Dynamic response of bi-directional gradient sandwich circular plates under multiple explosive loading

Abstract

Due to its advantages in terms of enhancing the performance of structures in the applications, the gradient design has recently attracted the interest of researchers in a number of engineering disciplines. A bionic bi-directional gradient sandwich model was established inspired by the Royal Water Lily venation structure. Four sandwich circular plate models with different gradient arrangements were constructed by adjusting cell wall thickness. The deformation modes, deflections of the face sheets, and energy absorption characteristics were analyzed under multiple explosions with different blast impulses. The results indicated that the deformation of the structure and the energy absorption values accumulated progressively under multiple blast loading. However, the incremental energy absorption of both the face sheets and the honeycomb core decreases as the number of explosions increases. Compared with a single explosion, the dynamic response of models can be effectively improved by changing its gradient arrangement without adding additional mass under the multiple blast loads. Besides, the model’s blast resistance remained largely unchanged, while its energy absorption capacity was significantly enhanced by 29.92% after the thickness of the front panel was reduced. Such results provide guidance for the design of bi-directional gradient sandwich circular plate under multiple blast loads.

Introduction

Blast loads are highly destructive to engineering structures, making the study of structural blast resistance and dynamic response crucial for protective design. Sandwich structures are widely used in blast protection applications due to the excellent energy absorption characteristics. The blast resistance and energy absorption effectiveness of sandwich structures vary depending on the core. Consequently, researchers have proposed various core designs for sandwich structures, such as pyramid lattice sandwich plates¹, perforated honeycomb-corrugation hybrid², circular honeycombs, hexagonal honeycombs, hexagonally arranged circular tubes³, and a three-dimensional sinusoidal curved-edge sandwich panel with a negative Poissonʼs ratio effect⁴. And the dynamic response and energy absorption characteristics of these sandwich plates under explosive loading have been studied. Zhang Xuhong et al.⁵ used a self-designed ballistic pendulum system to measure the impulse imparted to aluminum honeycomb sandwich plates by explosive loads, confirming that aluminum honeycomb sandwich plates exhibit superior energy absorption performance compared to conventional structures when undergoing large plastic deformation. Li Shiqiang et al.⁶ experimentally investigated the dynamic response of layered gradient honeycomb sandwich plates under explosive loading using a ballistic pendulum system and analyzed stress wave propagation in the gradient core using one-dimensional stress wave theory. Wang Hairen et al.⁷ designed a bi-directional gradient sandwich circular plate based on a bionic Royal Water Lily core structure, analyzed its dynamic response under explosive loading, and discussed design strategies for the gradient core. Ghanbari et al.⁸ proposed a bamboo-inspired bionic tubular design to provide optimal energy absorption performance under axial crushing loads. In reality, explosions often occur multiple times. Sandwich panels are also exposed to millions of low-energy, high-cycle impacts in service: hailstones on aircraft wings, gravel clouds beneath cars, rain droplets on wind-turbine blades, or ice shards on high-speed train roofs. These repetitive impacts, although individual energies are less than 1000 J, accumulate to produce core shear cracking and face-sheet delamination. Consequently, multiple explosions has long been a key research focus. Chen et al.⁹ conducted high-speed impact tests and numerical simulation studies on the damage characteristics of GLARE laminates under single and multiple impact loads to investigate the high-speed impact damage tolerance of glass fiber reinforced aluminum alloy (GLARE) laminates. Zhou You et al.¹⁰ built upon previous research to determine the influence of explosion number on the dynamic response of the tubes by numerically calculating the deformation and damage of thin-walled square tubes under single and repeated explosive loads. Zhu Ling et al.¹¹ established an elastoplastic numerical analysis model for the dynamic response of foam metal sandwich beams in the Abaqus-Explicit module, investigating stiffness variations during repeated loading and unloading cycles and the energy distribution laws between face sheets and the core. Tang et al.¹² analyzed the cumulative damage effect of ship hull beams in multiple underwater explosions by combining AUTODYN software with experiments, and investigated the influence of factors such as explosive yield, detonation distance, and explosion frequency on the cumulative damage effect of ship hull beams.In order to effectively simulate the damage and failure behavior of 3D printed lattice material sandwich structures under projectile impact. Chen et al.¹³ introduced plastic bonds into the near-field dynamic micropolar model and constructed a model and modeling method suitable for lattice material sandwich structures. The research found that under high-speed impact, the failure mode was characterized by fracture, hole penetration and fragment ejection. And it is accompanied by extensive plastic deformation. Yue et al.¹⁴ employed a lab-scale metal foam projectile impact testing technique to systematically measure the dynamic response and deformation modes of elastoplastic metal plates under repeated impulsive loading, paying particular attention to the occurrence of pseudo-shakedown (P-S) phenomena, and used finite element methods to explore the response characteristics and physical mechanisms of P-S. Zhang et al.¹⁵ established a high-speed impact finite element model for a new type of sandwich structure and found that under high-speed impact, the composite material panels of the structure would experience problems such as fiber tearing, fiber delamination, and matrix fracture. Moreover, embedding shear-hardened materials could significantly enhance the impact resistance of the structure.

Most existing research has predominantly focused on the response of structures to single explosive loads. However, structures are likely to be subjected to multiple explosive shocks in real-world scenarios, whose cumulative damage effects cannot be ignored. Currently, research on the dynamic response laws of bi-directional gradient sandwich structures under multiple explosive loading is insufficient. Wang et al.^7,16,19 proposed a bi-directional gradient honeycomb model inspired by Royal Water Lily veins, but only studied the mechanical properties of the sandwich circular plate under a single explosion load. Building on this work, this paper couples “multiple explosions” with “bi-directional gradients” for the first time and establishes circular sandwich-plate models with four dual-gradient sandwich circular plate models. Explosive loading was simulated by applying pressure, and the sandwich circular plates were subjected to numerical simulations in single and multiple explosion fields. The dynamic responses of the models, including deformation modes, deflections of face sheets and velocity curves, and energy absorption characteristics were analyzed. A comparative analysis of the dynamic response of the sandwich circular plates under single and multiple explosive loading and the “reverse overtaking” mechanism for energy absorbed by each layer and part was conducted. The design strategy of the honeycomb sandwich structure was discussed, and the results provide guidance for the design of sandwich circular plates subjected to repeated blast loading.

Finite element simulation

To analyze the dynamic response of dual-gradient sandwich circular plates under multiple blast loading, numerical simulations were performed using the finite element software of ABAQUS/Explicit for both single and multiple explosive load scenarios.

Finite element model

Numerical simulations of clamped sandwich circular plate as illustrated in Fig. 1 are conducted using finite element (FE) code ABAQUS/Explicit.

Fig. 1.

Core design strategy inspied by simplified Royal Water Lily model.

This sandwich circular plate model is consisted of face sheets and a honeycomb core. The model radius (R) is 100 mm. Each core layer height is 5 mm, resulting in a total core height (c) of 15 mm for the three layers. The thickness of the face sheets is h_f = h_b = 1 mm. The gradient core comprises three layers of equal height but different wall thicknesses, denoted as Layers A, B, and C. Layer A is closest to the blast-facing side (front face sheet), Layer C is closest to the back face sheet, and Layer B is the middle core layer. Both the face sheets and the core were meshed using S4R shell elements. The boundary condition was set as circumferentially clamped support. To enhance computational efficiency and accuracy, the element type was selected for finite membrane strains and explicit dynamic reduced integration. To prevent penetration between the face sheets and the core under blast loading, the ‘general contact’ algorithm in ABAQUS/Explicit was employed to simulate the interfacial interactions. The contact was defined with hard in the normal direction, a penalty friction with a coefficient of 0.2 in the tangential direction for all adjacent surfaces.

Model materials and parameters

The material selected for the face sheets of the bionic sandwich circular plate was 5052 aluminum alloy¹⁷, and the base material for the bionic core was 6060T4¹⁸ aluminum alloy. A bilinear constitutive model was adopted:

1

where: σ is stress; E is the elastic modulus; ε is strain; σ_y is the yield strength. Specific material parameters are listed in Table 1, where ρ is density, ν is Poisson’s ratio, and E_tan is the tangent modulus. Strain rate effects were not considered in the simulation, due to the relatively weak strain rate sensitivity of both aluminum alloys.

Table 1.

Material parameters of the aluminum alloys.

Material	ρ/(kg·m-3)	E/GPa	ν	σy/GPa	Etan/GPa
5052 aluminum alloy	2700	70	0.3	0.20	0.10
6060T4 aluminum alloy	2700	70	0.3	0.08	0.07

Relative density is one of the most important characteristic parameters for lattice materials, expressed as the ratio of apparent density ρ_p to base material density ρ_m. As shown in Fig. 1, the core is divided into 5 parts based on the radial vein bifurcation locations. The formula for calculating the relative density of different parts is:

2

where: N_i is the number of radial honeycomb walls in part i; R_i,max is the radius of part i (specifically, when i = 0, R_0,max = 0); t is the wall thickness, with superscript r denoting radial, c denoting circumferential, and H denoting the total core height.

To simplify calculations, the wall thickness in the radial and circumferential directions for each part was set to be identical. The density gradient of the core was divided using a strategy similar to reference¹⁹. Different relative density gradient distributions were achieved by adjusting the wall thickness values for each part through the proportional coefficient k for wall thickness.

This model features bi-directional gradients, both in-plane and out-of-plane. Two wall thickness proportional coefficients were set: k = 0.8 and k = 1.8. Here, k = 0.8 represents an in-plane negative gradient, meaning the in-plane gradient increases monotonically from the center to the boundary; k = 1.8 represents an in-plane positive gradient, meaning the in-plane gradient decreases monotonically from the center to the boundary. The core has three layers in the out-of-plane direction. Type I denotes an out-of-plane negative gradient, meaning the out-of-plane gradient increases monotonically from the front face sheet to the back face sheet; Type II denotes an out-of-plane positive gradient, meaning the out-of-plane gradient decreases monotonically from the front face sheet to the back face sheet. The in-plane positive and negative gradients were coupled with the out-of-plane positive and negative gradients, respectively. As shown in Table 2, using the middle core Layer B as a representative, the wall thicknesses for the four gradient arrangement models are listed. Taking k = 0.8-I as an example, the wall thicknesses of Part 1, Part 2, Part 3, Part 4, and Part 5 are denoted as h₁, h₂, h₃, h₄, h₅ for the in-plane direction, and k = 0.8 means h₁/h₂ = h₂/h₃ = h₃/h₄ = h₄/h₅ = k = 0.8. The wall thickness of core Layer A is 1/3 of that of Layer B, and the wall thickness of core Layer C is 3/5 of that of Layer B for the out-of-plane direction.

Table 2.

Model types and related parameters.

Model	Wall thickness hi of middle core layer B parts/mm					Out-of-plane relative density/%
	Part1	Part2	Part3	Part4	Part5	A	B	C
k = 0.8-I	0.0916	0.115	0.143	0.179	0.224	2	6	10
k = 0.8-II	0.0916	0.115	0.143	0.179	0.224	10	6	2
k = 1.8-I	0.774	0.430	0.239	0.133	0.0738	2	6	10
k = 1.8-II	0.774	0.430	0.239	0.133	0.0738	10	6	2

where, k = 0.8-I represents the in-plane negative gradient, out-of-plane negative gradient model; k = 0.8-II represents the in-plane negative gradient, out-of-plane positive gradient model; k = 1.8-I represents the in-plane positive gradient, out-of-plane negative gradient model; k = 1.8-II represents the in-plane positive gradient, out-of-plane positive gradient model.

Explosive loading

Energy is released instantaneously, compressing the surrounding environment and propagating forward as a velocity front following the explosion. Generally, most of high explosives results in ideal blast wave profile²⁰ as shown in Fig. 2. The blast wave causes the ambient atmospheric pressure (P₀) to instantaneously increase to the peak incident overpressure (P_so). The peak incident overpressure decays exponentially with time and return back to ambient air pressure in time t₀. The time from the peak overpressure returning to standard atmospheric pressure is called the positive phase duration. Subsequently, the peak overpressure generates suction, resulting in a negative pressure wave with a duration approximately 2–5 that of the positive phase (the negative phase duration is not considered in this study).

Fig. 2.

Time history curve of peak pressure for an ideal blast wave²⁰.

This research institution employs spherical explosives as the explosive filling. When a spherical charge is detonated in free space, the blast wave profile is described by the Friedlander equation:

3

where, P(t) is the pressure at time t (kPa); P_so is the peak incident pressure (kPa); t₀ is the positive phase duration (ms); A is the blast wave decay coefficient (dimensionless). The incident impulse associated with the blast wave is obtained by integrating the area under the pressure–time curve as follow:

4

Here, t_a is the arrival time (ms).

The empirical formula proposed by Henrych for calculating the overpressure from a spherical TNT explosion in air²¹ is used as Eq. (5):

5

A schematic of the explosion model is shown in Fig. 3. Here, represents the scaled distance; d is the distance (m) from the center of the spherical explosive charge to the target face sheet; W is the mass (kg) of TNT.

Fig. 3.

Schematic of spherical charge explosion.

According to KNR theory²² neglecting the resistance of the air behind the plate, the plate is idealized as an unsupported planar surface with mass/area M_p. Denote ambient atmospheric pressure, density and sound speed by p_A, ρ_A, and c_A, respectively. The mass/area of the plate is M_p = ρh, where ρ (ρ = ρ_f = ρ_b = 2700 kg m⁻³) and h (h = h_f = h_b = 1 mm) are its density and thickness. The incident wave impacting the plate generates a shock wave at the front face. When the incident wave reaches the plate, it is characterized by an overpressure history with an exponentially decaying form, i.e. , where t₀ is the time constant, typically taken as 0.1 ms, and the peak overpressure P_so occurs behind the shock wave front; it will be used to measure the strength of the incident blast wave and the period of the incident wave.

The reflected impulse I_R, taking into account the fluid–structure interaction, is²²:

6

where,

7

with

8

9

The ratio of the mass per unit area of the plate to the time scale β_s is:

10

with

11

The peak density behind the incident wave ρ₀ is:

12

The coefficient f_R is:

13

The speed of sound behind the incident wave c_s is:

14

The speed of sound under constant pressure c_A is:

15

Under standard atmospheric conditions, p_A = 0.1 MPa, and the air density under standard conditions is ρ_A = 1.25 kg m⁻³.

KNR theory is used to model the impact of a uniform plane wave on a flat plate in two ways. One method, called the applied pressure method, was selected for this study. This involves applying the following time-varying pressure history curve to the blast-facing surface of the plate:

16

Here,

17

Consequently, for an explosion distance of 200 mm and spherical TNT explosive masses of 15 g, 25 g, and 35 g, the calculated peak reflected over-pressures were 0.0063 GPa, 0.0091 GPa, and 0.0118 GPa. Meanwhile, the positive over-pressure durations were 0.0555 ms, 0.0535 ms, and 0.0522 ms, respectively. The peak pressure time history curves of the six explosion waves are shown in Fig. 4. The larger the explosive charge, the greater the reflected overpressure and the shorter the duration of the forward overpressure.

Fig. 4.

Six explosion pressure time history curves.

In the multiple explosion simulations, six analysis steps were established, which correspond to six explosion events. The duration of each analysis step was set at 0.6 ms to guarantee that the deformation of the sandwich structure attained a stable value. Multiple analysis steps were used to define the initial impact load for each shock, ensuring that the deformation and stress state subsequent to each shock served as the initial condition for the next.

To verify the accuracy of the mesh size, the model of k = 0.8-II under the 35g TNT condition was used as an example. Meshes with sizes of 3 mm, 2 mm, 1 mm, and 0.8 mm were tested, resulting in core element counts of 24,144, 46,370, 155,640, and 231,696, respectively. As shown in Fig. 5, the results converged at 1 mm. Therefore, a 1 mm mesh size was adopted in this study to achieve an optimal balance between computational efficiency and accuracy. Simultaneously, the energy balance of the sandwich circular plate was analyzed. As shown in Fig. 6, the sum of kinetic energy (E_k) and internal energy (E_i) in the structure equals the work done by external forces (E_w). The artificial strain energy (E_a) accounts for 4.12% of the total energy, which is less than 5%. Therefore, the system energy remained well balanced throughout the blast response process, indicating the reliability of the current numerical model.

Fig. 5.

Mesh size sensitivity.

Fig. 6.

Energy histories of the model with 1 mm mesh size.

Results analysis

Parametric analysis was conducted on the numerical simulation results obtained from the four density gradient sandwich circular plate FE models subjected to 1, 2, 4, and 6 explosions under three conditions with explosive masses of 15 g, 25 g, and 35 g, respectively. The response of different gradient sandwich structures was discussed under single and multiple explosive loads.

Deformation modes

The deformation mode process of the k = 0.8-I sandwich circular plate structure under the typical experimental condition of 35g TNT was selected for analysis. The results of multiple explosions are shown in Fig. 7. It can be observed that the deflection of the back face sheet of the model gradually increases as the number of explosions increases and the deflection at the center point is the maximum value. Moreover, the deflection of the back face sheet of the models decreases as the distance from the center point increases.

Fig. 7.

Deformation mode diagram of sandwich circular plate with k = 0.8-I under different numbers of explosions (35g TNT).

At the moment of explosion, an impulse I_R is uniformly applied to the front face sheet of the sandwich plate, imparting an initial velocity to the front face sheet given by²³:

18

where, the mass of the front face sheet . ρ_f is the bulk density of the face sheet; h_f is the face-sheet thickness.

Consequently, the common velocity acquired by the face sheets is:

19

where, the mass of the sandwich circular plate, . ρ is the density of face sheets; h is the face-sheet height; ρ_c is the core bulk density (159.15 kg·m^-3); c is the core height (15 mm). The calculated common velocity is 79.07 mm/ms.

Figure 8 indicates the velocity curves of the face sheets of the model with k = 0.8-I under a 35g TNT explosion. As shown in Fig. 8a, the dynamic response of the bi-directional gradient sandwich circular plate under blast loading can be divided into three distinct phases. The first phase is the fluid–structure interaction stage, which is completed in an extremely short duration. In this stage, the blast impulse is transmitted to the front face sheet, imparting an instantaneous velocity that rapidly increases to a peak value prior to the onset of deceleration. Meanwhile, the rest of the structure remains stationary. The second phase is the core compression stage. In this stage, the velocity of the back face sheet gradually increases, while the front face sheet still moves at a higher speed. The core is compressed until the front and back face sheets attain identical velocity due to inertia, making the end of this phase. The third phase is the overall motion stage of the sandwich plate, which the front and back face sheets interact for a certain period before reaching a common velocity again, and eventually come to rest through plastic bending and stretching. This response mechanism is consistent with the three-phase theory proposed by Fleck ²⁴. Figure 8b shows the velocity–time histories of the front and back face sheets under 6 blast loading. Each explosion follows the same three-stage theory.

Fig. 8.

Velocity–time curves of the mid-span of the back face sheet for the bi-directional gradient sandwich circular plate with k = 0.8-I (35g TNT).

Deflection of front and back face sheets

Figure 9 shows the maximum deflection at the midpoint of the back face sheet for the four gradient sandwich circular plates under single and multiple explosive loads with different impulse. The vertical axis represents dimensionless deflection , the horizontal coordinate represents dimensionless impulse (When the mass of the explosive is 15 g,Ī = 0.1654. When the mass of the explosive is 15 g,Ī = 0.2305. When the mass of the explosive is 15 g,Ī = 0.2907). Herein, δ represents the deflection at the midpoint of the back face sheet; R represents the radius of the sandwich circular plate. I_R represents the impulse when considering the fluid–structure coupling effect; M represents the total mass of the sandwich circular plate; and σ_y and ρ_f respectively represent the yield strength and density of the panel, respectively. Differences in the maximum back face-sheet deflection among the different gradient plates can be observed. The influence of the out-of-plane gradient still dominates the structural resistance to deformation⁷. Comparing the maximum deflection of back face sheet with a single out-of-plane gradient type, it reveals that the maximum back face-sheet deflection of Type II is smaller than that of Type I within the studied impulse, for models with k = 1.8. Since the blast resistance of sandwich structures is often evaluated based on the final central point deflection of the back face sheet²⁵, the blast resistance of Type II sandwich plates is slightly better than that of Type I (k = 1.8). Furthermore, the deflection of k = 0.8 after 6 explosions is smaller than k = 1.8, indicating better blast resistance under the same out-of-plane gradient.

Fig. 9.

Deflection of the back panel of the sandwich panel with different density gradients under various impulse after single and 6 explosions.

For the model with k = 1.8, the blast resistance of Type II is superior to Type I. Figure 10 shows the deflections of the face sheets and the core compression for k = 1.8 after 6 explosions. It reveals that deflections of the front and back face sheets and core compression of the k = 1.8-II model are smaller than those of the k = 1.8-I model when the dimensionless impulse is constant. Since the relative density of the k = 1.8-II core decreases from Layer A to C, while it increases for k = 1.8-I, the core layer near the front face sheet is thicker in the k = 1.8-II model, making the core layer less prone to deformation and thus having better anti-explosion performance.

Fig. 10.

Deflections of the front and back face and the compression of the core of two different out-of-plane gradient core layers (k = 1.8).

To further investigate the effect of out-of-plane gradient arrangement on the k = 0.8 models, Fig. 11 presents the displacement–time history curves of the face sheets for the k = 0.8-I and k = 0.8-II models under 6 explosions with 35g TNT. From Fig. 11a–f, it is evident that a rise in the bending deflection of the face sheets with an increase in the number of explosions correlates. However, the rate of increase in deflection diminishes with each subsequent explosion. Concurrently, the increment in core compression also decreases. As shown in Fig. 11, the k = 0.8 models exhibit superior blast resistance compared to the models of k = 1.8 under multiple explosive loading. The reason is as follows: the relative density of the model of k = 0.8 increases from the center (Part1) to the edge (Part 5) when the out-of-plane gradient is the same, making the midpoint of the sandwich circular plate more susceptible to compression. By Figs. 10 and 11, it also reveals that for models with the same out-of-plane gradient, the k = 0.8 sandwich circular plates exhibit greater core compression than the model with k = 1.8 under the same dimensionless impulse. When the core layer is compressed, it absorbs most of the energy, resulting in smaller back face-sheet deflection and better structural blast resistance.

Fig. 11.

Deflections of front and back faces and compression of the core of two different out-of-plane gradient cores with k = 0.8.

Energy absorption

Figure 12 shows the energy absorption of the four density gradient sandwich plates under single and 6 explosions. Longitudinal coordinates represent plastic dissipation¹⁹ , and the horizontal ordinate represents dimensionless impulse Ī. E_p is the plastic dissipation; M is the total mass of the sandwich plate; σ_f and ρ_f are the yield strength and the density of face sheet, respectively. It can be observed that the energy absorption value increases with the impulse. When the in-plane gradient is fixed, the energy absorption value of sandwich circular plates with Type I is greater than that of plates with Type II under both single and multiple blast loading. As shown in Fig. 12a, the k = 0.8-I model exhibits the best energy absorption characteristics (the first explosion). However, the energy absorption of the sandwich circular plates with different density gradients changes after the 6th explosions. As shown in Fig. 12b, the energy absorption rate of the k = 1.8-I increases significantly compared to other models with increasing impulse, making it the model with the optimal energy absorption effect.

Fig. 12.

Energy absorption of various density gradient sandwich plates under different impulse.

The energy absorption capacity of the core is an important performance indicator for evaluating sandwich structures. Figures 13 and 14 shows the energy absorption of the model of k = 0.8-I under multiple explosive loading with the dimensionless impulse of 0.1654, 0.2307 and 0.2905. In the in-plane direction, Fig. 13 shows that the energy absorption values of the parts of the k = 0.8-I, k = 1.8-I and k = 1.8-II are in the order: Part 4, Part 3, Part 5, Part 2, Part 1 under lower numbers of explosions. For the k = 0.8-II, the energy absorbed by the individual parts decreases in the order Part 5, Part 3, Part 4, Part 2, Part 1. Besides, the energy absorption values of all parts gradually increase as the number of explosions increases. Notably, the increase in energy absorption for Part 5 is the most significant. At the lowest dimensionless impulse level (Ī = 0.1654), the energy dissipated by Part 5 of the k = 0.8-I rises markedly with each additional detonation. When the dimensionless impulse is increased to 0.2307, this progressive growth enables Part5 to surpass Part3 under the later blasts; at the dimensionless highest impulse of 0.2905, Part 5 ultimately overtakes even Part4, becoming the dominant energy-absorbing component toward the end of the six-burst sequence. This phenomenon can be attributed to the fact that, plastic hinges gradually transfer from the edge (Part 5) towards the center (Part 1) with increasing explosion times. However, plastic hinges always exist at Part 5, leading to a decrease in bending strength at the edge. This, in turn, causes Part 5 to absorb a greater amount of energy through plastic deformation compared to other parts.

Fig. 13.

Energy absorption of in-plane parts under multiple explosive loading.

Fig. 14.

Energy absorption of out-plane parts under multiple explosive loading.

Out-of-plane response (Fig. 14) reveals a monotonic rise in energy absorption of every core with multiple blasts, accompanied by a progressive decline in absorption efficiency. The wall thickness of Layer A in the Type I is thinner, and Layer C is thicker, the part of the core layer with thinner wall thickness is compressed first, and the remaining part is subsequently compressed due to the smaller impulse when Ī = 0.1654. The same sequence is observed when Ī = 0.2307, except for the k = 1.8-II panel whose Layer C is now the thinnest, here the initial compressive front localises in C, giving it the highest early dissipation, while Layer A (adjacent to the front face) is engaged slightly later. As the number of detonations increases, Layer B is gradually recruited, so its contribution overtakes that of the already-densified Layer C. When Ī = 0.2905, the deformation strategy will vary depending on the layout: the core compacts in a progressive deformation of k = 0.8-I (thin Layer A, thick Layer C). Layer A absorbs energy first, but compression shifts into Layer B during the densification process, whose dissipation ultimately exceeds that of A. The thinnest-webbed Layer C collapses first, producing the largest initial dissipation energy of the k = 0.8-II (thick Layer A, thin Layer C), but continued blasting densifies C and transfers deformation to B, whose energy absorption eventually surpasses C. The model of k = 1.8-II (thick Layer A, thin Layer C) replicates the same trendwhere Layer C dominates early and Layer B overtakes it later. Consequently, optimum core grading must be selected with the anticipated number of explosions in mind: a thin-front/ thick-back layout maximises early absorption, whereas a more uniform or reversed gradient becomes superior once multiple reloadings drive the core toward full densification.

Figure 14 synthesizes the back face-sheet deflection and plastic dissipation histories of gradient sandwich panels subjected to multiple blast impulses at three normalized intensities (0.1654, 0.2307, 0.2905). Irrespective of impulse amplitude, the model of k = 0.8 consistently exhibit lower permanent deflections than the model of k = 1.8, whereas Type I gradients systematically outperform Type II in plastic energy absorption. Consequently, keeping charge mass and areal density fixed, only changing the gradient arrangement of the core will affect the dynamic response of the structure under multiple explosive loads.

Parametric analysis

Influence of core gradient

Four models with different gradient arrangements were constructed by modifying the mass distribution coefficient and relative density to change the thickness of the model’s shell elements (Table 2) to study the effect of different core gradients on the blast resistance and energy absorption characteristics of the sandwich circular plate. Figure 15 shows the back face-sheet displacement and energy dissipation curves for the four gradient arrang zement models under six explosions (Three types of explosive weight).

Fig. 15.

Dynamic response curves of cores with different gradients under multiple explosive loads.

Figure 15 systematically summarizes the deflection and plastic dissipation evolution laws of the back face sheet of the gradient sandwich circular plate after multiple explosions under three dimensionless impulse (0.1654, 0.2307, 0.2905): Regardless of the level of dimensionless impulse, the back face-sheet deflection of the k = 0.8 model is always lower than that of the k = 1.8 model, while the plastic dissipation energy of Type I arrangement is systematically higher than that of Type II. Therefore, under the condition that the equivalent of explosives and the model mass remain unchanged, only changing the gradient arrangement of the core layer will affect the dynamic response of the structure under multiple explosive loads.

Influence of dimensionless impulse

Figure 16 shows the time-history curves of the central point deflection of the back face sheet for the typical model k = 0.8-I model under dimensionless impulse (Ī = 0.1654, 0.2307, 0.2905). The Fig. 16 shows that, when the sandwich circular plate is multiply exploded under three loading conditions, the back-face deflection and the compression of the core gradually increase, but the increment shows a decreasing trend. This attenuation arises because the structure undergoes progressive stiffening: the core is gradually densified and absorbs an increasing fraction of the dissipation energy, so the growth rate of the back-panel deflection declines. The relationship between the back face-sheet deflection of different gradient arrangement sandwich plates and the dimensionless impulse is shown in Fig. 17. It reveals that the back face-sheet deflection of the sandwich circular plate also increases as the dimensionless impulse increases, showing a linear relationship with the applied dimensionless impulse. The relationship between dimensionless impulse and back face-sheet deflection obtained by fitting the data is:

20

where D is deflection of the back face sheet;I is the dimensionless impulse; a and b are fitting parameters. Table 3 shows the fitting parameter results for the different gradient arrangement sandwich circular plates. The dimensionless back-face deflection increases linearly with the dimensionless impulse, and the slope varies with the gradient type. A larger slope “a” signifies that the back face-sheet deflection of that particular layout is more sensitive to changes in impulse. The coefficient of determination R² is greater than 0.96 for all fits, demonstrating that the proposed model captures the experimental data with high fidelity. Therefore, it can be concluded that the back face-sheet deflection of the k = 0.8-II model is smaller than other models within the studied range of dimensionless impulse, meaning the model of k = 0.8-II have better blast resistance. At the lowest effective impulse (Ī = 0.1654), the back face-sheet deflections of all different gradient arrangements are practically indistinguishable. At the highest impulse (I = 0.2905), however, the model of k = 0.8 exhibit markedly smaller deflections than k = 1.8, and Type II layouts consistently outperform Type I. Consequently, a higher relative-density core adjacent to the front face enhances blast resistance under high blast loading, whereas a lower relative-density layer in the same region reduces it. Conversely, a lower relative-density core near the mid-plane improves performance, while a higher relative-density layer at that location degrades it.

Fig. 16.

Back face-sheet deflection and compression of the core of k = 0.8-I under multiple explosions for different dimensionless impulse.

Fig. 17.

Back face-sheet deflection of different gradient sandwich plates under different dimensionless impulse.

Table 3.

Linear-fit parameters (D = aĪ + b) and coefficients of determination for bace face-sheet deflection versus impulse.

Model	a	b	R2
k = 0.8-I	0.93	1.96	0.964
k = 0.8-II	0.89	2.31	0.999
k = 1.8-I	0.97	2.53	1
k = 1.8-II	0.99	1.24	0.999

Influence of face-sheet thickness distribution

The present study aims to investigate the optimization design of blast resistance and energy absorption for bi-directional gradient sandwich circular plate structures subjected to multiple explosive loading. Taking the k =  0.8-II model under Ī = 0.2307 as an example, the thicknesses of the face sheets were varied while keeping the total mass constant. Three distinct cases of face-sheet thickness allocation were constructed, as shown in Table 4.

Table 4.

Different face-sheet thickness combinations for the k =  0.8-II sandwich circular plate when Ī = 0.2307.

Model	Front face-sheet thickness hf/mm	Back face-sheet thickness hb/mm
Case 1	0.5	1.5
Case 2	1	1
Case 3	1.5	0.5

Figure 18 shows the dynamic deformation (back face-sheet deflection) and energy absorption characteristics of the k  = 0.8-II model under the multiple impulse for the different allocation cases. It can be seen that the back face-sheet deflection of the sandwich plate models for all three cases still increases with the number of explosions, and the energy absorption rate decreases under multiple explosive loading. This is because the overall bending stiffness of the sandwich plate increases under multiple explosive loading, which reduces the deformation growth rate. Furthermore, compared to Case 2, increasing or decreasing the front face-sheet thickness (Case 1, Case 3) leads to an increase in back face-sheet bending deflection. After 6 explosions, the deflections for Case 1 and Case 3 increased by −0.013% and 8% compared to Case 2 respectively; the energy absorption for Case 1 and Case 3 increased by 29.92% and decreased by 0.048% compared to Case 2 respectively. This indicates that reducing the front face-sheet thickness increases the energy absorption value by 29.92% while the deflection at the midpoint of the rear panel remains almost unchanged, significantly enhancing the structure’s energy absorption characteristics with little change in blast resistance under constant total sandwich plate mass. Therefore, optimizing the allocation of front and back face-sheet thickness can effectively achieve a balance between the multiple impact resistance and energy absorption performance of the sandwich circular plate.

Fig. 18.

Influence of face-sheet thickness allocation.

Generally, the back face-sheet deflection increases with the number of explosions. However, as shown in Fig. 18a, the back face-sheet deflection of the Case 3 model decreases after the 2nd explosion. This can also be seen in Fig. 19: the first explosion occurs before 0.6 ms. At 0.339 ms, the back face-sheet deflection reaches its maximum value of 18.415 mm under the first explosion; at 0.606 ms, the back face-sheet deflection reaches its maximum value of 17.725 mm under the second explosion. This indicates that the back face-sheet deformation did not increase but instead exhibited a rebound phenomenon after the second explosion.

Fig. 19.

Deflection of face sheets and core compression for Case 3 after two explosions.

The analysis suggests the reason is shown in Fig. 20, which displays the velocity time history curves for the face sheets of Case 3. It can be seen that the velocity of the back face sheet becomes negative for a period during the explosion process, indicating motion opposite to the blast direction, which leads to reduced deflection. Specifically, at 0.339 ms, the back face-sheet velocity begins to turn negative. After this point, the area enclosed between the zero axis and the back face-sheet velocity curve below the axis is larger than the area above it, meaning the rebound deflection of the back face sheet exceeded its continued downward deflection. Therefore, the maximum deflection produced by the second explosion is less than that of the first explosion.

Fig. 20.

Velocity time history curves for Case 3 face sheets.

On the other hand, Fig. 21 shows the deformation of Case 3 after two explosions. In Case 3, the front face sheet exhibits a greater thickness compared to the other cases. Correspondingly, the core layer of this case features a density gradient that increases both from the top to the bottom and from the center to the edge. This gradient distribution is accompanied by an increase in wall thickness, which is designed to enhance the structural integrity and performance under the specified loading conditions. Therefore, the wall thickness at the center is the thinnest and more prone to deformation. As can be seen from the Fig. 21: after the first explosion, deformation at the center is most significant, causing local deformation. Following the second explosion, the entire sandwich circular plate undergoes deformation. Specifically, its edges deflect downward, which in turn induces an upward bulging of the central part. This geometric alteration effectively reduces the central point deflection of the back face sheet.

Fig. 21.

Deformation of Case 3 after two explosions.

Conclusions

The dynamic response of bi-directional gradient sandwich circular plates under multiple explosive loading was investigated using FE code ABAQUS/Explicit based on the bionic Royal Water Lily. The bi-directional gradient proposed increases the design dimension and significantly improves the energy absorption performance under multiple explosion loads while maintaining the same mass. It can provide design basis and parameter reference for the design of military vehicles, explosion-proof buildings, ships resistant to multiple impacts and spacecraft protective structures. The main conclusions are as follows:

1. The applicability of Fleck’s three-stage theory was verified under multiple explosive loading. For model of equal mass, the deflection of the back face sheet increases with increasing explosive load.

2. Under multiple explosive loading, the deflection of the face sheets increases with the number of explosions, but the increment of increase gradually decreases. Comparing the deflection of sandwich plates based on single in-plane or out-of-plane gradients reveals that sandwich plates with k = 0.8 and Type II exhibit better blast resistance under multiple explosions. Under a single explosion, the model of k = 0.8-II shows the best energy absorption characteristics. However, after multiple explosions, the sandwich plate with the model of k = 1.8-I demonstrates optimal energy absorption.

3. As the number of explosions increases, the incremental energy absorption of the edge part (Part 5) and the middle core Layer (B) increases significantly, becoming the parts that absorb the most energy gradually.

4. The deflection of the face sheets increases with explosive mass. The energy absorption characteristics of the structure can be significantly enhanced without substantially sacrificing blast resistance by optimizing the face-sheet thickness allocation (e.g., appropriately reducing the front face-sheet thickness).

5. Overall, the number of explosions, impulse, and face-sheet thickness allocation all influence the dynamic response of the structure. Therefore, the selection of the core layer for the sandwich circular plate should consider the expected number of explosions based on practical circumstances to more effectively absorb energy and mitigate the hazards posed by blast loads.

Author contributions

Wang Hairen and Liu Yaoyao wrote the main manuscript text. Lei Jianyin and Li Zhiqi provided important assistance in finite element simulation and provided Figs. 1–3. Chen Yunhui, Zhang Jiahui, and Sun Xiyong made important contributions to the calculation of the explosion load impulse formula and wrote a Matlab program to calculate the effective impact load. Peng Jun and Li Yugen made important contributions to deflection and energy absorption, helping to draw Figs. 6–19.

Funding

This work is supported by the National Natural Science Foundation of China [grant number 12302490 and 12372363], the Shaanxi Provincial Department of Education Special Research Program Project a Grant No. of 25JS144, the Yulin University Graduate Innovation Project Fund Grant No.2025YLYCX65, Yulin University High level Talent Research Launch Fund with a Grant No. of 22GK10, Yulin City Scientist + Engineer Innovation Talent Team Project with a Grant No. of KJZG-2025-K + G-04. The financial contributions are gratefully acknowledged.

Data availability

Data is provided within the manuscript or supplementary information files.

Declarations

Competing interests

The authors declare that they have no competing interests.