Method for determining back convex deformation and ballistic limit velocity of orthogonal laminated board under impact of flat-head bomb

Abstract

The invention relates to a method for determining back convex deformation and ballistic limit velocity of an orthogonal laminated board under impact of a flat-head bomb, and belongs to the technical field of damage and protection. According to the method, transient back convex deformation, moving hinge propagation and ballistic limit velocity of the UHMWPE laminated board along with progressive penetration under the impact of the flat-head projectile can be determined, and any fitting parameter and empirical formula do not need to be used. The determination result of the method passes test verification, and the calculation result and the test result have good consistency, so that the method provided by the invention is high in determination precision. According to the method, transient analysis and the Lagrangian method are combined, the progressive penetration failure criterion is introduced, and finally the back convex deformation and the ballistic limit of the UHMWPE laminated board are determined.

Description

Technical Field

[0001] The invention relates to a method for determining the back convex deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-headed bullet, and belongs to the technical field of damage and protection. Background Art

[0002] Ultra-high molecular weight polyethylene (UHMWPE) fiber composites are increasingly used in armor protection systems due to their excellent mechanical properties, such as high specific strength and specific modulus, and extremely high plastic wave velocity, which can quickly absorb and disperse energy. A typical cross-ply UHMWPE laminate consists of a high volume fraction of UHMWPE fiber composites (>80%) and a low volume fraction of thermoplastic polymer matrix (<20%), and has excellent resistance to high-speed local impact.

[0003] The dynamic failure process of UHMWPE orthogonal laminates includes a local penetration stage and a convex deformation stage accompanied by progressive penetration. Convex deformation and ballistic limit velocity are important indicators for evaluating their ballistic performance. Although existing numerical models can provide high-fidelity ballistic response predictions, they are computationally expensive and difficult to use for large-scale iterative simulation optimization. Therefore, it is very important to develop an analytical model that can quickly and accurately estimate convex deformation and ballistic limit velocity.

[0004] During ballistic impact, the convex deformation of the laminate is closely related to the moving hinge propagating from the center to the outside. Existing analytical models based on momentum conservation or energy conservation often ignore the progressive penetration process of the laminate, and mostly use empirical formulas to calculate the evolution of lateral deformation. And the empirical formulas often need to combine experiments or simulations to combine relevant parameters, resulting in high complexity and insufficient accuracy in the prediction of convex deformation and ballistic limit velocity. Summary of the invention

[0005] In view of this, the present invention provides a method for determining the convex deformation and ballistic limit velocity of an orthogonal laminate under a flat-headed projectile impact, which can efficiently and accurately predict the transient center deflection, moving hinge position and ballistic limit velocity of a UHMWPE laminate under a flat-headed projectile impact. The term "moving hinge" involved in the present invention refers to the propagation of the bending wave along the in-plane direction of the laminate over time, that is, the lateral deformation profile of the laminate.

[0006] The present invention can accurately predict the transient convex deformation, moving hinge propagation and ballistic limit velocity of the UHMWPE laminate during the progressive penetration of a flat-headed bullet without using any fitting parameters and empirical formulas.

[0007] The technical solution of the present invention is:

[0008] A method for determining the convex deformation and ballistic limit velocity of an orthogonal laminate under impact of a flat-headed bullet, the method comprising the following steps:

[0009] The first step is to use an initial velocity of v 01 The flat-headed projectile is impacting the fixed-support orthogonal laminate 1. The impact position of the flat-headed projectile is the center of the orthogonal laminate. At this time, the orthogonal laminate is penetrated. The penetration process of the orthogonal laminate is divided into a local penetration stage and a back convex deformation stage accompanied by progressive penetration.

[0010] Use the initial velocity v 02 The flat-headed projectile directly impacts the fixed-support orthogonal laminate 2. The impact position of the flat-headed projectile is the center of the orthogonal laminate. At this time, the orthogonal laminate is penetrated. The penetration process of the orthogonal laminate is divided into a local penetration stage and a back convex deformation stage accompanied by progressive penetration.

[0011] …

[0012] Use the initial velocity v 0n The flat-headed projectile is impacting the fixed-support orthogonal laminate n. The impact position of the flat-headed projectile is the center of the orthogonal laminate. At this time, the orthogonal laminate is penetrated. The penetration process of the orthogonal laminate is divided into a local penetration stage and a back convex deformation stage accompanied by progressive penetration.

[0013] The second step is to use the initial velocity v of the flat-head bullet in the first step. 01 , the convex deformation parameter C1 of the orthogonal laminate under the impact of the flat-headed projectile and the residual velocity v of the flat-headed projectile are obtained r1 ;

[0014] According to the initial velocity v of the flat-head bullet in the first step 02 , the convex deformation parameter C2 of the orthogonal laminate under the impact of the flat-headed projectile and the residual velocity v of the flat-headed projectile are obtained r2 ;

[0015] …

[0016] According to the initial velocity v of the flat-head bullet in the first step 0n , the convex deformation parameter Cn of the orthogonal laminate under the impact of the flat-headed bullet and the residual velocity v of the flat-headed bullet are obtained rn ;

[0017] The third step is to get v from the second step. 01 、v r1 、v 02 、v r2 ,…,v 0n 、v rn Fitting is performed to obtain the ballistic limit velocity v bl .

[0018] In the first step, the material parameters of the flat-head bullet and the orthogonal laminate include the mass M of the flat-head bullet p, the density of the orthogonal laminate ρ, the elastic modulus of the fiber in the x direction E1 and the y direction E2 of the orthogonal laminate, the elastic modulus in the thickness direction E3 of the orthogonal laminate, and the Poisson's ratio v in the in-plane direction 12 , the in-plane shear modulus is G 12 , failure stress σ f , failure strain ∈ f ;

[0019] The geometric parameters of the flat-head bullet and the orthogonal laminate include the cross-sectional area A of the bullet, the side lengths a and b of the orthogonal laminate, and the thickness h0 of the orthogonal laminate;

[0020] In the second step, the convex deformation parameters of the orthogonal laminate include the central deflection w0, the mobile hinge position ζ in the x direction and the mobile hinge position η in the y direction; the x direction coincides with the 0° fiber direction of the orthogonal laminate, the y direction coincides with the 90° fiber direction of the orthogonal laminate, and the z direction is the thickness direction of the orthogonal laminate; the convex deformation parameters of the orthogonal laminate under the impact of the flat-head bullet are calculated by numerically solving the following differential equations:

[0021]

[0022] where L is the Lagrangian value, which is related to the thickness h of the orthogonal laminate during the back-convex deformation stage accompanying progressive penetration;

[0023] is the velocity of the flat-headed projectile in the convex deformation stage with progressive penetration, and its initial value is the residual velocity v of the flat-headed projectile in the local penetration stage. m , whose final value is the remaining velocity v of the flat-headed projectile r ;

[0024] Based on the Ritz method and boundary conditions, a trial function of the displacement field of the laminate is assumed, and the Lagrangian value L is calculated according to the strain energy Π of the orthogonal laminate and the total kinetic energy T of the system (the kinetic energy of the laminate and the kinetic energy of the bullet);

[0025] The thickness h of the orthogonal laminate during the back convex deformation stage with progressive penetration, whose initial value is h2 = h0-h1, where h0 is the initial total thickness of the orthogonal laminate and h1 is the penetration thickness during the local penetration stage;

[0026] When calculating the thickness of the orthogonal laminate in the back-convex deformation stage accompanied by progressive penetration, a homogenized layered model is performed on the remaining unpenetrated orthogonal laminate, and the progressive penetration process of the orthogonal laminate is predicted based on the mixed mode failure criterion and the tensile failure criterion. The homogeneous layer in contact with the flat-headed bullet is penetrated after the failure criterion is met, and the deflection of the center point and the position of the moving hinge no longer increase.

[0027] The velocity expression of the flat-head projectile in the local penetration stage is:

[0028]

[0029] Where: v0 is the initial velocity of the bullet, and its specific value is v 01 、v 02 ,...,v 0n ; t is the response time; C h is the compression wave propagation velocity in the laminate,

[0030] By integrating formula (1), we can get the penetration depth in the local penetration stage:

[0031]

[0032] Where: t1 is the end time of the local penetration stage;

[0033] The end time of the local penetration phase is:

[0034]

[0035] By combining equations (2) and (3), we can obtain the penetration thickness h1 of the flat-headed projectile in the local penetration stage and the residual velocity of the flat-headed projectile:

[0036] The orthogonal laminate in the convex deformation stage with progressive penetration is modeled as N homogeneous layers, each of which is composed of [0 / 90] and has a thickness of h. L , then the material mechanical properties parameters of the homogeneous layer are the same as the macroscopic mechanical properties parameters of the laminate;

[0037] The displacement field of the orthogonal laminate in the back convex deformation stage accompanied by progressive penetration can be expressed as follows:

[0038]

[0039] Where: w0(t) is the deflection of the center point of the orthogonal laminate, ζ(t) and η(t) are the positions of the moving hinges in the x and y directions, respectively. Field equations to satisfy boundary conditions for orthogonal laminates;

[0040] The strain energy of the orthogonal laminate at the 1 / 4 back-convex deformation stage accompanied by progressive penetration is:

[0041]

[0042] Where: h is the thickness of the orthogonal laminate during the convex deformation stage with progressive penetration, and its initial value is h2.

[0043] The total kinetic energy of the orthogonal laminate and flat-head projectile during the convex deformation stage of 1 / 4 progressive penetration is:

[0044]

[0045] Where: μ is the surface density of the orthogonal laminate;

[0046] Lagrangian value:

[0047] L=T-Π (8)

[0048] The mixed mode failure criterion is:

[0049]

[0050] Where: j represents the jth homogeneous layer in contact with the flat-headed bullet, and takes a value from 1 to N as the flat-headed bullet penetrates layer by layer; is the maximum equivalent stress of the jth layer, is the equivalent stress of the jth layer, and are the normal stress and shear stress of the jth layer respectively;

[0051] The tensile failure criterion is:

[0052]

[0053] Where: is the tensile strain of the jth layer;

[0054] After the homogeneous layer in contact with the flat-head bullet penetrates, the thickness of the orthogonal laminate in the back convex deformation stage accompanied by progressive penetration is updated as:

[0055] h=h2-jh L (11)

[0056] Beneficial Effects

[0057] 1. The present invention can predict the transient convex deformation, moving hinge propagation and ballistic limit velocity of UHMWPE laminates under the impact of flat-headed bullets accompanied by progressive penetration without using any fitting parameters and empirical formulas.

[0058] 2. The prediction results of the present invention have been verified by experiments, and the prediction results are highly consistent with the experimental results. Therefore, the prediction accuracy of the method proposed in the present invention is relatively high.

[0059] 3. The present invention combines transient analysis with the Lagrangian method and introduces the progressive penetration failure criterion, and finally gives the back convex deformation and ballistic limit prediction of the UHMWPE laminate. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 It is a schematic diagram of the failure process of the UHMWPE orthogonal laminate under the impact of the flat-head bullet in this example;

[0061] Figure 2is a schematic diagram comparing the center deflection determined in this example with the experimental results;

[0062] Figure 3 is a schematic diagram comparing the dimensionless moving hinge position determined in this example with the experimental results;

[0063] Figure 4 It is a schematic diagram comparing the ballistic limit velocity determined in this example with the experimental results. DETAILED DESCRIPTION

[0064] The present invention is further described below in conjunction with the accompanying drawings and specific embodiments.

[0065] This embodiment is aimed at the flat-head bullet impact grade In the case of a [0 / 90] cross-ply UHMWPE laminate of HB26, the method provided by the present invention is used to calculate the transient backplane center deflection w0(t), the moving hinge position ζ(t) and the ballistic limit velocity v of the laminate. bl (t) is calculated.

[0066] In the first step, the initial geometric parameters and material parameters of the flat-head bullet and the orthogonal laminate are determined;

[0067] With initial velocity v 01 = 365m / s flat-headed projectile directly impacting the fixed orthogonal laminate 1. The impact position of the flat-headed projectile is the center of the orthogonal laminate. At this time, the orthogonal laminate is penetrated. The penetration process of the orthogonal laminate is divided into a local penetration stage and a back convex deformation stage accompanied by progressive penetration, as shown in Figure 1 As shown;

[0068] In this embodiment, the geometric and material parameters of the flat-head bullet and the laminate are shown in Table 1.

[0069] Table 1

[0070] Bullet diameter D 20mm <![CDATA[Elastic moduli E1, E2 of the laminate]]> 51.1GPa Bullet cross-sectional area A 3.14E4 <![CDATA[Elastic modulus E3 of the laminate]]> 3.62Gpa <![CDATA[Bullet mass M p > 54g <![CDATA[Poisson's ratio v of the laminate 12 > 0.008 Laminate side length a 300mm <![CDATA[Laminate shear modulus G 12 > 0.19GPa <![CDATA[Laminate thickness h0]]> 10mm <![CDATA[Laminate failure stress σ f > 1.5GPa Laminate density ρ <![CDATA[980kg / m 3 ]]> <![CDATA[Laminate failure strain ∈ f > 0.8 <![CDATA[Homogeneous layer thickness h L > 0.14mm

[0071] The second step is to use the initial velocity v of the flat-head bullet in the first step. 01 , the convex deformation parameter C1 of the orthogonal laminate under the impact of the flat-headed projectile and the residual velocity v of the flat-headed projectile are obtained r1 ;

[0072] The penetration thickness of the flat-headed projectile in the local penetration stage is h1 = 3 mm, and the residual velocity is v m =331.4m / s;

[0073] The initial thickness of the orthogonal laminate during the convex deformation stage of progressive penetration is h2 = h0-h1 = 7 mm, which can be homogenized as A homogeneous layer;

[0074] The displacement field of the orthogonal laminate in the back convex deformation stage accompanied by progressive penetration can be expressed as follows:

[0075]

[0076] Combined with the thin plate bending theory, the strain energy Π of the laminate and the total kinetic energy T of the system (the kinetic energy of the laminate and the kinetic energy of the bullet) are obtained, and the Lagrangian value L is calculated. Substitute it into the Lagrangian equation, derive the dynamic response ordinary differential equations about the deflection of the center point of the laminate and the position of the moving hinge, and use MATLAB to solve it numerically. Considering the high-speed impact response time while ensuring the calculation accuracy, the time step is set to 0.01μs, and the initial value is set to

[0077] For each time step, calculate D1 and D2. When either value is greater than 1, update the number of penetrated layers j and update the thickness of the remaining complete laminate to h = h2-jh. L . Re-derive the dynamic response ordinary differential equations, and perform calculations and failure judgments for the next time step. The final remaining velocity of the bullet is v r1 = 0m / s, the laminate does not penetrate, and the calculation ends. Figure 2 and Figure 3 As shown, the central deflection curve and the dimensionless moving hinge curve determined by the embodiment of the present invention are in good agreement with the experimental results.

[0078] The third step is to get v from the second step. 01 、v r1 、v 02 、v r2 ,…,v 0n 、v rn Fitting is performed to obtain the ballistic limit velocity v bl .

[0079] Select a series of initial velocities in the range of 365m / s to 1500m / s as v 01 、v 02 ,…,v 0n , the corresponding residual speed v can be calculated r1 、v r2 ,…,v rn ,like Figure 4 As shown in FIG. 1 , the initial-residual velocity curve determined by the embodiment of the present invention is in good agreement with the experimental results. The ballistic limit velocity v determined by the embodiment of the present invention is bl =420m / s, the ballistic limit velocity measured experimentally is 415m / s, and the relative error is 1.2%. Therefore, the calculated result of the ballistic limit velocity is in good agreement with the experiment and can meet the needs of engineering applications.

[0080] In summary, the above are only preferred embodiments of the present invention and are not intended to limit the protection scope of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for determining the convex deformation and ballistic limit velocity of an orthogonal laminate under a flat-headed projectile impact, characterized in that The steps of the method include: In the first step, use the initial velocity v 01 The flat-headed projectile is impacting the fixed-support orthogonal laminate 1. The impact position of the flat-headed projectile is the center of the orthogonal laminate. At this time, the orthogonal laminate is penetrated. The penetration process of the orthogonal laminate is divided into a local penetration stage and a back convex deformation stage accompanied by progressive penetration. Use the initial velocity v 02 The flat-headed projectile directly impacts the fixed-support orthogonal laminate 2. The impact position of the flat-headed projectile is the center of the orthogonal laminate. At this time, the orthogonal laminate is penetrated. The penetration process of the orthogonal laminate is divided into a local penetration stage and a back convex deformation stage accompanied by progressive penetration. … Use the initial velocity v 0n The flat-headed projectile is impacting the fixed-support orthogonal laminate n. The impact position of the flat-headed projectile is the center of the orthogonal laminate. At this time, the orthogonal laminate is penetrated. The penetration process of the orthogonal laminate is divided into a local penetration stage and a back convex deformation stage accompanied by progressive penetration. The second step is to use the initial velocity v of the flat-head bullet in the first step. 01 , the convex deformation parameter C1 of the orthogonal laminate under the impact of the flat-headed projectile and the residual velocity v of the flat-headed projectile are obtained r1 ; According to the initial velocity v of the flat-head bullet in the first step 02 , the convex deformation parameter C2 of the orthogonal laminate under the impact of the flat-headed projectile and the residual velocity v of the flat-headed projectile are obtained r2 ; … According to the initial velocity v of the flat-head bullet in the first step 0n , the convex deformation parameter Cn of the orthogonal laminate under the impact of the flat-headed bullet and the residual velocity v of the flat-headed bullet are obtained rn ; The third step is to get v from the second step. 01 、v r1 、v 02 、v r2 ,...,v 0n 、v rn Fitting is performed to obtain the ballistic limit velocity v bl .

2. The method for determining the convex deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-headed bullet according to claim 1, characterized in that: In the first step, the material parameters of the flat-head bullet and the orthogonal laminate include the mass M of the flat-head bullet p , the density of the orthogonal laminate ρ, the elastic modulus of the fiber in the x direction E1 and the y direction E2 of the orthogonal laminate, the elastic modulus in the thickness direction E3 of the orthogonal laminate, and the Poisson's ratio v in the in-plane direction 12 , the in-plane shear modulus is G 12 , failure stress σ f , failure strain ∈ f ; The geometric parameters of the flat-head bullet and the orthogonal laminate include the cross-sectional area A of the bullet, the side lengths a and b of the orthogonal laminate, and the thickness h0 of the orthogonal laminate.

3. The method for determining the convex deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-headed bullet according to claim 2, characterized in that: In the second step, the convex deformation parameters of the orthogonal laminate include the central deflection w0, the mobile hinge position ζ in the x direction and the mobile hinge position η in the y direction; the x direction coincides with the 0° fiber direction of the orthogonal laminate, the y direction coincides with the 90° fiber direction of the orthogonal laminate, and the z direction is the thickness direction of the orthogonal laminate; the convex deformation parameters of the orthogonal laminate under the impact of the flat-head bullet are calculated by numerically solving the following differential equations: where L is the Lagrangian value, which is related to the thickness h of the orthogonal laminate during the back-convex deformation stage accompanying progressive penetration; is the flat-head bullet velocity in the convex deformation stage with progressive penetration, and its initial value is the flat-head bullet residual velocity v in the local penetration stage. m , whose final value is the remaining velocity v of the flat-headed projectile r .

4. The method for determining the convex deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-headed bullet according to claim 3, characterized in that: A trial function of the displacement field of the laminate is assumed based on the Ritz method and boundary conditions, and the Lagrangian value L is calculated based on the strain energy Π and the total kinetic energy T of the orthogonal laminate. The total kinetic energy T includes the kinetic energy of the laminate and the kinetic energy of the bullet.

5. The method for determining the convex deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-headed bullet according to claim 4, characterized in that: The thickness h of the orthogonal laminate during the back convex deformation stage with progressive penetration, whose initial value is h2 = h0-h1, where h0 is the initial total thickness of the orthogonal laminate and h1 is the penetration thickness during the local penetration stage; When calculating the thickness of the orthogonal laminate in the back-convex deformation stage accompanied by progressive penetration, a homogenized layered model is performed on the remaining unpenetrated orthogonal laminate, and the progressive penetration process of the orthogonal laminate is predicted based on the mixed mode failure criterion and the tensile failure criterion. The homogeneous layer in contact with the flat-headed bullet is penetrated after the failure criterion is met, and the deflection of the center point and the position of the moving hinge no longer increase. The velocity expression of the flat-head projectile in the local penetration stage is: Where: v0 is the initial velocity of the bullet, and its specific value is v 01 、v 02 ,...,v 0n ; t is the response time; C h is the compression wave propagation velocity in the laminate, The penetration depth in the local penetration stage is obtained by integrating formula (1): Where: t1 is the end time of the local penetration stage; The end time of the local penetration phase is: By combining equations (2) and (3), we can obtain the penetration thickness h1 of the flat-headed projectile in the local penetration stage and the residual velocity of the flat-headed projectile: The orthogonal laminate in the convex deformation stage with progressive penetration is modeled as N homogeneous layers, each of which is composed of [0 / 90] and has a thickness of h. L , then the material mechanical properties parameters of the homogeneous layer are the same as the macroscopic mechanical properties parameters of the laminate.

6. The method for determining the convex deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-headed bullet according to claim 5, characterized in that: The displacement field of the orthogonal laminate in the back convex deformation stage accompanied by progressive penetration can be expressed as follows: Where: w0(t) is the deflection of the center point of the orthogonal laminate, ζ(t) and η(t) are the positions of the moving hinges in the x and y directions, respectively. Field equations for the boundary conditions of the orthogonal laminate.

7. The method for determining the convex deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-headed bullet according to claim 6, characterized in that: The strain energy of the orthogonal laminate at the 1 / 4 back-convex deformation stage accompanied by progressive penetration is: Where: h is the thickness of the orthogonal laminate during the back-convex deformation stage with progressive penetration, and its initial value is h2.

8. The method for determining the convex deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-headed bullet according to claim 7, characterized in that: The total kinetic energy of the orthogonal laminate and flat-head projectile during the convex deformation stage of 1 / 4 progressive penetration is: Where: μ is the surface density of the orthogonal laminate; Lagrangian value: L=T-Π(8) The mixed mode failure criterion is: Where: j represents the jth homogeneous layer in contact with the flat-headed bullet, and takes a value from 1 to N as the flat-headed bullet penetrates layer by layer; is the maximum equivalent stress of the jth layer, is the equivalent stress of the jth layer, and are the normal stress and shear stress of the jth layer respectively.

9. The method for determining the convex deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-headed bullet according to claim 8, characterized in that: The tensile failure criterion is: Where: is the tensile strain of the jth layer.

10. The method for determining the convex deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-headed bullet according to claim 9, characterized in that: After the homogeneous layer in contact with the flat-head bullet penetrates, the thickness of the orthogonal laminate in the back convex deformation stage accompanied by progressive penetration is updated as: h=h2-jh L (11)。

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