Study of rupture behavior of 40Cr ring-scored bursting diaphragms

The rupture diaphragm is a key part of pressure control in the launcher and the reliability of the rupture pressure is directly related to the internal ballistic performance. In order to study the rupture behavior of the 40Cr annular scored diaphragm under internal ballistic conditions and the remaining thickness of the diaphragm’s scoring, the stress-strain curve of the diaphragm is obtained by fitting the internal ballistic loading curve and simulation, the relationship between the rupture pressure and the remaining thickness of the scoring is established and the accuracy of the simulation is verified by the blasting test.


Introduction
Bursting diaphragms are widely used in the field of artillery firing and various mechanical power systems with the function of regulating, opening, sealing, etc. Bursting diaphragms have different scoring forms, such as petal-shaped scoring, ring-shaped scoring, C-shaped scoring, etc., which are used in different scenarios with different material selections [1][2][3] .
Bursting diaphragms have also been studied by many national and international researchers.On the domestic front, Jiang et al. [4] proposed a two-damage-variable GTN material constitutive model, which can be widely used to simulate the fracture process of ductile metals under the mixed loading mode of tension-shear.Nie et al. [5] analyzed diaphragms with various thicknesses and groove lengths using the control variable method and obtained the rules for variations in diaphragm burst pressure and effective burst time.Internationally, Colombo et al. [6] investigated the rupture process of a scored thin plate using numerical simulation and the issues of the choice of numerical model and the validity of the nonlinear response are discussed in detail.Lee et al. [7] investigated the characteristics of hydrogen spontaneous combustion in tubes at different bursting diaphragm failure pressures using numerical simulation and obtained the conclusion that the spontaneous combustion characteristics of hydrogen have a strong correlation with the rupture pressure.In summary, although researchers in this country and abroad have carried out comprehensive studies on bursting diaphragms, there is still a lack of studies on the rupture behavior of diaphragms under ultra-high pressure conditions, especially under internal ballistic conditions.
In this paper, a ring-shaped scored diaphragm is used to carry out simulation calculations to analyze the plastic deformation behavior of the diaphragm as well as the evolution of the cracking process and to compare the breakage pressure law of the diaphragm with different depths of the scoring, with a view to providing a reference for the selection of the diaphragm and the design of the diaphragm.

Bursting diaphragm structure and material modeling
In a previous series of experiments, a new form of scoring had to be selected because the diaphragm with petal-shaped scoring did not work well, as shown in Figure 1 where the diaphragm opened prematurely.It did not reach the desired pressure and broke from the root due to high pressure.As the launch chamber pressure requirements are 150 MPa -200 MPa, belonging to the ultra-high-pressure launch, taking into account the need for materials with good mechanical properties and impact toughness, the diaphragm material selection is for the 40Cr modulated steel.In this paper, LS-DYNA is used to simulate the diaphragm and this module can solve the dynamic deformation problems in the field of large deformation such as metal forming, high-speed collision, high-speed impact, etc.The structure of the diaphragm is shown in Figure 2. The pressure loading rate has a large influence on the rupture response.Therefore, a more accurate diaphragm pressure loading curve is obtained by fitting the internal ballistic pressure data accumulated in the previous experiments, as shown in Figure 3, from which can be seen that the loading rate of diaphragm pressure is gradually increasing.Therefore, the strain rate of the corresponding diaphragm material is also gradually increasing. [8]gure 3. Internal ballistic pressure fitting data.
In order to more accurately describe the response of the diaphragm to dynamic loading, this paper adopts the Johnson-Cook material strength model, which is suitable for describing the mechanical behavior of the material at large strains, a wide range of strain rates and a wide range of temperatures. [9]The model is specifically expressed as: where A , B and C are for the material quasi-static yield strength, strain hardening coefficient and strain rate coefficient respectively; n is for the strain hardening index; m is for the temperature softening index; * δ% is for the dimensionless strain rate, the expression is * 0 p δ δδ < % %% , p δ % is for the current effective strain rate; 0 δ% is for the reference strain rate.* T is the temperature scale and the expression is * ( ) () , where T , r T and m T are the ambient temperature, the reference temperature and the melting point respectively.The material-specific parameters are shown in Table 1.Table 1.Partial material parameters [10] .

Analysis of diaphragm rupture behavior
In this paper, a pressure diaphragm with a remaining thickness of 6 mm is selected to analyze the rupture behavior of the diaphragm from the stress distribution at different moments, as shown in Figure 4.At the initial stage of pressure rise, due to the complete constraint of the outer ring of the diaphragm, the maximum stress is mainly distributed in the constraint outer edge and the center of the diaphragm, as shown in Figure 4 (a).With the accelerated growth of pressure, the diaphragm stress gradually spread to the indentation, as shown in Figure 4 (b).At this time, the diaphragm has not yet occurred more obvious plastic deformation, showing a certain strain rate enhancement, which verifies that the 40Cr on the strain rate has a certain sensitivity to the conclusion. [11]When the pressure is further increased, the maximum stress is concentrated in the indentation and the inner ring of the diaphragm, the diaphragm has undergone large plastic deformation, the root of the indentation begins to fracture and gradually expands outward until the entire inner ring of the diaphragm is completely fractured, the indentation of the diaphragm at the diaphragm stress is completely fractured.The diaphragm stress is released after the indentation is completely broken, as shown in Figures 4 (c) and 4 (d).

Pressure analysis of membrane rupture with different residual thicknesses
Taking the remaining thickness of the 6 mm diaphragm as an example, the diaphragm is in its pressure rupture process.After a significant elastic and plastic deformation stage, it is in 7.97×10-3 s.The equivalent force at the diaphragm indentation is 1367 Mpa, it begins to rupture in a localized instability, and develops rapidly, as shown in Figure 5.At this time, the maximum strain of the diaphragm is 0.0216.For the sake of convenience in the calculation, the rupture strain is taken as 0.02.In order to study the effect of different residual thicknesses of the diaphragm on the rupture pressure, the residual thickness is taken as a single variable, five groups of diaphragms are taken for simulation and the strain-pressure curves of the whole process of rupture of diaphragms with different indentation residual thicknesses are obtained, as shown in Figure 6.Under the simulated real internal ballistic loading curve, the strain value changes abruptly at the moment of rupture, which is corroborated by Figure 5.The pressure value corresponding to the curve's abrupt change point is the rupture pressure.The obtained rupture pressure of each remaining thickness is polynomially fitted, as shown in Figure 7.It can be seen that, in the simulation of the internal ballistic loading process, the remaining thickness of the diaphragm and the rupture pressure do not exhibit a linear relationship.However, with an increase in the remaining thickness of the diaphragm, the growth of the rupture pressure gradually slows down.This law holds a significant reference for selecting the appropriate thickness of the diaphragm.In the launch device, the required pressure control range is 150 Mpa -200 Mpa.Therefore, based on the fitting curve formula interpolation calculation, the remaining thickness of the diaphragm should be selected within the range of 3.5 mm -6 mm.

Experimental verification
In order to verify the accuracy of the simulation calculations, five repetitive bursting experiments were carried out by taking the diaphragm with a remaining thickness of 6 mm.The data obtained were compared with the simulated bursting pressure, as shown in Figure 8.The maximum bursting pressure obtained from the experiments was 208.17 MPa and the minimum bursting pressure was 185.73 MPa.The errors compared with the simulated data were 3.19% and 7.93% respectively.Considering the microscopic defects of metal materials, the accuracy of processing and testing equipment and the deviation of the loading curve, the errors are within the acceptable range, which verifies the accuracy of the simulation calculation.The experimental setup is shown in Figure 9 and the ruptured diaphragm is shown in Figure 10.

Conclusion
The three conclusions obtained from this paper are as follows.
(1) The pressure loading rate and the residual thickness of the indentation have a large influence on the rupture pressure of the diaphragm, when the rupture pressure is 150 MPa to 200 MPa, the residual thickness ranges from 3.5 mm to 6 mm.
(2) Under the simulated internal ballistic loading curve, the relationship between the remaining thickness of the diaphragm and the increase in its burst pressure is not linear.However, the increase in burst pressure gradually slows down as the remaining thickness of the diaphragm increases.
(3) Experimental results agree with numerical simulations of 5 replicate burst tests on a 6 mm r esidual thickness diaphragm.

Figure 2 .
Figure 2. Bursting Diaphragm Structure.In this paper, LS-DYNA is used to simulate the diaphragm and this module can solve the dynamic deformation problems in the field of large deformation such as metal forming, high-speed collision, high-speed impact, etc.The structure of the diaphragm is shown in Figure2.The pressure loading rate has a large influence on the rupture response.Therefore, a more accurate diaphragm pressure loading curve is obtained by fitting the internal ballistic pressure data accumulated in the previous experiments, as shown in Figure3, from which can be seen that the loading rate of diaphragm pressure is gradually increasing.Therefore, the strain rate of the corresponding diaphragm material is also gradually increasing.[8]

Figure 4 .
Figure 4. Diaphragm Rupture Evolution Process.In order to study the effect of different residual thicknesses of the diaphragm on the rupture pressure, the residual thickness is taken as a single variable, five groups of diaphragms are taken for simulation and the strain-pressure curves of the whole process of rupture of diaphragms with different indentation residual thicknesses are obtained, as shown in Figure6.Under the simulated real internal ballistic loading curve, the strain value changes abruptly at the moment of rupture, which is corroborated by Figure5.The pressure value corresponding to the curve's abrupt change point is the rupture pressure.The obtained rupture pressure of each remaining thickness is polynomially fitted, as shown in Figure7.It can be seen that, in the simulation of the internal ballistic loading process, the remaining thickness of the diaphragm and the rupture pressure do not exhibit a linear relationship.However, with an increase in the remaining thickness of the diaphragm, the growth of the rupture pressure gradually slows down.This law holds a significant reference for selecting the appropriate thickness of the diaphragm.In the launch device, the required pressure control range is 150 Mpa -200 Mpa.Therefore, based on the fitting curve formula interpolation calculation, the remaining thickness of the diaphragm should be selected within the range of 3.5 mm -6 mm.

Figure 5 .
Figure 5.The equivalent stress-time curve of the score.