An analysis method of energy-focused jet forming and stability

Abstract

The present invention provides a method for analyzing the forming and stability of a shaped charge jet, which first performs a jet forming analysis on a shaped charge jet formed by the detonation wave generated by the detonation of explosives crushing a charge liner, so as to establish the relationship between the liner structural parameters and the jet diameter, temperature and strain rate; then performs a jet stability analysis on the shaped charge jet to determine the jet instability criterion; and finally obtains the jet break time. The present invention establishes the relationship between the shaped charge liner structural parameters and the jet diameter, temperature and strain rate, proposes a jet stability criterion and a jet break time evaluation method that comprehensively considers strain hardening, strain rate effect, thermal effect and necking effect, constructs an analysis method for the effect of liner structural parameters on jet stability, and effectively analyzes jet stability.

Description

Technical Field

[0001] The invention relates to the technical field of shaped charge, and in particular to an analysis method for shaped charge jet forming and stability. Background Art

[0002] Shaped charge technology is widely used in modern weapon systems, and its core lies in the design of the liner. The liner effectively converts the energy generated by the explosion into a high-speed jet through a specific geometric shape and material properties. After the jet is formed, it is affected by many factors, including strain hardening, strain rate effect, thermal effect and necking effect. When the jet encounters unstable factors, it may break, which will cause the effectiveness of the jet to drop significantly, or even lose its ability to strike the target.

[0003] Therefore, in-depth research on the jet stability problem is crucial for optimizing the design of shaped charge. In this context, it is particularly urgent to establish a jet instability criterion. This criterion can not only provide a theoretical basis for the prediction of jet fracture, but also guide the structural optimization of the liner to achieve better jet performance. Summary of the invention

[0004] In view of the technical problems existing in the above-mentioned background technology, the present invention proposes an analysis method for the forming and stability of a shaped charge jet. The method has a reasonable conception, establishes the relationship between the structural parameters of the shaped charge liner and the jet diameter and strain rate, proposes a jet stability criterion and a jet breakage time evaluation method that comprehensively considers strain hardening, strain rate effect, thermal effect and necking effect, constructs an analysis method for the effect of the liner structural parameters on the jet stability, and effectively solves the problem of jet stability.

[0005] In order to solve the above technical problems, the present invention provides a method for analyzing the forming and stability of a shaped charge jet, which first performs a jet forming analysis on the shaped charge jet formed by the detonation wave generated by the detonation of the explosive crushing the charge liner, so as to establish the relationship between the structural parameters of the charge liner and the jet diameter, temperature and strain rate; then performs a jet stability analysis on the shaped charge jet to determine the jet instability criterion; finally, the jet breakup time is obtained.

[0006] The analysis method of energy-focused jet forming and stability, wherein the jet forming analysis comprises the following steps:

[0007] 1.1) Solve for the velocity of the jet

[0008] First, the jet velocity is obtained according to the steady model and pestle speed :

[0009] ;

[0010] ;

[0011] ;

[0012] In the above formulas (1)-(3), is the crush angle, is the semi-cone angle of the liner, is the crushing speed, Indicates the detonation velocity of the explosive;

[0013] Next, assuming that the scattering direction of the explosive products is consistent with the normal direction of the charge surface, the effective amount of explosives is determined according to the effective charge theory; assuming that the velocity of the detonation gas on the metal surface is linearly distributed, the crushing velocity is obtained. :

[0014] ;

[0015] In the above formula (4), is the internal energy of the explosive, For metal quality, is the mass of explosives;

[0016] 1.2) Solve the time of jet forming

[0017] The time it takes for the detonation wave to travel from the top to the bottom of the liner is recorded as :

[0018] ;

[0019] In the above formula (5), Indicates the height of the liner;

[0020] Assume that after the detonation wave contacts the top of the liner, the top of the liner converges toward the axis, and the convergence time is :

[0021] ;

[0022] ;

[0023] In the above formulas (6)-(7), is the distance from the top of the liner to the axis. is the inner radius of the liner top, is the deflection angle of the liner, is the semi-cone angle of the liner;

[0024] After the detonation, the bottom of the liner converges toward the axis, and the convergence time is :

[0025] ;

[0026] In the above formula (8), is the distance from the bottom of the liner to the axis. is the inner radius of the liner bottom;

[0027] When the bottom of the liner converges to the axis, , which indicates that the jet has been fully formed; therefore, the time from the top liner forming the jet to the complete formation is determined :

[0028] ;

[0029] 1.3) Solve for the diameter, length, and strain rate of the jet

[0030] The moment when the detonation wave reaches the bottom of the liner, that is, the moment when the detonation ends, is 0. Assuming that the jet is stretched uniformly after forming, the length of the jet at this time is for:

[0031] ;

[0032] After the jet is fully formed, the length of the jet is :

[0033] ;

[0034] Determining the strain of the jet , strain rate and temperature :

[0035] ;

[0036] ;

[0037] ;

[0038] In the above formulas (12)-(14), represents the initial area of ​​the jet, represents the area of ​​the jet at any time, is the work-to-heat conversion coefficient, is the jet stress, is the density of the jet, is the specific heat capacity of the jet, It is room temperature.

[0039] The method for analyzing the forming and stability of the energy-focused jet, wherein the stability analysis is to perform jet stability analysis starting from the moment of jet forming and assuming that the jet is subject to a small disturbance; specifically comprises the following steps:

[0040] 2.1) Constructing a stretching model of the jet

[0041] Plastic work The heat it generates The relationship between them is as follows:

[0042] ;

[0043] In formula (15), , the remaining part Still lurking in the metal;

[0044] Since elastic deformation energy is much smaller than plastic deformation energy, the former is neglected and the plastic work is :

[0045] ;

[0046] In the above formula (16), is stress, It is strain;

[0047] Since the free flight time of the jet is short and heat conduction is negligible, the work-heat conversion equation is:

[0048] ;

[0049] In the above formula (17), is the temperature change of the jet, is the density of the jet, is time; the jet is cylindrical, the velocity is distributed along the axial direction, the head has a fast velocity, and the tail has a slow velocity. The jet is simplified as a cylinder with a fixed support at the left end; the right end of the jet is a free end, with a velocity Right motion, initial cross-sectional area , the initial length is , thus determining:

[0050] Initial strain rate: ;

[0051] Displacement: ;

[0052] Cross-sectional area: ;

[0053] strain: ;

[0054] Due to the axial length of the jet Diameter, axial stress on the same section , axial displacement ,

[0055] Axial speed is the spatial coordinate and time Function, with coordinates The equation of motion of the jet is determined as follows:

[0056] ;

[0057] During the stretching motion, the jet is approximated as an incompressible fluid;

[0058] From the law of conservation of mass:

[0059] ;

[0060] ;

[0061] ;

[0062] In the above formulas (23)-(25), is the jet density, is the cross-sectional area, is the initial density, is the initial cross-sectional area of ​​the undisturbed jet, is the space coordinate of the Euler coordinate system, is the Lagrangian coordinate;

[0063] Combining equation (22) with equation (25) we can obtain:

[0064] ;

[0065] Convert the above formula (26) to Taking partial derivatives we get:

[0066] ;

[0067] When the jet surface is disturbed, the stress distribution of the jet cross section becomes three-dimensional, and the average axial stress is equivalent to:

[0068] ;

[0069] In the above formula (28), is the stress in the ideal tensile state, is the radius of curvature at the neck, is the radius of the neck;

[0070] When the disturbance of the jet is small, the above equation (28) is equivalent to:

[0071] ;

[0072] In formula (29), represents the radius of the jet at any position;

[0073] Simplifying the above formula (29) we can get The expression is:

[0074] ;

[0075] From this we get:

[0076] ;

[0077] In formula (31), ;

[0078] Assume that the constitutive equation of the material is:

[0079] ;

[0080] In formula (32), ;

[0081] Finally, the governing equation of the jet is obtained:

[0082] ;

[0083] 2.2) Setting the perturbation of the jet

[0084] ;

[0085] In formula (34), is the disturbance growth rate of the jet, is the disturbance wave number of the jet;

[0086] It is thus determined that:

[0087] ;

[0088] ;

[0089] ;

[0090] Assumptions ,but:

[0091] ;

[0092] 2.3) Superimpose the disturbance on the jet control equation (32)

[0093] Assumptions is the solution of equation (33), and the perturbation is superimposed on this solution to obtain:

[0094] ;

[0095] The spectral equation is obtained as follows:

[0096] ;

[0097] 2.4) Establishing the instability criterion of jet

[0098] First, define the following dimensionless numbers: ;

[0099] Then, the above equation (40) is dimensionless to obtain equation (41):

[0100] ;

[0101] According to the stability judgment rule, the instability condition can be obtained by analyzing equation (40):

[0102] ;

[0103] According to the properties of the inequality, inequality (42) can be solved to obtain:

[0104] ;

[0105] definition , according to the order of magnitude estimate, Simplified, ;

[0106] Finally, the instability criterion of the jet is determined:

[0107] ;

[0108] 2.5) Establishing the break time of the jet

[0109] when , corresponding to is the maximum value, that is, the fastest growth rate of disturbance, which must satisfy:

[0110] ;

[0111] From this we get:

[0112] ;

[0113] ;

[0114] The simultaneous equations (41) and (47) yield:

[0115] ;

[0116] ;

[0117] From this, the breakup time of the jet can be obtained:

[0118] ;

[0119] 2.6) Establishing the instability criterion and break time of the jet based on the Johnson-Cook model

[0120] When the constitutive model of the liner material uses the Johnson-Cook constitutive model as the thermo-viscoplastic constitutive model, we have:

[0121] ;

[0122] In the above formula (51), is the static yield strength, is the hardening modulus, For strain, is the hardening index, is the strain rate constant, is the jet temperature, is room temperature, is the melting point of the jet, is the thermal softening coefficient;

[0123] The jet instability criterion is obtained by the following formula (52):

[0124] ;

[0125] The break time of the jet is obtained by the following formula (53):

[0126] .

[0127] By adopting the above technical solution, the present invention has the following beneficial effects:

[0128] The analysis method for the forming and stability of a shaped charge jet of the present invention is reasonably conceived, and the relationship between the structural parameters of the shaped charge liner and the jet diameter and strain rate is established. A jet stability criterion and a jet breakage time evaluation method that comprehensively considers strain hardening, strain rate effect, thermal effect and necking effect are proposed, and an analysis method for the effect of the liner structural parameters on the jet stability is constructed, which effectively solves the problem of jet stability.

[0129] The present invention mainly realizes the precise control of the stability and forming process of the shaped charge jet. By establishing the relationship between the structural parameters of the liner and the jet diameter and strain rate, the morphology and stability of the jet can be effectively predicted and optimized, avoiding the jet instability problem caused by unreasonable structure in the traditional method. In addition, by proposing a jet stability criterion that comprehensively considers strain hardening, strain rate effect, thermal effect and necking effect, it helps to improve the jet's ability to resist disturbance during high-speed formation.

[0130] Compared with the existing technology, the present invention has significant advantages. Traditional methods mostly rely on experience or single stability analysis, while the present invention comprehensively considers multiple physical effects and establishes a more scientific and accurate evaluation model, which can effectively predict the stability and fracture behavior of the jet under a variety of complex working conditions. This innovation provides a more efficient and reliable design and optimization path for the energy-focused jet technology, greatly improving the prediction accuracy and operability of the jet stability.

[0131] The present invention not only improves the stability of the jet forming process and reduces the incidence of unstable jets, but also improves the overall performance of shaped charges and enhances their effect in practical applications, especially in the military field, and can significantly improve the strike accuracy and destructive power of shaped charge ammunition. In addition, the method also lays a theoretical foundation for the further development of jet forming technology and has broad application prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0132] In order to more clearly illustrate the specific implementation methods of the present invention or the technical solutions in the prior art, the drawings required for use in the specific implementation methods or the description of the prior art will be briefly introduced below. Obviously, the drawings in the following description are some implementation methods of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0133] Figure 1 It is a flow chart of the analysis method of energy-focused jet forming and stability of the present invention;

[0134] Figure 2 A schematic diagram of determining effective charge involved in the analysis method of shaped-energy jet forming and stability of the present invention;

[0135] Figure 3 It is a schematic diagram of the liner structure parameters involved in the analysis method of shaped-energy jet forming and stability of the present invention. DETAILED DESCRIPTION

[0136] The technical solution of the present invention will be described clearly and completely below in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0137] The present invention is further explained below in conjunction with specific implementation modes.

[0138] like Figure 1As shown, this embodiment provides a method for analyzing the forming and stability of a shaped charge jet. First, a jet forming analysis is performed on the shaped charge jet formed by the detonation wave generated by the detonation of the explosive crushing the charge liner to establish the relationship between the liner structure parameters and the jet diameter, temperature and strain rate; then a jet stability analysis is performed on the shaped charge jet to determine the jet instability criterion; finally, the jet breakup time is obtained.

[0139] The above-mentioned jet forming analysis is to obtain the basic physical quantities after jet forming, such as diameter, length, speed, strain rate, etc. Figure 3 As shown in Figure 2, these parameters have an important influence on the jet stability; specifically, the following steps are included:

[0140] 1.1) Solve for the velocity of the jet

[0141] First, the jet velocity is obtained according to the steady model , and pestle speed :

[0142] ;

[0143] ;

[0144] ;

[0145] In the above formulas (1)-(3), is the crush angle, is the semi-cone angle of the liner, is the crushing speed, Indicates the detonation velocity of the explosive;

[0146] Next, if Figure 2 As shown in the figure, assuming that the scattering direction of the explosive products is consistent with the normal direction of the charge surface, the effective amount of explosive is determined according to the effective charge theory; assuming that the velocity of the detonation gas on the metal surface is linearly distributed, the crushing velocity can be obtained :

[0147] ;

[0148] In the above formula (4), is the internal energy of the explosive, For metal quality, is the mass of explosives;

[0149] 1.2) Solve the time of jet forming

[0150] The time it takes for the detonation wave to travel from the top to the bottom of the liner is recorded as :

[0151] ;

[0152] In the above formula (5), Indicates the height of the liner;

[0153] Assume that after the detonation wave contacts the top of the liner, the top of the liner converges toward the axis, and the convergence time is :

[0154] ;

[0155] ;

[0156] In the above formulas (6)-(7), is the distance from the top of the liner to the axis. is the inner radius of the liner top, is the deflection angle of the liner, is the semi-cone angle of the liner;

[0157] After the detonation, the bottom of the liner converges toward the axis, and the convergence time is :

[0158] ;

[0159] In the above formula (8), is the distance from the bottom of the liner to the axis. is the inner radius of the liner bottom;

[0160] When the bottom of the liner converges to the axis, , which indicates that the jet has been fully formed; therefore, the time from the top liner forming the jet to the complete formation is determined :

[0161] ;

[0162] 1.3) Solve for the diameter, length, and strain rate of the jet

[0163] The moment when the detonation wave reaches the bottom of the liner, that is, the moment when the detonation ends, is 0. Assuming that the jet is stretched uniformly after forming, the length of the jet at this time is for:

[0164] ;

[0165] After the jet is fully formed, the length of the jet is :

[0166] ;

[0167] Determining the strain of the jet , strain rate and temperature :

[0168] ;

[0169] ;

[0170] ;

[0171] In the above formulas (12)-(14), represents the initial area of ​​the jet, represents the area of ​​the jet at any time, is the work-to-heat conversion coefficient, is the jet stress, is the density of the jet, is the specific heat capacity of the jet, is room temperature;

[0172] The above stability analysis is based on the jet forming moment as the starting point for jet stability analysis, and assumes that the jet is subject to a small disturbance; specifically, it includes the following steps:

[0173] 2.1) Constructing a stretching model of the jet

[0174] Plastic work The heat it generates The relationship between them is as follows:

[0175] ;

[0176] In formula (15), ≈0.9, the remainder (1-𝛽) Still lurking in the metal;

[0177] Since elastic deformation energy is much smaller than plastic deformation energy, the former can be ignored and plastic work :

[0178] ;

[0179] In the above formula (16), is stress, It is strain;

[0180] Since the free flight time of the jet is short and heat conduction is negligible, the work-heat conversion equation is:

[0181] ;

[0182] In the above formula (17), is the temperature change of the jet, is the density of the jet, For time;

[0183] The jet is cylindrical, and the velocity is distributed along the axial direction. The velocity at the head is fast, and the velocity at the tail is slow. The jet can be simplified as a cylinder with a fixed support at the left end; the right end of the jet is a free end, and the velocity is Moving to the right, initial cross-sectional area , the initial length is , from which we can determine:

[0184] Initial strain rate: ;

[0185] Displacement: ;

[0186] Cross-sectional area: ;

[0187] strain: ;

[0188] Due to the axial length of the jet Diameter, axial stress on the same section , axial displacement ,

[0189] Axial speed is the spatial coordinate and time Function, with coordinates The equation of motion of the jet is determined as follows:

[0190] ;

[0191] During the stretching motion, the jet is approximated as an incompressible fluid;

[0192] From the law of conservation of mass, we can get:

[0193] ;

[0194] ;

[0195] ;

[0196] In the above formulas (23)-(25), is the jet density, is the cross-sectional area, is the initial density, is the initial cross-sectional area of ​​the undisturbed jet, is the space coordinate of the Euler coordinate system, is the Lagrangian coordinate;

[0197] Combining equation (22) with equation (25) we can obtain:

[0198] ;

[0199] Convert the above formula (26) to Taking partial derivatives we get:

[0200] ;

[0201] When the jet surface is disturbed, the stress distribution of the jet cross section becomes three-dimensional, and the average axial stress is equivalent to:

[0202] ;

[0203] In the above formula (28), is the stress in the ideal tensile state, is the radius of curvature at the neck, is the radius of the neck;

[0204] When the disturbance of the jet is small, the above equation (28) is equivalent to:

[0205] ;

[0206] In formula (29), represents the radius of the jet at any position;

[0207] Simplifying the above formula (29) we can get The expression is:

[0208] ;

[0209] From this we get:

[0210] ;

[0211] In formula (31), ;

[0212] Assume that the constitutive equation of the material is:

[0213] ;

[0214] In formula (32), ;

[0215] Finally, the governing equation of the jet is obtained:

[0216] ;

[0217] 2.2) Setting the perturbation of the jet

[0218] ;

[0219] In formula (34), is the disturbance growth rate of the jet, is the disturbance wave number of the jet;

[0220] It is thus determined that:

[0221] ;

[0222] ;

[0223] ;

[0224] Assumptions ,but:

[0225] ;

[0226] 2.3) Superimpose the disturbance on the jet control equation (32)

[0227] Assumptions is the solution of equation (33), and the perturbation is superimposed on this solution to obtain:

[0228] ;

[0229] The spectral equation is obtained as follows:

[0230] ;

[0231] 2.4) Establishing the instability criterion of jet

[0232] First, define the following dimensionless numbers: ;

[0233] Then, the above equation (40) is dimensionless to obtain equation (41):

[0234] ;

[0235] According to the stability judgment rule, the instability condition can be obtained by analyzing equation (40):

[0236] ;

[0237] According to the properties of the inequality, inequality (42) can be solved to obtain:

[0238] ;

[0239] definition , according to the order of magnitude estimate, Simplified, ;

[0240] Finally, the instability criterion of the jet is determined:

[0241] ;

[0242] 2.5) Establishing the break time of the jet

[0243] when , corresponding to is the maximum value, that is, the fastest growth rate of disturbance, which must satisfy:

[0244] ;

[0245] From this we get:

[0246] ;

[0247] ;

[0248] The simultaneous equations (41) and (47) yield:

[0249] ;

[0250] ;

[0251] From this, the breakup time of the jet can be obtained:

[0252] ;

[0253] 2.6) Establishing the instability criterion and break time of the jet based on the Johnson-Cook model

[0254] When the constitutive model of the liner material uses the Johnson-Cook constitutive model as the thermo-viscoplastic constitutive model, we have:

[0255] ;

[0256] In the above formula (51), is the static yield strength, is the hardening modulus, For strain, is the hardening index, is the strain rate constant, is the jet temperature, is room temperature, is the melting point of the jet, is the thermal softening coefficient;

[0257] The jet instability criterion is obtained by the following formula (52):

[0258] ;

[0259] The break time of the jet is obtained by the following formula (53):

[0260] .

[0261] The present invention establishes the relationship between the structural parameters of the shaped charge liner and the jet diameter and strain rate, proposes a jet stability criterion and a jet breakage time evaluation method that comprehensively considers strain hardening, strain rate effect, thermal effect and necking effect, constructs an analysis method for the effect of the liner structural parameters on the jet stability, and effectively solves the problem of jet stability.

[0262] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for analyzing energy-focused jet forming and stability, characterized in that: First, the jet forming analysis is carried out on the shaped jet formed by the detonation wave generated by the explosive detonation crushing the liner, so as to establish the relationship between the liner structure parameters and the jet diameter, temperature and strain rate; then the jet stability analysis is carried out on the shaped jet to determine the jet instability criterion; finally, the jet breakup time is obtained; The injection molding analysis includes the following steps: 1.1) Solve for the velocity of the jet First, the jet velocity is obtained according to the steady model and pestle speed : ； ； ； In the above formulas (1)-(3), is the crush angle, is the semi-cone angle of the liner, is the crushing speed, Indicates the detonation velocity of the explosive; Next, assuming that the scattering direction of the explosive products is consistent with the normal direction of the charge surface, the effective amount of explosives is determined according to the effective charge theory; assuming that the velocity of the detonation gas on the metal surface is linearly distributed, the crushing velocity is obtained. ; ； In the above formula (4), is the internal energy of the explosive, For metal quality, is the mass of explosives; 1.2) Solve the time of jet forming The time it takes for the detonation wave to travel from the top to the bottom of the liner is recorded as ; ； In the above formula (5), Indicates the height of the liner; Assume that after the detonation wave contacts the top of the liner, the top of the liner converges toward the axis, and the convergence time is ; ； ； In the above formulas (6)-(7), is the distance from the top of the liner to the axis. is the inner radius of the liner top, is the deflection angle of the liner, is the semi-cone angle of the liner; After the detonation, the bottom of the liner converges toward the axis, and the convergence time is : ； In the above formula (8), is the distance from the bottom of the liner to the axis. is the inner radius of the liner bottom; When the bottom of the liner converges to the axis, , which indicates that the jet has been fully formed; therefore, the time from the top liner forming the jet to the complete formation is determined : ； 1.3) Solve for the diameter, length, and strain rate of the jet The moment when the detonation wave reaches the bottom of the liner, that is, the moment when the detonation ends, is 0. Assuming that the jet is stretched uniformly after forming, the length of the jet at this time is for: ； After the jet is fully formed, the length of the jet is : ； Determining the strain of the jet , strain rate and temperature : ； ； ； In the above formulas (12)-(14), represents the initial area of ​​the jet, represents the area of ​​the jet at any time, 𝛽 is the work-to-heat conversion coefficient, 𝜎 is the stress of the jet, 𝜌 is the density of the jet, 𝐶𝑝 is the specific heat capacity of the jet, and 𝑇𝑟 is the room temperature.

2. The method for analyzing energy-focused jet forming and stability according to claim 1, characterized in that: The stability analysis is to perform jet stability analysis starting from the jet forming moment and assuming that the jet is subject to a small disturbance; specifically, the following steps are included: 2.1) Constructing a stretching model of the jet Plastic work The heat it generates The relationship between them is as follows: ； In formula (15), ≈0.9, the remaining part Still lurking in the metal; Since elastic deformation energy is much smaller than plastic deformation energy, the former is neglected and the plastic work is : ； In the above formula (16), is the stress, It is strain; Since the free flight time of the jet is short and heat conduction is negligible, the work-heat conversion equation is: ； In the above formula (17), is the temperature change of the jet, is the density of the jet, For time; The jet is cylindrical, and the velocity is distributed along the axial direction, with a fast speed at the head and a slow speed at the tail. The jet is simplified into a cylinder with a fixed support at the left end; the right end of the jet is a free end, with a velocity of Moving to the right, initial cross-sectional area , the initial length is , thus determining: Initial strain rate: ; Displacement: ; Cross-sectional area: ; strain: ; Due to the axial length and diameter of the jet, the axial stress on the same section , axial displacement , axial speed is the spatial coordinate and time Function, with coordinates The equation of motion of the jet is determined as follows: ； During the stretching motion, the jet is approximated as an incompressible fluid; From the law of conservation of mass: ； ； ； In the above formulas (23)-(25), is the jet density, is the cross-sectional area, is the initial density, is the initial cross-sectional area of ​​the undisturbed jet, is the space coordinate of the Euler coordinate system, is the Lagrangian coordinate; Combining equation (22) with equation (25) we can obtain: ； Convert the above formula (26) to Taking partial derivatives we get: ； When the jet surface is disturbed, the stress distribution of the jet cross section becomes three-dimensional, and the average axial stress is equivalent to: ； In the above formula (28), is the stress in the ideal tensile state, is the radius of curvature at the neck, is the radius of the neck; When the disturbance of the jet is small, the above equation (28) is equivalent to: ； In formula (29), represents the radius of the jet at any position; Simplifying the above formula (29) we can get The expression is: ； From this we get: ； In formula (31), ; Assume that the constitutive equation of the material is: ； In formula (32), ; Finally, the governing equation of the jet is obtained: ； 2.2) Setting the perturbation of the jet ； In formula (34), is the disturbance growth rate of the jet, is the disturbance wave number of the jet; It is thus determined that: ； ； ； Assumptions ,but: ； 2.3) Superimpose the disturbance on the jet control equation (32) Assumptions is the solution of equation (33), and the perturbation is superimposed on this solution to obtain: ； The spectral equation is obtained as follows: ； 2.4) Establishing the instability criterion of jet First, define the following dimensionless numbers: , , , , , , ; Then, the above equation (40) is dimensionless to obtain equation (41): ； According to the stability judgment rule, the instability condition can be obtained by analyzing equation (40): ； According to the properties of the inequality, inequality (42) can be solved to obtain: ； definition , according to the order of magnitude estimate, Simplified, ; Finally, the instability criterion of the jet is determined: ； 2.5) Establishing the break time of the jet when , corresponding to is the maximum value, that is, the fastest growth rate of disturbance, which must satisfy: ； From this we get: ； ； The simultaneous equations (41) and (47) yield: ； ； From this, the breakup time of the jet can be obtained: ； 2.6) Establishing the instability criterion and break time of the jet based on the Johnson-Cook model When the constitutive model of the liner material uses the Johnson-Cook constitutive model as the thermo-viscoplastic constitutive model, we have: ； In the above formula (51), is the static yield strength, is the hardening modulus, For strain, is the hardening index, is the strain rate constant, is the jet temperature, is room temperature, is the melting point of the jet, is the thermal softening coefficient; The jet instability criterion is obtained by the following formula (52): ； The break time of the jet is obtained by the following formula (53): 。

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