Analysis of the mechanical response of solid rocket engine propellant under acceleration shock

The increased-range solid rocket engine of new ammunition often needs to withstand extremely high acceleration overload, leading to damage to the propellant’s structural integrity. To address this issue, this paper constructs a nonlinear viscoelastic constitutive model for the CMDB propellant by using low- and high-strain rate mechanical property tests. Combined with the secondary development of explicit dynamics and numerical simulation technology, the mechanical response of a certain type of solid rocket propellant under different acceleration impacts is analyzed. The results show that under acceleration impact loads, the propellant will compress, rebound, and recover over time, continuously cycling. Its axial displacement, maximum equivalent stress, and maximum equivalent strain exhibit irregular sinusoidal wave-like periodic cycles. Looking at the time when the peaks appear, the time when the maximum equivalent stress appears always lags behind the time when the maximum axial displacement peak appears. Due to the viscous effect of the viscoelastic material of the propellant, the time when the equivalent strain peak appears will lag behind the equivalent stress. Because of the material’s damping effect, both the peak values of the maximum equivalent stress and equivalent strain decrease over time. Under continuous high acceleration impact loads, this viscous damping phenomenon continuously diminishes, and the peak value of the propellant’s axial displacement gradually increases.


Introduction
New ammunition such as the United States STAFF and ERGM, Russia's (AT-10) 9M117, and AK-130 [1][2][3] , often requires their solid rocket engines to withstand nearly tens of thousands of overloads in instantaneous axial acceleration overloads during the launch process.This leads to excessive stress concentration at the bottom of the solid propellant charge [4] .As a typical solid particulate polymer, when the solid propellant is subjected to such a violent impact load, it may undergo processes like dehydration, particle fragmentation, and matrix tearing internally.This results in the formation of micro-cracks, micropores, and other damages.These damages, after continuous accumulation, can lead to irreversible macro-level destruction [5][6][7] .If the propellant charge deforms or even breaks, it will alter the predetermined thrust plan, cause severe degradation in the performance of the solid rocket engine, and lead to engine burn-through or even disintegration and explosion during operation [8] .
Guo et al. [9] used a static linear viscoelastic constitutive model to analyze the structural integrity of the propellant in a booster engine of a certain bottom-row rocket with a composite extended range, under 2 an axial overload of 13, 412.2 g.Sui et al. [10] adopted a static linear viscoelastic constitutive model to study the structural strength of the propellant in a gun-launched missile's booster engine under an axial overload of 6, 000 g.They provided the rules on how Poisson's ratio and thickness of the liner affect the stress distribution in the propellant.Wei and Wang [11] used a static linear viscoelastic model to research the influence of cross-sectional shape, aspect ratio, and modulus on deformation under high overload conditions for a certain gun-launched missile.They found that the maximum axial stress and strain occurred in the contact area between the propellant and the engine, with significant circumferential strain in the propellant.Wang et al. [12] employed a nonlinear viscoelastic constitutive model with cumulative damage to analyze the deformation and mechanical response of the propellant in a shell by using a modified double-base propellant, under a launch overload of 13, 900 g.
Scholars often adopt propellant constitutive models represented by the Prony series, which are derived from static relaxation tests.However, the mechanical properties of polymers differ significantly under impact loads and quasi-static conditions.Analyzing the structural integrity of propellants under overload conditions by using linear viscoelastic constitutive models derived from quasi-static relaxation tests will inevitably lead to significant computational errors.Using the Prony series representation of linear viscoelastic constitutive models under dynamic impact loads poses challenges due to its high order and the extreme difficulty in data fitting [13] .Furthermore, propellant materials, as typical high-polymer materials, exhibit pronounced nonlinear viscoelastic mechanical characteristics.The ZWT nonlinear viscoelastic constitutive model can span 8 orders of magnitude in strain rate to describe the dynamic mechanical properties of high-polymer materials within the viscoelastic deformation range, with strain rates between 10 -4 to 10 3 s -1 , and is easier for parameter fitting [14] .Therefore, this paper adopts the Zhu-Wang-Tang nonlinear viscoelastic constitutive model and combines it with the secondary development technology of finite element software to analyze the dynamic mechanical response of propellants under high acceleration impacts.

ZWT nonlinear viscoelastic constitutive model
The ZWT constitutive model is composed of a nonlinear elastic body and two Maxwell elements in parallel.One of the Maxwell elements is used to describe the viscoelastic response under low strain rates, and the other is used to describe the viscoelastic response under high strain rates.Its constitutive equation can be seen in Equation (1).
Where , , and are nonlinear elastic constants; and are the relaxation times under low and high strain rates respectively; and are the elastic moduli under low and high strain rates respectively.The three-dimensional form can be seen in Equation (2).
Where ij S is the second-order Kirchhoff stress tensor, ij E is the Green strain tensor, and   A is the three-dimensional isotropic elasticity matrix.Its three-dimensional incremental form can be seen in Equation (3).

Compression test with low strain rate
The test specimens are modified double-base propellant cylindrical samples designed according to GB/T 7314-2005, with a diameter of 10 mm and a gauge length of 15 mm.Tests are conducted by using the IOP Publishing doi:10.1088/1742-6596/2764/1/0120983 universal material testing machine, as shown in Figure 1.The compression test curve with a low strain rate is shown in Figure 2.

Compression test with high strain rate
For the compression test with a high strain rate, a split Hopkinson pressure bar (SHPB) apparatus is used.The actual setup of the experimental device is shown in Figure 3.In the test, cylindrical samples are used with a dimension of 10 mm in diameter and a gauge length of 5 mm.The stress-strain curve under high strain rates is shown in Figure 4.

Parameter fitting of the constitutive model
For the CMDB propellant, since the ZWT constitutive model is only applicable to its mechanical performance in the viscoelastic segment without damage before the yield point, this paper uniformly selects the strain within approximately 3.5% to fit the curve.Under a constant strain rate, the integral Zhu-Wang-Tang constitutive model can be rewritten in the form of Equation ( 4).
To validate the obtained constitutive model, an additional set of tests is added for both low and high strain rates.For the low strain rate, 5×10 -1 s -1 is chosen, and for the high strain rate, 2, 100 s -1 is chosen.The comparison between the validation test curves and the constitutive model is shown in Figure 5.Both fit well, demonstrating that the parameters of the constitutive model obtained in this experiment are applicable.The specific fitting parameters and their values in the ZWT equation are shown in Table 1.

User-defined material subroutine
Due to the absence of the ZWT constitutive model in the Abaqus software, it is necessary to carry out secondary development of the software.Using the dynamic explicit method requires deriving the constitutive model in incremental form and writing it into VUMAT by using the Fortran language.This process involves assigning values to the material property array, calculating stress increments and updating stresses, and looping through the element nodes.The subroutine is used to simulate the condition at a low strain rate of 5×10 -1 s -1 from the test to verify the correctness of VUMAT.The output stress-strain curve is shown in Figure 6, and the results are relatively consistent.

Finite element simulation under acceleration impact
As shown in Figure 7, the engine propellant is cylindrical with a length-to-diameter ratio of 400:110.The casing has a thickness of 3 mm, and there is a 2 mm gap between the propellant and the inner wall of the casing.The displacement of the bottom of the casing is restricted, and an acceleration load is applied to the model in the axial direction.During the modeling process, the model is simplified, assuming a uniform thickness of the casing and neglecting complex structures like the nozzle.Since the focus is on the propellant column and the deformation of the casing is much smaller than that of the propellant column, to reduce computational cost, the casing is assumed to be rigid.The model mesh consists of eight-node linear hexahedral C3D8R elements with reduced integration and hourglass control.
The smallest mesh size for the propellant column is 6 mm, with a total of 20, 100 elements.The material properties are shown in Table 2.

Result analysis
Taking an overload of 4, 000 g as an example, the mechanical response of a single propellant column is analyzed.As shown in Figure 8, the axial displacement, Mises equivalent stress, and strain of nodes at different positions on the central axial path of the propellant column at different time points are extracted and plotted as contour maps to observe the transmission of stress waves within the propellant column and its deformation process.Within 10 ms, there are five major peaks in the contour lines of axial displacement.The propellant column continuously compresses, rebounds, and recovers, cycling five times.We take the first three cycles as examples: The first axial displacement peak is 9.595 mm, occurring at 0.8 ms; The second is 9.704 mm, at 2.75 ms; The third is 10.411 mm, at 4.6 ms, with the peak positions all occurring at the top of the propellant Correspondingly, there are also five major peaks in the Mises equivalent stress contour map.The first peak is 33.323MPa at 1 ms; The second is 31.851MPa at 2.8 ms; The third is 31.27MPa at 4.65 ms, with peak stresses all appearing at the bottom of the propellant column.Similarly, there are five peaks in the strain contour, with the first strain peak at 5.78% at 1.05 ms, the second at 5.65% at 2.85 ms, and the third at 5.56% at 4.8 ms, with the peak positions all occurring at the bottom of the propellant column.
Looking at the occurrence time of the peaks, it can be seen that the stress peaks lag behind the axial displacement peaks.This implies that as the propellant column compresses severely, the equivalent stress at the bottom of the propellant column gradually increases.The loading acceleration load excitation can be equivalent to displacement excitation, with displacement acting as an external stimulus that triggers the propellant column's response.Hence, there is always a phase difference in time between stress peaks and displacement peaks.The peak time of strain also lags behind that of stress, indicating that strain changes lag behind stress changes, which is a key feature of dynamic viscoelasticity.From a peak value perspective, both stress and strain peaks show a gradual decay trend, which is caused by the viscous damping of viscoelastic materials, in line with their characteristics.However, the axial displacement shows a trend of gradual increase, indicating that as time progresses, the maximum compression amount of the propellant column is continuously growing, and meaning that it becomes more compressible.Under continuous impact loads at high overloads, the viscous damping effect of the viscoelastic material is gradually decaying.A cross-section of the propellant column reveals the cloud diagrams of axial displacement, equivalent stress, and strain, as shown in Figure 9, providing a further description of the propellant column's mechanical response under high overload impact conditions.Taking the cloud diagram of the first cycle as an example, it can be seen that all three diagrams present a consistent pattern.In the first stage, axial displacement, equivalent stress, and strain all show an "increasing" state with clear and uniform stratification from the inside of the propellant column.The second stage is similar to the first, with the stratification still evident, but all three show a "decreasing" state.The third stage presents a "shift" state, where the propellant column rebounds to a state close to its pre-compression condition, the stress rapidly unloads, and the maximum axial displacement, maximum equivalent stress, and maximum strain start to appear at the top of the propellant column and gradually shift to the bottom.

Conclusion
This study conducted low and high-strain rate compression tests on modified double-base propellants, constructed the ZWT nonlinear viscoelastic constitutive model for the propellant, and wrote a Vumat subroutine in Fortran language suitable for this constitutive model, thereby achieving secondary development for the Abaqus finite element software.Using finite element software, we analyzed the mechanical response of the propellant under different axial acceleration impact loads.The main conclusions drawn are as follows: (1) Under acceleration impact loads, the deformation process of the propellant can be divided into three cyclic phases: compression, rebound, and recovery.Correspondingly, its axial displacement, equivalent stress, and equivalent strain also exhibit irregular sinusoidal cyclical patterns.
(2) The peak time of the maximum equivalent stress for the propellant always lags behind that of the maximum equivalent strain.The peak time of the maximum equivalent stress consistently lags behind that of the maximum axial displacement.
(3) The peak values of the equivalent stress and equivalent strain of the propellant gradually decay over time.The peak of axial displacement increases progressively with time.

Figure 1 .
Figure 1.Universal material testing machine.Figure 2. Low strain rate test curve.

Figure 2 .
Figure 1.Universal material testing machine.Figure 2. Low strain rate test curve.

Figure 3 .
Figure 3. Split Hopkinson pressure bar.Figure 4. High strain rate test curves of stress-strain.

Figure 4 .
Figure 3. Split Hopkinson pressure bar.Figure 4. High strain rate test curves of stress-strain.

Figure 7 .
Figure 7. Schematic diagram of the single-segment model.

Figure 8 (Figure 8 .
Figure 8. Contour plot of propellant with single stage at 4, 000 g. From the perspective of axial displacement, the deformation of the propellant column can be roughly divided into three stages that continuously cycle.The first stage: The propellant column is severely compressed and the axial compression displacement at the top reaches its maximum; The second stage: The propellant column begins to rebound; The third stage: The propellant column rapidly recovers to a state close to its initial condition.

Figure 9 .
Figure 9. Distribution cloud of propellant with single stage at 4, 000 g.

Table 2 .
Material properties of parts.