A nonlinear viscoelastic constitutive equation with cumulative damage for modified double-base propellants①

MENG Hong-lei

(The 41st Institute of the Fourth Academy of CASC，Xi＇an 710025，China)

0 Introduction

Modified double-base propellants belong to composite materials composed of particles in a double-base propellant matrix.The double-base propellant matrices have viscoelastic properties.Damage in highly filled composites appears as microcracks and voids along the particle-matrix interface under loading.The products and evolution of damage are generally thought to contribute to nonlinear behaviour.Continuum damage mechanics is based on continuum mechanics and irreversible thermodynamics.In theory，the damage evolution equation can be obtained from dissipative potential functions.As dissipative potential functions are usually obtained experimentally，the damage equations are obtained directly by these methods.

Schapery introduced the concept of pseudo-strain，and proposed a nonlinear viscoelastic constitutive equation composed of pseudo-strain and a nonlinear coefficient that is a function of the damage variable［1-2］.The damage model of Schapery＇s constitutive equation is called the Work Potential Model，and is similar in form to the power-law of crack growth for viscoelastic materials.Kim developed and applied Schapery＇s constitutive equation to asphalt con-crete materials［3－9］.Sook-Ying established a nonlinear coelastic constitutive equation incorporating damage to characterize the high strain rate response of the propellant under impact loading［10］.Duncan also included cumulative damage into the viscoelastic constitutive equation.The nonlinear component of the model consisted of a strain rate term，a damage term and a nonlinear exponent.However，this model can only predict the response at constant strain rate［11］.

In this paper，a nonlinear viscoelastic constitutive equation with cumulative damage，similar in form to the constitutive equation of Schapery，is proposed for modified double base propellants.

1 Schapery＇s nonlinear viscoelastic constitutive equation with damage

The nonlinear viscoelastic constitutive equation with damage for modified double-base propellants presented in this paper is based on Schapery＇s nonlinear viscoelactic constitutive equation with damage.Therefore，it is first necessary to introduce Schapery＇s equation and parameter fitting method.

1.1 Elastic-viscoelastic correspondence principle

The linear viscoelastic problem in Laplace space is identical to the elastic problem，which is also called the elastic-viscoelastic correspondence principle or correspondence principle.Schapery proposed the concept of pseudostrain，and extended the elastic-viscoelastic correspondence principle to nonlinear viscoelastic problems.The pseudo-strain εRis defined as:

where ε is the physical strain and ERis the reference modulus，which is an arbitrary constant commonly taken to be one or the value of the initial elastic modulus.

The pseudo-strain is a term of the convolution integral and includes the effect of time.Using the concept of pseudo strain，the linear viscoelastic constitutive equation can be written as:

The linear viscoelastic constitutive equation(Eq.2)has a similar form to the elastic constitutive equation(Hooker＇s Law).By using the concept of pseudo-strain，the relationship between linear viscoelasticity and elasticity can be obtained without the use of Laplace and Fourier transforms.For linear viscoelastic material，the reference modulus(ER)is constant.The propellant material shows linear viscoelastic behaviour at small strain，but nonlinear behaviour when the strain exceeds a certain value.The modified double-base propellant will show nonlinear behavior when strain exceeds 0.5%under normal temperature condition.For nonlinear viscoelastic behaviour，ERisn＇t constant，which is related not only to the loading level，but also to the loading procedure.To describe the propellant＇s nonlinear viscoelastic character，the relationship between ERand the loading procedure needs to be determined.Recently，a number of studies have proposed that the nonlinear character is related to the damage inside the material.Here，an internal damage variable is integrated into the viscoelastic constitutive equation:

where S is an internal damage variable and C is a function of S.

C is related to the internal damage variable and is a property of the material that is independent of the loading conditions and loading history.At small strain，the material shows linear behaviour，so C=ER.However，when the strain increases，damage develops and the value of C decreases，which reflects the softening process of the material.

The relationship of stress and pseudo-strain are shown in Fig.1.For linear viscoelastic materials，the stresspseudo strain curves show a linear relationship，while for nonlinear viscoelastic materials the stress-strain curves reach a maximum and then slowly decrease.

Fig.1 Stress-pseudo-strain curves

For uniaxial constant strain rate tensile tests，the pseudo-strain can be written as:

where

Substituting Eq.(5)into Eq.(4)gives:

Eq.(6)gives the relationship between strain and pseudo-strain for uniaxial constant strain rate tensile tests.

1.2 Schapery＇s damage evolution equation

Before establishing Eq.(3)，the damage evolution equation needs to be determined.Generally，the damage evolution equation can be obtained by the irreversible thermodynamic dissipation power.As the dissipation power needs to be determined experimentally，so does the damage evolution equation.There are various forms of the damage evolution equation，such as Kachonov＇s damage model，the cumulative damage model，and Lebesgue Norm of strain.

Schapery proposed the work potential damage evolution model:

where WRis the pseudo-strain energy density function:

Substituting Eq.(8)into Eq.(7)gives:

Eq.(3)and(9)constitute Schapery＇s nonlinear viscoelastic constitutive equation with damage.

For Schapery＇s nonlinear viscoelastic model，two material-dependent properties need to be determined:the function C(S)and the constant a.The parameters of the equation are obtained from constant strain rate tests.Firstly，an approximate form of the damage model(Eq.(9))needs to be determined and an initial estimate of a madel.The C(S)function is then fitted using the approximate damage model，with the value of a varied to find the best fit of the C(S)curve.A more detailed explanation of the process can be found in the Reference［2］.

2 Nonlinear viscoelastic constitutive equation with cumulative damage

2.1 The constitutive equation

The parameter determination process used by Schapery is complex.As the cumulative damage model is relatively simple and the parameters are easy to obtain，a nonlinear viscoelastic constitutive model with cumulative damage is proposed.It is based on the form of Schapery＇s model and has advantages in parameter determination，application and numerical simulation.The cumulative damage model is:

where a and η are material damage parameters.

Combining Eq.(3)and(10)gives the nonlinear viscoelastic constitutive equation with cumulative damage:

2.2 Parameter determination method

Stress relaxation experiments and uniaxial constant strain rate tension tests were performed with strain rates ranging from 0.001 9 s－1to 0.068 s－1to fit the parameter S and the function C(S).The steps of the fitting procedure are:

(1)A relaxation experiment is undertaken to obtain the relaxation modulus.

(2)An initial value for a is chosen.

(3)The stress and strain values from the strain ratetension test are used to give a relationship between stress and pseudo strain using Eq.(6).

(4)The ratio of stress to pseudo strain is equal to C，and the relationship between C and pseudo-strain is determined from the strain ratetension test.

(5)a is given the initial value of η，and the relationship between pseudo strain and the internal damage variable S is determined using the strain ratefrom tension test and Eq.(11).

(6)Based on the relationships between C and εRand between S and εR，the relationship between C and S is established，then the C-S curve is plotted at strain ratetension test.

(7)Steps 1～6 are repeated for five strain rate tension tests，giving C-S curves at five strain rates.As a and η are estimate value，the C-S curves at different strain rates may not be the same.The a value only effects the fitting result of the C(S)function，and not the consistency of CS curves.However，the η value does effect the consistency of the C-S curves.Thus，substituting the initial value for a for the final value for a and changing the value of η pro-duces different C-S curves.

If the consistency of C-S curves isn＇t good，the value of η is changed until consistent C-S curves are obtained.From the curve fitting，the function of C(S)can be obtained.A flow diagram of the fitting procedure is shown in Fig.2.

Fig.2 Fitting procedure for the nonlinear viscoelastic constitutive equation

3 Experiments and parameter determination for modified double-base propellants

3.1 Materials

The material used in this study is modified doublebase propellant，which is a composite composed of particles embedded in a double base propellant matrix.

3.2 Relaxation modulus

The relaxation modulus is an important parameter for characterizing the linear viscoelastic and nonlinear viscoelastic behavior.There are two ways to obtain relaxation modulus.The first is the complex modulus test，where the relaxation is determined through interconversion of the complex modulus function.The second is the relaxation test.Relaxation is difficult in the relaxation test and requires a robust high-capacity testing machine.However，methods have been developed to obtain the relaxation modulus that do not require a high-capacity testing machine［12－15］.Here，Sorvari＇s method is used to obtain the relaxation modulus.

The relaxation modulus is expressed by the Prony series:where E(t)is the relaxation modulus as a function of time，E∞is the long-term equilibrium modulus，τiis the relaxation time，and Eiare the Prony series coefficients.

3.3 Relationship between stress and pseudo-strain

In this study，the reference modulus is chosen to be 1 MPa.Several constant strain rate tests at different strain rates were performed，the relationship between pseudostrain and stress is obtained from Eq.(6).The stress-pseudo-strain curves are then obtained，and are shown in Fig.3.

Fig.3 σ-εRcurves at different strain rates

3.4 Fitting of the C(S)function

If a is taken to be 10，according to the method introduced above，the optimum η value is 5.5.The C-S curves at the optimum η value are shown in Fig.5，which are consistent for different values of strain rate.The relationship between C and S is then obtained by fitting Eq.(13)u-

From the relationship between stress and pseudostrain，the relationship between C and pseudo-strain can be obtained from Eq.(11).

The C-εRcurves are shown in Fig.4.Initially，with increasing pseudo-strain the C value stays almost constant at 1.0.Then，as εRincreases，the damage evolution starts to decrease.sing a least squares method for the C and S data.The regression results are also shown in Fig.5.

Fig.4 C-εRcurves at different strain rates

Fig.5 C-S curves and regression results

The C(S)function can be represented by Eq.(13)，and the damage model by Eq.(14).

The complete nonlinear viscoelastic constitutive equations with cumulative damage are as following:

3.5 Model verification

Firstly，the relationships between stress and strain for several constant strain rate tension processes were predicted using the constitutive equation determined above.The comparison between the predictions and the experimental measurements are shown in Fig.6.The predictions are in reasonable agreement with the experimental measurements.

Fig.6 Comparison between predictions and experimental measurements at constant strain rate tension

Secondly，two uniaxial tension tests with complex strain loading profiles were performed.The first strain loading profile has two stages:(1)Constant strain rate tension at a strain rate of 0.003 s－1until the stress reach 5 MPa;then，(2)constant strain rate tension at a strain rate of 0.02 s－1until fracture.The second strain loading profile has three steps:(1)Constant strain rate tension at a strain rate of 0.004 5 s－1until the stress reach 5 MPa;(2)Maintain the strain for 10 seconds;(3)Constant strain rate tension at a strain rate of 0.011 s－1until fracture.The relationship between stress and strain with the complex strain loading profiles was simulated using the proposed constitutive equation，and the comparison between the predictions and the experimental measurements can be shown in Fig.7.The predictions show good agreement with the experimental measurements.

Fig.7 Comparison between predictions and experimental measurements with complex strain loading

4 Conclusions

A nonlinear viscoelastic constitutive equation with cumulative damage for modified double-base propellant was proposed.The parameter determination method was introduced，whereby the parameters were obtained through relaxation and constant strain rate tests.The model is then used to predict stress-strain curves for a constant strain rate tension process and for two complex strain tension processes.The proposed method is in good agreement with the experimental measurements.

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