Numerical investigations on blast resistance of sandwich panels with multilayered graded hourglass lattice cores

Abstract

This research focuses on the dynamic response of sandwich panels with multilayered graded hourglass lattice core subjected to blast loading. A three-layer lattice core configuration is proposed to improve the absorption efficient of kinetic energy resulted from blast shock wave. The relative density of each core layer is changed with sectional dimension of core truss members to regulate the energy absorption of each core layer. Three-dimensional numerical simulation analyses of dynamic response are carried out, and the applied impulsive pressure distribution on the surface of the panels is calculated using the CONWEP code. The panels are made of stainless steel AL6XN, which is assumed to follow bilinear strain hardening and strain rate-dependence. Peak back sheet deflection and energy absorption of core layers for four types of hourglass lattice panels are comparatively analyzed and the effects of load intensity on the peak deflection are discussed. Furthermore, the near-optimal configuration under blast loadings is proposed.

Keywords

Sandwich panel blast resistance hourglass lattice graded core numerical simulation

Introduction

Owing to the characteristics of low density, high strength-to-weight ratio, and excellent energy absorption, metallic sandwich structures with cellular core have attracted tremendous attention in recent years and become significant protective structure in transportation industry, aerospace structures, and marine structures fields [1,2]. Numerous researches have been carried out to study the dynamic behavior of sandwich structures suffering from blast loadings or local impact during the few decades. Fleck and Deshpande [3] developed a theoretical model to analyze the blast resistance of clamped sandwich beams, observing that the sandwich structures with soft core can provide the best blast resistance for the case of water blast compared to monolithic structures. Qiu et al. [4] obtained the same conclusions as Fleck and Deshpande [3] by conducting a dynamic response analysis of clamped sandwich beams subjected to shock loadings in numerical method. Schiffer et al. [5] performed analytical predictions and finite element calculations to predict the one-dimensional (1D) response of sandwich panels with elastic cores under water blast loadings. Imbalzano et al. analyzed the resistance performances of sandwich panels composed of auxetic and conventional honeycomb cores against local impact [6] and blast loadings [7] in numerical method. Rubino et al. [8 –10] investigated the collapse response of sandwich beam with a Y-frame core under distributed and local impact loadings, indicating that the Y-frame core sandwich beams have a higher energy absorption efficiency compared to the metal foam sandwich beams. Xiang et al. [11] studied the dynamic response of the tube core sandwich structures under the transverse blast loadings in numerical and experimental method.

The lattice core sandwich structures have better characteristics in absorbing impulsive kinetic energies than traditional sandwich structures with close cells (e.g. foam, corrugated, and honeycomb cores) due to its open topological configuration which can provide enough space for larger core plastic deformation caused by blast loadings and spread out the impact force more quickly [12,13]. For this reason, metallic lattice cores have attracted interest as core constructions of sandwich panels designed to support quasi-static [14 –17], dynamic impact loadings [18 –23] replacing traditional close cell core. Sandwich panels with lattice core are already increasingly interesting in aerospace, transport, and marine applications.

Functionally graded material (FGM), which was proposed in 1990s, is a kind of advanced inhomogeneous composite structures. Material constants of FGM are usually designed to vary continuously along a specific direction. FGM has widely application prospect in aerospace, nuclear, biological engineering, and many other fields where the superior performances of materials are required. To this end, a variety of researches have focused on this topic [24 –26]. In recent years, the concept of FGM design has been applied to the sandwich structures in order to enhance the impact resistance of structures. Apetre et al. [27] investigated several available sandwich beam theories for their suitability of application to 1D sandwich panels with functionally graded core. Ajdari et al. [28] analyzed the compressive behavior of functionally graded voronoi structure. Fan et al. [29] investigated the behavior of the functionally graded honeycomb structures with defects under in-plane crushing loading. Zheng et al. [30] conducted the theoretical and numerical analyses of 1D impact response of density-graded cellular rods with middle-low and middle-high density distributions, observing that the density distribution of cellular core has significant influence on the impact response and the dynamic energy absorption of sandwich structures. Zhang et al. [31] presented the dynamic behavior of sandwich steel panels with three kinds of corrugated core arrangements consisting of identical core density subjected to dynamic air pressure loadings. Li et al. [32] studied dynamic response of metallic sandwich panels with stepwise graded aluminum honeycomb cores under blast loadings by finite element method. Liu et al. [33,34] carried out a numerical study on the dynamic responses and blast resistance of all-metallic sandwich-walled hollow cylinders with graded aluminum foam cores using finite element method. Liang et al. [35] investigated blast resistance and design of sandwich cylinder with graded foam cores using numerical simulations.

Compared to pyramidal lattice, the hourglass lattice [36,37] has superior mechanical response, including: (1) superior resistance to buckling of the core truss; (2) superior resistance to local buckling of the face sheets; (3) simple and low-defect fabrication method – Snap Fit method [16,18]. Therefore, it is of great significance to design and analyze a multilayered graded hourglass lattice sandwich structure to enhance blast protective capability of sandwich structures. The objective of this study is to investigate the blast resistance of a sandwich panel with multilayered hourglass lattice core of graded relative density. The outline of this paper is organized as follows. The details of a sandwich panel with multilayered lattice core of graded relative density are provided in the next section. The finite element models of sandwich panel are established and validated in section ‘Finite element model’. The simulation results and analyses are discussed and presented in section ‘Results and analyses’. Conclusions are presented in the last section.

Geometric modeling

To enhance the blast resistance of the lattice sandwich structures, a panel with three-layer hourglass lattice core of graded relative density is proposed in this research, as shown in Figure 1. This sandwich panel contains a front sheet, a back sheet, two intermediate face sheets, and three core layers. Each core layer contains five hourglass unit cells, respectively, along the direction of length and width. The overall height of panel is 96 mm and the spans of all sheets are 232.4 $\times$ 232.4 mm². Front sheet and back sheet are 3 mm in thickness. Due to the less effect of the intermediate face sheets on carrying capacity of structures, the thickness of intermediate sheets is taken as 1.5 mm, half the thickness of front and back sheets. The relative density of each core layer for graded models is different by varying sectional dimension of core truss members. Fabrication process of hourglass lattice using Snap Fit method is sketched in Figure 2.

The geometry of representative unit cell of hourglass lattice is shown in Figure 3. Some critical geometric parameters are L = 45.2 mm, c = 1.546 mm, h = 27 mm, h_f =1 mm, and ω = 45°. The sectional dimension d of core truss is a variables changed with the relative density of each layer.

Assuming that an hourglass unit cell occupies a space of $V$ and the total volume of all core truss members is denoted by $V_{t}$ , the relative density of hourglass lattice core can be represented by [36]

ρ = \frac{V_{t}}{V} = \frac{2 {2 [(2 d^{2} h - \frac{1}{2} d^{2}) + 4 (a + \frac{d}{2}) d h_{f} + d^{2}] - 2 d^{3}}}{L^{2} (h + 2 h_{f})}

(1)

Four types of models with three-layer hourglass lattice core were proposed in this paper, as shown schematically in Figure 4, where C1, C2, and C3 represent the first layer, the second layer, and the third layer of cores, respectively. According to the relative densities distribution in core layers of models, four types of graded models were defined as ACA model, CAC model, ABC model, and CBA model, respectively. Here, character A stands for strong core, character B for moderate core, and character C for soft core. The density gradient $γ$ of core layer of structures is defined as

γ = \frac{ρ_{\min}}{ρ_{\max}}

(2)

where ρ_min and ρ_max are, respectively, the minimum relative density and the maximum relative density of core layers.

Seven types of configurations G1–G7 were given for each model. G0 is ungraded configuration. Tables 1 to 4 show the configuration parameters of all models, respectively, including the sectional dimension of core truss d_i (i = 1, 2, 3), relative density ρ_i (i = 1, 2, 3) of core layer, and the density gradient $γ$ . The subscript i in d_i and ρ_i denotes the ith core layer of sandwich panels. The total mass M of all models is the same as 4.731 kg.

Table 1.

Configuration parameters of model ACA.

Configuration	d₁ (mm)	d₂ (mm)	d₃ (mm)	$ρ_{1}$	$ρ_{2}$	$ρ_{3}$	$γ$
G0	2.400	2.400	2.400	0.0227	0.0227	0.0227	100%
G1	2.443	2.311	2.443	0.0235	0.0211	0.0235	90%
G2	2.489	2.212	2.489	0.0243	0.0195	0.0243	80%
G3	2.538	2.100	2.538	0.0252	0.0176	0.0252	70%
G4	2.589	1.974	2.589	0.0262	0.0157	0.0262	60%
G5	2.643	1.830	2.643	0.0272	0.0136	0.0272	50%
G6	2.701	1.660	2.701	0.0284	0.0113	0.0284	40%
G7	2.763	1.456	2.763	0.0296	0.0089	0.0296	30%

Table 2.

Configuration parameters of model CAC.

Configuration	d₁ (mm)	d₂ (mm)	d₃ (mm)	$ρ_{1}$	$ρ_{2}$	$ρ_{3}$	$γ$
G0	2.400	2.400	2.400	0.0227	0.0227	0.0227	100%
G1	2.354	2.49	2.354	0.0219	0.0243	0.0219	90%
G2	2.300	2.59	2.300	0.0209	0.0262	0.0209	80%
G3	2.236	2.701	2.236	0.0199	0.0284	0.0199	70%
G4	2.158	2.828	2.158	0.0186	0.0309	0.0186	60%
G5	2.060	2.976	2.060	0.0170	0.0340	0.0170	50%
G6	1.935	3.146	1.935	0.0151	0.0378	0.0151	40%
G7	1.768	3.348	1.768	0.0128	0.0425	0.0128	30%

Table 3.

Configuration parameters of model ABC.

Configuration	d₁ (mm)	d₂ (mm)	d₃ (mm)	$ρ_{1}$	$ρ_{2}$	$ρ_{3}$	$γ$
G0	2.400	2.400	2.400	0.0227	0.0227	0.0227	100%
G1	2.466	2.400	2.332	0.0239	0.0227	0.0215	90%
G2	2.538	2.400	2.255	0.0252	0.0227	0.0202	80%
G3	2.616	2.400	2.165	0.0267	0.0227	0.0187	70%
G4	2.701	2.400	2.060	0.0284	0.0227	0.0170	60%
G5	2.795	2.400	1.935	0.0302	0.0227	0.0151	50%
G6	2.899	2.400	1.783	0.0324	0.0227	0.0130	40%
G7	3.015	2.400	1.591	0.0349	0.0227	0.0105	30%

Table 4.

Configuration parameters of model CBA.

Configuration	d₁ (mm)	d₂ (mm)	d₃ (mm)	$ρ_{1}$	$ρ_{2}$	$ρ_{3}$	$γ$
G0	2.400	2.400	2.400	0.0227	0.0227	0.0227	100%
G1	2.332	2.400	2.466	0.0215	0.0227	0.0239	90%
G2	2.255	2.400	2.538	0.0202	0.0227	0.0252	80%
G3	2.165	2.400	2.616	0.0187	0.0227	0.0267	70%
G4	2.060	2.400	2.701	0.0170	0.0227	0.0284	60%
G5	1.935	2.400	2.795	0.0151	0.0227	0.0302	50%
G6	1.783	2.400	2.899	0.0130	0.0227	0.0324	40%
G7	1.591	2.400	3.015	0.0105	0.0227	0.0349	30%

Finite element model

Model details and material properties

Three-dimensional dynamic finite-element simulations were performed using the explicit time integration version of the commercially available FE code ABAQUS in this research. Non-uniformly distributed pressure which was imposed on the front sheet facing layer C1 was calculated using the CONWEP code [20].

The fully clamped boundary conditions were imposed on the four inner edges of face sheets. Eight-node brick elements with reduced integration (type C3D8R in ABAQUS notation) were employed. Such elements are capable of accurately capturing the dynamic stresses and strains. The front and back sheets were discretized with three layers of element through the thickness and the intermediate sheets were discretized with one layer of element. Numerical damping associated with volumetric straining in ABAQUS/explicit was switched off. The mesh density was selected when further mesh refinements did not appreciably improve the accuracy of the calculations remarkably. There are 23,176 elements of size 3 mm for sheets and 88,800 elements of size 1.2 mm for core in any calculations.

Both sheets and core truss of the multilayered lattice panel were made from super austenitic stainless steel AL6XN. The yielding of AL6XN was modeled by Mises criterion. A bilinear strain rate dependent constitutive model was employed to describe the true stress versus true strain relation of the material as

σ = {\begin{array}{l} E ε ε \leq \frac{σ_{Y}}{E} [1 + {({\dot{ε}}_{P} / {\dot{ε}}_{0})}^{m}] \\ σ_{Y} [1 + {({\dot{ε}}_{P} / {\dot{ε}}_{0})}^{m}] + E_{t} {ε - \frac{σ_{Y}}{E} [1 + {({\dot{ε}}_{P} / {\dot{ε}}_{0})}^{m}]} ε > \frac{σ_{Y}}{E} [1 + {({\dot{ε}}_{P} / {\dot{ε}}_{0})}^{m}] \end{array}

(3)

where E,

σ_{Y}

E_{t}

{\dot{ε}}_{p}

{\dot{ε}}_{0}

, and m are, respectively, Young’s modulus, initial yielding strength, tangent modulus, equivalent plastic strain rate, reference strain rate, and materials constant. The material parameters for stainless steel AL6XN [20] are: mass density

ρ

= 8060 kg/m³, Young’s modulus E = 200 GPa, Poisson ratio

μ

= 0.3, initial yielding strength

σ_{Y}

=410 MPa, tangent modulus

E_{t}

=2 GPa, reference plastic strain rate

{\dot{ε}}_{0}

= 4916 s⁻¹, and materials constant m = 0.154. A failure criterion was not included in the calculations, so neither fracture of the sheet nor core debonding from the front face was not capable of being captured.

Validation of numerical model

No experimental or numerical studies of the blast response of sandwich panels with hourglass lattice cores have been conducted in the existing literature. In this study, the finite element model and the finite element method were validated by an experimental study of pyramidal lattice sandwich panel subjected to blast loadings [20]. A single-layer pyramidal lattice sandwich panel with the same geometric parameters and loading parameters as Dharmasena et al. [20] was established. The thickness of face sheets is 0.76, 1.52, and 1.90 mm, corresponding to the mass density of 16.93, 29.26, and 35.47 kg/m², respectively. TNT charge mass is 150 g and the stand-off distance is 150 mm. One-quarter of the panel was modeled by applying symmetry boundary conditions over symmetrical sections so as to compare with the results of Dharmasena et al. [20]. The corresponding geometric model is shown in Figure 5.

Perfect bonding between the core and sheets was adopted here. The element types, materials properties, and loading modes are the same as section ‘Model details and material properties’. There are 19,602 elements of size 2 mm for face sheets and 23,660 elements of size 1 mm for core in simulation analysis. The geometric nonlinear is defined by setting the NLGEOM parameter as ON in option STEP in ABAQUS/explicit.

The deformation diagrams of sandwich panel obtained from simulation study are shown in Figure 6(a) to (c). Figure 6(d) to (f) presents the experimental results given in Dharmasena et al. [20]. Comparing Figure 6(b) with (e), Figure 6(c) with (f), it is evident that the deformations obtained from the simulations agree better with the experimental results. Due to the fact that failure criteria were not embodied in the simulation analysis, no material failure was observed in Figure 6(a), which is a little different from experimental study (as shown in Figure 6(d)).

The dimensionless peak deflection δ_max/S of back sheet obtained from simulation study and from experimental study [20] are compared in Figure 7 to validate the FEA, where δ_max is the peak deflection at center point of back sheet and S denotes half the length of sandwich panel. From Figure 7, it can be found that the numerical simulation results are in good agreements with experimental results of Dharmasena et al. [20].

In order to further assess the validity of the analysis on hourglass lattice cores in this paper, deformations of the sandwich panels with pyramid lattice cores and hourglass lattice, respectively, are shown in Figure 8 under the identical conditions. There exists the distinct similarity between these two deformations. Also, we can find that the residual peak deflection of hourglass lattice cores (Figure 8(b)) is smaller than that of pyramidal ones (Figure 8(a)).

Results and analyses

The blast resistance analyses of a sandwich panel with multilayered graded hourglass lattice core are carried out in this section. The geometry and dimension of the lattice unit are the same with that shown in Figure 3. The blast loadings are imposed on the front sheet (facing layer C1) of the structures by CONWEP code and the core layers was compressed sequentially.

Back sheet deflections of sandwich panels

Since personnel or important objects shielded from blast attacks are usually behind the protective structures (e.g. sandwich panels), the back face deflection of sandwich panels is herein considered as the main response of interest. In this section, simulations were performed under a specific load intensity of a TNT charge mass of 350 g and a stand-off distance of 200 mm. Figure 8 reveals distinctly the variation of the peak back sheet deflections $δ_{max}$ at center point of back sheet for four types of models as a function of the density gradient $γ$ .

It can be seen from Figure 9 that the relative density arrangements in cores had a significant influence on the peak deflection, peak deflection of different types of models varied with the density gradient $γ$ in fully different ways. Some studies [30 –34] indicated that the impact resistance of graded sandwich structures was superior to that of conventional ungraded sandwich structures. It can also be observed in this research that the peak back sheet deflection of models ABA and CBA (when $γ < 100 %$ ) was smaller than that of ungraded models $(γ$ =100%) and model CBA has the minimal peak deflection to any density gradient.

Numerical results of this research shows that, for three-layer hourglass lattice sandwich panels, the graded models with soft layer C3 (i.e. models CAC and ABC) has a lower blast resistance than ungraded models, on the contrary, the models with strong layer C3 (i.e. models ACA and CBA) has better blast resistance.

Comparing model CBA with model ACA, it can be noted from Figure 9 that the peak deflections of these two models are basically the same when $γ$ is 50%, and models CBA and ACA with $γ$ of 50% have the near-optimal blast resistance under the present blast loadings. Since the peak back sheet deflections of model CBA are always smaller than that of model ACA to any density gradient, the model CBA, of which core layer relative density gradually increase from layer C1 to layer C3, has the superior blast resistant, that is, the tapered acoustic impedance of hourglass lattice sandwich panels along the wave transmitting direction is conducive to the shock wave dissipation. In addition, it can also be found that the dynamic response of model ACA is more sensitive to the density gradient $γ$ than model CBA.

A pyramidal lattice sandwich panel of the same areal mass as hourglass lattice sandwich panels was investigated under blast loadings in Han et al. [38]. Peak back sheet deflections of sandwich panels with pyramidal lattice cores and sandwich panels with hourglass lattice cores were comparatively analyzed under the condition of identical blast loadings, as shown in Figure 10, where ACA-P, CBA-P is corresponding to pyramidal lattice core, and A, B, C still stand for, respectively, strong, moderate, and soft core. It can be seen from Figure 10 that the blast resistance of hourglass lattice models is generally better than that of pyramidal lattice models since the peak residual deflection of Models CBA is smaller than that of Models CBA-P for various density gradient. Models CBA-P (with pyramidal lattice core) and CBA (with hourglass lattice cores), of which relative density gradually increase from layer C1 to C3, have the smooth decrease of the peak back sheet deflections with the decrease of the density gradient $γ$ . For pyramidal lattice structures, models ACA-P has the superior blast resistance to models CBA-P; on the contrary, models ACA for hourglass lattice structures has the inferior blast resistance to models CBA. Additionally, configuration G5 ( $γ = 50 %$ ) for both pyramidal lattice models and hourglass lattice models are the near-optimal configuration under present blast loadings.

Energy absorption of layers

In order to comprehend the energy absorption mechanism of the sandwich structures under blast loadings, layer plastic energy absorption of four types of models under the same loading with the previous subsection are given in Figure 10 as a function of density gradient $γ$ . It can be easily found that there exists the largest difference of core layers energy absorption in ungraded model ( $γ = 100 %$ ), energy absorption of its layer C2 is too low to play inessential part in blast protection. For all of four models, plastic energy absorption of layer C1 decreases with density gradient $γ$ decreasing while that of layer C2 increases with the decrease of $γ$ $.$

It can be concluded by analyzing Figures 11 and 9 that there exist links between blast resistance and energy absorption of core layers. Figure 8 has revealed that models ACA and CBA have the better blast resistance than models CAC and CBA. From Figure 10, layer C2 of models ACA and CBA has more energy absorption and its plastic energy absorption is more than layer C3 when $γ < 80 %$ . However, for models CAC and ABC, energy absorption of layer C2 is always less than that of layer C3. The calculation results reveal that, for models ACA and CBA, the energy absorption of each core layer can be changed remarkably by varying density gradient $γ$ and layer C2 can be guided to absorb more energy so as to reduce the energy absorption of layer C3 and provide better protection for layer C3 and back sheet.

Figure 12 presents dynamic plastic deformation response of two graded models and ungraded models at time t = 0.3, 0.6, 0.9, and 1.2 ms, respectively. In order to further investigate the effect of model types on core layer deformation, the near-optimal configuration of each graded model (corresponding to the minimum peak back sheet deflection) are selected. Obviously, plastic deformations of layer C2 for ungraded model and model ABC are insignificant and far less than those of layer C1 and layer C3. For model CBA, the plastic deformation of layer C2 is larger while layer C3 has smaller plastic deformation. In addition, layer C2 of models CBA has already been densified at the time t = 0.9 ms.

For a sandwich structure with lattice core under blast load, the residual deflection of back sheet is closely related to the plastic buckling deformation (energy absorption) of various core layers. The more full buckling deformation occurs in layer C2 of model CBA in which the relative density of layer facing back sheet is the largest, which leads to the larger plastic deformation of layer C1 and the smaller plastic deformation of layer C3 (facing the back sheet). Thus, it can be concluded that layers C1 and C2 of the graded model CBA absorb more energy than those of other models. This results in a better protection on layer C3 and back sheet and makes models CBA exhibit excellent blast resistance capacity.

Zhang et al. [39] investigated the blast resistance of a multilayered sandwich panel with ungraded pyramidal lattices subjected to near field underwater explosion by an experimental method and found the similar results as this paper that the middle layers of sandwich panels had smaller plastic deformation than the front and rear layers. Zhang et al. [31] investigated the impact response of sandwich panels with three kinds of graded corrugated core consisting of identical core density by experimental method and also obtained the same conclusion as this paper that the sandwich panels with smoothly and monotonically increasing relative density of core from the front sheet to the rear sheet have the smallest peak rear sheet deflection.

The effect of load intensity on the peak deflection

In order to determine the influences of blast loading intensity on dynamic response of sandwich panels with multilayered graded hourglass core, different TNT charge mass of, respectively, 200 g, 350 g, and 500 g are imposed on the graded models ACA at a specific stand-off distance of 200 mm, and the corresponding peak back sheet deflection are shown in Figure 13 as a function of density gradient $γ$ . It can be seen from Figure 13 that the peak deflection increases distinctly with the load intensity increasing and different load intensity corresponds to different optimal configurations. For all of charge mass, the peak deflection decreases first and then increase with density gradient $γ$ decreasing. Configuration G6 ( $γ$ =40%) is near-optimal for 200 g charge mass and configuration G5 ( $γ$ =50%) is near-optimal for 350 g and 500 g charge mass. Moreover, it can be concluded easily that the greater the charge mass is, the more sensitive to density gradient $γ$ the peak deflection of panels is.

Conclusions

Numerical simulations are carried out for sandwich panels with multilayered graded hourglass lattice core subjected to blast loadings using ABAQUS//Explicit. Existing experimental measurements for single-layer pyramidal lattice panel are used to validate the numerical approach. The peak deflections of back sheet center point and the energy absorption of core layers are focused on. The influences of blast loading intensity are analyzed. Major findings from the numerical simulations can be summarized as follows:

The density gradient of multilayered graded hourglass lattice sandwich structures has a significant influence on the blast resistance of structures and the graded hourglass lattice sandwich panels have the superior blast resistance to conventional ungraded sandwich structures.

For three-layer hourglass lattice sandwich panels, the graded models with soft layer C3 (i.e. models CAC and ABC) has a lower blast resistance than ungraded models; on the contrary, the model with strong layer C3 (i.e. models ACA and CBA) has better blast resistance.

The model CBA, of which core layer relative density gradually increase from layer C1 to layer C3, has the superior blast resistant and configuration G5 ( $γ$ =50%) of model CBA is the near-optimal configuration under the present loading.

The blast resistance of models with hourglass lattice cores is generally better than that of models with pyramidal lattice cores.

The greater the charge mass is, the more sensitive to density gradient $γ$ the peak deflection of panels is.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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