Research on the influences of confining pressure and strain rate on NEPE propellant:Experimental assessment and constitutive model

Hui Li ,Jin-sheng Xu ,* ,Ji-ming Liu ,Ting-yu Wng ,Xiong Chen ,Hong-wen Li

a School of Mechanical Engineering,Nanjing University of Science and Technology,Nanjing,210094,China

b Technology Center,Jinxi Industries Group Corporation,Taiyuan,030027,China

Keywords: Confining pressure Strain rate NEPE propellant Constitutive model Damage

ABSTRACT In order to study the influences of confining pressure and strain rate on the mechanical properties of the Nitrate Ester Plasticized Polyether(NEPE)propellant,uniaxial tensile tests were conducted using the selfmade confining pressure system and material testing machine.The stress-strain responses of the NEPE propellant under different confining pressure conditions and strain rates were obtained and analyzed.The results show that confining pressure and strain rate have a remarkably influence on the mechanical responses of the NEPE propellant.As confining pressure increases (from 0 to 5.4 MPa),the maximum tensile stress and ultimate strain increase gradually.With the coupled effects of confining pressure and strain rate,the value of the maximum tensile stress and ultimate strain at 5.4 MPa and 0.0667 s-1 is 2.03 times and 2.19 times of their values under 0 MPa and 0.00333 s-1,respectively.Afterwards,the influence mechanism of confining pressure on the NEPE propellant was analyzed.Finally,based on the viscoelastic theory and continuous damage theory,a nonlinear constitutive model considering confining pressure and strain rate was developed.The damage was considered to be rate-dependent and pressuredependent.The constitutive model was validated by comparing experimental data with predictions of the constitutive model.The whole maximum stress errors of the model predictions are lower than 4%and the corresponding strain errors are lower than 7%.The results show that confining pressure can suppress the damage initiation and evolution of the NEPE propellant and the nonlinear constitutive model can describe the mechanical responses of the NEPE propellant under various confining pressure conditions and strain rates.This research can lay a theoretical foundation for analyzing the structural integrity of propellant grain accurately under working pressure loading.

1.Introduction

As one of the main power system of modern propulsion,solid rocket motors(SRMs)are widely used in rocket projectiles,missiles and launch vehicles.The structural integrity of the solid propellant grains are an important guarantee for the safety ignition of SRMs.However,the loading strain,environmental temperature and loading modes have an obvious influence on the mechanical properties of solid propellant [1-6].During the working process,the solid propellant grains are in a triaxial compression state due to the gas pressurization.According to the previous numerical results[7],the hoop of propellant grains are subjected to tensile strain,while the radical of the propellant grains are compressive strain.The mechanical properties of polymers are always exhibit the pressure-dependence [8,9].As a kind of typical viscoelastic polymer,solid propellant also has a strong pressure-dependence of its mechanical properties.However,up to now,the mechanical parameters used to analyze the structural integrity of solid propellant grains are still obtained under room pressure (atmospheric pressure),which can’t accurately reflect the mechanical responses of propellant grains during the ignition pressurization.Therefore,in order to ensure the normal operation of SRMs,it is necessary to conduct more indepth researches on the pressure-dependence of the mechanical properties of solid propellant.Meantime,a nonlinear constitutive model considering the effect of pressure should be constructed to lay a foundation for a better numerical analysis of the structural integrity of propellant grains.

During the past few decades,a few researchers have made some progresses about the effect of confining pressure on the mechanical properties of solid propellant.Traissac et al.[10] investigated tensile and shear mechanical responses of the Hydroxy-Terminated Polybutadiene (HTPB) propellant at different temperatures,strain rates and superimposed pressures.The ultimate properties were found to be strongly pressure sensitive.Besides,a saturation pressure over which no further influence of the testing pressure was observed,which was determined by the testing conditions.Park and Schapery [11,12] studied the mechanical properties of HTPB propellant under various confining pressure conditions and strain rates.The results showed that axial stress increases with an increasing of confining pressure.Based on the damage theory,they proposed a viscoelastic constitutive equation considering pressure and strain rate,which can describe the mechanical responses of HTPB propellant under different confining pressure conditions and strain rates.Yet,the identification processes of damage parameters are complex.?züpek[13,14]found that superimposed pressure has no noticeable effect on Polybutadiene-Acrylonitrile (PBAN) propellant behavior until dewetting is observed.The effect of pressure on the strain was not presented in her research.According to viscoelastic theory and continuum damage mechanics,three constitutive models considering the confining pressure effect were constructed to describe the stress-strain behavior of the solid propellant.However,due to the assumption regarding rateindependent damage,these models overpredict the stress responses after the onset of dilatation.Zhang et al.[15,16]researched the mechanical properties of a double-base (DB) propellant under hydrostatic compressive loading using a triaxial compressive device.The yield strength and compressive strength have obtained more remarkable enhancement under hydrostatic pressure than room pressure.Wang[17]showed that there is a threshold value in the influence of environmental pressure on the mechanical behavior of NEPE propellant,and the maximum shear stress intensity of the propellant is more sensitive to pressure at lower temperature.Tun? and ?züpek [18,19] proposed a nonlinear viscoelastic constitutive model with damage to be used in a three dimensional finite element analysis.The pressure was considered through a damage function presented by ?züpek[13].Kumar et al.[20,21] regarded that the solid propellant is compressible at large deformation.Based on the experimental results of Park [11,12],compressibility was modelled by considering dilatational part of strain energy density as the dilation with quadratic and quartic terms or as the hyperbolic function of dilation[22].In general,these constitutive models incorporated pressure proposed by Tun?[18,19]and Kumar[20,21]have made a big success in some degrees.Zhang et al.[23] developed an experimental system with wide temperature-ambient pressure range to study the effect of superimposed pressure on the mechanical behavior of HTPB propellant.There are no apparent dewetting point in the stress-strain curves under superimposed pressure,which is different from previous results.They stated that the performance differences of propellant can be attributed to the initial stress state and composition(particle size or proportion of AP).It indicated that different kinds of solid propellants maybe behave different mechanical behaviors under confining pressure condition.

Since the particularity of the confining pressure tests,compared with other test conditions on the mechanical properties of solid propellant,the reports about mechanical responses and constitutive models of solid propellant under confining pressure conditions are rarely published.Various compressible hyperelastic models are widely used to construct the constitutive model to model the effect of pressure [17-20].These models involve the calculation of volumeJ.The accuracy of the calculated volumeJis important for the accuracy of model predictions.However,the iterative calculation processes of volume are complex and difficult to calculate accurately.Therefore,this approach isn’t suitable for large deformable propellant,such as NEPE propellant.In addition,previous researches mainly focused on the HTPB propellant.The mechanical responses of solid propellant have been modified in recent years.It is obviously that previous research results can’t provide an accuracy theoretical foundation for the analysis of the structural integrity of solid propellant grains of SRMs under working pressure loading.NEPE propellant is the highest energy propellant applied in the SRMs in the word.It can be capable a large deformation.However,there is almost no current work on the constitutive model which was developed to predict the mechanical responses of NEPE propellant at the coupled effects of strain rate and confining pressure.Therefore,it is necessary to study the mechanical properties of NEPE propellant under confining pressure and establish a constitutive model that can describe the large deformation and the effects of confining pressure and strain rate.

The main objective of this paper is to research the effects of confining pressure and strain rate on the mechanical responses of NEPE propellant.Firstly,uniaxial tensile tests under various confining pressure conditions and strain rates were conducted using the self-made confining pressure system and material testing machine.The stress-strain responses of the NEPE propellant under different confining pressure conditions and strain rates were obtained and analyzed.Afterwards,the influence mechanism of confining pressure on the mechanical responses of NEPE propellant were analyzed.Finally,a nonlinear constitutive model considering confining pressure and strain rate was constructed to describe stress-strain properties of the NEPE propellant.It is believed that the research will promote the interpretation of mechanical properties of solid propellant under confining pressure.It will lay a theoretical foundation for the analysis of the structural integrity of solid propellant grains of SRMs under working pressure loading.

2.Experimental section

2.1.Experimental system

The experimental system is modified by the material testing machine,which consists of high pressure chamber,confining pressure testing chamber and a computer system,as shown in Fig.1.Nitrogen is chosen as high pressure gas for its safety.The confining pressure testing chamber is installed in the material testing machine,as shown in Fig.2.In addition,a small window and tempered glass are installed on the side of the confining pressure testing chamber,which is convenient to observe the tensile process of solid propellant under the confining pressure condition and replace the test specimen after the test.The accuracy of the pressure gauge is up to 0.01 MPa.The pressure response curve of the confining pressure testing chamber recorded by the pressure gauge is shown in Fig.3.It can be seen that the confining pressure testing chamber can increase from 0 MPa to the pressure value(e.g.2 MPa)required for the test in about 70s,and maintain a stable confining pressure value for a long time.As shown in Fig.1,the pressure relief solenoid valve is used to exhaust quickly after the test.

Fig.1.Schematic diagram of the confining pressure testing system.

Fig.2.Confining pressure testing chamber.

Fig.3.The pressure response curve of confining pressure testing chamber.

2.2.Experimental material and specimen

The experimental material used in this study is a Nitrate Ester Plasticized Polyether (NEPE) propellant.Its main components are adhesive(Polyethylene Glycol,6%-8%),plasticizer(Nitroglycerine/1,2,4-Butanetriol Trinitrate,17%-21%),AP particles(20%-30%),Al particles (20%-30%),RDX particles (18%-20%) and catalyst (1%-3%).According to the GJB 770B-2005 Test method of solid propellant,the NEPE propellant was designed as a dumbbellshaped,as shown in Fig.2.

2.3.Uniaxial tensile tests

Previous research results showed that once a limit pressure referred to the saturation pressure (pressure threshold) is exceeded,a further increase of pressure does not lead to a significantly increase in ultimate properties of the solid propellant [10].Meanwhile,Wang pointed out the pressure threshold is about 6 MPa[17].In addition,before starting the normal testing,some pretests were conducted to assess the confining pressure system and make a preliminary exploration of the effect of confining pressure on solid propellant.Therefore,in order to better reflect the influence of pressure on the NEPE propellant,four groups of confining pressure conditions with relative atmospheric pressure 0 MPa (room pressure or atmospheric pressure 0.1 MPa),0.5 ± 0.05 MPa,2 ± 0.05 MPa and 5.4 ± 0.05 MPa were selected for testing.

In addition,due to the NEPE propellant exhibits a good elongation performance,if the tensile speed is too low,the experiments will be lasted more than 1 h.Thence,uniaxial tensile tests of 10 mm/min,50 mm/min and 200 mm/min were performed under each pressure condition,and the corresponding strain rates were 0.00333 s-1,0.0167 s-1and 0.0667 s-1,respectively.

When the pressure reached the experimental value in the testing chamber,the test was conducted.For each test,a preloaded force of 2 N was applied to the propellant sample by means of the material testing machine,which keeps the sample in the same position in the pressure chamber.Each test condition was performed at least three times and the stress-strain curves in the following sections were obtained by the average of the three replicas.Besides,the ambient temperature during the tests was 25°C.

2.4.Stress relaxation test

In order to obtain the linear viscoelastic parameters of the NEPE propellant,the stress relaxation test was carried out.Due to the NEPE propellant perform a good elastic behavior,the propellant specimen was firstly being stretched up to 8%strain with a speed of 100 mm/min,and then the displacement of the propellant specimen was maintained constant for 1000 s,herein the force variation with time was recorded by the computer system of material testing machine.The experimental temperature is same with the uniaxial tensile tests.

3.Experimental results and analysis

3.1.Stress-strain curves

The engineering stress-strain curves of the NEPE propellant under different confining pressure conditions and strain rates are shown in Fig.4.As shown in Fig.4(a)and(b),confining pressure has a remarkable influence on the mechanical properties of the NEPE propellant.For a given strain rate,NEPE propellant shows the same nonlinear behavior under different confining pressure conditions.Comparing to the room pressure,NEPE propellant can be capable of a larger deformation with applied confining pressure.With increasing the confining pressure,the maximum tensile stress and ultimate strain increases,which is different from Zhang’s research results for HTPB propellant [23].For a same strain rate test,the whole stress-strain curves are close to overlap at the initial tensile stage under different confining pressure conditions.It indicates that the initial elastic modulus don’t be influenced that much by confining pressure.

As it is shown in Fig.4(c) and (d),NEPE propellant performs a typical viscoelastic behavior.The mechanical properties of the NEPE propellant still exhibit rate-dependence under different confining pressure conditions.With the increase of strain rate,the maximum tensile stress and initial elastic modulus also increases.However,comparing to 0 MPa,the ultimate strain at the strain rate of 0.0667 s-1is obviously higher than that of 0.00333 s-1under 5.4 MPa.

3.2.Influence of coupling confining pressure and strain rate

In order to directly reflect the coupled influences of confining pressure conditions and strain rates on the mechanical properties of the NEPE propellant,Fig.5 shows the variation of maximum tensile strength,ultimate strain and initial elastic modulus of the NEPE propellant with confining pressure conditions and strain rates.It can be seen from Fig.5(a)and(b)that the maximum tensile stress and ultimate strain increase as the pressure increases.With the coupled effects of confining pressure and strain rate,the value of the maximum tensile stress at 5.4 MPa and 0.0667 s-1is 2.03 times of its value under 0 MPa and 0.00333 s-1.The variation of the ultimate strain at room pressure is very small,while the ultimate strain increase with strain rate under confining pressure conditions.The value of ultimate strain at 5.4 MPa and 0.0667 s-1is 2.19 times of its value under 0 MPa and 0.00333 s-1.Fig.5(c)shows that the initial elastic modulus have a slightly enhancement with applied confining pressure than that of room pressure,which can also be observed in Refs.[13,23].Comparing to 0 MPa and 0.00333 s-1,the initial elastic modulus still increases 33%under 5.4 MPa and 0.0667 s-1.Thereby,it can be concluded that under the coupled effects of confining pressure and stain rate,the initial elastic modulus,maximum tensile stress and ultimate strain of the NEPE propellant will have a significant increase.

3.3.The analysis of influence mechanism of confining pressure on NEPE propellant

Combined with Han’s [24] mesoscopic numerical simulation analysis of NEPE propellant under tensile loading shown in Fig.6,the influence mechanism of confining pressure on the NEPE propellant can be obtained.Firstly,due to the NEPE propellant is a kind of soft propellant and the damage is very small during the initial tensile stage,the NEPE propellant perform a good elastic behavior and is in an incompressible state.The confining pressure has a little effect on the NEPE propellant.Thence,the stress-strain curves trend are very close in the beginning under various confining pressure conditions.

Fig.4.The stress-strain curves of NEPE propellant under different confining pressure conditions and strain rates.(a) 0.00333 s-1;(b) 0.0667 s-1;(c) 0 MPa;(d) 5.4 MPa.

Then,with increasing the loading(strain of 15%-30%),the filler particles and matrix begin to debond,called as dewetting [25].However,the number of dewetted particles and the dewetted area in the propellant are small,and the dewetting speed of particles is relatively slow at this stage.The suppression effect of the confining pressure on the dewetting of the large particles is small.For a give strain,the tensile stress under confining pressure is slightly higher than that of room pressure.

Under room pressure,as the tensile loading continues to increase,the dewetting speed of large particles is accelerated.At the same time,the small particles around the large particles also begin to debond.The number of dewetted particles begins to increase.A large area of dewetted particles form cavities.Finally,the cavities are connected to cause the propellant to fail.While under the confining pressure condition,the surrounding pressure suppresses the dewetting of filler particles.The dewetting speed of particles is slower than room pressure.The particles need larger force to debond from the matrix.Thence,the dewetted process is extended as the confining pressure increases,as shown in Fig.4(a)and(b).In reality,under confining pressure,tensile process of the NEPE propellant is a resistance process,in which confining pressure resists the dewetting of large particles caused by the tensile loading.In another word,confining pressure reduces the deweting speed of the particles and suppresses the damage initiation and growth.If the confining pressure is certain,its resistance to deweeting and the formation of cavities is also certain.When the resistance of the confining pressure to dewetting of particles and the formation of cavities in the propellant is exceeded,cavities will be formed and connected,and the NEPE propellant finally fails.Therefore,the maximum tensile stress and ultimate strain increase with increasing the confining pressure.

In generally speaking,the influence of confining pressure on the NEPE propellant is to delay or suppress the development of dewetted damage,so the NEPE propellant shows higher mechanical properties under confining pressure condition.It means that the damage evolution curves of the NEPE propellant should be dependent on the confining pressure.This influence mechanism can be incorporated into constitutive model as will be our work in the next section.

4.Constitutive model

4.1.Constitutive model theory

From the stress-strain curves of the NEPE propellant shown in Fig.4,it can be found that the stress-strain curves meet the linear viscoelastic theory at the initial stage of stretching.After the strain reaches a critical value,these curves behave non-linear behavior,which is mainly due to the tensile loading makes the micro-defects inside the propellant be unstable to weaken performance and damage.Thence,it is appropriate to use a viscoelastic constitutive model with damage to describe the large deformation of the NEPE propellant under different loading conditions.The constitutive model considering pressure and strain rate can be expressed as[26,27]:

Fig.5.The variation of maximum stress and ultimate strain of NEPE propellant at various experimental conditions.(a)Maximum tensile strength;(b)ultimate strain and(c) initial elastic modulus.

where the functiong（P，˙ε）is softening function caused by damage,which is dependent on the environmental pressure and strain rate;h（ε，t）is applied to describe the linear viscoelastic mechanical properties.

4.1.1.Linear viscoelastic model

The linear viscoelastic model is the simplest expression for characterizing linear viscoelastic mechanical properties of solid propellant without damage.Based on the Boltzmann superposition law principle,the integral type linear viscoelastic can be written as:

where τ is integral variable,andE（t）denotes the relaxation elastic modulus,which can be presented by a Prony series as follows:

whereE∞is the long term equilibrium modulus,Eiis the elastic modulus corresponding to the characteristic time τi.

4.1.2.Softening function and damage variable

The soften performance of NEPE propellant under large deformation is caused by damage (dewetting,vacuole formation and microcracks) generated inside the solid propellant.There are usually two methods to present the damage.One is to measure the formation and evolution of microdefects in the material using acoustic emission(AE),CT scanning or X-ray.Another is to study the change of macroscopic mechanical properties during the loading process,such as stress,strain energy et al.It is generally considered that the second way is simple and convenient.Therefore,the softening function can be expressed through a macroscopic damage variable as follows:

whereD（P，˙ε）is damage variable,which depends on confining pressurePand strain rate ˙ε.For an undamaged stateD（P，˙ε）=0,propellant material behaves a linear viscoelastic behavior.

Damage variableDis usually defined through the using of an effective stress or net stress σ′as [28,29]:

Essentially,for NEPE propellant,the net stress can be regarded as linear viscoelastic stress if there is no original damage.And σ is the experimental stress.Because of the damage initiation inside propellant materials at a large deformation,experimental stress is lower than the linear viscoelastic stress under tensile process[30].Therefore,in this work,the damage variable is defined as [30,31]:

where σ exp denotes the stress with damage,which is experimental results;σlinearis the stress without damage,which is calculated through the linear viscoelastic model Eq.(2).The damage variableD（P，˙ε）reflects the reduction ratio of the stress due to damage growth.

Fig.6.The mesoscopic numerical simulation analysis of NEPE propellant by Han [24].(a) Strain of 0.18;(b) strain of 0.38 and (c) strain of 0.59.

For the NEPE propellant,its microstructure is composed of a large number of solid particles with different sizes (such as AP particles,RDX particles and Al particles) embedded in the matrix.The matrix and solid particles are bonded to each other by the binder.It can be considered that the NEPE propellant is composed of many small units.The strength of each small unit is different and there are many weak links inside these units.Based on statistical theory,they should follow the probability distribution functionf（x）.During the uniaxial loading process,the damage inside the propellant develops continuously,and the damage degree of the material is random.Therefore,it can be studied by statistical methods,which can reflect the damage degree of the material on the macroscopic level and characterize whether the small units are damaged on the microscopic level.Weibull distribution function[32] can be adopted to characterize the local failure strength[33-35].Considering that the damage variable is dependent on the tensile strain and critical strain of damage initiation under each testing condition,the damage probability function density of units can be presented as follows:

where ε is the tensile strain;εcis critical strain of damage initiation;mand η are shape parameter and scale parameter,respectively,which are used to characterize the distribution characteristics of the internal defects.

Based on the Lemaitre’s [36] damage law,the damage variable can be defined as the ratio of the number of damage units under tensile loading.The damage evolution curves are different under various strain rates and confining pressure conditions.Assuming that the model parameters are also dependent on the pressure and strain rate,and integrating Eq.(7),damage variable can be given as:

wherek=1/η.

The nonlinear viscoelastic constitutive model considering the confining pressure and strain rate is:

4.2.Constitutive model calibration

4.2.1.Identification of the viscoelastic parameters

Since the confining pressure have a slightly influence on the initial elastic stage of the NEPE propellant,the viscoelastic parameters were obtained by stress relaxation test under room pressure.The result was fitted by 3 terms of Prony series,as shown in Fig.7.The related parameters are listed in Table 1.

Fig.7.The curve of relaxation modulus.

4.2.2.Identification of the damage parameters

In the present work,since the damage parameters are ratedependent and pressure-dependent,the identification processes are divided two steps.Firstly,the experimental data of strain rate of 0.00333 s-1were used to identify the pressure-dependent damage parameters.The critical strain εcof damage initiation was determined through an empirical method,that is,the comparison of the experimental results and linear viscoelastic model results.When the linear viscoelastic model (Eq.(2)) overpredicts the experimental stress,the corresponding strain is regarded as the critical strain of damage initiation.The damage evolution curves under different confining pressure conditions were obtained by using Eq.(6).And then these curves were fitted by Eq.(8),as shown in Fig.8.The fitted damage parameters under different confining pressure conditions are listed in Table 2.

Fig.8.The experimental damage evolution curves and the fitted curves under various confining pressure conditions and 0.00333 s-1.

Table 1 Prony series coefficients of relaxation modulus.

Table 2 The fitted parameters of damage function under various confining pressure conditions and 0.00333 s-1.

Fig.9.The fitted results of the damage parameters εc.

Fig.10.The damage evolution curves of the NEPE propellant at different strain rates and 0 MPa.

Fig.11.The value of m at different strain rates and confining pressure conditions.

Fig.12.The value of linear fitting function parameters a and b under different confining pressure conditions.

As shown in Fig.8,the critical strain εcof damage initiation increases with increasing confining pressure.This is because that the matrix and filler particles are compacted by the pressure,so the NEPE propellant behave stiffer than room pressure.Besides,the rate of damage growth of the NEPE propellant decreases with an increasing of confining pressure(the slope of the damage evolution curves decreases as the confining pressure increases),which is coincident with the analysis in section 3.3.When the NEPE propellant is at strain of 0.6,the value of damage is 0.26 under room pressure,while it is only 0.08 at 5.4 MPa.This indicates that the confining pressure has a very obvious suppression effect on the damage evolution of the NEPE propellant caused by the tensile loading.In addition,when the propellant is close to failure,the value of damage is about 0.45,which may be further used to predict the failure point of the solid propellant under different confining pressure conditions.

It can be found from Table 2 that the fitted damage function parameters all change with the confining pressure.Therefore,the damage parameters can be fitted and expressed using the functional relationship with pressure,respectively.The fitted results are plotted in Fig.9.

Secondly,the rate-dependent damage parameters were identified.The damage evolution curves at 0 MPa and different strain rates are obtained and fitted using similar method,as shown in Fig.10.From Ma’s[37]research results,the critical strain of damage initiation decreases slowly with the strain rate.In order to simply the fitted process,critical strain of damage initiation is regarded as a constant at a same confining pressure condition.From Fig.10,it can be observed that for a given strain,with increasing the strain rate,the damage increases.The damage evolution curves have a similar shape.However,the experimental damage evolution curves are not ideal data,the fitted processes should consider the ratedependence of the damage evolution curves when developing constitutive model.Therefore,there is a few errors between the experimental damage curves and fitted damage curves.In general,it can be assumed thatmandkare rate-dependent.

Accordingly,the damage parametermandkat different confining pressure conditions can be determined.Due to the fitted process is same betweenmandk,onlymis shown in the work.The damage parametermare fitted using a linear function with strain rate,seen in Fig.11.The slope decreases with increasing the confining pressure.It indicates that confining pressure can reduce the difference of damage evolution curves at different strain rates.Furthermore,the linear fitting function parameter slopeaand interceptbat different confining pressure condition can be obtained,which are related to confining pressure.Therefore,they can be fitted and expressed using the functional relationship with pressure,respectively.The fitted results are shown in Fig.12.

It is worth noting that the fitting functions selected have considered the existence of pressure threshold in this paper.It means that with a continuous increase of the higher pressure,the damage parameters will not change much.

Finally,the damage parameters considering confining pressure and strain rate can be expressed as:

4.3.Constitutive model validation

In this section,the experimental data under different experimental conditions were used to validate the accuracy of the nonlinear constitutive model.The comparison between the experimental data and the prediction stress-strain curves using the constitutive model Eqs.(8) and (9) and corresponding model parameters Eqs.(10-12)are shown in Fig.13.It can be found that the constitutive model considering confining pressure and strain rate can well capture the mechanical responses of NEPE propellant under different experimental conditions.Although the deformation is large,the predicted results are still good.

In addition,in order to verify the ability of the constitutive model and damage parameters to capture the mechanical responses of NEPE propellant under higher confining pressure condition.A uniaxial tensile test with confining pressure of 7 MPa and strain rate of 0.00333 s-1was conducted.The comparisons of stress-strain curve and damage evolution curve between the experimental data and the model predictions are shown in Fig.14.It can be observed that the predictions of the constitutive model are in good agreement with the experimental data,which proves that the constitutive model still has good predictive ability for higher confining pressure condition.It also proves that the fitting functions are reliable.Besides,it can be found that comparing to 5.4 MPa,the maximum tensile strength has a little increase at the 7 MPa.The ultimate strain of 7 MPa is lower than 5.4 MPa.It shows that the pressure threshold has been researched.Certainly,it needs more experiments to determine in further studies.

In engineering application,researchers are more concerned about the maximum tensile strength and the corresponding strain of solid propellant.Fig.15 provides the errors on the predicted values by the constitutive model with the experimental data.The error is defined as:

Fig.15 proves that,for different strain rates and confining pressure conditions (see Fig.13),clearly,the constitutive model predictions are in close agreement with the test data.The whole stress errors of the model predictions are lower than 4% and corresponding strain errors are lower than 7%.It demonstrates that the developed nonlinear constitutive model has a good predicted ability,which can be used to analyze the structural integrity of propellant grains under working pressure loading.

Fig.13.Comparison of experiment data and constitutive model predictions.(a)0.00333 s-1;(b) 0.0167 s-1 and (c) 0.0667 s-1.

5.Conclusion

Fig.14.Comparison of experiment data and constitutive model predictions of 7 MPa and 0.00333 s-1.

In this paper,based on the self-made confining pressure system and material testing machine,uniaxial tensile tests of the NEPE propellant were conducted.The influence and influence mechanism of confining pressure on the mechanical properties of the NEPE propellant were analyzed and discussed.Based on viscoelastic theory and continue damage theory,a nonlinear constitutive model was proposed to capture mechanical responses of the NEPE propellant,which takes into the effects of strain rate and confining pressure.Conclusions of this study can be summarized as follows:

(1) Confining pressure and strain rate have a significant influence on the mechanical responses of the NEPE propellant.As the confining pressure increases (from 0 to 5.4 MPa),the maximum tensile stress and ultimate strain increase gradually.With the coupled effects of confining pressure and strain rate,the value of the maximum tensile stress and ultimate strain at 5.4 MPa and 0.0667 s-1is 2.03 times and 2.19 times of their values under 0 MPa and 0.00333 s-1,respectively.The initial elastic modulus have a slightly enhancement with applied confining pressure than that of room pressure.Therefore,for the analysis of structural integrity of propellant grains of SRMs,the effect of the ignition pressurization on the mechanical properties of propellant materials should be considered.

(2) Comparing to the room pressure,the dewetting speed decreases and the dewetting processes are extended under confining pressure.The critical strain εcof damage initiation increases with increasing the confining pressure.The influence mechanism of confining pressure on the NEPE propellant is to suppress the damage initiation and evolution inside the propellant.

(3) Through assuming that the damage is rate-dependent and pressure-dependent,a nonlinear constitutive model considering confining pressure and strain rate was developed and validated.Good agreement is observed between the experimental data and model predictions.The whole maximum stress errors of the model predictions are lower than 4% and the corresponding strain errors are lower than 7%.The model can be further used for numerical simulation of SRMs.This research can lay a theoretical foundation for analyzing the structural integrity of propellant grains accurately under working pressure loading.

Fig.15.The errors on the predicted values by the constitutive model with the experimental data.

Declaration of competing interest

The authors confirm that there are no known conflicts of interest associated with this publication.

Acknowledgements

The research was supported by the National Natural Science Foundation of China (Grant No.51606098) and Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX20_0303).