A method for estimating the diameter of a hole pierced by a cylindrical projectile impacting a thin metal plate at high velocity

Abstract

This invention discloses a method for estimating the perforation diameter of a cylindrical projectile impacting a thin metal plate at high velocity. The method first analyzes the physical parameters involved in a cylindrical projectile impacting a metal target at high velocity. Using dimensional analysis and the π theorem, seven dimensionless quantities are derived. Ultimately, a perforation diameter estimation model is determined. Finally, the model parameters are fitted using the least squares method using sample test data. Compared to values calculated using existing empirical formulas, the perforation diameter estimated by this method has a smaller relative error from the actual value. The method is applicable to different projectile and target materials and thicknesses, offering wide applicability and high accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of hypervelocity damage assessment, and in particular to a method for estimating the diameter of a hole formed by a cylindrical projectile impacting a metal sheet at a hypervelocity. Background Art

[0002] Hypervelocity impacts (also known as hypersonic impacts) typically refer to impacts at velocities exceeding 5 mAh. Hypervelocity impacts of projectiles against thin metal targets (where the target thickness to projectile diameter ratio is less than 2) can cause target plate perforation, delamination, the generation of fragment swarms, and even material phase transitions such as melting and vaporization. The assessment and understanding of perforation in thin metal plates caused by hypervelocity damage are fundamental to studying the subsequent projectile-target impact process, debris cloud formation, and evaluating the projectile's overall damage capability and damage mechanisms, and are therefore of great significance.

[0003] The correlation between target plate penetration diameter and projectile parameters can be quantitatively described using empirical formulas. Research on penetration stems from space debris protection, and numerous empirical formulas have been developed to describe the diameter of holes in target plates caused by hypervelocity impacts with spherical projectiles. However, these formulas are mostly applicable to lightweight aluminum alloy projectile targets. However, currently, kinetic projectiles are mostly cylindrical projectiles with high aspect ratios, composed of heavy metals such as tungsten alloys. Due to the varying target materials and projectile shapes, previous empirical formulas for spherical projectile penetration are not applicable to estimating the diameter of holes in heavy metal cylindrical projectiles caused by hypervelocity impacts with heavy metal targets.

[0004] Empirical formulas for describing the diameter of a hole in a metal target plate caused by a cylindrical projectile at hypervelocity impact are very limited. Rolsten et al. (1964) first proposed an empirical formula applicable to aluminum and stainless steel targets with a thickness of 0.25-1.6 mm (Rolsten's empirical formula, Equation (1), R.F. Rolsten, J.N. Wellnitz, and H.H. Hunt. An example of hole diameter in thin plates due to hypervelocity impact. Journal of Applied Physics. 1964; 35; 556). However, the results of their calculations were found to be higher than the experimental values. In 1991, Schonberg et al. constructed an empirical formula applicable to cylindrical projectiles with an aspect ratio of 1 under specific velocity and target thickness / projectile diameter ratio conditions (Schonberg empirical formula, formula (2), Schonberg WP. Hypervelocity Impact physics. NASA CR-4343. Washington, DC. 1991). However, when the projectile inclination angle approaches 0°, the two formulas used by Schonberg to describe the major and minor axis apertures of the perforation ellipse are different. Therefore, they cannot be applied to low inclination angles and normal penetration conditions. In addition, under certain impact conditions, the minimum perforation formula value is greater than the maximum perforation formula value. Zhang Wei and Scott et al. constructed empirical formulas for cylindrical projectiles in 2001 and 2004, respectively (Zhang Wei's empirical formula, formula (3), Zhang Wei, Pang Baojun, Luo Dekun, Zhang Zehua. Study on the material state characteristics of debris clouds formed by cylindrical projectiles impacting protective screens. China Space Science and Technology. 2001; 3; 53-59; Scott's empirical formula, formula (4), Scott A. Hill. Determination of an empirical model for the prediction of penetration hole diameter in thin plates from hypervelocity impact International Journal of Impact Engineering. 2004; 30; 303-321). However, when used to predict the penetration hole diameter of tungsten alloy projectiles under hypervelocity impact, the errors are large. In 2021, Ma Kun et al. proposed a relatively simple empirical formula for cylindrical projectiles (Ma Kun's empirical formula, formula (5), Ma Kun, Chen Chunlin, Feng Na, et al. Perforation and fragment group expansion characteristics of cylindrical 93 tungsten projectiles impacting thin steel plates at high speed. Acta Armamentarii. 2021; 42; 2350-2359). However, because the influence of the sound velocity and strength of the projectile and target materials was not considered, it is only applicable to tungsten alloy projectiles impacting Q345 steel thin targets in a specific speed range.

[0005] Rolsten empirical formula:

[0006]

[0007] Schonberg empirical formula:

[0008]

[0009] Zhang Wei's empirical formula:

[0010]

[0011] Scott's empirical formula:

[0012]

[0013] Ma Kun's empirical formula:

[0014]

[0015] Where, d h is the target plate diameter after impact, v0 is the projectile impact velocity, d p is the projectile diameter, ρ p , ρ t are the densities of the projectile and target plate, c p 、c t The distribution is the sound velocity of the projectile and target plate material, T t is the target plate thickness, L p is the length of the projectile, and θ is the inclination angle of the projectile hitting the target.

[0016] In summary, existing empirical formulas for perforating metal targets with cylindrical projectiles at high velocities mostly account for aluminum materials only, without considering parameters such as target material strength. Consequently, they are only applicable to lightweight aluminum alloys and projectiles with low aspect ratios. These empirical formulas, when used to calculate perforation predictions for high-density alloy projectiles at high velocities, exhibit large relative errors and poor applicability. Therefore, developing an empirical formula for estimating the perforation diameter of a heavy metal cylindrical projectile at high velocities into thin metal plates has become a pressing technical challenge in this field. Summary of the Invention

[0017] In light of this, the present invention provides a method for estimating the diameter of a hole punctured by a cylindrical projectile upon a hypervelocity impact with a thin metal plate. The calculated result has a smaller relative error than the actual value, a higher correlation coefficient, and a wider range of applicability. This method can provide parameters for estimating the diameter of a hole punctured by a cylindrical projectile upon a hypervelocity impact with a metal target, assessing its comprehensive damage capability, and designing projectiles, significantly reducing engineering testing costs. This method offers the advantages of simplicity, practicality, reliable results, and efficient computation.

[0018] The method for estimating the diameter of a hole pierced by a cylindrical projectile at a high velocity into a metal sheet according to the present invention is characterized in that the following model is used to estimate the diameter of the hole:

[0019]

[0020] Among them, d h is the diameter of the target plate after impact, v0 is the impact velocity of the projectile, d p is the projectile diameter, c p is the sound velocity of the projectile material, c t is the sound velocity of the target material, ρ p is the projectile density, ρ t is the target plate density, L p is the length of the projectile, T t is the target plate thickness, Y t is the target plate material strength, Y p is the strength of the projectile material; a1~a 10 are model parameters.

[0021] The best method is to collect sample test data and use the least square method to obtain the model parameters a1~a 10 .

[0022] The Levenberg-Marquardt optimization method is used to fit the model parameters and obtain the model parameters a1~a 10 .

[0023] The preferred modeling method for perforation diameter estimation is as follows:

[0024] S1, determine the physical parameters involved in the diameter estimation model of the hole pierced by a cylindrical projectile at high speed into a metal sheet, including: projectile impact velocity v0, projectile diameter d p , projectile length L p , projectile density ρ p , sound velocity of projectile material c p , elastic material strength Y p , target plate thickness T t , target plate density ρ t , target material sound velocity c t and target plate material strength Y t ;Establish a function formula for the diameter of the hole when a cylindrical projectile penetrates a metal target at high speed;

[0025] S2, analyze the dimensions of the physical parameters determined in S1, including: projectile impact velocity LT -1 , projectile diameter L, projectile length L, projectile density ML -3 , sound velocity of projectile material LT -1 , projectile material strength ML -1 T -2, target plate thickness L, target plate density ML -3 , target material sound velocity LT -1 and target plate material strength ML -1 T -2 ; Determine the basic dimensions as mass M, length L and time T;

[0026] S3, select the impact velocity, projectile density and projectile diameter as three independent dependent variables as reference physical quantities, and obtain the relationship expression between the basic dimension and the reference physical quantity dimension;

[0027] S4, the perforation diameter model is determined in the form of a multivariable power function. Based on dimensional analysis theory and the π theorem, combined with the form of previous empirical formulas and fitting accuracy comparison and iterative optimization, the dimensionless terms are finally determined to include: the ratio of the compressible term to the inertia term of the projectile and target; the density ratio of the projectile to the target; the aspect ratio of the projectile; the target thickness to projectile diameter ratio; the ratio of the strength term of the projectile and target to the inertia term of the projectile; and the final perforation diameter estimation model is obtained.

[0028] The present invention also provides a method for evaluating the damage effect of a cylindrical projectile impacting a metal sheet at a high velocity, and the method for estimating the perforation of a metal sheet by a cylindrical projectile at a high velocity is used to estimate the perforation diameter.

[0029] Beneficial effects:

[0030] The present invention first analyzes the physical parameters involved in a cylindrical projectile's hypervelocity impact with a metal target. Using dimensional analysis and the π theorem, seven dimensionless quantities are derived, ultimately defining a perforation diameter estimation model. Finally, using sample test data, the model parameters are fitted using the least squares method. Compared to values calculated using existing empirical formulas, the perforation diameter estimated by the present invention exhibits a smaller relative error from the actual value. The method is applicable to different projectile-target materials, varying target thickness-to-projectile diameter ratios, and even larger aspect ratios, offering broad applicability and high accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 Flowchart of the method of the present invention.

[0032] Figure 2 This is a comparison chart of the empirical formula proposed in the present invention and the Scott empirical formula value and the experimental value.

[0033] Figure 3 The empirical formula proposed by the present invention and the empirical formula of Scott and Ma Kun show changing trends with different projectile parameters.

[0034] Figure 4 This is the trend of the empirical formula proposed by the present invention and the empirical formula of Scott and Ma Kun with different target plate parameters. DETAILED DESCRIPTION

[0035] The present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0036] The present invention provides a method for estimating the perforation of a metal sheet by a cylindrical projectile with a high velocity impact, as shown in the flow chart. Figure 1 As shown, the specific steps include:

[0037] Step 1: Establish a model for the diameter of a hole formed by a cylindrical projectile penetrating a metal target at high speed:

[0038] Under hypervelocity impact, the material in the contact area between the projectile and the target undergoes dynamic fragmentation, and the target plate forms a perforation and further evolves. The degree of dynamic fragmentation of the projectile and the target is directly related to the strength of the material. From the perspective of stress waves, the target plate perforation process is a process of interaction between the strong shock wave generated by the projectile-target impact and the sparse wave generated by the projectile-target free surface. The propagation of the wave depends on the shock wave velocity, which is related to the impact velocity and the sound velocity of the material. Therefore, in addition to the projectile-target geometric parameters that directly affect the perforation result, the above-mentioned material parameters must also be considered. Therefore, the present invention determines the physical parameters involved in the process of the cylindrical projectile penetrating the metal target plate at hypervelocity, including: the projectile impact velocity v0, the projectile diameter d p , projectile length L p , projectile density ρ p , sound velocity of projectile material c p , elastic material strength Y p , target plate thickness T t , target plate density ρ t , target material sound velocity c t and target plate material strength Y t ;

[0039] Based on the above parameters, we first analyze the dimensions of the physical parameters involved in the hypervelocity penetration of the cylindrical projectile into the metal target plate, which are as follows: -1 , projectile diameter L, projectile length L, projectile density ML -3 , sound velocity of projectile material LT -1 , projectile material strength ML -1 T -2 , target plate thickness L, target plate density ML -3 , target material sound velocity LT -1 and target plate material strength ML -1 T -2 ;

[0040] Then, through the method of dimensional analysis, the relationship between the target plate perforation diameter and the projectile-target impact parameters was obtained, and a correlation formula was established, which is specifically:

[0041] 1) Based on the physical quantities involved in the projectile-target impact process, the perforation function of the cylindrical projectile penetrating the metal target plate at high speed is established as follows:

[0042] dh =f(v0,c p ,c t ,ρ p ,ρ t ,d p ,L p ,T t ,Y p ,Y t ) (6)

[0043] Equation (6) contains only three basic dimensions: mass M, length L, and time T;

[0044] 2) Selecting impact velocity, projectile density, and projectile diameter as three independent dependent variables as reference physical quantities, the relationship between the basic dimension and the reference physical quantity dimension can be expressed as:

[0045]

[0046] 3) Based on dimensional analysis and the π theorem, seven dimensionless quantities are given as the function of the diameter of the hole when a cylindrical projectile penetrates a metal target at high speed:

[0047]

[0048] Formula (6) can be written as a dimensionless function as follows:

[0049]

[0050] 4) Construct a new model for the diameter of a cylindrical projectile penetrating a metal target plate at high speed. The model adopts a multivariable power function form. Based on the above-mentioned dimensional analysis theory and the π theorem, combined with the form of the previous empirical formula and the comparison of fitting accuracy, iterative optimization is performed to finally determine the dimensionless terms. The dimensionless terms include: the ratio of the compressibility term to the inertia term of the projectile and target; the density ratio of the projectile to the target; the aspect ratio of the projectile; the thickness-to-projectile ratio; and the ratio of the strength term to the projectile inertia term. is the ratio of the inertial term to the compressible term, is the ratio of the inertia term to the material strength. The inertia term facilitates aperture expansion, while the compressibility term and the strength term act inversely on aperture growth. The above parameters work together to determine the final perforation diameter. The final perforation diameter model formula is as follows:

[0051]

[0052] a1~a 10 are the model parameters to be fitted.

[0053] Step 2: Obtain valid test data on the perforation of a metal target plate by a cylindrical projectile impacting at high speed:

[0054] By collecting relevant reports, papers, and test results, the test data of cylindrical projectiles impacting metal target plates at high speeds under different impact conditions are integrated, and the impact velocity, projectile-target geometric parameters, and projectile-target material parameter data are statistically analyzed.

[0055] Step 3: Fit model parameters using the least squares method:

[0056] The principle of least squares method is to determine the model parameters by minimizing the sum of squares Q of the residuals between the sample test value and the empirical formula estimate, that is,

[0057]

[0058] In the formula, (x i ,y i ) is the sample test value, f(x i ) is the estimated value of the empirical formula. The Levenberg-Marquardt optimization method is used to fit the model parameters to obtain the model parameters to be fitted.

[0059] Step 4: Empirical formula model estimation accuracy test:

[0060] The degree of consistency between the empirical formula model and the sample data is determined by examining the values of the correlation coefficient R and the root mean square RMS.

[0061] Specific embodiment: The perforation diameter of cylindrical projectiles made of various materials impacting target plates of different materials at high speed

[0062] The collected test data and material performance parameters of the cylindrical projectile impacting the metal sheet at high speed are shown in Table 1.

[0063] Table 1 Data and parameters of the perforation test of cylindrical projectiles impacting metal targets at high speed

[0064]

[0065]

[0066] Based on the above-mentioned projectile target geometric parameters, material parameters, and projectile impact velocity data, the model parameters obtained by fitting the empirical formula (10) described in the present invention are shown in Table 2. The correlation coefficient R of the empirical formula fitting described in the present invention is 0.991, the root mean square error RMS is 0.0641, the maximum error is 7.251%, and for more than 93.33% of the data, the error is less than 4.705%.

[0067] Table 2 Fitting parameter values

[0068]

[0069] Figure 2This is a comparison chart of the empirical formula proposed in the present invention and the Scott empirical formula values with the experimental values. It can be seen that the Scott empirical formula value is greater than the experimental value, and the empirical formula proposed in the present invention has a smaller fitting error.

[0070] Table 3 compares the errors between the empirical formulas of the present invention, Scott's empirical formula, and Ma Kun's empirical formula, and the experimental values (other formulas have large errors or are not applicable and are not listed). The results show that the empirical formula proposed in this invention has a smaller relative error than the actual value, is applicable to several different projectile and target materials, and can be used for fitting medium-thick targets, with a wider range of applications.

[0071] Figure 3 、 Figure 4 Figure 1 shows the empirical formula proposed by the present invention and the empirical formula of Scott and Ma Kun as they change with different projectile and target parameters (other formulas have large errors or are not applicable and are not listed). As can be seen from the figure, the empirical formula proposed by the present invention can better describe the trend and physical laws of perforation value as it changes with different projectile and target parameters.

[0072] Table 3 Comparison of errors in empirical formulas for cylindrical projectiles impacting metal targets at high speed

[0073]

[0074]

[0075] In summary, the above are only preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for estimating the diameter of a hole pierced by a cylindrical projectile at high velocity into a metal sheet, characterized in that: The following model is used to estimate the perforation diameter: Among them, d h is the diameter of the target plate after impact, d p is the diameter of the projectile, v0 is the impact velocity of the projectile, c p is the sound velocity of the projectile material, c t is the sound velocity of the target material, ρ p is the projectile density, ρ t is the target plate density, L p is the length of the projectile, T t is the target plate thickness, Y t is the target plate material strength, Y p is the strength of the projectile material; a1~a 10 are model parameters.

2. The method according to claim 1, wherein By collecting sample test data, the model parameters a1~a are obtained using the least squares method. 10 .

3. The method according to claim 2, wherein The Levenberg-Marquardt optimization method was used to fit the model parameters and the model parameters a1~a 10 .

4. The method according to claim 1, wherein The modeling method of the perforation diameter estimation model is as follows: S1, determine the physical parameters involved in the diameter estimation model of the hole pierced by a cylindrical projectile at high speed into a metal sheet, including: projectile impact velocity v0, projectile diameter d p , projectile length L p , projectile density ρ p , sound velocity of projectile material c p , elastic material strength Y p , target plate thickness T t , target plate density ρ t , target material sound velocity c t and target plate material strength Y t ;Establish a function formula for the diameter of the hole when a cylindrical projectile penetrates a metal target at high speed; S2, analyze the dimensions of the physical parameters determined in S1, including: projectile impact velocity LT -1 , projectile diameter L, projectile length L, projectile density ML -3 , sound velocity of projectile material LT -1 , projectile material strength ML -1 T -2 , target plate thickness L, target plate density ML -3 , target material sound velocity LT -1 and target plate material strength ML -1 T -2 ; Determine the basic dimensions as mass M, length L and time T; S3, select the impact velocity, projectile density and projectile diameter as three independent dependent variables as reference physical quantities, and obtain the relationship expression between the basic dimension and the reference physical quantity dimension; S4, the perforation diameter model is determined in the form of a multivariable power function. Based on dimensional analysis theory and the π theorem, combined with the form of previous empirical formulas and fitting accuracy comparison and iterative optimization, the dimensionless terms are finally determined to include: the ratio of the compressible term to the inertia term of the projectile and target; the density ratio of the projectile to the target; the aspect ratio of the projectile; the target thickness to projectile diameter ratio; the ratio of the strength term of the projectile and target to the inertia term of the projectile; and the final perforation diameter estimation model is obtained.

5. A method for evaluating the damage effect of a cylindrical projectile impacting a metal sheet at high velocity, characterized in that: The perforation diameter is estimated using the method according to any one of claims 1 to 4.

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