A Simplified Method for Evaluating the Dynamic Response of the Metal Sandwich Structure under Explosion Load

Abstract

The metal sandwich structure is characterized by light weight, high stiffness, and high-impact energy absorption capacity, which is considered a good alternative protective structure in explosion scenarios. In this paper, four design forms of the metal sandwich structure are introduced, and the dynamic response of the metal sandwich structure under explosion load is researched. Then, a new analytical method is proposed to predict the dynamic response of the metal sandwich structure under explosion load, and numerical simulations are carried out to verify the accuracy of the proposed analysis method. The proposed analytical method is designed for aerial explosions without considering the fluid–structure interaction. In addition, a series of parameter studies are carried out, including the influence of the face sheet aspect ratio, the core height, and the thickness of the face sheet and core on the dynamic response of the sandwich structure under explosion load. The research in this paper has reference value for the anti-explosion design of the metal sandwich structure and will provide a useful reference for the design and optimization of the metal sandwich structure.

1. Introduction

The metal sandwich structure has the advantages of high stiffness, light weight, good impact resistance, good vibration, and noise reduction, and is more and more widely used in the construction of hulls and superstructures in the naval industry. A sandwich structure usually consists of two face sheets with a core in the middle. The material of the core structure can be made of metal, wood, plastic, or composite material, and the structure of the core can be net, corrugated, lattice, or honeycomb [1,2]. Sandwich panels are considered a good alternative to steel panels for offshore platforms or ships, which can enhance the anti-explosion performance of the structure.

Many scholars have studied the dynamic response of the sandwich structure under explosion load. Cheng et al. [3] treated the 3D sandwich panel as an equivalent 2D orthotropic panel and derived equivalent stiffness properties of sandwich structures with various types of cores by using the finite element analysis method. Tilbrook et al. [4] studied the crashworthiness of sandwich structures with the impact velocity ranging from quasi-static to 200 m/s and found that the stress of the rear plate basically remains unchanged when the impact velocity increases. Zhu et al. [5] carried out a series of experimental studies on the impact resistance of the honeycomb core sandwich structure and conducted related numerical simulations. Ebrahimi and Vaziri [6] studied the mechanical response and fracture of metal sandwich panels subjected to multiple impulsive pressure loads. Hou et al. [7] exemplified corrugated sandwich panels with trapezoidal and triangular cores to determine the relationship between the structural parameters and the crashworthiness under low-velocity local impact and planar impact. Li et al. [8] investigated the dynamic response of corrugated sandwich panels under air blast loading by using a ballistic pendulum system, and the residual deflection of the back face sheet and the deformation/failure modes of the sandwich panel under different impulse levels were analyzed. The dynamic response of metallic trapezoidal corrugated core sandwich panels under air blast loading was studied by Zhang et al. [9] Their study mainly focused on the influences of several parameters on the anti-explosion performance, for example, the stand-off distance, face sheet thickness, core web thickness, core height, and corrugation angle. Xia et al. [10] found that the spaced and connected tube-core sandwich structures have their own advantages under close range and contact explosion, respectively. The tube-core sandwich panel is considered as a promising anti-explosion structure, which has the advantages of low price, convenient welding, and good anti-explosion performance. Xiang et al. [11] pointed out that the sandwich beam with a smaller diameter of the tube-core can bear a large explosion load through the experiment and numerical simulation of the circular and square tube-core sandwich beams. Zhang et al. [12] investigated the dynamic response of I-core sandwich panels under combined blast and fragment loading by the LS-DYNA software, and the results demonstrated that the damage caused by combined blast loading was more severe than that by bare blast loading.

The dynamic response of the complex structure under explosion load can be obtained by numerical analysis and model tests, but numerical analysis needs a lot of time. Therefore, some scholars pay more attention to the simplified analysis method. As a fast evaluation method, the simplified analysis method has reference value for the design of protective structures. Fleck and Deshpande [13] proposed an analytical method for calculating the blast resistance of clamped sandwich beams. Qiu et al. [14] used the analytical formulas to determine optimal geometries of the sandwich plates that maximize the shock resistance of the plates for a given mass, and the optimization reveals that sandwich plates have a superior shock resistance relative to monolithic plates of the same mass. Qin and Wang [15] studied the dynamic response of a fully clamped metallic sandwich beam under impulsive loading and found that core strength and membrane force induced by large deflections have significant effects on the dynamic response of sandwich beams, increasing the transient deflections. Yuan et al. [16] derived the analytical solution for the large deflection of a fully clamped metal sandwich beam under impulsive loading, and the analytical formulae are used to determine the optimal geometries of a sandwich beam that maximize the resistance to impulsive loading at a given mass.

In this paper, four design forms of the metal sandwich structure are proposed, and the dynamic response of the metal sandwich structure under explosion load is investigated. An innovative simplified analysis method based on the energy dissipation theory and the assumption of a rigid body–completely plastic material model is proposed to predict the dynamic response of the metal sandwich structure under explosion load. The simplified analysis method allows providing results within a much faster time than with numerical modeling, so it is very useful for optimization of structures, the overall goal. The proposed analytical method is designed for aerial explosions without considering the fluid–structure interaction. I-type, U-type, V-type, and Uc-type folding sandwich panels are mainly considered, and the responses of the above four types of folding sandwich panels can be obtained by analytical methods. In the analytical method, the sandwich panel is equivalent to a homogeneous filled sandwich, and the characteristics of different forms of sandwiches are personalized by the equivalent density of the sandwich, which is the originality of the analytical method. The analytical method ignores the deformation effect of the sandwich, and the accuracy of the simplified analysis method is verified by numerical simulation. Moreover, a series of parameter studies are carried out, including the influence of the face sheet aspect ratio, the core height, and the thickness of the face sheet and core, and some useful conclusions are obtained. The research in this paper has reference value for the anti-explosion design and optimization of the metal sandwich structure.

2. Simplified Analysis Method

2.1. Typical Metal Sandwich Structure

The metal sandwich structure consists of two face sheets with a core in the middle. According to the core manufacturing method, it can be divided into two categories: a discrete sandwich panel and a continuous sandwich panel. The core of the discrete sandwich panel is independent, while the core of the continuous sandwich panel is a whole structure manufactured by bending forming.

In this study, four typical metal sandwich structures were selected as research objects, namely I-type, V-type, U-type, and Uc-type sandwich structures, as shown in Figure 1. The core of I-type and U-type sandwich structures is discrete structure, while the core of V-type and Uc-type sandwich structures is continuous structure. The structural parameters are shown in Figure 1, where h_c is the height, t_f and t_c are the thickness of the face sheet and core, d_f is the distance of the core in the upper face sheet, and d_p is the distance of the core in the lower face sheet.

2.2. The Dynamic Response Process of the Sandwich Structure under Explosion Load

According to the observation of the experiment, the response process of the sandwich structure can be divided into three stages [13,14,17]. Stage I is the initial stage, when the upper face sheet is activated, and the core and back face sheet remain stationary. Then, the core is compressed, and the velocity of the upper face sheet decreases gradually due to the reaction force of the core. Although Figure 2a shows a spherical wave, it is actually the deformation of the sandwich plate under far-field explosion load by default. In the background of far-field explosion, it is considered that the spherical wave reaching the structure surface can be approximately regarded as a plane wave, so the sandwich plate surface is subjected to a uniformly distributed impact load. Stage II is the buckling stage until the upper face sheet and the core reach the same velocity, and energy is dissipated by buckling. In Stage III, the whole sandwich structure undergoes plastic bending deformation, and the impact energy is gradually dissipated. The general description of the above three stages is shown in Figure 2.

2.3. Analytical Prediction Method for Dynamic Response of Metal Sandwich Structure under Explosion Load

2.3.1. Stage I: Initial Speed of the Upper Face Sheet

In general, the sandwich structure will undergo elastic deformation and plastic deformation when it is subjected to explosion load. The degree of elastic deformation is much smaller than that of plastic deformation. The proposed analytical method is designed for aerial explosions without considering the fluid–structure interaction. In addition, the action time of the explosion load is short, and the structure begins to deform in the plastic range. Therefore, it is reasonable to use a rigid plastic material model and ignore the elastic deformation.

When a fully clamped metal sandwich structure is subjected to impulse wave load in the vertical direction, the incident impulse per unit area can be expressed as Equation (1) [18]:

$$I_{+}=A_i\frac{\sqrt[3]{m_e^2}}{r}$$

(1)

where, $I_{+}$ is the incident positive impulse per unit area, A_i is the coefficient, A_i = 200~250, m_e is the Trinitrotoluene (TNT) equivalent weight, and r is the perpendicular distance between the detonation source and the sandwich structure.

When an explosion wave passes through the sandwich panel, the incident and reflected impulses have the following proportions [19]:

$$I=2I_{+}$$

(2)

where, I is the reflected impulse.

Under the action of impulse, the face sheet has an initial velocity, and the initial velocity of the upper face sheet in Stage I can be obtained:

$$v_i=\frac{I}{{\rho }_f{t}_f}$$

(3)

where, $v_i$ is the initial velocity of the upper face sheet in Stage I, ${\rho }_f$ is the density of the sandwich panel, and t_f is the thickness of the upper face sheet.

According to the law of kinetic energy, when the influence of the boundary is ignored, the initial kinetic energy of the sandwich structure can be obtained:

$$E_i=\frac{1}{2}S{\rho }_f{t}_f{v}_i^2=\frac{I^{2}S}{2{\rho }_f{t}_f}$$

(4)

where, $E_i$ is the initial kinetic energy of the sandwich structure, and S is the area of the upper face sheet.

2.3.2. Stage II: The Compression of the Core

In Stage II, the core of the sandwich structure begins to be compressed. At the same time, due to the reaction force of the core, the speed of the upper face sheet gradually decreases, while the speed of the core and the back face sheet gradually increases. After the core compression process is completed, the face sheet reaches the same speed as the core. According to the law of momentum conservation, the speed of the structure, $v_k$, can be obtained:

$$v_k=\frac{{\rho }_f{t}_f{v}_i}{2{\rho }_f{t}_f+{\rho }_c{t}_c}=\frac{I}{2{\rho }_f{t}_f+{\rho }_c{t}_c}$$

(5)

where, ${\rho }_c$ is the equivalent density of the core, t_c is the thickness of the core, ${\rho }_c=\overline{\rho }\cdot {\rho }_f$, $\overline{\rho }$ is the relative density coefficient, and h_c is the height of the core.

For the four typical sandwich panels proposed in this paper, as shown in Figure 1, the relative density coefficient values are different, which are summarized in

Table 1. Relative density coefficients.

Type	Relative Density Coefficient
I-type	$\overline{\rho }=t_c/d_p$
U-type	$\overline{\rho }=\left[t_c\left(d_p-d_f\right)\right]/\left[h_c\left(d_p+d_f\right)\cos \theta \right]$
V-type	$\overline{\rho }=t_c/\left(h_c\cos \theta \right)$
Uc-type	$\overline{\rho }=\left[t_c\left(2d_f\cos \theta +d_p-d_f\right)\right]/\left[h_c\left(d_p+d_f\right)\cos \theta \right]$

.

Similar to the initial kinetic energy (E_i), the kinetic energy of the whole structure (E_k) can be obtained when the speed reaches the same (v_k):

$$E_k=\frac{1}{2}S\left(2{\rho }_f{t}_f+{\rho }_c{t}_c\right)v_k^2=\frac{I^{2}S}{2\left(2{\rho }_f{t}_f+{\rho }_c{t}_c\right)}$$

(6)

Moreover, during the compression process, part of the shockwave energy is absorbed by the sandwich structure and transformed into the plastic deformation energy of the sandwich structure. The energy absorbed by the sandwich structure, $E_a$, can be described as:

$$E_a=E_i-E_k$$

(7)

The deformation of the rectangular sandwich structure, $w\left(x,y\right)$, after compression can be assumed as follows:

$$w\left(x,y\right)=w_{mn}\sin \left(\frac{\pi }{2}+\frac{m\pi x}{a}\right)\sin \left(\frac{\pi }{2}+\frac{n\pi y}{b}\right)$$

(8)

The relationship between the absorbed energy, $E_s$, and the compression deformation can be obtained as:

$$\begin{array}{l}{E}_s={\sigma }_c{\displaystyle {\int }_{-a/2}^{a/2}{\displaystyle {\int }_{-b/2}^{b/2}{w}_{mn}\sin \left(\frac{\pi }{2}+\frac{m\pi x}{a}\right)\sin \left(\frac{\pi }{2}+\frac{n\pi y}{b}\right)}}dxdy\\ =w_{mn}\frac{4ab}{{\pi }^2}{\sigma }_c\end{array}$$

(9)

where, a is the length of the structure, and b is the width of the structure.

Assuming that the absorbed energy is completely transformed into deformation energy, then:

$$E_a=E_s=w_{mn}\frac{4ab}{{\pi }^2}{\sigma }_c$$

(10)

where, $E_a$ is the deformation energy, ${\sigma }_c=\tilde{\rho }\cdot {\sigma }_f$ is the transverse compressive strength of the core, $\tilde{\rho }$ is the yield strength coefficient of the material, and ${\sigma }_f$ is the yield strength of the material.

For different panel configurations, different yield strength coefficient values are given as follows:

$$\tilde{\rho }=\left\{\begin{array}{l}\hspace{1em}\hspace{1em}\overline{\rho }, \hspace{1em}\hspace{1em}\mathrm{For} \mathrm{I}, \mathrm{U} \mathrm{and} \mathrm{V} \mathrm{type} \mathrm{panel}\\ \frac{t_c\left(d_p-d_f\right)}{h_c\left(d_p+d_f\right)\cos \theta }, \mathrm{For} \mathrm{Uc} \mathrm{type} \mathrm{panel}\end{array}\right.$$

(11)

where, d_f and d_p are the distances of a unit core on the upper and rear face sheet, respectively, as shown in Figure 1.

Integrating the above equations, the compression of the core can be obtained as follows:

$$w_{mn}=\frac{{\pi }^2}{4ab{\sigma }_c}\left(E_i-E_k\right)$$

(12)

2.3.3. Stage III: Plastic Deformation of the Metal Sandwich Structure

(1)

Plastic deformation mode

The rectangular metal sandwich structure consumes the kinetic energy after the core crushing through plastic bending and stretching, due to the orthotropic nature of the metal corrugated sandwich structure, so it is difficult to determine the plastic deformation mode. Therefore, for the rectangular sandwich panel with the core parallel to the short side direction, to analyze its overall deformation under explosion load, the overall plastic deformation mode is assumed to refer to [18], and the previously proposed plastic deformation mode of the entire panel is adopted in our research, as shown in Figure 3 [18].

The four sides of the sandwich panel are rigidly fixed. Under the action of the explosion load, the deformed sandwich panel is divided into four rigid areas, namely two rigid areas I and two rigid areas II. There are plastic hinge lines between each rigid region and between each rigid region and the boundary. At the same time, it is considered that the deformation mode does not change with time during the deformation process of the sandwich panel. It can be seen from Figure 3 that for the fully clamped metal sandwich structure, the plastic deformation only occurs at the plastic hinge. The other parts of the structure are considered ideally rigid and are divided into four regions, named Region I and Region II, respectively, which are in different directions. To describe the deformation of the structure, the dimensionless factor $\xi =b/a$ is defined as the aspect ratio of the structure. Thus, the angle, $\phi$, can be obtained by the following equation:

$$\tan \phi =\sqrt{3+{\xi }^2}-\xi$$

(13)

The transverse deformation, $w_i$, can be calculated by two regions, respectively:

Region I:

$$w_i=\frac{\left(b/2\right)\tan \phi -x^{\prime }}{\left(b/2\right)\tan \phi }w=\frac{b\tan \phi -2x^{\prime }}{b\tan \phi }w$$

(14)

Region II:

$$w_i=\frac{\left(b/2\right)-y}{\left(b/2\right)}w=\frac{b-2y}{b}w$$

(15)

(2)

Energy dissipation of the structure

Considering the effect of the bending moment and membrane force of the metal sandwich structure, the internal energy dissipation per unit length of the hinge is [17]:

$$D=\left(M+N{w}_i\right){\theta }_i$$

(16)

where, D is the internal energy dissipation per unit length of the hinge, M is the bending moment per unit length, N is the membrane force, ${\theta }_i$ is the angle of the hinge i, and w_i is the transverse deformation of the hinge i.

The virtual energy of the internal force of the sandwich structure in the process of dynamic plastic deformation is:

$$E_p={\displaystyle \sum _{i=1}^n{\displaystyle {\int }_{l_i}\left(M+N{w}_i\right){\theta }_{i}d{l}_i}}$$

(17)

where, $E_p$ is the virtual energy of the internal force of the sandwich structure, l_i is the length of the hinge i, and n is the number of hinges.

The total energy dissipation at each plastic hinge can be expressed as:

$$\begin{array}{l}{E}_p=2{\displaystyle {\int }_{l_{AB}}\left(M+N{w}_i\right){\theta }_{AB}d{l}_{AB}}+2{\displaystyle {\int }_{l_{AD}}\left(M+N{w}_i\right){\theta }_{AD}d{l}_{AD}}\\ \hspace{1em} +4{\displaystyle {\int }_{l_{AE}}\left(M+N{w}_i\right){\theta }_{AE}d{l}_{AE}}+{\displaystyle {\int }_{l_{EF}}\left(M+N{w}_i\right){\theta }_{EF}d{l}_{EF}}\end{array}$$

(18)

Moreover, ${\theta }_i$ is defined by the following functions:

$$\begin{array}{c}{\theta }_{AB}=\frac{2w}{b\tan \phi }, {\theta }_{AD}=\frac{2w}{b}, {\theta }_{EF}=\frac{4w}{b}\\ {\theta }_{AE}={\theta }_{AB}\cos \phi +{\theta }_{AD}\sin \phi =\frac{2w}{b\sin \phi }\end{array}$$

(19)

The energy consumption of each hinge is:

$$\begin{array}{l}{E}_{AB}={\displaystyle {\int }_{l_{AB}}\left(M+N{w}_i\right){\theta }_{AB}d{l}_{AB}}\\ \hspace{1em} ={\displaystyle {\int }_{l_{AB}}M\frac{2w}{b\tan \phi }d{l}_{AB}}\\ \hspace{1em} =\frac{2Mw}{\tan \phi }\end{array}$$

(20)

$$\begin{array}{l}{E}_{AD}={\displaystyle {\int }_{l_{AD}}\left(M+N{w}_i\right){\theta }_{AD}d{l}_{AD}}\\ \hspace{1em} ={\displaystyle {\int }_{l_{AD}}M\frac{2w}{b}d{l}_{AD}}\\ \hspace{1em} =\frac{2aMw}{b}\end{array}$$

(21)

$$\begin{array}{l}{E}_{AE}={\displaystyle {\int }_{l_{AE}}\left(M+N{w}_i\right){\theta }_{AE}d{l}_{AE}}\\ \hspace{1em} ={\displaystyle {\int }_{l_{AE}}\left(M+N\frac{b\tan \phi -2x^{\prime }}{b\tan \phi }\right)\frac{2w}{b\sin \phi }d{l}_{AE}}\\ \hspace{1em} ={\displaystyle {\int }_0^{\frac{b}{2}\tan \phi }\left[\frac{2Mw}{b\sin \phi }+Nw\frac{2w\left(b\tan \phi -2x^{\prime }\right)}{b^2\tan \phi \sin \phi }\right]\frac{d{x}^{\prime }}{\sin \phi }}\\ \hspace{1em} =\frac{Mw}{\sin \phi +\cos \phi }+\frac{N{w}^2}{2\sin \phi \cos \phi }\end{array}$$

(22)

$$\begin{array}{l}{E}_{EF}={\displaystyle {\int }_{l_{EF}}\left(M+N{w}_i\right){\theta }_{EF}d{l}_{EF}}\\ \hspace{1em} ={\displaystyle {\int }_{l_{EF}}\left(M+Nw\right)\frac{4w}{b}d{l}_{EF}}\\ \hspace{1em} =\frac{4w}{b}\left(M+Nw\right)\left(a-b\tan \phi \right)\end{array}$$

(23)

Integrating the above equations, the total dissipation energy of each hinge can be obtained as:

$$\begin{array}{l}{E}_p=2\cdot \frac{2Mw}{\tan \phi }+2\cdot \frac{2aMw}{b}+4\left(\frac{Mw}{\sin \phi +\cos \phi }+\frac{N{w}^2}{\sin \phi \cos \phi }\right)+\frac{4w}{b}\left(M+Nw\right)\left(a-b\tan \phi \right)\\ \hspace{1em} =\left(\frac{4a}{b}+\frac{2\cos \phi }{\sin \phi }-\tan \phi \right)N{w}^2+\left(\frac{8a}{b}+\frac{8\cos \phi }{\sin \phi }\right)Mw\end{array}$$

(24)

(3)

Yield function

The yield surface condition depends on the cross-sectional shape, strength, and thickness of the skin and the core. Jones [20] proposed a yield theory to estimate the permanent transverse deflections of beams and arbitrarily shaped plates subjected to large dynamic loads, and the yield condition follows the following equation:

$$\frac{\left|M\right|}{M_0}+\frac{\left|N\right|}{N_0}=1$$

(25)

where, $M_0$ is the maximum bending moment and $N_0$ is the maximum membrane force.

The maximum bending moment and the maximum membrane force of the metal sandwich structure in two directions are calculated as follows:

For the sandwich structure with discrete panels, the maximum bending moment, $M_{0x}$, perpendicular to the X-axis (the direction of the core) is:

$$M_{0x}={\sigma }_f{t}_f\left[\left(h_c-w_{mn}\right)+t_f\right]$$

(26)

The maximum membrane force, $N_{0x}$, is:

$$N_{0x}=2{\sigma }_f{t}_f$$

(27)

In the direction parallel to the core, the maximum bending moment, $M_{0y}$, is:

$$M_{0y}={\sigma }_f{t}_f\left[\left(h_c-w_{mn}\right)+t_f\right]+{\sigma }_c{\left(h_c-w_{mn}\right)}^2/4$$

(28)

In addition, the maximum membrane force, $N_{0y}$, is:

$$N_{0y}=2{\sigma }_f{t}_f+{\sigma }_c\left(h_c-w_{mn}\right)$$

(29)

In contrast, for the sandwich structure with a continuous core, the maximum bending moment and the maximum membrane force perpendicular to the X-axis (the direction of the core) are the same as those of the above-mentioned configuration. In the direction parallel to the core, the maximum bending moment, $M_{0y}$, is:

$$M_{0y}={\sigma }_f{t}_f\left[\left(h_c-w_{mn}\right)+t_f\right]+\frac{{\sigma }_c{\left(h_c-w_{mn}\right)}^2}{4}+{\sigma }_f{t}_f(\frac{d_f}{d_f+d_p})\left[\left(h_c-w_{mn}\right)+t_f\right]$$

(30)

In addition, the maximum membrane force, $N_{0y}$, is:

$$N_{0y}=2{\sigma }_f{t}_f+{\sigma }_c\left(h_c-w_{mn}\right)+2{\sigma }_f{t}_c\frac{d_f}{d_f+d_p}$$

(31)

For the dynamic response of the sandwich structure, to obtain its deformation as accurately as possible, it is difficult to obtain the closed form solution of the final deformation. The circumscribed yield square and the inscribed yield square of the yield function are used as the yield conditions to solve the problem, as shown in Figure 4.

Circumscribed yield square:

$$\left|N\right|=N_0, \left|M\right|=M_0$$

(32)

Inscribed yield square:

$$\left|N\right|=0.5N_0, \left|M\right|=0.5M_0$$

(33)

Theoretically, taking the final deformation of the circumscribed yield square as the yield function of the structure is underestimated, but using the external yield square will be higher than the actual deformation. Therefore, the average value is taken as the final calculated value of the sandwich structure to reduce the deviation.

Using the circumscribed yield square as the yield surface, the total dissipation is:

$$E_p=\left(\frac{4a}{b}+\frac{2\cos \phi }{\sin \phi }-\tan \phi \right)N_0{w}^2+\left(\frac{8a}{b}+\frac{8\cos \phi }{\sin \phi }\right)M_{0}w$$

(34)

Using the inscribed yield square as the yield surface, the total dissipation is:

$$E_p=\left(\frac{2a}{b}+\frac{\cos \phi }{\sin \phi }-\frac{\tan \phi }{2}\right)N_0{w}^2+\left(\frac{4a}{b}+\frac{4\cos \phi }{\sin \phi }\right)M_{0}w$$

(35)

The initial kinetic energy is consumed in the whole deformation process, that is:

$$E_p=E_k$$

(36)

Substitute Equations (36) and (37) into (38), respectively, then:

$$\left(\frac{4a}{b}+\frac{2\cos \phi }{\sin \phi }-\tan \phi \right)N_0{w}^2+\left(\frac{8a}{b}+\frac{8\cos \phi }{\sin \phi }\right)M_{0}w=E_k$$

(37)

$$\left(\frac{2a}{b}+\frac{\cos \phi }{\sin \phi }-\frac{\tan \phi }{2}\right)N_0{w}^2+\left(\frac{4a}{b}+\frac{4\cos \phi }{\sin \phi }\right)M_{0}w=E_k$$

(38)

The following values can be obtained:

$$w_1=\frac{\sqrt{{\left[\left(\frac{8a}{b}+\frac{8\cos \phi }{\sin \phi }\right)M_0\right]}^2+4E_k{N}_0\left(\frac{4a}{b}+\frac{2\cos \phi }{\sin \phi }-\tan \phi \right)}-\left(\frac{8a}{b}+\frac{8\cos \phi }{\sin \phi }\right)M_0}{2N_0\left(\frac{4a}{b}+\frac{2\cos \phi }{\sin \phi }-\tan \phi \right)}$$

(39)

$$w_1=\frac{\sqrt{{\left[\left(\frac{4a}{b}+\frac{4\cos \phi }{\sin \phi }\right)M_0\right]}^2+4E_k{N}_0\left(\frac{2a}{b}+\frac{\cos \phi }{\sin \phi }-\frac{\tan \phi }{2}\right)}-\left(\frac{4a}{b}+\frac{4\cos \phi }{\sin \phi }\right)M_0}{2N_0\left(\frac{2a}{b}+\frac{\cos \phi }{\sin \phi }-\frac{\tan \phi }{2}\right)}$$

(40)

Substitute the parameters M_0x, N_0x, M_0y, and N_0y into Equations (41) and (42), respectively, and remove the negative values, then:

$$w_{1x}=\frac{\sqrt{{\left[\left(\frac{8a}{b}+\frac{8\cos \phi }{\sin \phi }\right)M_{0x}\right]}^2+4E_k{N}_{0x}\left(\frac{4a}{b}+\frac{2\cos \phi }{\sin \phi }-\tan \phi \right)}-\left(\frac{8a}{b}+\frac{8\cos \phi }{\sin \phi }\right)M_{0x}}{2N_{0x}\left(\frac{4a}{b}+\frac{2\cos \phi }{\sin \phi }-\tan \phi \right)}$$

(41)

$$w_{1y}=\frac{\sqrt{{\left[\left(\frac{8a}{b}+\frac{8\cos \phi }{\sin \phi }\right)M_{0y}\right]}^2+4E_k{N}_{0y}\left(\frac{4a}{b}+\frac{2\cos \phi }{\sin \phi }-\tan \phi \right)}-\left(\frac{8a}{b}+\frac{8\cos \phi }{\sin \phi }\right)M_{0y}}{2N_{0y}\left(\frac{4a}{b}+\frac{2\cos \phi }{\sin \phi }-\tan \phi \right)}$$

(42)

$$w_{2x}=\frac{\sqrt{{\left[\left(\frac{4a}{b}+\frac{4\cos \phi }{\sin \phi }\right)M_{0x}\right]}^2+4E_k{N}_{0x}\left(\frac{2a}{b}+\frac{\cos \phi }{\sin \phi }-\frac{\tan \phi }{2}\right)}-\left(\frac{4a}{b}+\frac{4\cos \phi }{\sin \phi }\right)M_{0x}}{2N_{0x}\left(\frac{4a}{b}+\frac{2\cos \phi }{\sin \phi }-\tan \phi \right)}$$

(43)

$$w_{2y}=\frac{\sqrt{{\left[\left(\frac{4a}{b}+\frac{4\cos \phi }{\sin \phi }\right)M_{0y}\right]}^2+4E_k{N}_{0y}\left(\frac{2a}{b}+\frac{\cos \phi }{\sin \phi }-\frac{\tan \phi }{2}\right)}-\left(\frac{4a}{b}+\frac{4\cos \phi }{\sin \phi }\right)M_{0y}}{2N_{0y}\left(\frac{4a}{b}+\frac{2\cos \phi }{\sin \phi }-\tan \phi \right)}$$

(44)

Finally, the analytical solution of the final plastic deformation of corrugated sandwich structures with different panel configurations under explosion load is obtained:

$$w=\frac{w_{1x}+w_{1y}+w_{2x}+w_{2y}}{4}$$

(45)

3. Numerical Simulation

The finite element software ABAQUS (2016, Dassault Systèmes, Paris, France) was used for numerical simulation, and the finite element models are shown in Figure 5 and Figure 6, respectively, including the metal sandwich structure model and the air model. In the numerical simulation, the boundary conditions of the sandwich structure are rigid and fixed. The flow field is simulated by the 4-node acoustic element AC3D4, with a total of 409,409 elements, while the mesh number of the 4 different sandwich structures is different. The mesh size of the finite element model of the sandwich structure is 20 mm. The length and width of the sandwich structure model are 3000 and 2000 mm, and other parameters are shown in Figure 1. The air model was established by using the acoustic model, and the fluid structure coupling between the air domain and the model was defined by using the tie constraint. The model contact surface is the master surface, and the air contact surface is the slave surface. The model adopts a penalty function to define contact.

The four-node shell element (S4R) was used for element types of the metal sandwich structure model. Considering the computational efficiency and the sensitivity of the mesh size, the overall finite element length is 25.0 mm. The ideal elastic–plastic material model was adopted in the metal sandwich structure model to describe the steel mechanical behavior, and the detailed material properties are shown in

Table 2. Mechanical properties of the material (mild steel).

Property	Units	Mild Steel
Mass density	kg/m3	7850
Young’s modulus	GPa	210
Poisson’s ratio	-	0.3
Yield stress	MPa	235
Fracture strain	-	0.3

. The Cowper–Symonds model was adopted to solve the rate dependency of steel, and the material parameters q = 5 and D = 40.4 were used in the Cowper–Symonds model. The four-node linear tetrahedral acoustic element (AC3D4) was used for element types of the air model, a cylinder was used in the middle of the air model, and the two ends were hemispherical. The air density is 1.25 kg/m³, and the bulk modulus is 142 kPa. In numerical simulation, the acoustic–structure coupling algorithm was used to set the coupling boundary between the flow field and the structure.

4. Comparison between the Proposed Analysis Method and Numerical Simulation Method

To verify the accuracy of the proposed analysis method in predicting the dynamic response of metal sandwich structures under different explosion loads, the analysis results were compared with the numerical simulation results. The dimensions of the four metal sandwich structures are listed in

Table 3. Dimensions of the I-, U-, V-, and Uc-type metal sandwich structures.

Shape	a/mm	b/mm	df/mm	dp/mm	hc/mm	tf/mm	tc/mm
I	3000	2000	50	50	50	2	2
U	3000	2000	25	75	50	2	2
V	3000	2000	0	25	50	2	2
Uc	3000	2000	25	75	50	2	2

.

For convenience, a dimensionless factor called the impulse coefficient was introduced:

$$\underset{\_}{I}=\frac{I}{M\sqrt{{\sigma }_f/{\rho }_f}}$$

(46)

where, $M=2{\rho }_f{t}_c+h_c{\rho }_c$ is the mass per unit area of the metal sandwich structure.

The comparisons of the analytical and numerical simulation results of the maximum plastic deformation are shown in Figure 7. Since the displacement of the center point of the rear face sheet can reflect the maximum plastic deformation of the whole metal sandwich structure, the center point of the rear face sheet was selected as the typical displacement measurement point in the finite element simulation. Some measuring points were selected on the middle line, and the displacements of measuring points were calculated by the analytical method and compared with the finite element method.

It can be seen from Figure 7 that the displacement-impulsion curves of the metal sandwich structure have a similar trend, and the maximum plastic deformation increases gradually with the increase of the explosion load. Under the same explosion load, the deformation of the V-type sandwich structure is the smallest. The higher the number of cores, the better the explosion resistance. The maximum deformation of the Uc-type sandwich structure is smaller than that of the U-type sandwich structure, because the continuous sandwich core equivalently increases the thickness of the sandwich panel and improves the explosion resistance. According to the research, the analysis results match well with the numerical simulation results, and the maximum deviation of the two methods is less than 10%. Therefore, the proposed analytical method is feasible.

In order to detect the overall deformation of the metal sandwich structure, the same impulsion ($\underset{\_}{I}=0.185$) was introduced to analyze the displacement of the metal sandwich structure along the center line in the X and Y directions. The displacement distribution along the center line in the X and Y directions is shown in Figure 8 and Figure 9. It can be seen from Figure 8 and Figure 9 that the analytical method proposed in this paper can accurately predict the maximum structural deformation under explosion load. It should be pointed out that the disadvantage of this method is that the deformation mode of the structure is simplified (as shown in Figure 3), so the analytical result is a polyline, and the green curve is smooth. However, this simplified method can describe the deformation mode of the structure to a certain extent, for there is a platform in the X direction, while no platform in the Y direction, so the shapes of the curves for two results are different. The comparison results show that the analytical results of the deformation of the metal sandwich structure are close to the numerical simulation results, but the deformation near the middle part of the sandwich panel is larger. Due to the simplification of the rigid plastic material model, deviation from the actual deformation mode is inevitable. In general, the analytical method can predict the deformation of the structure and has good engineering application value.

The trend of the simplified analysis results along the center in the X and Y directions of the sandwich structure is basically consistent with the numerical simulation results, and the two peaks match well, but the analytical results of the edge area are less than the numerical simulation results. In the central part, there is a platform area in the X direction, but it is not obvious in the Y direction. This is because according to the simplified deformation mode of the sandwich structure in this paper (as shown in Figure 3), there is a platform in the long side direction and no platform in the short side direction. The calculated displacement value of the inscribed yield square is close to the actual deformation of the sandwich structure, but its maximum value is overestimated. It can be seen from Figure 8 and Figure 9 that the curves of internal results, external results, and analytical results have similar shapes, while their maximum values are different. The reason is that these three curves are the results calculated by the proposed analytical method with three different yield criteria (as shown in Figure 4), and thus the shapes of the black curve and green curve are different. However, when comparing these analytical curves, it can be found that the internal results’ curve has the smallest error with the simulation (especially for V-type), and it is reasonable to introduce the internal yield square to predict the deformation. The displacement value of the circumscribed yield square is less than the numerical simulation value for the entire structure. Therefore, it is reasonable to introduce the yield square in the prediction of the deformation of the structure under explosion load.

5. Parameter Study

By changing a series of parameters, the influence of different parameters on the performance of the metal sandwich structure under explosion load was studied. The effects of different parameters include aspect ratio, core height, angle, and thickness.

5.1. Effect of the Aspect Ratio

The study was carried out by changing the length while maintaining the same width (b).

Table 4. Dimensions of the sandwich panel with different aspect ratios.

a/mm	b/mm	$\mathit{\lambda }$
2000	2000	1
2500	2000	1.25
3000	2000	1.5
3500	2000	1.75
4000	2000	2

lists the analyzed cases. For convenience, the dimensionless parameter $\lambda =a/b$ is defined as the aspect ratio.

The comparison between the analysis results and the numerical simulation results is shown in Figure 10, and the displacement curves show a logarithmic increase trend. As the aspect ratio increases, the growth rate of the panel deformation slows down, because the width plays a leading role in the deformation of the longer sandwich structure. It can be predicted from the final trend of the curve that when the aspect ratio increases to a certain value, the plastic deformation will no longer occur, and the panel will eventually approach the cylindrical bending.

In the selected ratio range, the numerical simulation results of the I-type and U-type sandwich structures are slightly larger than the analytical results, and the results show that the deviation between them decreases with the increase of the aspect ratio. For the V-type sandwich structure, it can be seen that the two curves are basically consistent, and the analytical results are very close to the numerical results. For the Uc-type sandwich panel, the analytical results are smaller than the numerical results when the ratio is small. With the increase of aspect ratio, the analytical results are larger than the numerical results. It can be seen from the above comparison results that the proposed analytical method has high accuracy and is suitable for solving the plastic deformation of the four kinds of sandwich structures.

5.2. Effect of the Core Height

The other four core heights were considered, and the explosion resistance was studied by analytical and numerical methods. Considering that the width of the panel is the main indicator of transverse deformation, for convenience, a dimensionless parameter $\delta =h_c/b$ is defined as the ratio of core height to core width.

5.2.1. I-Type Sandwich Panel

Due to the special structural form of the I-type sandwich structure, it was divided into different types of units and discussed, respectively.

(1)

Panel with square units

The unit type of the sandwich panel cross-section is square, and the core height is the distance between adjacent cores. To ensure the integrity of the metal sandwich structure, the length is slightly adjusted. Therefore, the rectangular unit structures with different edges were calculated by analytical and numerical simulation methods. The parameters are shown in

Table 5. Geometric parameters of the I-type corrugated sandwich panel with square units.

hc/mm	a/mm	b/mm	df/mm	dp/mm	tf/mm	tc/mm
40	3000	2000	40	40	2	2
50	3000	2000	50	50	2	2
60	3000	2000	60	60	2	2
70	3010	2000	70	70	2	2
80	3040	2000	80	80	2	2

.

(2)

Panel with Rectangular Units

The spacing between adjacent cores is kept constant, and the influence of different core heights on the structural plastic response under explosion load is discussed. The parameters are shown in

Table 6. Geometric parameters of the I-type corrugated sandwich panel with rectangular units.

hc/mm	a/mm	b/mm	df/mm	dp/mm	tf/mm	tc/mm
40	3000	2000	50	50	2	2
50	3000	2000	50	50	2	2
60	3000	2000	50	50	2	2
70	3000	2000	50	50	2	2
80	3000	2000	50	50	2	2

. Figure 11 shows the displacement curves of the I-type sandwich panel with square and rectangular units. It can be seen from the figure that the displacement of the I-type sandwich structure with different unit types has a similar trend with the increase of the core height. The higher the core, the greater the bending resistance, which is the reason for the greater deformation resistance.

The main dimensions of the I-type sandwich structure with square units remain unchanged. Therefore, for the sandwich structure with a higher core height, the number of cells will be reduced. The sandwich structure with the core height of 80 mm is twice the height of the sandwich structure with the core height of 40 mm. However, due to the reduction of the number of cells, the displacement of the sandwich structure with the core height of 80 mm is not significantly reduced compared with the displacement of the sandwich structure with the core height of 40 mm.

In contrast, for the I-type sandwich structure with rectangular units, only the height of the core changes. Therefore, when the height of the core increases, the number of units for the structure with rectangular units is more than the number of units for the structure with square units. Compared with the I-type sandwich panel with square units, the displacement value span of the panel with rectangular units is larger, and the height of the core has a great influence on the displacement of the I-type sandwich panel.

In the selected height range, the proposed method matches well with the numerical simulation results, and the numerical results are slightly higher. When the height is small, the two results are very close. The greater the height, the greater the deviation between the numerical simulation results and the analytical results, but the maximum deviation does not exceed 7%.

5.2.2. Other Sandwich Panels

To ensure the structural design rationality of the U-type, V-type, and Uc-type sandwich panels, different values of d_f and d_p were selected according to different heights, and the parameters are shown in

Table 7. Geometric parameters of the U-type corrugated sandwich panel with different core heights.

hc/mm	a/mm	b/mm	df/mm	dp/mm	tf/mm	tc/mm
40	3000	2000	62.5	87.5	2	2
50	3000	2000	59.4	90.6	2	2
60	3000	2000	56.2	93.8	2	2
70	3000	2000	53	97	2	2
80	3000	2000	50	100	2	2

,

Table 8. Geometric parameters of the V-type corrugated sandwich panel with different core heights.

hc/mm	a/mm	b/mm	df/mm	dp/mm	tf/mm	tc/mm
40	3000	2000	0	50	2	2
50	3000	2000	0	50	2	2
60	3000	2000	0	75	2	2
70	3000	2000	0	100	2	2
80	3000	2000	0	100	2	2

and

Table 9. Geometric parameters of the Uc-type corrugated sandwich panel with different core heights.

hc/mm	a/mm	b/mm	df/mm	dp/mm	tf/mm	tc/mm
40	3000	2000	62.5	87.5	2	2
50	3000	2000	59.4	90.6	2	2
60	3000	2000	56.2	93.8	2	2
70	3000	2000	53	97	2	2
80	3000	2000	50	100	2	2

, respectively. The comparison between the analytical results and numerical simulation results of the three types of sandwich panels is shown in Figure 12.

For the U-type sandwich panel (Figure 12a), the displacement decreases with the increase of the core height. The analytical results are very close to the numerical results, and the deviation is very small (nearly 5%). The curves of the U-type sandwich panel are approximately linear, even if the values of d_f and d_p are different. This is because the number of cells is unchanged, which is similar to the number of the I-type sandwich panels (Figure 11).

For the V-type sandwich panels with different core heights (Figure 12b), the displacement values have obvious fluctuations, but the two methods have good consistency. The deviation is large at the heights of 60 and 70 mm, and this is because the number of core grids varies with core height and d_p. It can be seen that the changes of the height and number of the core grids have an effect on the plastic response of the sandwich panel.

For the Uc-type sandwich panel, the deviation is small when the height is small, the deviation increases with the increase of the height, and the maximum deviation is 4.6%. The analysis results are linearly distributed, the fluctuation of numerical results is small, and the overall trend is consistent.

5.3. Effect of the Angle

The effect of the angle between the face sheet and the core on the plastic deformation resistance under explosion load was studied. Since the angle of the I-type sandwich panel is permanently 90°, only the U-type, V-type, and Uc-type sandwich panels are discussed in this section. Considering the different types of sandwich panels and the rationality of the angle, different angle ranges were selected.

For the U-type sandwich panel, it can be regarded as a transition between I-type and V-type sandwich panels. The angles used are shown in

Table 10. Geometric parameters of the U-type sandwich panel with different angles.

$\mathit{\theta }/°$	a/mm	b/mm	hc/mm	df/mm	dp/mm	tf/mm	tc/mm
60	3000	2000	50	21.1	78.9	2	2
65	3000	2000	50	27.7	72.3	2	2
70	3000	2000	50	31.8	68.2	2	2
75	3000	2000	50	36.6	63.4	2	2
80	3000	2000	50	41.2	58.8	2	2

, and other geometric parameters were determined accordingly.

For the V-type sandwich panel, the included angle is between 45° and 70°. To ensure the integrity of the structure, some minor adjustments were made to the length of the panel, and the geometric parameters are shown in

Table 11. Geometric parameters of the V-type sandwich panel with different angles.

$\mathit{\theta }/°$	a/mm	b/mm	hc/mm	df/mm	dp/mm	tf/mm	tc/mm
45	3000	2000	50	0	100	2	2
50	3020.4	2000	50	0	83.9	2	2
55	3010	2000	50	0	70	2	2
60	3000.4	2000	50	0	57.7	2	2
65	2982.4	2000	50	0	46.6	2	2
70	2984.8	2000	50	0	36.4	2	2

. Referring to the selection angle of the U-type sandwich panel, the selection angle of the Uc-type sandwich panel is shown in

Table 12. Geometric parameters of the Uc-type sandwich panel with different angles.

$\mathit{\theta }/°$	a/mm	b/mm	hc/mm	df/mm	dp/mm	tf/mm	tc/mm
60	3000	2000	50	21.1	78.9	2	2
65	3000	2000	50	27.7	72.3	2	2
70	3000	2000	50	31.8	68.2	2	2
75	3000	2000	50	36.6	63.4	2	2
80	3000	2000	50	41.2	58.8	2	2

. The comparison between the analytical results and numerical simulation results of the three different types of sandwich panels is shown in Figure 13, where it can be seen that the analytical results match well with the numerical simulation results.

For the U-type sandwich panel (Figure 13a), the change of angle has little effect on the displacement, only a slight increasing trend. When the core height and the number of cells are the same, the effect of the angle on plastic deformation is not significant. For the V-type sandwich panel (Figure 13b), the influence of the angle on the displacement response under explosion load is more obvious than that of the U-type sandwich panel. The displacement decreases with the increase of the inclination angle, because the number of cells of the V-type sandwich panel will change due to different angles. When the main dimensions of the sandwich panel are the same, the smaller the angle of the sandwich panel, the higher the number of cells, and the stronger the strength and stiffness of the sandwich panel. In the selected angle range, the numerical simulation results are slightly larger than the analytical results, and the maximum deviation is about 7.3% at the angle of 45°. As the inclination angle increases, the deviation gradually decreases. For the Uc-type sandwich panel, the angle has less influence on the displacement, which is the same as the U-type sandwich panel. However, increasing the core angle of the U-type sandwich panel and decreasing that of the Uc-type sandwich panel, the total mass of the sandwich panel will decrease because the length of the core becomes shorter.

5.4. Effect of the Face Sheet and Core Thicknesses

In practical application, the thicknesses of the face sheet and the core have a great influence on the structural weight of the sandwich panel, so the structural weight of the sandwich panel was also studied in this paper. In this part, two cases are discussed, one is that the thickness of the face sheet is the same as that of the core, and the other is that the thickness of the two structures is different.

5.4.1. Case I: The Thickness of the Face Sheet Is the Same as the Core

The thicknesses of the face sheet and the core were selected as 1, 1.5, 2, 2.5, and 3 mm, respectively, and the dynamic response of the sandwich panel was studied by using the proposed analytical method and numerical simulation method.

For convenience, a dimensionless parameter $\gamma =\left(2t_f+t_c\right)/d$ is defined, that is the ratio of the total thickness of the face sheet and core to the height of the sandwich panel.

The comparison results obtained by the proposed analytical method and numerical simulation method are shown in Figure 14. The results of the two methods match well, and the results of different types of sandwich panels have similar trends. The maximum deviation appears when the thickness is the minimum, but the deviation is less than 10%. Moreover, with the increase of the thickness, the bending stiffness and the explosion resistance are improved. Therefore, the structural plastic deformation and the reduction of the deformation velocity show an exponential decreasing trend.

5.4.2. Case II: The Thickness of the Face Sheet Is Different from the Core

For I-type, U-type, and Uc-type panels, the thickness of the face sheet was selected as 1, 1.5, 2, 2.5, 3, 3.5, and 4 mm, and the thickness of the core was 2 mm. For the V-type sandwich panel with more cores, the thickness of the core was 1.5 mm, and the thickness of the face sheet was selected as 1, 1.5, 2, 2.5, 3, and 3.5 mm, respectively. It should be emphasized that the selected thickness is close to the actual situation, and for convenience, a dimensionless parameter $\tau =t_f/t_c$ is defined, that is the ratio of the thickness of the face sheet to the thickness of the core.

Figure 15 shows the comparison results of the proposed analytical method and numerical simulation method for different types of sandwich panels. It can be seen that with the increase of the thickness of the panel, the explosion resistance of the sandwich panel gradually improves. The changing trend of the curve is similar to that of Case I, which decreases gradually with the increase of the ratio.

It should be noted that the explosion resistance of the sandwich panel can be improved by increasing the structural thickness, but this will make the structure heavier. Therefore, in practical application, both the increase of material thickness and the increase of material weight should be considered. In general, the results of the two methods match well, and the proposed method can be used to predict the plastic deformation of the metal sandwich panels with different size parameters under explosion load.

6. Conclusions

In this paper, the response of the metal sandwich structure under explosion load was studied, and the dynamic response of the sandwich panel under explosion load can be divided into three stages. Based on the energy conservation theory and the assumption of the rigid body–completely plastic material model, a new analytical method was proposed, and numerical simulations were carried out to verify the analytical method. A series of parameter studies were carried out, and the results showed that the deformation increased with the increase of the aspect ratio, but the rate of displacement gradually decreased with the increase of the aspect ratio. The deformation decreased with the increase of the core height and the thickness of the face sheet, and the angle had little effect on the U-type and Uc-type sandwich panels, but for the V-type sandwich panel, the smaller the angle, the smaller the plastic deformation. The angle between the core and the face sheet determines the number of the sandwich panel cells, and the number of cells had a great influence on the anti-explosion performance. The more cells, the better the anti-explosion performance, and the weight of the sandwich structure increased accordingly. In the analytical model, the fluid–structure interaction was not considered. The analytical results matched well with the numerical simulation results, which suggests that the fluid–structure interaction does not play a significant role in air, and the proposed analysis method can be used for the preliminary design and rapid evaluation of the anti-explosion performance of the metal sandwich panel, which can reduce the calculation time and improve the calculation efficiency.