Material parameter replacement method based on equivalent flexural response process of target plate under load impact condition

Abstract

The invention discloses a material parameter replacement method based on equivalent flexural response process of a target plate under a load impact condition, and belongs to the field of structural mechanics and protection. According to the method, a nonlinear differential equation of a target plate under the action of a dynamic load is established, and a parameter equivalent mapping relation between target plates made of different materials under the same boundary condition is deduced in combination with geometric dimensions, material characteristics and boundary conditions. The method specifically comprises a conversion formula of target plate thickness, elastic modulus, density and yield strength, and ensures that a replaced equivalent target plate has a consistent flexural response process with an original target plate under the same load. The method can be applied to the field of aerospace and armored protection material design, the cost of high-value equivalent materials is reduced, the equivalent performance of the target plate is determined, and the design efficiency is improved.

Description

Technical Field

[0001] The present invention belongs to the technical fields of structural mechanics and protection engineering, and particularly relates to a method for replacing material parameters equivalent to the flexural response process under the condition that a target plate is subjected to load impact. Background Art

[0002] In the fields of protection engineering and structural mechanics, the equivalent design of metal target plates is the core issue for improving equipment performance and reducing manufacturing costs. Traditional methods mainly rely on two types of technical routes: one is based on the classical thin plate small deformation theory, and the relationship between equivalent thickness and modulus is deduced through the principle of linear superposition. However, this method ignores the coupling effect of geometric nonlinear effects and material plastic deformation during the flexural process, resulting in a sharp increase in errors of the equivalent formula when the deformation amount exceeds the plate thickness. The other is to use finite element numerical simulation technology, which can partially characterize nonlinear behavior, but there are two major bottlenecks: serious computational time consumption (single working condition analysis takes several hours to several days), and lack of a universal analytical solution, making it difficult to guide the rapid matching of material parameters. More notably, existing technologies have not established an explicit analytical relationship between material parameters (target plate thickness (δ), elastic modulus (E), density (ρ), and yield strength (σ 0 )) and the nonlinear deflection response, forcing engineers to adopt an iterative trial-and-error process of "design - simulation - correction" when replacing materials, resulting in an extended typical R & D cycle. The above technical defects seriously restrict the development of major requirements such as military armor lightweighting and protective structure design. There is an urgent need to construct a parameter conversion theoretical system based on nonlinear mechanical responses, realize the collaborative optimization of material parameters and geometric parameters, meet the consistency requirements of flexural responses under impact loads, and achieve the equivalent design of target plates. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for replacing material parameters equivalent to the flexural response process under the condition that a target plate is subjected to load impact, ensuring that the equivalent target plate after replacement has the same flexural response process as the original target plate under the same load.

[0004] The technical solution for realizing the purpose of the present invention is as follows:

[0005] A method for replacing material parameters equivalent to the flexural response process under the condition that a target plate is subjected to load impact, characterized in that, by combining the geometric dimensions and material properties of the known target plate 1, the equivalent parameters of the dynamic response of the unknown target plate 2 under the same boundary conditions are specifically obtained through the following equivalent formula:

[0006]

[0007] In the formula, δ 2 , ρ 2 , E 2 , σ 02are the thickness, density, elastic modulus and yield strength of the unknown target plate 2; ρ 1 , δ 1 , a 1 , E 1 , v 1 , σ 01 are respectively the density, thickness, half of the side length in the target plate length direction, elastic modulus, Poisson's ratio and yield strength of the known target plate 1.

[0008] Compared with the prior art, the remarkable advantages of the present invention are:

[0009] By establishing a non-linear differential equation of the target plate under dynamic load, combining geometric dimensions, material properties and boundary conditions, the present invention deduces the parameter equivalent mapping relationship between different material target plates under the same boundary conditions, and thus obtains a material parameter replacement method equivalent to the flexural response process of the target plate under load impact conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Figure 1 is the derivation process of this method and the calculation flow of equivalent target parameters.

[0011] Figure 2 is the establishment diagram of the target plate coordinate system.

[0012] Figure 3 is the schematic diagram of the flexural change of the target plate. DETAILED DESCRIPTION OF THE INVENTION

[0013] The following further introduces the present invention with reference to the drawings and specific embodiments.

[0014] By establishing a non-linear differential equation of the target plate under dynamic load, combining geometric dimensions, material properties and boundary conditions, the present invention deduces the parameter equivalent mapping relationship between different material target plates under the same boundary conditions. Specifically, it includes the conversion formulas of the target plate thickness δ, elastic modulus E, density ρ and yield strength σ 0 to ensure that the equivalent target plate after replacement has the same dynamic deflection change D as the original target plate under the same load. The specific steps are as follows:

[0015] Step 1: Establish a flexural response model of the target plate under external load

[0016] Consult relevant literature, and according to the knowledge of elastic-plastic mechanics, establish a coordinate system for the target plate as Figure 2 . Among them, the origin is the center of the target plate, the X-axis is parallel to the target plate length direction, the Y-axis is parallel to the target plate width direction, and the Z-axis is perpendicular to the target surface direction. The target plate will generate deflection deformation under external load, and after deformation, it is as Figure 3As shown in the figure. Under the explosion shock load, the target plate undergoes large deflection deformation. At this time, the membrane force effect plays a dominant role, and the bending moment effect under small deformation can be ignored. According to the actual shape of the target plate deformation, its deformation can be described by the following formula:

[0017]

[0018] In the formula, u, e, and w are the displacements in the X-axis, Y-axis, and Z-axis directions; x, y, and z are the displacement variables in the X-axis, Y-axis, and Z-axis; a is half of the side length of the target plate in the X-axis direction, and b is half of the side length of the target plate in the Y-axis direction; u 0 , e 0 are the maximum displacements in the X-axis and Y-axis directions; D is the deflection of the center point of the target plate in the Z-axis direction.

[0019] 2. Under the plane stress state of the target plate, the strain components in the X-axis and Y-axis directions can be expressed as:

[0020]

[0021] Among them: ε x is the normal strain component of a certain point (x, y) of the target plate on the X-axis, ε y is the normal strain component of a certain point (x, y) of the target plate on the Y-axis, and γ xy is the shear strain component of a certain point (x, y) of the target plate.

[0022] 3. The total potential energy after the target plate deformation includes: the elastoplastic bending strain energy U b , the elastoplastic membrane strain energy U m and the boundary plastic hinge line deformation energy U 1 . The elastoplastic membrane strain energy U m is equal to the sum of the elastic membrane deformation energy U me and the plastic membrane deformation energy U mp , that is, U m =U me +U mp .

[0023] 4. The elastoplastic bending strain U b is as follows:

[0024]

[0025] In the formula, E is the elastic modulus of the target plate material, δ is the thickness of the target plate, and v is the Poisson's ratio of the material.

[0026] 5. The elastic membrane deformation energy U me is calculated by the following formula:

[0027]

[0028] According to the definition of strain energy, the deformation energy U of the plastic film mp is calculated as follows:

[0029]

[0030] where σ 0 is the yield strength of the target plate material, G is the shear modulus of the target plate material, ε p is the maximum elastic strain, the shear strain is the maximum shear stress

[0031] 6. The bending deformation energy U of the plastic hinge line at the four peripheral boundaries 1 can be calculated by the following formula:

[0032]

[0033] In the formula, U 1 is the rotational plastic energy of the boundary plastic hinge line; θ is the rotational angle of the boundary region; M p is the plastic rotational moment per unit length of the target plate.

[0034]

[0035] Calculation gives:

[0036]

[0037] 7. The kinetic energy of the target plate is:

[0038]

[0039] In the formula, ρ is the density of the target plate, is the rate of change of the displacement in the X-axis direction with respect to time, is the rate of change of the displacement in the Y-axis direction with respect to time, is the rate of change of the deflection of the target plate with respect to time.

[0040] 8. Assuming that the explosive shock wave load P(t) acts vertically and uniformly on the entire plate surface, the generalized force is determined by the virtual work ΔW done by the explosive shock wave load P(t) along the Z-axis. Since P(t) acts vertically on the plate surface, the generalized force is only related to the deflection in the Z-axis direction and has nothing to do with the displacements in the X-axis and Y-axis directions. Then

[0041]

[0042] Integrating both sides gives:

[0043]

[0044] Among them, W is the work done by the external explosion shock load on the target plate.

[0045] 9. Simplify the Lagrangian differential equation of motion

[0046] The Lagrangian function is:

[0047] L = U b + U m + U 1 - W

[0048] Among them: U m is the elastic-plastic membrane strain energy, equal to the sum of the elastic membrane deformation energy U me and the plastic membrane deformation energy U mp That is, U m = U me + U mp

[0049] The Lagrangian differential equation of motion is:

[0050]

[0051] t is time.

[0052] 10. Substitute E v and L to solve for the dynamic response equation of the deflection change at the center of the target plate as:

[0053]

[0054] Among them, is the acceleration of the deflection deformation of the target plate.

[0055] Step 2: Conversion of the equivalent formula

[0056] 1. If the dynamic responses of the two target plates are the same, that is, the displacement change D(t) of the deflection with time is the same, then the coefficients of the above differential equation are equal, where D(t) is the value when D is a function of time t. That is:

[0057] For target plate 1:

[0058]

[0059] For target plate 2:

[0060]

[0061] Among them: ρ 1 、δ 1 、a 1 、b 1 、 E 1 、v 1 、P1 (t 1 ),σ 01 , D 1 They are the density, thickness, half of the side length in the X-axis direction, half of the side length in the Y-axis direction, acceleration of flexural deformation, elastic modulus, Poisson's ratio, explosion impact load, yield strength, and deflection change of the target plate 1. 2 , δ 2 、a 2 、b 2 , E 2 、v 2 , P 2 (t 2 ),σ 02 , D 2 They are respectively the density, thickness, half of the side length in the X-axis direction, half of the side length in the Y-axis direction, acceleration of flexural deformation, elastic modulus, Poisson's ratio, explosion impact load, yield strength, and deflection change of the target plate 2.

[0062] 2. According to the target plate equivalent prerequisites: P 1 (t 1 )=P 2 (t 2 ), a 1 =a 2 =b 1 =b 2 , the dynamic response of target plate 1 and target plate 2 should be consistent, that is, the deflection change of target plate 1 and target plate 2 should be consistent, that is, D 1 =D 2 ,have:

[0063] ρ 1 δ 1 a 1 b 1 =ρ 2 δ 2 a 2 b 2

[0064]

[0065] Ignore the influence of the small parameter Poisson's ratio and assume that = v 2 , and then solve the equivalent target parameters to get:

[0066]

[0067] Implementation Cases:

[0068] Assume that the thickness of a square target plate is 20 mm, the side length a is 400 mm, and the density is 2.7 g / cm 3, with an elastic modulus of 72 GPa, a yield strength of 200 MPa, a Poisson's ratio of 0.3, and an ideal fixed boundary condition on all four sides. With the same side length and fixed boundary condition of the target plate, substituting the parameters into the above equivalent formula, the parameters of the dynamic response equivalent target plate of this target plate are as follows:

[0069] The thickness is: 16.5859 mm

[0070] The density is 3.2558 g / cm 3

[0071] The elastic modulus is 86.8208 GPa

[0072] The yield strength is 145.4060 MPa.

Claims

1. A material parameter replacement method based on the equivalent bending response process of a target plate under load impact conditions, characterized in that: Combined with the known geometric dimensions and material properties of the target plate 1, the dynamic response equivalent parameters of the unknown target plate 2 under the same boundary conditions are obtained by the following equivalent formula: In the formula, δ2, ρ2, E2, σ 02 are the thickness, density, elastic modulus and yield strength of the unknown target plate 2; ρ1, δ1, a1, E1, v1, σ 01 The density, thickness, half of the side length in the length direction of the target plate 1, elastic modulus, Poisson's ratio, and yield strength are known respectively.

2. The material parameter replacement method according to claim 1, characterized in that: The equivalent formula establishes a dynamic response equation of the deflection change at the center of the target plate to obtain the dynamic response equations of the known target plate 1 and the unknown target plate 2 respectively. According to the target plate equivalent premise, the dynamic responses of the known target plate 1 and the unknown target plate 2 are made consistent, and the dynamic response equivalent parameters of the unknown target plate 2 are obtained after the combination.

3. The material parameter replacement method according to claim 1, characterized in that: The dynamic response equation is: Where ρ is the density of the target plate, E is the elastic modulus of the target plate material, δ is the thickness of the target plate, v is the Poisson's ratio of the material, a is half the length of the target plate in the length direction, b is half the length of the target plate in the width direction, D is the deflection of the target plate center point perpendicular to the target surface, and p(t) is the explosion shock wave load; The target plate equivalent prerequisites are: p1(t1)=p2(t2), a1=a2=b1=b2, D1=D2; Where P1(t1) and P2(t2) are the explosion impact loads on the known target plate 1 and the unknown target plate 2 respectively, a1 and a2 are half of the length of the side of the known target plate 1 and the unknown target plate 2 respectively, b1 and b2 are half of the length of the side of the known target plate 1 and the unknown target plate 2 respectively, and D1 and D2 are the deflection changes of the known target plate 1 and the unknown target plate 2 respectively.

4. The material parameter replacement method according to claim 3, characterized in that: The dynamic response equation is obtained as follows: Establish the target plate elastic-plastic bending strain U b , elastic-plastic membrane strain energy U m , the rotational plastic performance of the boundary plastic hinge U1, the Lagrangian function of the work W done by the external explosion impact load on the target plate, where the elastic-plastic membrane strain energy U m Equal to the elastic membrane deformation energy U me and plastic membrane deformation energy U mp The sum of U m =U me +U mp . And according to the Lagrangian motion differential equation, the dynamic response equation of the deflection change at the center of the target plate is obtained; The Lagrangian function is: L=U b +U m +U1-W The Lagrangian differential equation of motion is: t is time, E v is the kinetic energy of the target plate, D is the deflection of the target plate center point perpendicular to the target surface, is the rate of change of the target plate deflection with time.

5. The material parameter replacement method according to claim 4, characterized in that: Elastic-plastic bending strain U of target plate b for: Where E is the elastic modulus of the target material, δ is the thickness of the target, and v is the Poisson's ratio of the material.

6. The material parameter replacement method according to claim 4, characterized in that: Elastic-plastic membrane strain energy U m for: Plastic membrane deformation energy U mp for: Where E is the elastic modulus of the target plate material, δ is the thickness of the target plate, v is the Poisson's ratio of the material; σ0 is the yield strength of the target plate material, and G is the shear modulus of the target plate material.

7. The material parameter replacement method according to claim 4, characterized in that: The rotational plastic performance U1 of the boundary plastic hinge is: Where σ0 is the yield strength of the target material.

8. The material parameter replacement method according to claim 4, characterized in that: The work W done by the external explosion impact load on the target plate is: Where P(t) is the explosion shock wave load.

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