Method for determining the back bulging deformation and the ballistic limit velocity of a flat-nosed projectile impacting an orthogonal laminate

Abstract

The present application relates to a kind of flat-nosed projectile impact under orthogonal laminated plate back convex deformation and ballistic limit speed determination method, belong to damage and protection technical field.The present application can determine the transient back convex deformation of UHMWPE laminated plate, moving hinge propagation and ballistic limit speed under the impact of flat-nosed projectile with progressive penetration, without using any fitting parameter and empirical formula.The determination result of the present application has been verified by test, and the calculation result is well consistent with the test result, so the method proposed in the present application has higher determination precision.The present application combines transient analysis and Lagrange method, and introduces progressive penetration failure criterion, and finally determines the back convex deformation of UHMWPE laminated plate and ballistic limit.

Description

TECHNICAL FIELD

[0001] The present application relates to a method for determining the back bulging deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-nosed projectile, and belongs to the technical field of damage and protection. BACKGROUND

[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 speed, which can quickly absorb and disperse energy. A typical cross-ply UHMWPE laminate is composed of a high volume fraction of UHMWPE fiber composite (>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 back bulging deformation stage accompanied by progressive penetration. Back bulging deformation and ballistic limit velocity are important indicators for evaluating the ballistic performance of the laminate. 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 important to develop an analytical model that can quickly and accurately estimate the back bulging deformation and ballistic limit velocity.

[0004] During the ballistic impact process, the back bulging deformation of the laminate is closely related to the moving hinge that propagates 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 use empirical formulas to calculate the evolution of lateral deformation. The empirical formulas often require the combination of experiments or simulations to combine related parameters, resulting in high complexity and insufficient accuracy in predicting back bulging deformation and ballistic limit velocity. SUMMARY

[0005] Therefore, the present application provides a method for determining the back bulging deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-nosed projectile, which can efficiently and accurately predict the transient central deflection, moving hinge position and ballistic limit velocity of a UHMWPE laminate under the impact of a flat-nosed projectile. The term "moving hinge" in the present application refers to the propagation of bending waves in the in-plane direction of the laminate, i.e., the lateral deformation profile of the laminate.

[0006] The present application can accurately predict the transient back bulging deformation, moving hinge propagation and ballistic limit velocity of a UHMWPE laminate during the progressive penetration process of a flat-nosed projectile without using any fitting parameters or empirical formulas.

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

[0008] A method for determining the back bulging deformation and ballistic limit velocity of an orthogonal laminate under the impact of a flat-nosed projectile, the steps of the method comprising:

[0009] In the first step, use the initial velocity v 01 The flat-headed projectile directly impacts 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-nosed projectile is impacting the fixed-support orthogonal laminate n. The impact position of the flat-nosed 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 based on 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-head projectile and the residual velocity v of the flat-head 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-head projectile and the residual velocity v of the flat-head 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-head projectile and the residual velocity v of the flat-head projectile 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 cross-ply laminate p, the fiber x-direction elastic modulus E1 and y-direction elastic modulus E2 of the cross-ply laminate, the cross-ply laminate thickness-direction elastic modulus E3, the in-plane direction Poisson's ratio v 12 , the in-plane direction shear modulus G 12 , the failure stress s f , the failure strain e f ;

[0019] The geometry parameters of the flat-nosed projectile and the cross-ply laminate include the cross-sectional area A of the projectile, the side length a and b of the cross-ply laminate, and the thickness h0 of the cross-ply laminate;

[0020] In the second step, the back-cupping deformation parameters of the cross-ply laminate include the center deflection w0, the moving hinge position zeta in the x-direction, and the moving hinge position eta in the y-direction; the x-direction coincides with the 0° fiber direction of the cross-ply laminate, the y-direction coincides with the 90° fiber direction of the cross-ply laminate, and the z-direction is the thickness direction of the cross-ply laminate; the back-cupping deformation parameters of the cross-ply laminate under the impact of the flat-nosed projectile are calculated by numerically solving the following differential equations:

[0021]

[0022] wherein L is the Lagrange value, and h is the thickness of the cross-ply laminate in the back-cupping deformation stage accompanying the progressive penetration;

[0023] v is the velocity of the flat-nosed projectile in the back-cupping deformation stage accompanying the progressive penetration, the initial value of which is the residual velocity v m of the flat-nosed projectile in the local penetration stage, and the final value of which is the residual velocity v r of the flat-nosed projectile;

[0024] The trial function of the laminate displacement field is assumed based on the Ritz method and the boundary condition, and the Lagrange value L is calculated according to the strain energy p of the cross-ply laminate and the total kinetic energy T of the system (the kinetic energy of the laminate and the kinetic energy of the projectile);

[0025] The thickness h of the cross-ply laminate in the back-cupping deformation stage accompanying the progressive penetration, the initial value of which is h2 = h0 - h1, wherein h0 is the initial total thickness of the cross-ply laminate, and h1 is the penetration thickness in the local penetration stage;

[0026] In the calculation of the thickness h of the cross-ply laminate in the back-cupping deformation stage accompanying the progressive penetration, the residual un-penetrated cross-ply laminate is homogenously and layer-wise modeled, and the progressive penetration process of the cross-ply laminate is predicted according to the mixed mode failure criterion and the tensile failure criterion; the homogenously layered cross-ply laminate in contact with the flat-nosed projectile is penetrated after the failure criterion is met, and the center point deflection and the moving hinge position no longer increase;

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

[0028]

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

[0030] The penetration depth of the local penetration stage can be obtained by integrating equation (1):

[0031]

[0032] wherein: t1 is the termination time of the local penetration stage;

[0033] The termination time of the local penetration stage is:

[0034]

[0035] By combining equations (2) and (3), the penetration thickness h1 of the local penetration stage of the flat-nosed bullet and the residual velocity of the flat-nosed bullet can be obtained:

[0036] The orthogonal laminate in the back bulging deformation stage accompanied by gradual penetration is modeled as N homogeneous layers, each of which is composed of [0 / 90] and has a thickness of h L The material mechanics performance parameters of the homogeneous layer are the same as the macroscopic mechanics performance parameters of the laminate;

[0037] The displacement field trial function expression of the orthogonal laminate in the back bulging deformation stage accompanied by gradual penetration is:

[0038]

[0039] wherein: w0(t) is the deflection of the center point of the orthogonal laminate, ζ(t) and η(t) are the moving hinge positions in the x and y directions, respectively, is the field equation to meet the boundary conditions of the orthogonal laminate;

[0040] The strain energy of the 1 / 4 orthogonal laminate in the back bulging deformation stage accompanied by gradual penetration is:

[0041]

[0042] wherein: h is the thickness of the orthogonal laminate in the back bulging deformation stage accompanied by gradual penetration, and the initial value is h2.

[0043] The total kinetic energy of the 1 / 4 orthogonal laminate in the back bulging deformation stage accompanied by gradual penetration and the flat-nosed bullet is:

[0044]

[0045] where: μ is the face density of the orthotropic laminate;

[0046] Lagrangian value:

[0047] L = T - Π (8)

[0048] The mixed mode failure criterion is:

[0049]

[0050] where: j represents the jth layer of homogeneous layer contacted by the ogive, and the value is 1 to N with the layer-by-layer penetration of the ogive; 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 penetration of the homogeneous layer contacted by the ogive, the thickness of the orthotropic laminate in the back bulging deformation stage with progressive penetration is updated as:

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

[0056] Beneficial effects

[0057] 1. The UHMWPE laminate under the impact of the ogive can be predicted by the present application, and the transient back bulging deformation, moving hinge propagation and ballistic limit velocity are accompanied by progressive penetration, without using any fitting parameters and empirical formula.

[0058] 2. The prediction result of the present application has been verified by experiments, and the prediction result and the experimental result have good consistency, so that the prediction precision of the method proposed by the present application is high.

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

[0060] Figure 1 is the failure process diagram of the UHMWPE orthotropic laminate under the impact of the ogive of the present example;

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

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

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

[0064] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0065] This embodiment is aimed at the flat-head bullet impact brand In the case of HB26 [0 / 90] cross-ply UHMWPE laminate, 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 impacting the clamped orthogonal laminate 1 as an example. 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 2. 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 20 mm Laminate modulus of elasticity E1, E2 51.1 GPa Bullet cross-sectional area A 3.14E4 laminated board modulus of elasticity E3 3.62 Gpa Bullet mass M p ]] 54g Laminate Poisson's ratio v 12 ]]> 0.008 Laminate edge length a 300 mm Laminate shear modulus G 12 ]] 0.19 GPa Laminate thickness h0 10 mm Laminate failure stress s f ]]> 1.5 GPa Laminate density p 980 kg / m 3 ]] Laminate failure strain f ]]> 0.8 Homogeneous layer thickness h L ]]> 0.14 mm

[0071] The second step is based on 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-head projectile and the residual velocity v of the flat-head 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 with progressive penetration is h2 = h0 - h1 = 7 mm, which can be homogenized as a homogeneous layer;

[0074] The orthogonal laminated plate displacement field test function expression of the back bulging deformation stage along with the progressive penetration is:

[0075]

[0076] The strain energy Π of the laminated plate and the total kinetic energy T (the kinetic energy of the laminated plate and the kinetic energy of the bullet) of the system are obtained in combination with the thin plate bending theory, and the Lagrange value L is calculated. The Lagrange equation is brought into the Lagrange equation, and the dynamic response ordinary differential equation set about the center point deflection of the laminated plate and the moving hinge position is derived, and the numerical solution is obtained by using matlab. The time step is set to 0.01 μs and the initial value is set to

[0077] For each time step, D1 and D2 are calculated, when any value is greater than 1, the penetrated layer number j is updated, and the thickness of the remaining complete laminated plate is updated as h=h2-jh L . The dynamic response ordinary differential equation set is re-derived, and the calculation and failure judgment of the next time step are carried out. Finally, the residual speed of the bullet is v r1 =0 m / s, the laminated plate does not penetrate, and the calculation is completed. As shown in Figure 2 and Figure 3 , the center deflection curve and the dimensionless moving hinge curve determined by the embodiment of the application are in good agreement with the experimental results.

[0078] Thirdly, v 01 , v r1 , v 02 , v r2 , …, v 0n , v rn obtained in the second step are fitted to obtain the ballistic limit speed v bl .

[0079] A series of initial speeds in the range of 365 m / s to 1500 m / s are selected as v 01 , v 02 , …, v 0n , and the corresponding residual speeds v r1 , v r2 , …, v rn are calculated. As shown in Figure 4 , the initial-residual speed curve determined by the embodiment of the application is in good agreement with the experimental results. The ballistic limit speed v bl =420 m / s determined by the embodiment of the application, and the ballistic limit speed measured by the experiment is 415 m / s, and the relative error is 1.2%. Therefore, the calculation result of the ballistic limit speed is in good agreement with the experiment, and can meet the needs of engineering application.

[0080] To sum up, the above is only the preferred embodiment of the present application, and is not used to limit the protection scope of the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A method of determining the ogival deformation and ballistic limit velocity of a flat-nosed projectile impacting a cross-ply laminate, characterized by The steps of the method include: Firstly, the flat-nosed projectile with initial velocity of v 0i impinges the clamped orthotropic laminates i, and the impact position is the center of the orthotropic laminates, at this time the orthotropic laminates is penetrated, and the penetration process of the orthotropic laminates is divided into local penetration stage and back bulging deformation stage with gradual penetration; i = 1, 2, 3, …, n; n is the number of orthotropic laminates; Second step, according to the initial velocity v of the first step flat-nosed bullet 0i , the back convex deformation parameters Ci of the flat-nosed bullet impacting the orthogonal laminated board and the residual velocity of the flat-nosed bullet are obtained ri ; i = 1, 2, 3, …, n; n is the number of orthogonal laminated boards; Thirdly, the v 0i , v ri obtained in the second step is fitted to obtain the ballistic limit speed v bl ; The material parameters of the flat-nosed projectile and the orthogonal laminate in the first step include the mass M of the flat-nosed projectile p , the density p of the orthogonal laminate, the elastic modulus E1 and E2 in the x and y directions of the orthogonal laminate, the elastic modulus E3 in the thickness direction of the orthogonal laminate, the Poisson's ratio v in the in-plane direction 12 , the shear modulus G in the in-plane direction 12 , the failure stress s f , and the failure strain e f ​ The flat-nosed projectile and the orthogonal laminated plate geometric parameters include the cross-sectional area A of the projectile, the side lengths a and b of the orthogonal laminated plate, and the thickness h0 of the orthogonal laminated plate; Based on the Ritz method and the boundary condition, the trial function of the displacement field of the laminated plate is assumed, and the Lagrange value L is calculated according to the strain energy Π and the total kinetic energy T of the orthogonal laminated plate, wherein the total kinetic energy T includes the kinetic energy of the laminated plate and the kinetic energy of the projectile; The expression of the displacement field trial function of the orthogonal laminated plate in the back bulging deformation stage accompanying the progressive penetration is: wherein: w0(t) is the central point deflection of the cross-ply, ζ(t) and η(t) are the moving hinge positions in the x and y directions, respectively, field equations that satisfy the boundary conditions of the cross-ply; The strain energy of the orthogonal laminated plate in the back bulging deformation stage accompanying the progressive penetration is: In the formula, h is the thickness of the orthogonal laminated plate in the back bulging deformation stage accompanying the progressive penetration, and the initial value is h2; The total kinetic energy of the orthogonal laminated plate and the flat-nosed projectile in the back bulging deformation stage accompanying the progressive penetration is: In the formula, μ is the surface density of the orthogonal laminated plate; The Lagrange value is: L=T-Π (8) The mixed mode failure criterion is: where: j represents the jth layer of homogeneous layer contacted by the flat-nosed projectile, and takes the value of 1 to N with the layer-by-layer penetration of the flat-nosed projectile; 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.

2. The method according to claim 1, wherein: In the second step, the back bulging deformation parameters of the orthogonal laminated plate include the central deflection w0, the moving hinge position ζ in the x direction, and the moving hinge position η in the y direction; the x direction coincides with the 0° fiber direction of the orthogonal laminated plate, the y direction coincides with the 90° fiber direction of the orthogonal laminated plate, and the z direction is the thickness direction of the orthogonal laminated plate; the back bulging deformation parameters of the orthogonal laminated plate under the impact of the flat-nosed projectile are calculated by numerically solving the following differential equation set: Wherein, L is the Lagrange value, which is related to the thickness h of the orthogonal laminated plate in the back bulging deformation stage accompanying the progressive penetration; is the flat-nosed projectile velocity at the stage of the back bulging deformation accompanying the progressive penetration, the initial value of which is the residual velocity of the flat-nosed projectile at the stage of the local penetration v m , and the final value of which is the residual velocity of the flat-nosed projectile v r .

3. The method according to claim 2, wherein: The thickness h of the orthogonal laminated plate in the back bulging deformation stage accompanying the progressive penetration has an initial value h2=h0-h1, wherein h0 is the initial total thickness of the orthogonal laminated plate, and h1 is the penetration thickness in the local penetration stage; In the calculation of the thickness of the orthogonal laminated plate in the back bulging deformation stage accompanying the progressive penetration, the remaining unpenetrated orthogonal laminated plate is homogenized and layered modeled, and the progressive penetration process of the orthogonal laminated plate is predicted according to the mixed mode failure criterion and the tensile failure criterion; the homogenized layer in contact with the flat-nosed projectile penetrates after satisfying the failure criterion, and the central point deflection and the moving hinge position no longer increase; The expression of the flat-nosed projectile velocity in the local penetration stage is: In the formula: v0 is the initial speed of the bullet, and the specific value is v 01 , v 02 ,..., v 0n ; t is the response time; C h is the compression wave propagation speed in the laminate, The penetration depth in the local penetration stage is obtained by integrating formula (1): In the formula, t1 is the termination time of the local penetration stage; The termination time of the local penetration stage is: By simultaneous equations (2) and (3), the penetration depth h1 of the flat-nosed projectile in the local penetration stage and the residual velocity of the flat-nosed projectile can be obtained The orthotropic laminates with the back-cupping deformation stage accompanying the progressive penetration are modeled as N homogenous layers, each of which is composed of [0 / 90] and has a thickness of h L The material mechanics performance parameters of the homogenous layers are the same as the macro mechanics performance parameters of the laminates.

4. The method according to claim 1, wherein: The tensile failure criterion is: In the formula: is the tensile strain of the jth layer.

5. The method according to claim 4, wherein: After the homogenized layer in contact with the flat-nosed projectile penetrates, the thickness of the orthogonal laminated plate in the back bulging deformation stage accompanying the progressive penetration is updated as: h = h2 - jh L (11)

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