A method for predicting dynamic response of foam sandwich panel under near blast loading

Abstract

This invention discloses a method for predicting the dynamic response of foam sandwich panels under near-explosion loading, belonging to the field of explosion-resistant structures with porous foam cores. The method involves: based on the uneven impact of the panel under near-explosion loading, which leads to a strong degree of localization, a deflection surface function reflecting this localization is constructed. The displacement and strain functions of the panel and the core layer are derived using this deflection surface function. The deformation resistance of the panel and core layer is calculated, and the velocity equations for the fluid-structure interaction stage and the motion equations for the core layer compression stage and the structural dynamic response stage are derived. A dynamic model is constructed based on these velocity and motion equations. Given known parameters, the displacement and velocity curves at the center points of the front and rear panels are calculated using the dynamic model, thus achieving the prediction of the dynamic response of the foam sandwich panel. This invention also reduces experimental preparation and site requirements, and improves experimental safety.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of anti-blast structure of porous foam sandwich, and relates to a prediction method for dynamic response of foam sandwich panel under near-blast load. BACKGROUND

[0002] In the research on the anti-blast ability of the porous foam sandwich panel, in order to more simply and effectively judge the anti-blast ability of the sandwich panel, scholars gradually carried out experimental research and theoretical analysis on the dynamic response of the sandwich panel under air blast load. However, the use of explosion experiment to judge the anti-blast ability of the sandwich panel needs a long preparation time and has strict site requirement restrictions, and there are high experimental costs and even safety hazards.

[0003] Zhu Feng et al. carried out experimental research on the square metal sandwich panel with a foam aluminum core under a medium and long distance explosion load, analyzed the structural response, and derived a theoretical formula for the final deformation deflection of the sandwich panel based on the principle of energy conservation using small deflection or large deflection theory. The theoretical formula assumes that the front panel is subjected to uniform impact and the calculation result is that the deflection of the center point of the front panel of the sandwich panel is always greater than that of the back panel, and is not applicable to the dynamic response analysis and prediction of the sandwich panel under near-blast load. SUMMARY

[0004] In order to solve the problems that the traditional theoretical formula can only predict the deformation of the foam sandwich panel under a medium and long distance explosion load, the experimental research on the sandwich panel under near-blast load needs a long preparation time and has strict site requirement restrictions, and there are high experimental costs and even safety hazards. The main purpose of the present application is to provide a prediction method for the dynamic response of a foam sandwich panel under near-blast load. The impact on the panel under near-blast load is non-uniform, resulting in a strong localization degree characteristic. A deflection surface function reflecting the localization degree characteristic is constructed, and the deflection and strain functions of the panel and the deflection and strain functions of the core are derived using the deflection surface function. The deformation resistance of the panel and the core is calculated, the velocity equation of the fluid-structure interaction stage and the motion equation of the core compression stage and the structural dynamic response stage are derived; a dynamic model is constructed based on the velocity equation and the motion equation; given the known parameters, the displacement-time curve and the velocity-time curve of the center points of the front and back panels can be obtained according to the dynamic model, that is, the dynamic response prediction of the foam sandwich panel under near-blast load is realized.

[0005] The purpose of the present application is realized by the following technical solutions.

[0006] The present application discloses a prediction method for the dynamic response of a foam sandwich panel under near-blast load, which comprises the following steps:

[0007] Step 1, a deflection surface function reflecting the localization degree of the panel under near-blast load is established;

[0008] Step 2, obtaining displacement and strain functions of the panel and the core layer by using the warping function;

[0009] Step 3, calculating the deformation resistance of the panel and the core layer; the deformation resistance respectively contains the core layer compression force, the bending resistance and the elongation resistance coefficient of the front and rear panels, and the bending resistance and the elongation resistance coefficient of the core layer;

[0010] Step 4, deriving the velocity equation of the fluid structure interaction stage, and deriving the motion equation of the core layer compression stage and the structure dynamic response stage,

[0011] Step 5, constructing a dynamic model based on the velocity equation and the motion equation obtained in Step 4; given known parameters, the displacement-time curve and the velocity-time curve of the center points of the front and rear panels can be obtained according to the dynamic model, that is, the dynamic response prediction of the foam sandwich panel under the near-blast load is realized;

[0012] The known parameters include: blast load parameters, sandwich panel material parameters and geometric parameters;

[0013] Blast load parameters: specific impulse i / (Pa·s) of the midpoint of the front panel

[0014] Sandwich panel material parameters: panel density ρ f / (kg / m 3 ), panel tensile yield strength σ s / (Pa), core foam density ρ c / (kg / m 3 ), core foam compression yield strength σ c / (Pa), core foam tensile yield strength σ

[0015] Geometric parameters: side length of the sandwich panel 2a / (m) and 2b / (m), front panel thickness h f / (m), foam core layer thickness h c / (m), back panel thickness h b / (m).

[0016] As a preferred, the warping function in Step 1 is:

[0017]

[0018] In the formula, u, v, and w are respectively the displacement of any point in the rectangular panel in the x, y, and z directions, w0 is the displacement of the midpoint of the panel in the z direction; 2a and 2b are the side length of the rectangular panel.

[0019] As preferred, the displacement and strain functions of the panel derived in step 2 are as follows:

[0020] Let the displacement of the front panel center point be Y(t), then the displacement functions of the front panel in each direction are:

[0021]

[0022] The strain functions of the front panel are obtained according to equation (2):

[0023]

[0024] are the normal strain in x direction, the normal strain in y direction and the shear strain in xy plane of the front panel respectively;

[0025] Let the displacement of the back panel center point be y(t), then the displacement functions of the back panel in each direction are:

[0026]

[0027] The strain functions of the back panel are obtained according to equation (4):

[0028]

[0029] are the normal strain in x direction, the normal strain in y direction and the shear strain in xy plane of the back panel respectively;

[0030] Let the displacement of the core center point be the same as the displacement of the back panel center point y(t), then the displacement functions of the core in each direction are:

[0031]

[0032] The strain of the core is obtained according to equation (6):

[0033]

[0034] are the normal strain in x direction, the normal strain in y direction and the shear strain in xy plane of the core respectively.

[0035] As preferred, the deformation resistance of the panel and the core calculated in step 3 are as follows:

[0036] The front panel moves downward to compress the core, and the compression energy of the core is E c :

[0037]

[0038] σ c is the yield strength of the foam compression;

[0039] Core compression energy E c Differentiating Y(t) gives the core compression force Q d :

[0040]

[0041] Plastic twist deformation energy of front panel

[0042]

[0043] where: is the plastic limit moment of the front panel; h f is the front panel thickness, σ s is the panel yield strength;

[0044] The front panel flexural resistance R1 is then:

[0045]

[0046] Extension deformation energy of front panel

[0047]

[0048] The front panel extension resistance coefficient f1 is then:

[0049]

[0050] Plastic twist deformation energy of back panel

[0051]

[0052] where: is the plastic limit moment of the back panel; where h b is the back panel thickness;

[0053] The back panel extension flexural resistance R2 is then:

[0054]

[0055] Extension deformation energy of back panel

[0056]

[0057] The back panel extension resistance coefficient f2 is then:

[0058]

[0059] Plastic twist deformation energy of core

[0060]

[0061] where: is the plastic limit bending moment of the core layer; h c is the foam thickness, is the foam tensile yield strength;

[0062] then the bending resistance of the core layer R c :

[0063]

[0064] the elongation deformation energy of the core layer

[0065]

[0066] then the elongation resistance coefficient of the core layer f c :

[0067]

[0068] As a preference, the initial velocity v0of the midpoint of the front panel in the fluid-structure interaction phase in step 4:

[0069]

[0070] where v0is the initial velocity of the midpoint of the front panel, i is the specific impulse of the midpoint of the front panel; p f is the density of the panel;

[0071] The equivalent mass is solved according to the principle of conservation of kinetic energy:

[0072]

[0073]

[0074]

[0075] where m1, m2, m c are the equivalent masses of the front panel, the back panel and the core layer respectively; p c is the density of the core layer;

[0076] The motion equation of the front panel in the core compression phase:

[0077]

[0078] Initial conditions: where Y2(t) is the displacement of the center point of the front panel in the core compression phase;

[0079] The motion equation of the back panel in the core compression phase:

[0080]

[0081] Initial conditions: where y2(t) is the displacement of the center point of the back panel during the core compression stage;

[0082] Calculate the time t1 at which the velocity of the center point of the front panel is the same as that of the back panel:

[0083]

[0084] Core compression force during the structural dynamic response stage:

[0085] Q d ' = 0.058Q d (31)

[0086] Equation of motion of the front panel during the structural dynamic response stage:

[0087]

[0088] Initial conditions: Y3(t1) = Y2(t1), where Y3(t) is the displacement of the center point of the front panel during the structural dynamic response stage;

[0089] Equation of motion of the back panel during the structural dynamic response stage:

[0090]

[0091] Initial conditions: y3(t1) = y2(t1), where y3(t) is the displacement of the center point of the back panel during the structural dynamic response stage.

[0092] As a preference, in the dynamic model described in step 5, the displacements of the front panel and the back panel are respectively:

[0093]

[0094]

[0095] where t2 and t3 are respectively the time at which the motion of the center points of the front panel and the back panel is terminated.

[0096] Advantages:

[0097] 1. The application discloses a method for predicting the dynamic response of a foam sandwich panel under near-blast loading. The face plate is impacted unevenly under near-blast loading, resulting in strong localization characteristics. A flexural surface function reflecting the localization characteristics is constructed, and the flexural surface function is used to derive the displacement and strain functions of the face plate and the displacement and strain functions of the core layer, and the deformation resistance of the face plate and the core layer is calculated. The velocity equation of the fluid-structure interaction stage and the motion equation of the core compression stage and the structure dynamic response stage are derived. A dynamic model is constructed based on the velocity equation and the motion equation. Given the known parameters, the displacement-time curve and the velocity-time curve of the center points of the front and rear face plates can be obtained according to the dynamic model, that is, the dynamic response of the foam sandwich panel under near-blast loading is predicted. The flexural surface function of the face plate under near-blast loading is established, the displacement and strain functions are established, the influence of the bending moment and membrane force effect of the face plate and the core layer on the deformation is considered, the motion equation is derived to obtain the single-degree-of-freedom rigid-plastic dynamic model, the problem that the dynamic response of the foam sandwich panel under near-blast loading is difficult to predict can be effectively solved, the experimental preparation requirements and site requirements of the sandwich panel under near-blast loading can be reduced, and the safety of the explosion load experiment is improved.

[0098] 2. The application discloses a method for predicting the dynamic response of a foam sandwich panel under near-blast loading. The single-degree-of-freedom rigid-plastic dynamic model allows the deflection of the center point of the front face plate to be smaller than that of the rear face plate and can output the displacement-time curve and the velocity-time curve of the center points of the front and rear face plates, so that the dynamic response of the foam sandwich panel under near-blast loading can be accurately predicted conveniently and quickly. BRIEF DESCRIPTION OF DRAWINGS

[0099] Figure 1 It is a flow chart of the method for predicting the dynamic response of a foam sandwich panel under near-blast loading.

[0100] Figure 2 It is a schematic view of a rectangular face plate and coordinates in step 1.

[0101] Figure 3 It is a schematic view of a sandwich panel in step 2; wherein figure (a) is a three-dimensional schematic view of the sandwich panel, and figure (b) is a sectional view of the sandwich panel.

[0102] Figure 4 It is a schematic view of three stages of the dynamic response of a sandwich panel in step 4.

[0103] Figure 5 It is a schematic view of a dynamic model in step 5.

[0104] Figure 6 Figure (a) in the figure is a finite element model used to extract specific impulse in the example; and figure (b) is a pressure-time curve diagram used to extract specific impulse in the example.

[0105] Figure 7 Fig. (a) is a comparison chart of finite element simulation results and dynamic model calculation results of the center point displacement history curve of the USP-1 sandwich panel in the example; Fig. (b) is a comparison chart of finite element simulation results and dynamic model calculation results of the velocity history curve of the center point of the USP-1 sandwich panel. DETAILED DESCRIPTION

[0106] In order to enable personnel in the technical field to better understand the scheme of the present application, the technical scheme in the embodiments of the present application will be clearly and completely described below in combination with the drawings in the embodiments of the present application.

[0107] EMBODIMENT

[0108] As shown in the Figure 1 The implementation steps of the method for predicting the dynamic response of a foam sandwich panel under near-blast loading are as follows:

[0109] S1: Establish a flexure surface function that can reflect the degree of localization of the faceplate under near-blast loading. The schematic diagram of the rectangular faceplate and the coordinates is as shown in Figure 2 .

[0110]

[0111] S2: Obtain the displacement and strain functions of the faceplate and the displacement and strain functions of the core layer using the flexure surface function. The schematic diagram of the sandwich panel is as shown in Figure 3 .

[0112] The displacement of the front faceplate center point is set as Y(t), and the displacement of the front faceplate in each direction is:

[0113]

[0114] The strain function of the front faceplate is obtained according to formula (2):

[0115]

[0116] The displacement of the back faceplate center point is set as y(t), and the displacement of the back faceplate in each direction is:

[0117]

[0118] The strain function of the back faceplate is obtained according to formula (4):

[0119]

[0120] The displacement of the core layer center point is the same as that of the back faceplate, which is y(t), and the displacement of the core layer in each direction is:

[0121]

[0122] The strain of the core layer is obtained according to formula (6):

[0123]

[0124] S3: Calculate the deformation resistance of the panel and the core layer. The deformation resistance respectively includes the core layer compression force; the bending resistance and elongation resistance coefficient of the front and back panels; and the bending resistance and elongation resistance coefficient of the core layer.

[0125] The front panel moves downward to compress the core layer, and the core layer compression energy E c :

[0126]

[0127] The core layer compression energy E c Derivate Y(t) to obtain the core layer compression force Q d :

[0128]

[0129] The plastic twisted wire bending deformation energy of the front panel

[0130]

[0131] In the formula: The plastic limit bending moment of the front panel.

[0132] Then the bending resistance R1 of the front panel is:

[0133]

[0134] The elongation deformation energy of the front panel

[0135]

[0136] Then the elongation resistance coefficient f1 of the front panel is:

[0137]

[0138] The plastic twisted wire bending deformation energy of the back panel

[0139]

[0140] In the formula: The plastic limit bending moment of the back panel.

[0141] Then the bending resistance R2 of the back panel is:

[0142]

[0143] The elongation deformation energy of the back panel

[0144]

[0145] The back panel elongation resistance coefficient f2 is:

[0146]

[0147] The plastic wire bending deformation energy of the core layer

[0148]

[0149] In the formula: The plastic limit bending moment of the core layer.

[0150] The core layer bending resistance R is: c :

[0151]

[0152] The elongation deformation energy of the core layer

[0153]

[0154] The elongation resistance coefficient f of the core layer is: c :

[0155]

[0156] S4: Derive the velocity equation of the fluid-structure interaction stage, and the motion equation of the core layer compression stage and the structure dynamic response stage. Figure 4 is a schematic diagram of the three stages of the sandwich panel dynamic response in step 4.

[0157] The initial velocity calculation of the front panel midpoint in the fluid-structure interaction stage in step 4:

[0158]

[0159] In the formula, v0 is the initial velocity of the front panel midpoint, and i is the specific impulse of the front panel midpoint.

[0160] According to the principle of kinetic energy conservation, the equivalent mass is:

[0161]

[0162]

[0163]

[0164] In the formula, m1, m2, m cEquivalent mass of front panel, back panel and core layer respectively.

[0165] Equation of motion of front panel in core compression stage:

[0166]

[0167] Initial condition: Y2(0) = 0, where Y2(t) is the displacement of the center point of the front panel in the core compression stage.

[0168] Equation of motion of back panel in core compression stage:

[0169]

[0170] Initial condition: y(0) = 0, where y2(t) is the displacement of the center point of the back panel in the core compression stage.

[0171] In order to get the time t1 when the motion speed of the front panel is the same as that of the back panel, the following equations are solved simultaneously:

[0172]

[0173] Core compression force in structural dynamic response stage:

[0174] Q d ' = 0.058Q d (31)

[0175] Equation of motion of front panel in structural dynamic response stage:

[0176]

[0177] Initial condition: Y3(t1) = Y2(t1), where Y3(t) is the displacement of the center point of the front panel in the structural dynamic response stage.

[0178] Equation of motion of back panel in structural dynamic response stage:

[0179]

[0180] Initial condition: y3(t1) = y2(t1), where y3(t) is the displacement of the center point of the back panel in the structural dynamic response stage.

[0181] S5: Obtain the final dynamic model according to step four. The schematic diagram of the dynamic model is shown in Figure 5 The displacements of the front panel and the back panel are respectively:

[0182]

[0183]

[0184] In the formula, t2 and t3 are the time points at which the motion of the front panel and the back panel, respectively, is terminated.

[0185] S6: input explosion load parameters, sandwich panel material parameters, and geometric parameters.

[0186] A 2D model of ANSYS / AUTODYN was used for finite element simulation, and a fluid-structure coupling algorithm was adopted, with a free outflow boundary set outside the air. In this case, the seven working conditions were all subjected to the loading of 55 g TNT explosion at a blast distance (the distance from the midpoint of the front panel to the center of the explosive) of 100 mm, and the size of the TNT was Φ35 mm x 37.2 mm. In the 2D model, the panel was made of 304 stainless steel, and a measurement point was set at the midpoint of the panel to extract the pressure history curve. The finite element model and the pressure-time curve are shown in FIG. 2. Figure 6 The extracted explosion load parameter, i.e., the specific impulse at the midpoint of the front panel, was 4837 Pa·s.

[0187] In the case, the front and back panels were both made of 304 stainless steel, and the core layer was PVC foam H250. The input sandwich panel material parameters were: panel density ρ f = 7900 kg / m 3 , panel tensile yield strength σ s = 3.1 x 10 8 Pa, sandwich foam density ρ c = 250 kg / m 3 , sandwich foam compressive yield strength σ c = 1.152 x 10 7 Pa, and sandwich foam tensile yield strength σ c e = 9.2 x 10 6 Pa.

[0188] The geometric parameters of the sandwich panel were input as follows: the side length of the sandwich panel 2a = 0.15 m and 2b = 0.144 m. The thicknesses of the front and back panels and the core layer are shown in Table 1.

[0189] S7: output the calculation results. The results output by the working condition USP-1 are as follows:

[0190] The core layer compression force Q d = 1.3143 x 10 5 N, and Q d ' = 7.6228 x 10 3 N. The bending resistance of the front panel R1 = 1.3504 x 10 3 N, and the elongation resistance coefficient of the front panel f1 = 1.4522 x 10 6N / m, bending resistance of the back panel R2 = 1.3504 × 10 3 N, the elongation resistance coefficient of the back panel f2 = 1.4522 × 10 6 N / m, core bending resistance R c =4.1246×10 3 N, elongation resistance coefficient of the core layer f c =4.3722×10 5 N / m, equivalent mass of the front panel m1 = 0.0203 kg, equivalent mass of the back panel m2 = 0.0203 kg, equivalent mass of the core layer m c =0.0065kg, end time of core compression stage t1 = 3.8178 × 10 -5 s, the time when the front panel stops moving t2 = 1.3936 × 10 -4 s, the time when the back panel stops moving, t3 = 2.1277 × 10 -4 s.

[0191] S8: Extract the calculation results to predict the dynamic response of the sandwich panel.

[0192] The dynamic model calculation results of the displacement-time curve and velocity-time curve of the midpoint of the USP-1 sandwich panel were extracted and compared with the finite element simulation results. Figure 7 As shown in the figure, during the rapid displacement increase phase, the results calculated by the dynamic model and the finite element simulation results basically coincide, with the maximum displacement calculated by the dynamic model located at the center of the oscillation range of the finite element simulation results. The velocity history curves of the dynamic model calculation results and the finite element simulation results also show the same pattern. The comparison between the experimental results and the dynamic model calculation results of the midpoint deflection of the front and rear panels under seven working conditions, as well as the errors, are shown in Table 2. The errors are all within 15%, indicating that the dynamic model can predict the dynamic response of the foam sandwich panel under near-explosion loads relatively well.

[0193] Table 1 Experimental and Finite Element Simulation Conditions

[0194]

[0195] Table 2 Comparison of theoretical calculation results and experimental results

[0196]

[0197] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for predicting the dynamic response of a foam cored sandwich panel under near blast loading, characterized by: The method comprises the following steps, Step 1, establishing a flexural surface function capable of reflecting the degree of localization of the panel under near-blast load; The flexural surface function is as follows: In the formula, u, v and w respectively represent the displacement of any point in the rectangular panel in the x, y and z directions, and w0 represents the displacement of the midpoint of the panel in the z direction; 2a and 2b represent the side length of the rectangular panel. Step 2, obtaining the displacement and strain functions of the panel and the displacement and strain functions of the core layer by using the flexural surface function; Step 3, calculating the deformation resistance of the panel and the core layer; the deformation resistance comprises the compression force of the core layer, the bending resistance and elongation resistance coefficients of the front and rear panels, and the bending resistance and elongation resistance coefficients of the core layer; Step 4, deriving the velocity equation of the fluid-structure interaction stage, and deriving the motion equations of the core layer compression stage and the structure dynamic response stage, Step 5, constructing a dynamic model based on the velocity equation and the motion equation obtained in step 4; given known parameters, the displacement-time curve and the velocity-time curve of the center points of the front and rear panels can be obtained according to the dynamic model, that is, the dynamic response of the foam sandwich panel under near-blast load is predicted; The known parameters comprise blast load parameters, sandwich panel material parameters and geometric parameters; The blast load parameters: the specific impulse i / (Pa·s) of the midpoint of the front panel Sandwich panel material parameters: panel density p f (kg / m 3 ), panel tensile yield strength s s (Pa), sandwich foam density p c (kg / m 3 ), sandwich foam compressive yield strength s c (Pa), sandwich foam tensile yield strength s (Pa) Geometric parameters: sandwich panel side length 2a / (m) and 2b / (m), front panel thickness h f (m), foam core layer thickness h c (m), back panel thickness h b (m); In the dynamic model, the displacements of the front panel and the rear panel are respectively as follows: In the formula, t2 and t3 respectively represent the motion termination time of the center points of the front panel and the rear panel.

2. The method according to claim 1, wherein the displacement and strain functions of the panel and the core layer derived in step 2 are as follows: The displacement of the center point of the front panel is set as Y(t), and the displacement functions of the front panel in each direction are as follows: According to formula (2), the strain functions of the front panel are as follows: The displacement of the center point of the rear panel is set as y(t), and the displacement functions of the rear panel in each direction are as follows: respectively, are the normal strain in the x direction, the normal strain in the y direction, and the shear strain in the xy plane of the front panel. According to formula (4), the strain functions of the rear panel are as follows: The displacement of the center point of the core layer is the same as that of the center point of the rear panel, that is, y(t), and the displacement functions of the core layer in each direction are as follows: respectively, are the positive strain in the x direction, the positive strain in the y direction, and the shear strain in the xy plane of the back panel; According to formula (6), the strain of the core layer is as follows:

3. The method according to claim 2, wherein the deformation resistance of the panel and the core layer calculated in step 3 is as follows: are the positive normal strains in the x and y directions and the shear strain in the xy plane, respectively. The bending resistance of the front panel R1 is as follows: The elongation resistance coefficient of the front panel f1 is as follows: The front panel moves downward to compress the core layer, and the core layer compression energy E c : σ c σf is the compressive yield strength of the foam; and E c Q = dY / dt d : Plastic strand bending deformation energy of front panel : wherein: Mplast is the plastic moment of the front panel; h f h is the front panel thickness, σ s is the panel yield strength; The elongation and bending resistance of the rear panel R2 is as follows: Front panel elongation deformation energy : The elongation resistance coefficient of the rear panel f2 is as follows: Plastic strand bending deformation energy of back panel wherein: Mplis the plastic moment of the back panel; wherein h b is the back panel thickness; 4. The method according to claim 3, wherein the initial velocity v0 of the front panel midpoint in the fluid-structure interaction stage in step 4 is as follows: Back panel elongation deformation energy According to the principle of conservation of kinetic energy, the equivalent mass is as follows: Plastic strand bending deformation energy of core layer : wherein: Mplis the plastic limit moment of the core; h c h is the foam thickness, σy is the tensile yield strength of the foam; R = 2.5 * (t + 2 * h) / t c : elongation deformation energy of the core layer : then the elongation resistance coefficient f of the core layer is c : The motion equation of the front panel in the core layer compression stage is as follows: The motion equation of the rear panel in the core layer compression stage is as follows: where v0 is the initial velocity of the panel midpoint, i is the specific impulse of the panel midpoint; p f is the panel density; The time t1 at which the motion velocity of the center point of the front panel is the same as that of the rear panel is calculated as follows: where m1, m2, m c are the equivalent masses of the front panel, back panel and core layer, respectively; p c is the core layer density; The core layer compression force in the structure dynamic response stage is as follows: Initial conditions: where Y2(t) is the displacement of the front panel center point during the core compression phase. The motion equation of the front panel in the structure dynamic response stage is as follows: Initial conditions: y(0) = 0, where y2(t) is the displacement of the center point of the back panel during the core compression phase. The motion equation of the rear panel in the structure dynamic response stage is as follows: ​ Q' d = 0.058Q d (31) ​ Initial condition: Y3(t1) = Y2(t1), where Y3(t) is the displacement of the center point of the front panel in the structural dynamic response stage. ​ Initial conditions: y3(t1) = y2(t1), where y3(t) is the displacement of the center point of the back panel in the structural dynamic response phase.

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