The Resistance of Sandwich Cylinders With Density-Graded Foam Cores to Internal Blast Loadings

Sandwich structures, consisting of two strong and thin face sheets and a thick and soft cellular core, have been widely used in the automotive, aerospace, and defense industries. It is a remarkable fact that the sandwich structure subjected to blast loading can effectively absorb impact energy and simultaneously mitigate transmission stress [1–7]. Much attention has, therefore, been paid to the design and dynamic response of various sacrificial core sandwich structures subjected to blast loadings.

Fleck and Deshpande [8] developed a systematic design framework to study the blast resistance of clamped sandwich beams. The structural response is split into three phases: fluid–structure interaction (FSI), core compression, and combination of plastic bending and longitudinal stretching. It is found that the sacrificial layer plays a significant role in energy absorption [9–11]. To predict the crushing behavior of sacrificial layers during the core compression phase, the rigid perfectly-plastic locking (R-PP-L) model, which is primitively put forward to understand the crushing behavior of wood [12], was employed to describe the dynamic performance of cellular materials [13–15]. Furthermore, the graded cellular materials, such as continual density-graded foam and stepwise-graded foam, were proposed to synchronously enhance the energy absorption and shock mitigation of sacrificial claddings. Using the Voronoi method, the mesoscopic models of cellular materials were developed and the dynamic crushing of graded foams was investigated [16–19]. It has been confirmed that the blast resistance of the stepwise-graded foam can be effectively improved by placing the higher-density foam at the impact side and the lower-density foam at the support side [20–23]. Double-shock mode and three-shock mode may occur in the dynamic response of the continual density-graded foam [24–26]. Moreover, the continual density-graded foam has excellent designability which can meet various crashworthiness requirements [27–29]. It is suggested that the three phases may not accurately predict the total momentum [30]. Considering the fluid–structure interaction and cavitation phenomenon, Yin et al. [31] studied the 1D response of graded cellular claddings to water blast loading. Feng et al. [32] addressed the effects of the impulsive intensity, core strength, core density gradient, and boundary condition on the cavitation evolution during the dynamic fluid–structure interaction. Xiang et al. [33] took the influence of the structural bending effect on the foam core into consideration. Yang et al. [34] found that the density gradient distributions, such as positive, negative, middle–high, and middle–low gradients, have significant effects on the dynamic response of sandwich structures subjected to blast loadings.

As a typical class of sandwich structures, much attention has also been paid to the dynamic behaviors of sandwich cylinders subjected to blast loadings. The dynamic response of sandwich cylinders subjected to external blast loadings was investigated [35,36]. Shen et al. [37] experimentally, numerically, and analytically investigated the response of homogeneous foam core sandwich tubes subjected to internal explosive loadings. Karagiozova et al. [38] developed an analytical model for the deformation of the homogeneous foam core sandwich cylinder. The process of the dynamic foam compaction and stress transmission to the outer wall of a sandwich-walled cylinder was analyzed. Liang et al. [39–41] built up the mesostructure of layered foam sandwich structures based on the 2D Voronoi algorithm. According to the deformation process, an analytical model was proposed about the dynamic response of layered foam sandwich cylinder. It implies that the blast resistance is moderately sensitive to the core density distribution. Zhang et al. [42,43] investigated the deformation process of tubes and the influence of the physical and geometrical parameters of the structure on the deformation. These previous investigations suggest that the density distribution has significant influence on the dynamic response of cellular materials, and the rational design of the density distribution can improve the shock resistance of the sandwich structure. However, little research has been devoted to the dynamic response of sandwich cylinders with continuous density-graded foam cores subjected to internal blast loadings and the effects of the wall thickness distribution and density gradients.

The motivation of this work is to investigate the dynamic response of sandwich cylinders with density-graded foam cores subjected to internal blast loadings and clarify the effects of density gradient and wall thickness distribution. The paper is organized as follows. In Sec. 2, four kinds of density-graded foam cores of the sandwich cylinders are designed. In Sec. 3, the analytical model is developed to predict dynamic response of sandwich cylinders subjected to internal blast loadings. In Sec. 4, finite element (FE) simulations of sandwich cylinders with mesoscopic foam cores are carried out. In Sec. 5, comparisons between the analytical predictions and FE simulations are conducted and the effects of cylinder wall thickness distribution and foam core density distribution on the blast resistance are explored. Finally, concluding remarks are presented.

Consider a sandwich cylinder with a density-graded foam core subjected to internal blast loading, as shown in Fig. 1(a). The inner and outer wall thicknesses of the sandwich cylinder are $hin$ and $hout$⁠, respectively. The thickness of foam core is $hcore$⁠. The internal and external radii of the foam core are $Rin$ and $Rout$⁠, respectively. The density of the foam core is continuously graded along the radius direction. Herein, four density gradients are designed as follows:

1. Positive and negative gradients

The deformation process of the sandwich cylinder subjected to internal blast loading is assumed to be split into three phases: phase I (FSI), phase II (combined deformation of core and inner wall), and phase III (sandwich stage of motion) [8,38,41], as shown in Fig. 2 In phase I, the inner wall imposed by blast loading dilates at a uniform velocity. Subsequently, the inner wall expands and compresses the foam core in phase II. After the foam core is fully densified, the outer wall starts to deform with the foam core and the inner wall in phase III. In what follows, the analytical model is developed to predict the dynamic response of sandwich cylinder with a density-graded foam core subjected to internal blast loading.

After phase I, a shock wave with a discontinuity appears and propagates through the foam core along the radial direction. It is assumed that the strength of the outer wall is much higher than that of the foam core. The outer wall starts to deform after the foam core is fully densified. The density gradient distributions of the foam cores may result in various propagation modes.

When the shock front attains the distal end of the foam core at $t=tII$⁠, the foam core is fully densified. Both inner wall and foam core start to move with a common velocity $vII=v1(tII)$⁠. Subsequently, the dynamic response undergoes the next phase.

Part 1 starts to move with an initial velocity $v1(0)=v0$⁠, while part 2 starts to move with an initial velocity $v2(0)=0$⁠, and part 3 remains stationary, i.e., $v3=0$⁠.

After the foam core attains full densification, the dynamic response undergoes the next phase.

When the shock front 3 reaches the outer wall at time $t=tII$⁠, it means that the foam core is fully densified, and the inner wall and the densified foam core move with a common velocity $vII=v1(tII)$⁠. Subsequently, the dynamic response undergoes the next phase.

Once the foam core is fully densified, the outer wall collides with both the densified foam core and the inner wall, leading to deformation of the outer wall, which can be further divided into two stages [41]. In stage 1, the densified foam core and the inner wall move together and compress the outer wall. The velocity of the two decreases until is equal to the velocity of the inner wall. In stage 2, the inner wall separates from the densified foam core, while the densified foam core engages with the outer wall. Finally, their velocities attain zero, i.e., the dynamic response ends.

It is noted that the velocity of the inner wall first decreases to zero, and then the common velocity of the densified foam core and the outer wall decreases to zero.

All computations of the dynamic response of sandwich cylinders with density-graded foam core to internal blast loading are performed by using the commercial software abaqus/explicit (6.14). The building processes of the mesoscopic foam models in a given ring area A₀ can be divided into three steps, as shown in Fig. 4.

The isotropic bilinear hardening material model with density $ρw=2700kg/m3$⁠, Young's modulus $Ew=70GPa$⁠, Poisson's ratio $νw=0.3$⁠, yield strength $σsw=170MPa$⁠, and tangent modulus $ETw=350MPa$ is used to describe the mechanical behaviors of the walls. The internal blast loading is assumed to be a triangular pressure pulse with a peak pressure $p0=600MPa$ and the loading time $τ0=15μs$⁠. The internal and external radii of the foam core are $Rin=0.035m$ and $Rout=0.055m$⁠, respectively. The thickness of foam core is $hcore=0.02m$⁠. The intermediate density of the foam core is $ρ¯i=0.2$⁠. The inner and outer wall thicknesses range from 1.5 mm to 3.5 mm. General contact algorithm with friction coefficient $0.02$ is employed to model the contact among the walls and the foam core [26]. The outer and inner walls are modeled by linear hexahedron elements (C3D8R), while the cell walls of the foam core are modeled by shell elements (S4R).

The foam compression of the same size $(100mm×100mm)$ with different relative densities is conducted by FE simulations. Using the energy efficiency method [44], the densification strain $εD$ and the plateau stress $σp$ of the foam core are calculated, and then the parameters a₁, a₂, and a₃ are obtained by fitting Eqs. (7) and (8). Unless otherwise stated, $a1=1.28$⁠, $a2=2.0$⁠, and $a3=0.73$⁠. Mesh sensitivity investigations reveal that further refinements cannot improve the calculation accuracy appreciably.

Figure 5 shows the deformation process and the stress distribution of mesoscopic sandwich cylinders with different gradients (⁠$γ=1$ and $hin=hout=2.5mm$⁠). It can be found that for the positive gradient (Fig. 5(a)), the high stresses of the foam core at the proximal end of low density commence during the deformation process, which leads to the localized deformation of foam core at the low-density end. It means that only one shock front in the foam core propagates from the proximal end near the inner wall to the distal end near the outer wall during the deformation process. For the negative gradient (Fig. 5(b)), the high stresses of the foam core at the low- and high-density ends commence during the deformation process, which leads to the localized deformation of foam core at the low- and high-density ends. The double shock fronts in the foam core commence. One shock front propagates from the proximal end near the inner wall to the distal end near the outer wall, while the other shock front propagates in the opposite direction. There are high stresses at the proximal end of the low density and the distal end of the low density of the foam core with middle–high density gradient (Fig. 5(c)). The double shock fronts commence and propagate in the opposite directions. However, for the middle–low density gradient (Fig. 5(d)), the high stresses arise at the proximal end of the high density and the middle of the low density of the foam core. Three shock fronts commence. The first shock front propagates from the proximal end near the inner wall to the middle and the second shock front propagates from the middle to the proximal end near the inner wall, while the third shock front propagates from the middle to the distal end near the outer wall. Moreover, it can be seen from Fig. 5 that with increasing of the deformations of the density-graded foam core sandwich cylinders, the foam cores would separate from the inner walls after the foam cores have been fully densified.

Figure 7 shows the comparisons between the analytical predictions and FE simulations of displacement response for the inner and outer walls for the density-graded foam core sandwich cylinders (⁠$γ=1$ and $hin:hout=1:1$⁠) and homogeneous foam core sandwich cylinder $(hin/hout=1:1)$⁠. For the positive density-graded foam core sandwich cylinder in Fig. 7(a), the deformation of the inner wall gradually increases with time in the initial stage, while the outer wall remains static. The outer wall starts to deform and the displacement gradually increases with time after the foam core has been fully densified. After the kinematic energy is absorbed, inner and outer walls attain the maximum deformation and then the deformations end one after another. Moreover, the outer wall stops deformation later than the inner wall. The comparisons suggest that the analytical predictions agree well with FE simulations in the initial stage, while the analytical model overestimates the final deformations of the inner and outer walls. It is likely that the dynamic response in FSI phase is simplified by assuming that the impulse is totally translated into the momentum of the inner wall. The similar dynamic responses of negative density-graded foam core sandwich cylinder, middle–high density-graded foam core cylinder, middle–low density-graded foam core cylinder, and homogeneous foam core cylinder are shown in Figs. 7(b)–7(e).

Comparisons of non-dimensional velocity versus time curves of analytical predictions and FE simulations of the density-graded foam core sandwich cylinders (⁠$γ=1$ and $hin:hout=1:2$⁠, 2:1) and homogeneous foam core sandwich cylinders (⁠$hin:hout=1:2$⁠, 2:1) are shown in Figs. 8–11. The analytical predictions are in good agreement with the FE simulations. Moreover, the dynamic responses of density-graded and homogeneous sandwich cylinders (⁠$hin:hout=1:2$⁠, 2:1) are similar to those $(hin:hout=1:1)$⁠.

Figure 12 shows comparisons of the non-dimensional wall displacements of FE simulations versus analytical predictions of the foam core sandwich cylinders. The analytical model agrees well with FE simulations of the inner wall displacements, while overestimates FE simulations of the outer wall displacements.

Figure 13 plots FE simulations of energy absorption distributions of density-graded foam core sandwich cylinders $(γ=1)$ and homogeneous foam core sandwich cylinder with different thickness distributions of the walls. It can be found that the energy absorption of cylinders with different thickness distributions of the walls has a same trend. Most of the energy is absorbed by the foam core, followed by the inner wall and the outer wall. Additionally, the thickness distributions of the walls have significant effects on the energy absorption distributions. The proportions of energy absorption of the inner walls increase with increasing the inner wall thickness, while the proportions of the foam cores and the outer walls decrease.

The analytical predictions of the relationship between the outer wall displacement versus the ratio of the inner wall thickness to the outer wall thickness of density-graded and homogeneous foam core sandwich cylinders with the same mass are shown in Fig. 14. It can be identified that the displacements decrease with increasing the thickness ratios. It is demonstrated that the larger the inner wall thickness is, the better the blast resistance is. Furthermore, the positive density gradient foam core sandwich cylinder has the lowest displacement among all cases. It indicates that the design of the positive density gradient foam core has higher energy absorption than other density gradient designs and leads to the highest blast resistance in the present study. Similar conclusion has been obtained by Shen et al. [25].

Figure 15 shows the comparisons of analytical predictions of the non-dimensional outer wall displacement versus density gradient for density-graded foam core sandwich cylinders $(hin:hout=1:1)$⁠. The displacement of positive density gradient design decreases with the density gradient. Moreover, the positive design of the foam core has the lower displacement than other cases. In other words, the sandwich cylinder with positive foam core has the better blast resistance than other cases. However, the displacements of negative, middle–low, and middle–high foam core sandwich cylinders first increase, and then decrease with increasing the gradients. Comparing to the homogeneous foam core, the rational gradient designs of foam core may bring about the better blast resistance. On the other hand, these indicate that the specific displacement can give rise to multiple design choices for the gradient foam cores.

The dynamic response of metal sandwich cylinders filled with the density-graded foam cores subjected to internal blast loading is investigated. The analytical model consisting of FSI, combined deformation of core and inner wall, and sandwich stage of motion is proposed to predict the dynamic response of metal sandwich cylinders filled with four kinds of density-graded foam cores, such as positive, negative, middle–high and middle–low gradient distributions. It is shown that the analytical predictions are in good agreement with FE simulations. The blast resistance and the energy absorption distributions of sandwich cylinders with the same mass and the same foam core can be achieved by tuning the inner and outer thicknesses. Comparing to the homogeneous foam core, the sandwich cylinder with positive density gradient foam core has the best resistance to the inner blast loadings among four density-graded designs of foam cores. Moreover, the rational density-graded design can enhance the blast resistance of density-graded foam core sandwich cylinders.

The authors are grateful for financial supports of the National Natural Science Foundation of China (Grant Nos. 12472391 and 11972281), Opening Project of National Key Laboratory of Shock Wave Physics and Detonation Physics (Grant No. JCKYS2019212008), Aeronautical Science Foundation of China (Grant No. 201941070001), Project funded by China Postdoctoral Science Foundation (Grant No. 2021M702537), the Natural Science Foundation of Hubei Province of China (Grant No. 2021CFB029), Opening Projects of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology, Grant No. KFJJ22-07M), Hubei Province Key Laboratory of Systems Science in Metallurgical Process (Wuhan University of Science and Technology, Grant No. Y202204), and Laboratory of Aerospace Entry, Descent and Landing Technology (Grant No. 23-11-064-W).

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.