# The Analytical Method to Compute the Strain on the Soft PSD in Double-Pulse SRM

Authors: Chunguang Wang; Guiyang Xu
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

In order to obtain the analytical method to compute the circumferential strain on a soft pulse separation device (PSD), deformation processes of the middle section of the soft PSD, the medicine propellant grain and the case are simplified into a two-dimensional plane strain state. It is found that the main factors affecting the circumferential strain of the soft PSD are the circumferential strain of the inner surface of the propellant grain and the gap between the soft PSD and the propellant grain. In order to study the failure mechanism of the soft PSD in the double-pulse solid rocket motor (SRM), a two-dimensional axisymmetric finite element method (FEM) model of the stress process of the soft PSD is established. The variation of the strain of the soft PSD with the internal pressure load is obtained. It is found that the excessive circumferential strain is the main reason for the failure of the soft PSD. Comparing the analytical calculations with the FEM results, it can be found that the analytical method value is slightly higher than the FEM value, so the analytical method results can be used to initially estimate the circumferential strain of the soft PSD and then predict the rationality and feasibility of the design scheme. In order to further study the failure mechanism of the soft PSD, a micro-CT test of in situ stretching of the soft PSD material is carried out and the variation of porosity and elongation of the material is studied. The test results showed that when the material elongation is large, the microinterface debonding rapidly expands into a penetrating damage, and the PSD structure fails. The conclusions obtained in this paper can provide a useful reference for the design of double-pulse SRM.


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## Body

## 1. Introduction

A pulse solid rocket motor (SRM) uses the pulse separation device (PSD) to divide the combustion chamber of SRM into several parts to distribute the thrust and the pulse interval time reasonably and improve the performance of various types of tactical missile systems comprehensively. PSD is a difficult and key technology for the development of a dual SRM [1].In reference [2], the calculation method of the stress intensity factor of the prefabricated defects of the metal diaphragm is obtained by the circular plate large deflection theory and the fracture mechanics theory, and the design basis of the metal diaphragm type compartment in the double SRM is obtained. According to the bearing capacity, PSD can be divided into two typical pulse separation devices: hard PSD and soft PSD. Literature [3–6] studied the characteristics of a double-pulse SRM internal flow field. During the second-pulse operation, a backward-facing step flow occurred in the first-pulse combustion chamber due to the contraction of the interstage pulse channel. Thereby, a gas vortex is generated, which exacerbates the convective heat transfer of the first pulse and the particle flushing. In reference [7], the characteristics of the internal flow field of an inner bore clapboard pulse SRM on the condition of pure gas phase and gas-particle phase is studied. In reference [8], the second-pulse ignition transient process of a double-pulse SRM is studied by the self-developed multiphysics coupled solver. In reference [9], the material EPDM is used as a soft PSD for a double-pulse SRM. According to the theory of continuous medium mechanics of rubber materials, a viscous superelastic constitutive model describing the EPDM soft PSD under finite deformation is established. In reference [10], the pressurization process of PSD was calculated by FEM and analyzed during the operation of the first grain. Some basic reasons for the damage of PSD during compression were analyzed.In SRM work, a PSD has a high-pressure, high-temperature, high-stress, and large deformation working environment. In particular, a soft PSD has a large deformation under high-temperature and high-pressure gas. However, a soft PSD is far from reaching the material’s elongation, and it is ineffective. The high-temperature gas ignites the second-pulsed propellant grain through the PSD, and the two stages of pulses work simultaneously. The pressure inside the combustion chamber rises rapidly, causing SRM to explode. The structural failure of the soft PSD has become an important issue hindering the development of double-pulse SRM [11].In this paper, the research work on the failure of a soft PSD in the working process of a double-pulse SRM is carried out. In order to obtain the analytical method to compute the circumferential strain of the soft PSD, the deformation process of the intermediate section of the PSD, the propellant grain and the case are simplified into a two-dimensional plane strain state. This analytical method can be used as a formula basis for the design of a double-pulse SRM. The two-dimensional axisymmetric FEM model of the stress process of the soft PSD is established, and the variation of the strain of the soft PSD with the internal pressure load is obtained to verify the accuracy of the analytic method. In order to further study the failure mechanism of the soft PSD, the micro-CT test of in situ stretching of the soft PSD material is carried out, and the variation of porosity and elongation of the material is studied.


## 2. System Description and Critical Failure Mechanism of Soft PSD

### 2.1. System Description of PSD

The structure of the soft PSD double-pulse SRM is shown in Figure1. It includes a first pulse, a second pulse, a soft PSD, and a weak area. The working principle of the soft PSD is shown in Figure 2. During the first-pulse SRM operation, the soft PSD is pressure-bearing through the second-pulsed propellant grain. At the same time, it acts as a heat insulator and seal. The reliability of the seal and pressure is directly related to the safety of the second grain. As a result of the pressure from the first grain, the soft PSD will produce radial deformation with the second grain, resulting in circumferential strain. When the second-pulse grain is working, the soft PSD first breaks in the weak area. Then, the broken section is reversed and turned open. Therefore, a reasonable design not only requires the second pulse to work reliably at a predetermined pressure and a predetermined weak area but also requires the weak area section to be smoothly reversed and turned open, thereby ensuring reliable ignition and a stable combustion flow of the second-pulse grain [10].Figure 1
Soft PSD double-pulse SRM.Figure 2
Working principle of PSD.The thickness of soft PSD is generally about 8 mm. There is a mold release agent between the soft PSD and the second grain to ensure that the grain and soft PSD are separated. In addition, a natural gap will be formed between the soft PSD and the grain due to radial contraction during the curing and cooling process of the propellant. Due to the fluctuation of the processing process, the gap is generally between 1 and 5 mm.


### 2.2. Critical Failure Mechanism of Soft PSD

Through the cold gas test and the single test of the first pulse, it is found that the failure of PSD is generally caused by excessive circumferential strain. Thereby, it is very important to obtain the calculation method to compute the strain for PSD. Since SRM has a certain length, the strain can be approximated as a plane strain problem at the middle section of the soft PSD.The calculation of the circumferential strain in the middle of the soft PSD can be simplified to formula (1). εθ is the circumferential strain of the soft PSD. a is the inner diameter of the propellant grain, a∗=a−δ0 is the diameter of the soft PSD, and δ0 is the gap between the soft PSD and the propellant grain; the simplified form is shown in Figure 3.
(1)εθ=ΔRa−δ0−δg.Figure 3
Plane strain.Among them,ΔR contains two parts: δ0 and δg. δg is the radial displacement generated by the surface of the propellant grain under internal pressure.δg, which is the radial displacement of the inner surface of the propellant grain, can be obtained by the calculation formula (2).The specific derivation process can be found in the literature [12, 13].
(2)δg=1+υm2−1E1−2υa+mRp−mR+1−2υm2aq,where m is the ratio of the outer diameter to the inner diameter of the propellant grain, p is the internal pressure load of the SRM generated by first grain burning, υ is the Poisson’s ratio of the propellant grain, E is the elastic modulus of the propellant grain, and q is the external pressure generated by the constraint of the case. The calculation method is shown as follows [12, 13]:
(3)q=41−υ2P21+υ1+1−2υm2+2−υkm2−1ER/Ekh,where υk is the Poisson’s ratio of the case and Ek is the modulus of elasticity of the case. If the case material is composite, then υk is the ring longitudinal Poisson’s ratio and Ek is the circumferential modulus.Formulas (2) and (3) are brought into formula (1) to obtain the approximate calculation formula of the circumferential strain of the soft PSD under the action of internal pressure load, as shown in formula (4):
(4)εθ=δ0a+εθg,(5)εθg=1+υm2−1E1−2υ+m2p−m2+1−2υm2q.It can be seen from formula (4) that for the double-pulse SRM, which has determined the geometry of the propellant type, the main factors affecting the circumferential strain of the soft PSD are the circumferential strain of the propellant grain inner surface and the gap between the soft PSD and the propellant grain.


## 2.1. System Description of PSD

The structure of the soft PSD double-pulse SRM is shown in Figure1. It includes a first pulse, a second pulse, a soft PSD, and a weak area. The working principle of the soft PSD is shown in Figure 2. During the first-pulse SRM operation, the soft PSD is pressure-bearing through the second-pulsed propellant grain. At the same time, it acts as a heat insulator and seal. The reliability of the seal and pressure is directly related to the safety of the second grain. As a result of the pressure from the first grain, the soft PSD will produce radial deformation with the second grain, resulting in circumferential strain. When the second-pulse grain is working, the soft PSD first breaks in the weak area. Then, the broken section is reversed and turned open. Therefore, a reasonable design not only requires the second pulse to work reliably at a predetermined pressure and a predetermined weak area but also requires the weak area section to be smoothly reversed and turned open, thereby ensuring reliable ignition and a stable combustion flow of the second-pulse grain [10].Figure 1
Soft PSD double-pulse SRM.Figure 2
Working principle of PSD.The thickness of soft PSD is generally about 8 mm. There is a mold release agent between the soft PSD and the second grain to ensure that the grain and soft PSD are separated. In addition, a natural gap will be formed between the soft PSD and the grain due to radial contraction during the curing and cooling process of the propellant. Due to the fluctuation of the processing process, the gap is generally between 1 and 5 mm.


## 2.2. Critical Failure Mechanism of Soft PSD

Through the cold gas test and the single test of the first pulse, it is found that the failure of PSD is generally caused by excessive circumferential strain. Thereby, it is very important to obtain the calculation method to compute the strain for PSD. Since SRM has a certain length, the strain can be approximated as a plane strain problem at the middle section of the soft PSD.The calculation of the circumferential strain in the middle of the soft PSD can be simplified to formula (1). εθ is the circumferential strain of the soft PSD. a is the inner diameter of the propellant grain, a∗=a−δ0 is the diameter of the soft PSD, and δ0 is the gap between the soft PSD and the propellant grain; the simplified form is shown in Figure 3.
(1)εθ=ΔRa−δ0−δg.Figure 3
Plane strain.Among them,ΔR contains two parts: δ0 and δg. δg is the radial displacement generated by the surface of the propellant grain under internal pressure.δg, which is the radial displacement of the inner surface of the propellant grain, can be obtained by the calculation formula (2).The specific derivation process can be found in the literature [12, 13].
(2)δg=1+υm2−1E1−2υa+mRp−mR+1−2υm2aq,where m is the ratio of the outer diameter to the inner diameter of the propellant grain, p is the internal pressure load of the SRM generated by first grain burning, υ is the Poisson’s ratio of the propellant grain, E is the elastic modulus of the propellant grain, and q is the external pressure generated by the constraint of the case. The calculation method is shown as follows [12, 13]:
(3)q=41−υ2P21+υ1+1−2υm2+2−υkm2−1ER/Ekh,where υk is the Poisson’s ratio of the case and Ek is the modulus of elasticity of the case. If the case material is composite, then υk is the ring longitudinal Poisson’s ratio and Ek is the circumferential modulus.Formulas (2) and (3) are brought into formula (1) to obtain the approximate calculation formula of the circumferential strain of the soft PSD under the action of internal pressure load, as shown in formula (4):
(4)εθ=δ0a+εθg,(5)εθg=1+υm2−1E1−2υ+m2p−m2+1−2υm2q.It can be seen from formula (4) that for the double-pulse SRM, which has determined the geometry of the propellant type, the main factors affecting the circumferential strain of the soft PSD are the circumferential strain of the propellant grain inner surface and the gap between the soft PSD and the propellant grain.


## 3. Compartment Stress Analysis

### 3.1. Calculation Model

In order to verify the accuracy of the analytic method, FEM calculation is carried out. According to the stress state of the soft PSD, the structure of the double-pulse SRM is simplified, and a two-dimensional axisymmetric model is established to analyse the structural response characteristics of the double-pulse SRM under first-pulse grain burning. As shown in Figure4, the SRM is composed of a metal case, a second-pulsed propellant grain, and a soft PSD. The gap between the propellant grain and the PSD is valued according to the process detection data.Figure 4
Simplified physical model of second-pulse combustion chamber.


### 3.2. Calculation Method

The following assumptions are made regarding the operating characteristics of the SRM:(a)
The useful temperature range for the EPDM material is about -55°C~+150°C. When first-pulse grain is burning, high-temperature combustion gas is stationary on the surface of the soft PSD and the heat transfer speed to the soft PSD is not fast. The temperature of the surface of the soft PSD will rise by 80°C, and its overall average temperature will rise by 30°C, hardly affecting its mechanical properties. According to the experimental results, the thermal load is not the cause of failure of the soft PSD. As a result, the temperature rise and ablation of the soft PSD are negligible, and the performance change of the PSD material caused by the temperature change is not considered
(b)
Quasi-static analysis is performed for a given load without considering the inertial effect of structural mass. The effect of strain rate on the mechanical properties of EPDM and propellant grain is not considered too
(c)
The friction between the soft PSD and the structural parts, such as the case and the second-pulsed propellant grain, is ignored
(d)
The pressure of the first-pulse grain burning pressurization is evenly applied to the soft PSDThe internal pressure load is taken as the maximum working pressure of the SRM, which reaches to 9.0 MPa, and a fixed constraint is applied to the front skirt of case end face to eliminate the rigid body displacement, as shown in Figure5.Figure 5
Second-pulsed propellant grain stress.In order to simulate the pressure building process of the SRM internal pressure, the pressure load curve is loaded as shown by the curve change trend in Figure6.Figure 6
Internal pressure curve of SRM.


### 3.3. Material Properties

According to engineering development experience, the internal pressure of the SRM is usually established within 0.3 s, and the propellant grain and thermal insulation layer (including the compartment) can be simulated as a linear elastic material. The performance parameters of each material of the SRM are shown in Table1.Table 1
Performance parameters of the SRM.
Structure
Modulus
Poisson’s ratio
Case
200 GPa
0.3
Second-pulsed propellant grain
5.2 MPa
0.495
Soft PSD
10.0 MPa
0.495


### 3.4. Computing Grid

The geometric model is meshed under ABAQUS/CAE for a total of 60,000 finite element meshes, as shown in Figure7. For volume approximate incompressible material characteristics of the propellant grain and short fiber reinforced EPDM, the hybridization unit option is selected in the unit setting under ABAQUS/CAE. And the Plane Axisymmetric Hybridization unit (CAX4H) is selected to avoid the volume self-locking and finite element analysis by the high Poisson ratio, which is closest to 0.5. Linear Reduced Integral Planar Axisymmetric Unit (CAX4R) is used for the case. Figure 8 is a partial magnification of the mesh.Figure 7
Model overall grid.Figure 8
Local grid.The interface of the case and grain is the bonding condition. The interface of PSD and the grain is the contact in the gap condition. The interface of PSD and the case is the bonding condition.


### 3.5. Result Analysis

After the calculation is completed, the strain in the three directions of the soft PSD is shown in Figure9.Figure 9
PSD strain contours.
(a)
Radial strain
(b)
Axial strain
(c)
Circumferential strainAccording to the calculation results, it can be found that(1)
The radial strain in the middle of the soft PSD is negative, meaning that the central portion is radially compressed. And there is a high tensile strain zone at both ends due to adhesion to the case, but the area is extremely small
(2)
The two segments of the PSD are bound to the case and end face of the grain. The interface of the case and grain is the bonding condition. The axial deformation of the propellant/case combination structure causes the axial strain of the soft PSD. It is worth noting that the main axial strain still occurs at the two ends of the soft PSD bound to the case, and most of the soft PSD has a small axial strain, especially the middle section
(3)
The main strain of the soft PSD is the circumferential strain. The largest part of the circumferential strain is in the axial middle section of the soft partition, reaching a maximum of 21.05%The maximum strain point of the high-strain region of the soft PSD is recorded as point A, as shown in Figure10.Figure 10
The maximum strain point of the soft PSD (point A).Circumferential strain versus time curves of point A, the propellant grain inner surface, and the case of the same axial position as point A are shown in Figure11.Figure 11
Circumferential strain versus time curve of point A, the propellant grain inner surface, and the case.It can be seen from Figure10 that when the first-pulse grain working pressure acts on the soft PSD, the trend of the circumferential strain of the PSD, the propellant grain, and the case are similar to the loading trend of the internal pressure. The development of the circumferential strain can be divided into the following three stages:
(1)
The first stage: the gap between the soft PSD and the propellant grain is quickly filled, and the circumferential strain of the soft PSD is mainly due to the filling gap, and its value accounts for 26.5% of the total strain of the whole process
(2)
The second stage: the propellant grain and the soft PSD as a whole limit radial displacement of the soft PSD after the soft PSD is in contact with the propellant grain. Under further action of the pressure, the whole grain and the case are deformed under pressure, and the circumferential strain is further increased
(3)
The third stage: due to the approximate incompressibility of the propellant grain, the radial deformation of the grain is getting smaller and smaller, and the circumferential strain gradually approaches the maximum value under the limitation of the case


## 3.1. Calculation Model

In order to verify the accuracy of the analytic method, FEM calculation is carried out. According to the stress state of the soft PSD, the structure of the double-pulse SRM is simplified, and a two-dimensional axisymmetric model is established to analyse the structural response characteristics of the double-pulse SRM under first-pulse grain burning. As shown in Figure4, the SRM is composed of a metal case, a second-pulsed propellant grain, and a soft PSD. The gap between the propellant grain and the PSD is valued according to the process detection data.Figure 4
Simplified physical model of second-pulse combustion chamber.


## 3.2. Calculation Method

The following assumptions are made regarding the operating characteristics of the SRM:(a)
The useful temperature range for the EPDM material is about -55°C~+150°C. When first-pulse grain is burning, high-temperature combustion gas is stationary on the surface of the soft PSD and the heat transfer speed to the soft PSD is not fast. The temperature of the surface of the soft PSD will rise by 80°C, and its overall average temperature will rise by 30°C, hardly affecting its mechanical properties. According to the experimental results, the thermal load is not the cause of failure of the soft PSD. As a result, the temperature rise and ablation of the soft PSD are negligible, and the performance change of the PSD material caused by the temperature change is not considered
(b)
Quasi-static analysis is performed for a given load without considering the inertial effect of structural mass. The effect of strain rate on the mechanical properties of EPDM and propellant grain is not considered too
(c)
The friction between the soft PSD and the structural parts, such as the case and the second-pulsed propellant grain, is ignored
(d)
The pressure of the first-pulse grain burning pressurization is evenly applied to the soft PSDThe internal pressure load is taken as the maximum working pressure of the SRM, which reaches to 9.0 MPa, and a fixed constraint is applied to the front skirt of case end face to eliminate the rigid body displacement, as shown in Figure5.Figure 5
Second-pulsed propellant grain stress.In order to simulate the pressure building process of the SRM internal pressure, the pressure load curve is loaded as shown by the curve change trend in Figure6.Figure 6
Internal pressure curve of SRM.


## 3.3. Material Properties

According to engineering development experience, the internal pressure of the SRM is usually established within 0.3 s, and the propellant grain and thermal insulation layer (including the compartment) can be simulated as a linear elastic material. The performance parameters of each material of the SRM are shown in Table1.Table 1
Performance parameters of the SRM.
Structure
Modulus
Poisson’s ratio
Case
200 GPa
0.3
Second-pulsed propellant grain
5.2 MPa
0.495
Soft PSD
10.0 MPa
0.495


## 3.4. Computing Grid

The geometric model is meshed under ABAQUS/CAE for a total of 60,000 finite element meshes, as shown in Figure7. For volume approximate incompressible material characteristics of the propellant grain and short fiber reinforced EPDM, the hybridization unit option is selected in the unit setting under ABAQUS/CAE. And the Plane Axisymmetric Hybridization unit (CAX4H) is selected to avoid the volume self-locking and finite element analysis by the high Poisson ratio, which is closest to 0.5. Linear Reduced Integral Planar Axisymmetric Unit (CAX4R) is used for the case. Figure 8 is a partial magnification of the mesh.Figure 7
Model overall grid.Figure 8
Local grid.The interface of the case and grain is the bonding condition. The interface of PSD and the grain is the contact in the gap condition. The interface of PSD and the case is the bonding condition.


## 3.5. Result Analysis

After the calculation is completed, the strain in the three directions of the soft PSD is shown in Figure9.Figure 9
PSD strain contours.
(a)
Radial strain
(b)
Axial strain
(c)
Circumferential strainAccording to the calculation results, it can be found that(1)
The radial strain in the middle of the soft PSD is negative, meaning that the central portion is radially compressed. And there is a high tensile strain zone at both ends due to adhesion to the case, but the area is extremely small
(2)
The two segments of the PSD are bound to the case and end face of the grain. The interface of the case and grain is the bonding condition. The axial deformation of the propellant/case combination structure causes the axial strain of the soft PSD. It is worth noting that the main axial strain still occurs at the two ends of the soft PSD bound to the case, and most of the soft PSD has a small axial strain, especially the middle section
(3)
The main strain of the soft PSD is the circumferential strain. The largest part of the circumferential strain is in the axial middle section of the soft partition, reaching a maximum of 21.05%The maximum strain point of the high-strain region of the soft PSD is recorded as point A, as shown in Figure10.Figure 10
The maximum strain point of the soft PSD (point A).Circumferential strain versus time curves of point A, the propellant grain inner surface, and the case of the same axial position as point A are shown in Figure11.Figure 11
Circumferential strain versus time curve of point A, the propellant grain inner surface, and the case.It can be seen from Figure10 that when the first-pulse grain working pressure acts on the soft PSD, the trend of the circumferential strain of the PSD, the propellant grain, and the case are similar to the loading trend of the internal pressure. The development of the circumferential strain can be divided into the following three stages:
(1)
The first stage: the gap between the soft PSD and the propellant grain is quickly filled, and the circumferential strain of the soft PSD is mainly due to the filling gap, and its value accounts for 26.5% of the total strain of the whole process
(2)
The second stage: the propellant grain and the soft PSD as a whole limit radial displacement of the soft PSD after the soft PSD is in contact with the propellant grain. Under further action of the pressure, the whole grain and the case are deformed under pressure, and the circumferential strain is further increased
(3)
The third stage: due to the approximate incompressibility of the propellant grain, the radial deformation of the grain is getting smaller and smaller, and the circumferential strain gradually approaches the maximum value under the limitation of the case


## 4. Analysis of Strain Sensitivity Factors

In order to verify the accuracy of the analytical method results, several FEM calculations were carried out, based on the basic assumptions, loads, and boundaries described above.The strain results of the soft PSD and the inner hole of the propellant grain are shown in Table2, under a different elastic modulus of the case and the gaps between the grain and soft PSD. It is worth noting that when calculating the m of the propellant grain, it is necessary to consider the influence of the deformation of the propellant grain due to the internal pressure. The propellant grain inner diameter after the deformation is equal to the design value of the inner diameter plus the deformation of the propellant grain inner diameter obtained by the FEM calculation. The specific calculation formula is shown in formula (6), where δFEA is the increase in the propellant grain inner diameter.
(6)m=Ra+δFEA.Table 2
Calculation results of circumferential strain.
Input
Structural response
Gap/mm
Case modulus (GPa)
εθ of case
εθ of grain
FEM (%)
Analytical (%)
FEM (%)
Analytical (%)
1
1.5
150
21.67
23.71
15.99
21.29
2
5.0
150
25.85
28.29
14.89
19.74
3
1.5
200
21.05
23.21
15.46
20.79
4
5.0
200
25.26
27.81
14.39
19.27
5
1.5
220
20.82
23.03
15.27
20.61
6
5.0
220
25.05
27.64
14.21
19.09
Note: the circumferential strain of the soft PSD takes the maximum value at the end of the loading, and the circumferential strain of the case takes at the same axial position as the maximum circumferential strain of the soft PSD.It can be found from Table2 that when the gap increases, the circumferential strain of the soft PSD increases, and when the stiffness of the case increases, the circumferential strain of the soft PSD decreases. And the effect of the gap between the soft PSD and the propellant grain on the circumferential strain of the soft PSD is obvious.In order to verify the accuracy of the analytical method, another new example is carried out. The results of the calculation and comparison are shown in Table3.Table 3
Calculation results of circumferential strain of another example (a double-pulse SRM with twice the diameter;R/h stays the same).
Input
Structural response
Gap (mm)
Case modulus (GPa)
εθ of case
εθ of grain
FEM (%)
Analytical (%)
FEM (%)
Analytical (%)
1
1.5
150
21.99
24.05
16.25
21.60
2
5.0
150
26.21
28.67
15.14
20.04
3
1.5
200
21.36
23.54
15.71
21.10
4
5.0
200
25.61
28.19
14.63
19.56
5
1.5
220
21.13
23.36
15.52
20.92
6
5.0
220
25.40
28.02
14.45
19.38Comparing the analytical method results with the FEM results, we can find that the results are basically the same, and the analytical values are slightly higher than the FEM values. This is because the propellant grain section strain is treated as a plane strain, ignoring the axial strain caused by the compression of the propellant grain, resulting in a small calculated value of the axial stress, and according to the three-dimensional stress-strain relationship in the elastic mechanics, the calculated value of the axial stress is too small, resulting in a large calculation of the circumferential strain, during the derivation of formula (4).Considering the strain of the soft PSD alone, the calculated value of the formula is about 10% higher than the FEM value, which is more conservative. Therefore, the strain of the soft PSD can be estimated by formula (4) to determine the rationality and feasibility of the design in the engineering design. For further in-depth evaluations, further FEM is used for more detailed calculations.


## 5. Experimental Study on Failure Mechanism of Rubber Materials for Soft PSD

The soft PSD is generally made of special rubber, and it is mainly composed of two parts: a matrix EPDM and a chopped fiber filler. The rubber material not only has high-temperature resistance, low density, and high elongation but it also has a certain structural strength. As mentioned above, the soft PSD is deformed under the action of high-temperature and high-pressure gas; however, it is far from reaching the material’s elongation, and the soft PSD is ineffective, causing the two-stage pulse to work at the same time, and the case is detonated and exploded [14].In order to further explore the failure mechanism of the soft PSD material, a micro-CT test of in situ stretching is carried out to study the correlation between the mesodamage degree of the soft PSD material and the strain. The change in the meso-level of the test piece is characterized by porosity. The reconstruction and variation of the two-dimensional porosity image is shown in Figure12.Figure 12
Porosity of soft PSD material varies with elongation.It can be found from Figure12 that when the elongation is between 0 and 10%, the porosity remains substantially unchanged, indicating that no debonding occurs between the fibers and the filler; when the elongation is between 10% and 30%, the porosity increases linearly, and the fiber and the solid filler have a small amount of debonding with the matrix; when the elongation is greater than 30%, the porosity increases rapidly; when the elongation is greater than 50%, the debonding region has already produced a relatively large and through damage.The main reasons why the soft PSD fails at lower elongation are as follows: when the circumferential strain of the soft PSD material is increased to a certain extent, debonding between the chopped fibers and the rubber matrix begins to occur. As the strain continues to increase, the fiber is peeled off and the porosity increases until the through damage occurs, and the soft PSD fails.


## 6. Conclusions

(1)
The soft PSD strain is mainly caused by circumferential deformation by the load of the first-pulse grain internal pressure. The largest part of the circumferential strain is in the axial middle section of the soft PSD. It is considered that the circumferential strain is the main cause of the failure of the soft PSD
(2)
The circumferential strain of the soft PSD is related to the gap between the soft PSD and the propellant grain and the stiffness of the case. In the SRM manufacturing process, the shrinkage deformation of the soft PSD material and the propellant grain should be considered, and the gap between the propellant grain and the soft PSD should be as small as possible after the propellant grain is cured. In addition, under the same conditions, if the case has high stiffness, it is advantageous for the integrity of the PSD
(3)
In this paper, the deformation process of the soft PSD is simplified to the two-dimensional plane strain problem, and the calculation method to compute the internal surface strain of the soft PSD is obtained. The calculation results of this method are in good agreement with the FEM results and can be used for the strain estimation of the soft PSD to preliminarily determine the rationality and feasibility of the design scheme
(4)
In situ tensile/micro-CT scanning experiments are carried out on the soft PSD material to study the variation of porosity and elongation. When the material elongation is large, the micro-interface debonding rapidly expands into a penetrating damage, and the soft PSD structure fails. This phenomenon can better explain why the SRM disintegrates when the soft PSD material is far below the elongation at break, under the first-pulse grain operation



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