperature,can be horizontally shifted along the frequency(or the reduced time)axis then overlapped on the curve at the reference temperature.Afterwards,a fully overlapped curve could be formed at the reference temperature,which is called the master curve of that typical mechanical parameter.And the shift distance along the logarithmic reduced time axis is called the time-temperature shift factorαT.Therefore,the master curve can be used to describe and predict the mechanical properties of highly particle- filled elastomers such as solid propellant in wide intervals of temperatures and strain rates.

where P represents the typical viscoelastic behavior of materials.

The master curve of the uniaxial maximum tensile stress for HTPB propellant under the test conditions was constructed as shown in Fig.2,based on TTSP.To obtain the smoother master curve,the data were firstly multiplied by temperature ratio T0/T(T0=298K)prior to shifting.The related expressions of the master curve are as shown in Equations(2)and(3).Therefore,Equation(4)was further developed to describe the variation of the dynamic uniaxial tensile strength of HTPB propellant with strain rate and temperature in this study:

whereσYis the strength of the propellant,k1,k2,k3,k4,k5,k6,l1(=k5×k1)and l2(=k5×k2+k6)are material constant,and their values are shown in Table 2,T0is the reference temperature.

According to Equation(4)and Table 2,the dynamic uniaxial tensile strength criterion of HTPB propellant can be written as follows:

whereσutis the uniaxial tensile stress of the propellant obtained from the analysis of structural integrity with the finite element method.

3.2.Uniaxial compressive strength criterion

Up to now,there has been no standard available for defining the compressive strength of solid propellant.In addition,the trend of the uniaxial compressive stress-strain curves of HTPB propellant is very complex with increasing strain rate and decreasing temperature[25].Therefore,with the test results,a new method as shown in Fig.3 was proposed in this study to determine the dynamic uniaxial compressive strengthσucsof HTPB propellant.

The master curve of the uniaxial compressive strength for HTPB propellant was constructed as shown in Fig.4,based on TTSP.The reference temperature T0is also defined as 298 K.As can be seen clearly,the trend of the logarithmic shift factor log（αT）is also linear with temperature,which is consistent with that obtained in uniaxial tension(Fig.2).Moreover,all master curves of the propellant in uniaxial loading are nonlinear in form(Figs.2 and 4).Then Equation(4)can be also employed to describe the variation of the dynamic uniaxial compressive strengthσucsof HTPB propellant with strain rate and temperature in this study.The values of material constants are shown in Table 2.

According to Equation(4)and Table 2,the dynamic uniaxial compressive strength criterion of HTPB propellant can be written as follows:

whereσucis the uniaxial compressive stress of the propellant obtained from the analysis of structural integrity with the finite element method.

3.3.Uniaxial tensile-compressive strength ratio

To further investigate the effect of stress state on the uniaxial strength of solid propellant,the uniaxial tensile-compressive strength ratio(σumt/σucs)of HTPB propellant under the test conditions were obtained as shown in Table 3.It can be found that the value of this parameter is all smaller than 1,which indicates that it is easier for the propellant to fail due to the dynamic tensile loading under the same strain rate and temperature.In addition,the effect of temperature on this parameter is more remarkable.The value ofthis parameter is about 0.4at room temperature,and it reduces to 0.2-0.3 at low temperatures.In other words,the dynamic uniaxial compressive strength of HTPB propellant is 3-5 times of that in tension.

Table 2 Values of constants of the master curves for HTPB propellant.

4.Biaxial strength criterion

In general,the biaxial strength of materials includes the biaxial tensile strength σbmt,biaxial compressive strength σbcsand biaxialtensile-compressive strengthσbmtc.As stated in section 2,it is very difficult to conduct the dynamic biaxial tests on materials.Therefore,the dynamic biaxial tensile strength of HTPB propellant was only investigated here with the typical test results.Whereas,other dynamic biaxial strengths of the propellant were acquired with theoretical method.

Table 3 Uniaxial tensile-compressive strength ratio of HTPB propellant under the test conditions.

4.1.Biaxial tensile strength

The dynamic biaxial tensile tests were conducted on HTPB propellant through the use of a new uniaxial INSTRON testing machine,a new designed gripping apparatus and the strip biaxial tensile sample(as shown in Fig.1(c))[24].Based on the recorded load and displacement along the vertical loading direction(Y direction in Fig.1(c)),the related tensile stress-strain curves of HTPB propellant under the test conditions were obtained.

According to TTSP,the master curve of the maximum tensile stress along the vertical loading direction for HTPB propellant was constructed as shown in Fig.5,in which the value of the reference temperature T0is also defined as 298 K.It can be seen that the trend of the logarithmic shift factor log（αT）is linear with temperature,which is consistent with that obtained in uniaxial tension(Fig.2).In addition,the master curve of the propellant is also nonlinear in form.Then they can be also described with Equations(2)-(4),and the values of the material constants are shown in Table 2.

As stated in our previous work[24],the movement of the propellant part for the strip biaxial tensile sample was restrained in the transverse direction(x direction as shown in Fig.1(c))during loading,and the dimension of the sample in the thickness direction(z direction as shown in Fig.1(c))is far smaller than that in other directions.Therefore,the following expressions can be written（σ1≥ σ2≥ σ3）:

whereνis the Poisson's ratio of solid propellant.

According to the above discussion,the variation of the biaxial tensile strength for the strip sample with strain rate and temperature can be expressed as follows by taking Equation(4)into Equation(7a):

4.2.Biaxial strength criterion based on the unified strength theory

When the uniaxial tensile strengthσumtand compressive strengthσucswere used as the basic mechanical parameters,the unified strength theory can be expressed as follows with the principal stress（σ1≥ σ2≥ σ3）[27]:

where σ1,σ2and σ3are the three principal stress,α（σumt/σucs）is the uniaxial tensile-compressive strength ratio,b is the material constant to describe the effect of intermediate principal stress on the strength.

Taking Equation(5)into Equation(9),the variation of the biaxial strength for HTPB propellant with strain rate and temperature can be written as follows:

According to Table 3,the value of the parameterαis about 0.4 at room temperature,and its value is 0.2-0.3at low temperatures.Therefore,the theoretical biaxial strength criterion of HTPB propellant under dynamic loading(1-100 s-1)can be constructed with Equation(10)when the optimal value of the parameter b was determined by fitting the strength of the propellant under the typical stress state.

According to Equation(7),the principal stress of the unified strength theory can be expressed with Equation(9a)when conducting dynamic biaxial tensile test on HTPB propellant with the strip sample.Then taking Equation(7)into Equation(9a),the following expression can be given:

Taking the dynamic biaxial tensile strength along the vertical loading direction and dynamic uniaxial tensile strength of HTPB propellant into the left side and right side of Equation(11),respectively.Moreover,the value of the Poisson's ratioνis defined as 0.5.Then,the value of the parameter b is determined as 0.05,0.25 and 0.50at the temperature of 298,243 and 223K,respectively.Now,the theoretical biaxial strength criterion of HTPB propellant under dynamic loading(1-100 s-1)was constructed.

Substituting the values of the above parameters(αand b)into Equation(9),the limit lines of the principal stress plane for HTPB propellant under dynamic loading were plotted with the unified strength theory,as shown in Fig.6.During this process,data normalization method was employed.In other words,the dynamic uniaxial tensile strengthσumtof HTPB propellant is defined as 1,and the dynamic biaxial tensile strengthσbmt(stress ratio 1:1)is defined as the point(1.0,1.0).Furthermore,the dynamic uniaxial compressive strength σucsof the propellant is defined as 1/α,and the dynamic biaxial compressive strengthσbcs(stress ratio 1:1)is defined as the point(1/α,1/α).It can be seen that the scope of the limit line becomes larger when decreasing temperature,which indicates that the strength of the propellant increases with decreasing temperature.Based on Fig.6,the failure of HTPB propellant under dynamic biaxial loading can be assessed.

5.Conclusions

Uniaxial and biaxial strength criterion of HTPB-based composite solid propellant under dynamic loading(1-100 s-1)were investigated for the first time by conducting uniaxial tensile tests,uniaxial compressive tests and biaxial tensile tests with a new uniaxial INSTRON testing machine,different new designed gripping apparatus and samples with different configurations.The following conclusions can be drawn.

(1)The dynamic uniaxial tensile-compressive strength ratio of HTPB propellant is all smaller than 1.In addition,the effect of loading temperature on this ratio is more remarkable.Its value is about 0.4 at room temperature,and it reduces to 0.2-0.3 at low temperatures.These results are helpful to understand the dynamic uniaxial mechanical properties of the propellant.Moreover,they also indicate that it is still easier for a composite solid propellant to fail because of the tensile stress rather than the compressive stress under dynamic loading.And this failure properties of the propellant is more obvious at low temperatures.Thus,the dynamic uniaxial tensile strength criterion of the propellant can be employed as a uniaxial failure criterion.

(2)Based on the related master curves and the unified strength theory,the theoretical biaxial strength criterion of HTPB propellant under dynamic loading was constructed.Moreover,the limit lines of the principal stress plane for the propellant under dynamic loading were also plotted.The results indicate that the capacity of the propellant resistance to destroy improved when decreasing temperature.In addition,the effects of strain rate and temperature were considered in the developed dynamic biaxial strength criterion.Therefore,it is more useful to further assess the failure of the propellant under biaxial loading and the structural integrity of propellant grain during ignition of SRM.

(3)Because it is very difficult to conduct the dynamic biaxial test on materials,the dynamic biaxial tensile strength of HTPB propellant was only investigated with the new test method in this study.Whereas,otherdynamic biaxial strengths of the propellant were acquired with theoretical method.Therefore,it is necessary to propose more suitable test methods to conduct other biaxial loading tests(such as the dynamic biaxial compressive tests and dynamic biaxial tensile compressive tests)on solid propellant.Then the validity of the developed dynamic biaxial strength criterion of HTPB propellant can be further verified.

Acknowledgments

The authors gratefully acknowledge the financial support of the National 973 Program in China(No.61338)and the National Funds in China(Nos.11772352,61407200203 and 51328050101).