Computational analysis and optimization of sandwich panels with homogeneous and graded foam cores for blast resistance

Thin–Walled Structures 147 (2020) 106494

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Computational analysis and optimization of sandwich panels with homogeneous and graded foam cores for blast resistance Erdong Wang a, Qing Li b, Guangyong Sun a, b, * a b

State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha, 410082, China School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW, 2006, Australia

A R T I C L E I N F O

A B S T R A C T

Keywords: Sandwich panels Functionally-graded foam core Shock stress wave Design optimization Blast resistance

Structural responses, deformation modes, blast resistance and energy absorption of foam core signify some major functional characteristics for design of sandwich panels. This study aimed to address these issues by investigating uniform and graded foam core configurations. First, an experimental study was performed and the testing results of blast-loaded sandwich panels were analyzed. Second, a numerical model was developed and validated by comparing the simulation results with the experimental results in terms of deformation modes and back facesheet deflection. Third, the blast resistance of sandwich panels was comprehensively studied based upon the developed numerical models. Due to the high attenuation ability of the shock induced stress wave, the foam core with descending gradient of layer density across the thickness direction provided the highest blast resistance of all the core configurations considered here and its advantage could be further improved by enlarging the density dif­ ference of the core layer. While keeping total facesheet thickness unchanged, a relatively thick back facesheet is beneficial to enhance the blast resistance under relative low blast intensity. Finally, an optimization study was performed to improve the blast resistance of graded core sandwich panels. For the single objective optimization, the maximum back facesheet deflection of the optimum design decreased by 24.58% in comparison with that for the initial baseline design. For the multiobjective optimization, the optimal designs obtained from the Pareto solution can significantly enhance weight efficiency without compromising the resistance.

1. Introduction Sandwich structures typically comprise two thin strengthening facesheets and a thick core in between. Essentially, the design idea behind this hybrid structure is to take advantage of both stiff facesheet and soft core to generate superior overall performance to solid facesheet that has the same weight [1,2]. In sandwich structures, honeycombs or metallic foams are commonly used as soft core materials attributable to their excellent characteristic properties such as lightweight and energy absorption capacity, etc. [3–8]. Compared with honeycomb materials, metallic foams are thought to be a class of more desirable core materials for sandwich structures as they are less directional dependent. Functionally graded materials (FGMs) have exhibited considerable potential to be a more effective core configuration of sandwich structure in comparison with uniform counterpart [9,10]. Recently, bioinspired structures have drawn growing attention; many biological materials, especially those used for energy absorption such as sheep horns and horse hooves, contain some forms of graded structures for better

adapting to complex natural environment [11]. For example, tubular density increases in the inner-to-outer radial direction across the wall thickness in horse hoof, as shown in Fig. 1 (a). Such a gradient config­ uration results in varying mechanical properties in specific regions to accommodate the functionality of hoof, such as withstanding the high stress, that could cause failure of horse hoof [11]. Similarly, graded aluminum foam materials have been produced through advanced melt foaming technology, aiming at generating desirable mechanical prop­ erties [12], as shown in Fig. 1 (b). Further, stepwise graded aluminum foams (as seen in Fig. 1 (c)), where the material properties vary in a layer by layer fashion, exhibiting great simplicity and efficiency to be core materials, which have been however under-studied in literature to date [10]. For this reason, the present paper focuses on the sandwich struc­ tures with either homogeneous or stepwise graded foam cores. With growing concern in blast threats, studies on sandwich panels have been refueled to make it a class of proper protective structure for enhancing blast resistance [13–16]. Substantial efforts have been devoted to the investigation into their deformation/failure modes and

* Corresponding author. State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha, 410082, China. E-mail address: [email protected] (G. Sun). https://doi.org/10.1016/j.tws.2019.106494 Received 3 July 2019; Received in revised form 17 September 2019; Accepted 6 November 2019 Available online 28 November 2019 0263-8231/© 2019 Elsevier Ltd. All rights reserved.

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More recently, design optimization of blast-loaded sandwich struc­ tures with aluminum foam core has gained rising attention as a result of the development of associated numerical technique. For instance, Qi et al. [14] carried out a multiobjective design optimization (MDO) of the uniform foam core sandwich structures under variable blast load in­ tensity. Lim et al. [31] used the Kriging technique to seek for an optimal design of blast-loaded hybrid sandwich plates with uniform aluminum alloy foam core. Jing et al. [32] performed a MDO of stepwise graded aluminum foam core sandwich panels under blast loading. In their study, the density of each core layer was treated as a design variable while keeping thickness of each layer unchanged. In Zhang et al.’s study [33], the thickness and density of each core layer were found to greatly affect the ball impact responses of the functionally graded aluminum foam blocks. To the best of authors’ knowledge, there are few studies on the design optimization of graded foam core sandwich structures under blast loading by taking the layer thickness and density as design variables. Inspired by the aforementioned knowledge gaps, this study aimed to first explore the coupling effects of different core gradient directions, core layer thickness/density, and facesheet thickness configurations on the blast resistance of sandwich panels. A series of in-house blast ex­ periments were conducted for foam core sandwich panels using a fourcable ballistic pendulum system [20]. The corresponding numerical model was established in LS-DYNA and its credibility was validated against the experiment results. The structural response and deformation modes of sandwich structure were analyzed in detail. Second, a comprehensive parameter study was carried out to explore the effects of blast loading, stand-off distance, foam core gradient and facesheet thickness on the blast resistance and energy absorption of different sandwich panels. The propagation of shock-induced stress wave through the graded foam core was scrutinized in detail. Finally, the single and multiobjective optimizations were respectively carried out to generate the optimal designs for the graded foam core sandwich panels so that more efficient protective structures were obtained.

dynamic responses under blast loading, theoretically and experimen­ tally [17–20]. For example, Fleck and Deshpande et al. [17] proposed a rigid-plastic model to study the blast resistance of sandwich beams theoretically, in which three-stage responses were identified, namely fluid-structure interaction, core crushing and the retardation phases. Theobald et al. [18] conducted air-blast tests to investigate the inelastic response of unbonded metallic foam and honeycomb core sandwich panels under blast loading. The effects of facesheet thickness, core thickness and core type on the blast resistance were characterized. Liu et al. [19] experimentally investigated the blast wave attenuation and deformation process of sandwich panels with closed-cell aluminum foam core. Sun and Wang et al. [20] employed a four-cable ballistic pendulum to conduct blast tests on foam based sandwich, in which the influence of blast impulse, facesheet material and core gradient on the blast resistance of sandwich structure was analyzed in detail. In addition to theoretical and experimental methods, numerical modeling signifies an effective approach to gaining further compre­ hensive understanding in the dynamic behavior and blast resistance of sandwich structure. In this regard, Li et al. [21] investigated the dy­ namic responses and blast resistance of spherical and cylindrical sand­ wich shells with uniform foam core against blast loading. They found that the blast resistance of spherical sandwich shells is higher than that of the cylindrical sandwich shells. Jing et al. [22] carried out the nu­ merical study on cylindrical sandwich shells with metallic foam cores under blast loading, in which structural response, facesheet deflection and energy absorption of blast-loaded sandwich shells were quantified. Li et al. [23] investigated the impact responses of inner blast-loaded sandwich spherical shells with graded aluminum foam cores. They found that the tapered configuration of foam density from inside to outside was an optimal configuration to improve structural resistance to blast. While substantial studies on blast resistance of uniform foam core sandwich have been conducted [24–26], limited reports on the graded foam core sandwiches against blast loading were available. In literature, Yin et al. [27] and Fang et al. [28,29] pointed that the grading directions of foam core have great effect on the low-velocity impact and bending behavior of foam-filled structures. Zhang et al. [30] and Sun et al. [10] found that the ratio of front-to-back facesheet thickness significantly affected the energy absorption of sandwich structures when subjected to low-velocity impact. The coupling effects of core gradient (direction and magnitude) and the ratio of front-to-back facesheet thickness on the performance of blast-loaded sandwich structure have not been reported yet up to now. Further, how the graded foam core affects the resistance performance and what the corresponding mechanisms are behind remain to be clarified.

2. Experimental investigation A four-cable ballistic pendulum system was employed to conduct the in-house blast experiments. The effects of different facesheet materials and core gradients on the deformation modes and dynamic responses of sandwich panels were investigated in our previous study [20]. For completeness of this paper, the in-house experimental procedure and results are briefly outlined as follows.

Fig. 1. The evolution of bionic structures: (a) the graded tubal distribution over the wall thickness in horse hoof [11], (b) smooth foam gradient [12], and (c) layer-wise foam gradient. 2

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2.1. Experimental procedure

3. Numerical study

Foam core sandwich panels with a side length of 300 mm and a width of 300 mm were prepared for the blast tests. The 304 stainless steel, 5182 aluminum alloy and twill weave laminated CFRP were respectively used to be the facesheets and each with a thickness of 1.0 mm. Both the uniform (homogeneous) and graded density foams were used to be the core materials. The density and thickness of uniform foam core are 0.45 g/cm3 and 30 mm, respectively. The graded foam was made of three layers with different foam densities, and the thickness of each core layer was 10 mm (total layup 30 mm). The densities of graded core layer were ranged from 0.20 g/cm3 to 0.70 g/cm3. To provide a more dedi­ cated analysis on the deformation/failure modes of the single sandwich component, no adhesive was utilized between the sandwich compo­ nents, as adopted in literature [18]. The blast experiment set-up is sketched in Fig. 2. The sandwich specimens were clamped along their periphery through two square steel clamping fixtures; and the exposed area was set to be ​ 250 ​ mm � 250 ​ mm. Different masses of ball-shape rock emulsion explosive were utilized to generate blast loading. The distance between the sandwich specimens and explosive charges were fixed at 100 mm in all the tests. The displacement-time response of ballistic pendulum was recorded during explosion by a laser displacement transducer, and then the blast impulse could be calculated from it.

3.1. Numerical modeling Explicit finite element codes LS-DYNA 971 was used to simulate the dynamic behavior of foam core sandwich panels. The uniform core sandwich panels with steel front facesheet and steel back facesheet under different blast impulses were used to validate the established numerical model. The mechanical behaviour of closed-cell aluminium foams was modeled with Material Type 63, which is dedicated to model crushable foam with optional damping and tension cutoff [34]. The compressive stress-strain curves of different foam densities, obtained from the in-house quasi-static compression tests, are presented in Fig. 3. The input elastic modulus of the foam materials in LS-DYNA was based on our previous material tests in Ref. [10]. The Poisson’s ratio of aluminum foam is usually set to be zero due to no significant expansion in uniaxial compression tests, while the tensile stress cutoff (TSC) is adopted as the initial yield stress of foam material [35]. The mechanical behaviour of steel facesheet (304 Stainless Steel) was modeled using Material Type 3 with a density of 8060 kg/m3, Young’s modulus of 210 GPa, Poisson’s ratio of 0.3, yield stress of 300 MPa, and tangent modulus of 1.7 GPa [36]. The Cowper-Symonds model was used to consider the strain rate effect [34]: � ε_ �ð1=PÞ σd ¼1 þ σY C

2.2. Experimental results

(1)

where σd is the dynamic yield stress, σ Y is the yield stress, ε_ is the strain rate; the values of C ​ and P are material constants as 106 s 1 and 9, which are derived from Radford et al. respectively [36]. As the CONWEP algorithm developed by Kingery and Bulmash [37] has proved to be of high accuracy in structural response analysis under explosive loading, it was thus used to generate the blast loading in this study. The influence of incident pressure and reflection pressure is considered in the CONWEP algorithms, and the explosion load is defined as [37]: � pðtÞ ¼ pr ðtÞcos2 θ þ pi ðtÞ 1 þ cos2 θ 2cosθ (2)

Under the tested explosive charge range, no failure was observed on the sandwich specimens with steel (ST) facesheets and uniform density (UD) core (labelled as ST-UD45-ST). Based upon analysis of the tested sandwich specimens, the corresponding deformation and failure were classified into two modes. Specifically, large plastic deformation with slightly localized core compression (Mode I) was occurred when sub­ jected to a low level of emulsion explosive charge (50 g and 60 g). When the explosive mass was at a relatively high level (70 g), significantly localized compression was observed and the core compression was more than half of the original thickness (Mode II). The experimental results (e. g. deflection) and deformation/failure modes of tested sandwich speci­ mens will be compared with the numerical results in the following section.

where θ is the angle of incidence, pi ðtÞ is the incident pressure and ​ pr ðtÞ is the reflected pressure. The CONWEP algorithm were embedded in LS-DYNA code with the keyword of *LOAD_BLAST [34]. The parameters required for the *LOAD_BLAST include the equivalent mass of TNT, the coordinates of detonation point, and type of blast. The blast loading was generated in

Fig. 3. Experimental quasi-static compressive stress vs. strain curves of the used aluminium foams with different densities.

Fig. 2. A sketch of four-cable ballistic pendulum system. 3

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conjunction with the keyword *LOAD_SEGMENT or *LOAD_SHELL [34]. In the numerical modeling, the shape of charge was set up to be spherical and the detonated distance was 100 mm away from the sandwich front panel, which was consistent with the dedicated in-house blast experi­ ments. The conversion coefficient of the emulsion explosive to spherical TNT charge for CONWEP is about 0.7 [20]. Note that the units in the numerical simulation were cm, g, μs, and Mbar, while these units were presented as mm, g, μs and MPa in the following discussions respectively. Fig. 4 illustrates the whole the numerical model, in which only 3/4 is presented to show the internal details. Shell elements with five inte­ gration points across the thickness were used to model the thin face­ sheets. The foam cores were meshed using hexahedral solid element with single integration point. A mesh size of 2.0 mm was adopted for the facesheets and foam core according to the mesh convergence analysis. The clamping frames (fixture) were also modeled with a coarser mesh of 5.0 mm. There were 16 bolts with a diameter of 18 mm located around the rectangular fixture and the sandwich specimens were clamped be­ tween the two fixtures by tightening these bolts during the blast tests [20]. The heads and nuts of the installation bolts were also modeled to simulate the constraint (boundary) conditions. Specifically, the degrees of freedom (DOFs) of the nodes of the bolt heads and nuts were all constrained. The pre-tightening force of these bolts was simulated by applying the nodal pressure of the nodes of the front clamping fixture just underneath the bolt heads. Note that the nodal pressure was set as the initial yield strength (i.e., 1.35 MPa) of the lowest density foam. In addition, the DOFs of the nodes of the front and back clamping fixtures were constrained except the moving DOF along blast direction. The automatic surface to surface contact algorithm with a static and a dynamic friction value of 0.28 and 0.20 was used for all possible surfaces including: (i) contact between the bolts (bolt heads and nuts) and clamping fixtures (front and back fixtures), (ii) contact between the clamping fixtures and facesheets (front and back), and (iii) contact be­ tween the facesheets and foam cores [21]. To avoid self-penetration for the sandwich structures, automatic single surface contact was adopted for the sandwich components. As the foam core was compressed severely when subjected to strong blast loading, interior contact was applied to the foam brick elements to avoid the occurrence of negative volumes in simulation [34].

difference between the experimental and numerical boundary condi­ tions. In a similar case study, Li et al. [21] compared the numerical re­ sults with the time history of the experimental displacement in the center point of back facesheet for blast-loaded cylindrical sandwich panel; and they found that the above simulation approach can provide fairly accurate prediction before reaching the peak displacement. Since the dynamic peak response is usually one of the key criteria in the sandwich design; therefore the above modeling approach is considered effective. To validate the numerical model, the three sandwich specimens with a uniform foam core and steel facesheets were selected from the ex­ periments to compare both the deformation modes and back facesheet deflection in this study. Since the time history of the displacement at the center point of the back facesheet was not captured in the experiments, the experiment and simulation results of peak deflection in the back facesheet could not be compared directly. Fortunately, the springback of 304 stainless steel facesheet was marginal after blast loading [36,38]; thus the simulated deformation modes and maximum deflection of back facesheet can be to a considerable extent compared with the post-tested results, thereby verifying the prediction accuracy. Taking the sandwich specimen ST-UD45-ST-50g as an example, the deformation modes of the sandwich specimen from the simulation and experiment are compared in Fig. 5, which demonstrated that the simu­ lation can well capture the deformation patterns. Specifically, the compression deformation mainly occurred on the central region of the foam core; the radial plastic hinges of back facesheet were extended from the four corners to the base of the dome-shaped center. Further­ more, the peak deflections of back facesheets obtained from the simu­ lations also agreed well with the experimental results, as shown in Fig. 6. Therefore, the numerical approach was validated and can be used for the subsequent simulation studies. As for the sandwich structure in practice, facesheets and core are usually connected by the adhesive bonding. In this study, the bond connection was simulated by using the automatic surface to surface tiebreak contact. The failure criterion of tiebreak can be expressed in terms of normal and shear components as [34]: ​

jσ n j jσ s j þ �1 NFLS SFLS

(3)

where σn is the normal stress; σ s is the shear stress; NFLS and SFLS are the normal and shear failure strengths, respectively. In this study, the epoxy adhesive FORTIS AD825 had the NFLS of 15 MPa and SFLS of 2.47 MPa for the bond connection [21]. Moreover, the effect of the adhesive bonding between sandwich components was also investigated. Taking the sandwich specimen STUD45-ST-70g as an example, the experimental maximum back

3.2. Validation of numerical model In the numerical model, the foam core sandwich panel is clamped in between the two square clamping frames which cannot swig during blast loading. But in the experiment, the sandwich panel fixed to the impact pendulum can swig along with the pendulum so that the blast impulse could be measured [20]. For this reason, there was actually certain

Fig. 4. Numerical model of foam core sandwich panel under blast loading. 4

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Fig. 5. Comparison of deformation modes between the experiment and simulation: (a) the central section of the whole sandwich; (b) back facesheet.

Fig. 7. Energy time history of the system for the foam core sandwich panel.

Fig. 6. Comparison of experimental testing and numerical simulating results.

numerically. The central velocity and displacement histories of the front and back facesheet are presented in Fig. 8. To give a clear insight on the deformation modes of foam core sandwich panels, the deformation process at different time moments is also shown in the contour plots in Fig. 9. At t ¼ 30 μs, the blast wave started interacting with the front facesheet of sandwich panel (see in Figs. 8 (b) and Fig.9). The velocity of front facesheet induced by the blast loading quickly reached the peak at the time of 50 μs and then decreased due to foam core compression. The velocity of the back facesheet remained almost zero until the time point 100 μs; after which the velocity started increasing. Note that there is almost no deformation for the back facesheet from 30 μs to 100 μs (see in Fig. 9). The overall deflection of the whole sandwich structure increased from time point 100 μs–150 μs, progressively. At teq ¼ 150 μs, the ve­ locities of the front and back facesheets are nearly equal and it is observed that the central region of foam core began to separate from the front facesheet. After that, the core moved downwards together with the back facesheet. The deflection of back facesheet reached the maximum when the velocity of back facesheet reduced to zero (t ¼ 930 μs). At t ¼ 1100 μs, the central region of foam core began to separate from the back face­ sheet and then moved upwards (see in Fig. 9). The foam core contacted the front facesheet at t ¼ 2000 μs and pushed the facesheet moving up­ wards subsequently. The rebound of front facesheet and foam core ended at t ¼ 2760 μs. Then the sandwich panel began to vibrate freely until rest.

facesheet deflection was 27.06 mm; while the numerical results without and with consideration of the adhesive bonding were 25.13 and 21.29 mm respectively. Thus, it can be concluded that adding the ad­ hesive bonding between sandwich components can significantly enhance the structural blast resistance. To further confirm the accuracy and reliability of the numerical models, the energy conservation of sandwich panel was also investi­ gated with facesheet thickness of 1.0 mm, core thickness of 30 mm and core density of 0.45 g/cm3, subjected to the TNT weight of 80 g and the stand-off distance of 100 mm. The energy time history of the entire sandwich system was plotted in Fig. 7. It can be seen that the sum of kinetic energy, internal energy and hourglass energy is equal to the total energy at all time points. The ratio of the hourglass energy to the total energy is only 0.71%, which is far less than the threshold 5% typically used in evaluating the accuracy of numerical modeling [39]. Thus, the system energy during the entire blasting process remained in a good balance, indicating that the numerical results are accurate and reliable. 4. Results and discussion 4.1. Structural responses In order to gain further understanding of the blast resistance, the sandwich panel was investigated for quantifying the structural responses 5

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Fig. 8. Central velocity-time history (a) and displacement-time history (b) for the front and back facesheets.

As discussed above, the structural response process could be classi­ fied into three specific stages, namely, Stage I (t ¼ 30–100 μs), the compression of central foam core; Stage II (t ¼ 100–930 μs), significantly localized decent at central area of panel with global deformation; Stage III (t ¼ 930–2760 μs), free vibration of the sandwich structure. Similar structural response was also found in Ref. [21] for spherical sandwich shells under blast loading. Histories of numerical deflection from the centres of front and back facesheets were also plotted in Fig. 10. At the different time points, good symmetry of deflection about the facesheet centre was observed simply due to the symmetry of loading and boundary conditions. Obviously, the maximum deflections were all well positioned at the centres of face­ sheets. Interestingly, the time for the two facesheets to reach the maximum deflection differed, i.e. 660 μs for the front and 930 μs for the back facesheets, respectively.

difference in the front and back facesheets decreased from the centre to the boundary. This indicated that the blast resistance is remarkably more sensitive in the near-explosion zone. Fig. 12 (b) plots the de­ flections of the front and back facesheets under the different TNT masses. The curve fitting between the peak deflection and TNT mass m can be obtained as: δfront ¼

4:013 þ 0:331m;

δback ¼ 4:313 þ 0:272m;

R2 ¼ 0:999

R2 ¼ 0:998

(4) (5)

where δfront and δback are the peak deflections of front facesheet and back facesheet, respectively. It can be seen that the peak deflections are lin­ early proportional to the TNT mass. The fitting equations could be useful to predict the peak deformations of sandwich panel under different TNT masses. The energy absorption of each sandwich component and its pro­ portion in the total energy is compared in Fig. 13. The sandwich panels subjected to the TNT mass of 60, 80 and 100 g are numbered as Group 1, Group 2 and Group 3 here respectively. It is clear that the energy ab­ sorption of each sandwich component increased with the increase of TNT mass, since the deformation became more severe under a high explosive mass. The percentage of energy absorption of front facesheet and back facesheet shows slight increase, while the percentage of foam core decreased with increasing TNT mass. It can also be seen that the most of blast energy was absorbed by the foam core, and its percentage is more than 80% for all these three groups. Thus, improving the energy absorption efficiency of the foam core may be one of the most effective ways to enhance the overall blast resistance of sandwich panel.

4.2. Effect of the blast loading In order to explore the resistance capacity of sandwich panels against different levels of blast loading, the sandwich specimens with facesheet thickness of 1.0 mm, core thickness of 30 mm and core density of 0.45 g/ cm3 were considered for the stand-off distance of 100 mm and different masses of TNT at 60, 80 and 100 g. The deflection of back facesheet is usually of the main interest as it is commonly used to measure the protection of personnel and/or objects behind it from blast attacks [40]. As the central points of facesheets usually generated the greatest deflection, the time histories of the corresponding deflection in the back facesheet were illustrated in Fig. 11 for different TNT charges. It can be clearly seen that the peak deflection of back facesheet increased with increasing TNT mass, specifically 20.45, 26.34 and 31.31 mm for the TNT mass of 60, 80 and 100 g, respectively. The peak deflection of back facesheet reduced by 15.87% for 80 g and 34.69% for 60 g with respect to the TNT mass of 100 g. Furthermore, the time to reach the peak deflection in the back facesheet was about 900, 930 and 910 μs for the TNT mass of 60, 80 and 100 g respectively. With the increase of TNT mass, it shows that the time at the peak point of the back facesheet exhibit a slight increase trend. A similar characteristic was also reported by Li et al. [21] in blast loading on the spherical sandwich shells with metallic foam core. As the dynamic maximum deflection of back facesheet is the main concern under blast loading, the deformation details are further studied here. Fig. 12 plots the deformation curves for the different TNT masses. It is seen that the deflection curves exhibit good symmetry around the centre of facesheets (Fig. 12 (a)). The main deformation region concentrated around the central area of facesheets. When the TNT mass increased from 60 g to 80 g and from 80 g to 100 g, the deflection

4.3. Effect of the stand-off distance This section explored the effect of stand-off distance (SOD) on the blast responses, in which the same configuration of sandwich specimens was adopted as in Section 4.2. The TNT mass was fixed at 80 g, and the stand-off distances varied at 80, 100 and 120 mm. The time histories of central point deflection of back facesheet were plotted in Fig. 14 for the different SODs. The peak deflections of back facesheets are 28.84, 26.34 and 22.59 mm for these three stand-off distances, respectively. With the increase of stand-off distance from 80 to 100 mm, and from 100 to 120 mm, the peak deflections of back facesheet decreased by 8.67% and 21.67%, respectively. In other words, the deformation degree of sandwich panel decreased as the stand-off distance increased. Fig. 15 plots the deflection distributions of front and back facesheets with the different stand-off distances. Note that the deflection curves 6

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Fig. 9. Deformation process of sandwich panel at different time points. Note that the unit in the contour plots is mm.

were plotted at the moment when the deflection of back facesheet reached the maximum. Again, the deflection curves show good sym­ metry. When the stand-off distance decreased from 120 to 100 mm, the peak deflection of front facesheet was increased by 4.14 mm, i.e. from 18.41 to 22.55 mm; and when the stand-off distance decreased from 100 to 80 mm the peak deflection increased by 6.91 mm, i.e. from 22.55 to 29.46 mm. In other words, the deformation increase in the latter case (6.91 mm, from 100 to 80 mm) was evidently greater than that in the former (4.14 mm, from 120 to 100 mm) while in both the cases, the stand-off distance decreased evenly by 20 mm. This is due to the fact that the shock wave pressure induced by blast loading was exponentially attenuated as the stand-off distance increased; and the attenuation amplitude was large in the near-explosion zone for the same stand-off distance [41]. As for the back facesheet, the peak deflection increased by 3.75 mm with the decrease of stand-off distance from 120 mm to 100 mm, and increased by 2.50 mm when the stand-off distance changed from

100 mm to 80 mm. The increase pattern of peak deflection obtained from the back facesheet is opposite to that obtained from the front facesheet, which may be related to the degree of plastic deformation of the foam core under different SODs. The energy absorption of each sandwich component and its portion in the total energy is plotted in Fig. 16 for the different stand-off dis­ tances of 80, 100 and 120 mm. Evidently, the energy absorption of each sandwich component decreased with increasing SOD. Again, the blast energy was mainly absorbed by the foam core and its proportion in the total energy was more than 80%. The percentage of energy absorption of front and back facesheets shows marginal decreased, while the per­ centage of foam core increased. Therefore, the reason why the defor­ mation degree of back facesheet in the former case (SOD changed from 120 mm to 100 mm) is greater than the latter case (SOD decreased from 100 mm to 80 mm) was because the proportion of energy absorption of foam core in the total energy reduced with decreasing stand-off distance.

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Fig. 10. Time-histories of deflection measured from the central sections of (a) front facesheet and (b) back facesheet.

Fig. 11. Time histories of deflection at the back facesheet center with different TNT masses.

4.4. Effect of the foam core gradient 4.4.1. Core gradient, deflection and energy absorption In order to investigate the effect of foam core gradient on the blast resistance, the density gradient of foam core across the thickness di­ rection (from front to back facesheet) and lateral direction (from center to peripheral region) were both considered here. The schematic of the core density gradient in the thickness and lateral directions is presented in Fig. 17. Each color represents one density of foam core, and their volumes are the same. In this study, three densities, i.e. 0.61 g/cm3 (H), 0.45 g/cm3 (M) and 0.29 g/cm3 (L) were studied. As the graded foam cores have three different densities, there are six different configurations of placing foam layers. The sandwich specimens with different layer configurations were identified by a unique label for convenience of analysis, e.g., TG-LMH represents the graded density of foam-core layers with a density configuration of 0.29 g/cm3 (L), 0.45 g/cm3 (M) and 0.61 g/cm3 (H) from the front to the back facesheets across the thick­ ness direction (T); LG-LMH represents the density configurations of foam-core layers 0.29 g/cm3 (L), 0.45 g/cm3 (M) and 0.61 g/cm3 (H) from the center to peripheral in the lateral direction (L). The sandwich specimens with different core layer configurations under the TNT mass of 80 g and stand-off distance of 100 mm were discussed here. The peak deflections of back facesheet for the different core configurations in the thickness and lateral directions are graphed in Fig. 18. It can be seen that the peak deflection of back facesheet of TG-

Fig. 12. Deformation curves of sandwich panels with different TNT masses.

HML was smaller than the other configurations, and so was the LG-HML. The peak deflections of back facesheet of TG-HML and LG-HML were 23.72 mm and 25.66 mm, which decreased by 9.95% and 2.58%, respectively, in comparison with the uniform core configuration. Therefore, it can be concluded that a higher blast resistance can be obtained by selecting a proper core gradient. In addition, a decreasing pattern of core density in the thickness direction (TG-HML) was the best of all the gradient configurations. The energy absorption of each sandwich component with the different core configurations in the thickness and lateral directions is shown in Fig. 19. As for the total energy absorption, it can be found that 8

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Fig. 16. Energy absorption of each sandwich component and its proportion in the total energy. Note that the energy absorption is indicated by solid lines, while the energy absorption proportion by dotted lines.

Fig. 13. Energy absorption of each sandwich component and its proportion (%) in the total energy. Note that the energy absorption is indicated by solid lines, while the energy absorption proportion by dotted lines.

the sandwich absorbed the maximum energy when placing the low density core as the upper layer (in the thickness direction) or the inner layer (in the lateral direction); whilst absorbed the least energy when placing high density core as the upper layer (in the thickness direction) or the inner layer (in the lateral direction). The energy absorption of back facesheet shows the same trend as the whole sandwich structure. Note that the energy absorption of back facesheet of TG-HML or LG-HML was smallest in all the core configurations, thus the peak deflection had the lowest value. Therefore, the energy absorption could be changed by placing the core layers differently to improve the resistance. As discussed above, the sandwich with a decreasing foam density from the top layer to bottom layer in the thickness direction (TG-HML) has the highest blast resistance. Here, the effect of the density difference of core layer on the blast resistance was further investigated for the core configuration of TG-HML. The density difference was defined in terms of the absolute value between the two adjacent layers. In addition to the abovementioned density difference of 0.16 g/cm3, the density difference of 0.08 g/cm3 and 0.25 g/cm3 were also studied for the same loading conditions. Note that the density of middle layer was kept at 0.45 g/cm3 for the different core density differences. The maximum deflection-density differences in the back facesheet and the declension percentages of peak deflection of back facesheet (blue dotted lines marked by stars) are compared with the benchmark uniform density (UD) specimen in Fig. 20. It is evident that the peak deflection of the back facesheet decreased with the increase in the core density difference. As for the core density difference of 0.25 g/cm3, the peak deflection of back facesheet decreased approximately by 20% in comparison with that of the UD specimen. Thus, increasing the core density difference for the specimen TG-HML can effectively enhance the blast resistance, thus benefiting the design of graded core sandwich structures in protective engineering. The blast resistance of the sandwich specimen with the core density differences of 0.25 g/cm3 was simulated under seven different TNT masses (i.e. 50, 60, 70, 80, 90, 100 and 110 g) and the same stand-off distance of 100 mm. The maximum back facesheet deflection vs. TNT mass was compared for the two specimens (TG-HML and UD) in Fig. 21. It can be seen that both the maximum deflections in the back facesheet increased with increasing TNT charge. Under the considered range of TNT charge, specimen TG-HML exhibited clear advantage in blast resistance over the UD counterpart that has an equal weight. Compared with the UD configuration, the reduction percentage of peak deflection of TG-HML was also shown in Fig. 21 (blue dotted lines marked by stars). It can be seen that the reduction percentage of deflection increased first and then decreased, reaching a maximum at the TNT mass of 70 g. As for the sandwich panel under the low level of TNT charge (� 50 ​ g), the

Fig. 14. Time histories of center point deflection of back facesheet with different stand-off distances.

Fig. 15. Deflection distributions of front and back facesheets with different SODs.

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Fig. 17. Schematic of the density gradient of foam core in (a) thickness direction and (b) lateral direction.

Specimens TG-LMH and TG-HML were used as the two different cases, the stress distributions through the core thickness at the selected time points are plotted in Fig. 22 (b) and (c), respectively. Clearly, the compressive stresses of the core layer appear at different time points, illustrating that the graded core was compressed in a layer by layer fashion under blast loading. At the initial loading stage (t ¼ 30–70 μs), the stress mainly locates on the Core layers #1 and #2; while the stress was almost zero at Node 16, indicating that the stress had not trans­ mitted to the back facesheet at this stage yet. After that, plastic defor­ mation started occurring on the back facesheet due to the effect of the Core layer #3 acting on the facesheet. As loading increased (t ¼ 90–150 μs), the compressive stress of Core layer #2 was becoming larger than those of Core layer #1 for the TG-LMH and TG-HML. The Core layer #3 also produced a relative large compressive stress. At t ¼ 110 μs, the average compressive stress of the Core layer #3 for the TG-LMH is about 12 MPa, which is much higher than that for the TG-HML (5 MPa). As a result, more severe deformation would occur on the back facesheet of TG-LMH than that of TG-HML. Generally speaking, the smaller the compressive stress produced by Core layer #3, the less the deformation generated on the back facesheet. For the specimens TG-LMH and TG-HML under the same level of blast loading, the stress in Core layer #1 in TG-HML is higher than that in TGLMH. However, the stress in Core layer #3 in TG-HML is smaller than that in TG-LMH. That is to say, the core layer configuration in TG-HML could effectively reduce the stress to transmit to the back facesheet. Therefore, a better blast-resistance performance could be obtained for the specimen TG-HML. With the purpose of investigating how the core configuration affects the attenuation capacity of impact force, the attenuation ratio of different sandwich specimens UD, TG-HML, and TG-LMH are also plotted in Fig. 23. As per Li et al. [40], the attenuation ratio of contact force can be expressed by the following equation:

Fig. 18. The peak deflection of back facesheet for the core layer configurations in the thickness direction and lateral direction.

deflection reduction percentage was less than 10%. Therefore, it is better to utilize the TG-HML sandwich specimen when the structure was subjected to the medium or high level of TNT charge (� 70 ​ g). 4.4.2. Stress wave propagation through foam core As foam material is able to absorb large blast energy through compression, the explosion shock wave could be attenuated [39,40]. Sometimes, the back facesheet contacts with the protected objects directly, where a less impact force would be expected to transmit through to the back facesheet for higher blast resistance. Therefore, the sandwich panel with high resistance would usually anticipate having a high attenuation capacity of blast load [23]. As discussed in Section 4.4.1, the sandwich structure with a decreasing foam density from the top to bottom layer across the thickness direction (TG-HML) exhibited certain advantages in blast-resistance over the other sandwich config­ urations. To better understand the mechanism behind this, propagation of the shock induced stress wave in thickness direction of the foam core was studied here. The graded core with three densities, i.e. 0.61 g/cm3 (H), 0.45 g/cm3 (M) and 0.29 g/cm3 (L), were also taken as an example here. The TNT mass and stand-off distance was also fixed at 80 g and 100 mm, respectively. The central nodes of the foam core in the thickness direc­ tion are presented in Fig. 22 (a), where only 1/4 model is shown here.

λ¼

FP

FD FP

(6)

where FP represents the maximum contact force between the front facesheet and the upper core layer; FD represents the maximum contact force between the back facesheet and the lower core layer. Apparently, the attenuation ratio of TG-HML is greater than those of UD and TGLMH. This is due to the fact that soft foam (low density) is beneficial in attenuating the blast impulse transmitted to the back facesheet [40]. Thus, sandwich TG-HML performs better in enhancing the blast resis­ tance than the other configurations.

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Fig. 19. Energy absorption of the different core configurations: (a) in thickness direction; (b) in lateral direction.

Fig. 20. Comparison of the maximum deflection-density difference in the back facesheet and the declension percentage of peak deflection with the uniform density (UD) specimen.

Fig. 21. Comparison of the maximum back facesheet deflection at the different TNT charge mass for the TG-HML and UD, and the reduction percentage of peak deflection.

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Fig. 22. Analysis of stress wave: (a) the central nodes of foam core in thickness direction; and the stress distributions through the core thickness in specimen TG-LMH (b) and TG-HML (c).

space and potentially achieve better performance [10]. For this reason, the thickness variation in facesheet was considered here to investigate its effect on the blast resistance. Three thickness pairs of front and back facesheets, specifically 0.5 and 1.5, 1.0 and 1.0, 1.5 and 0.5 mm, were used while the total thickness was all remained at 2.0 mm. In other words, the front-to-back facesheet thickness ratios are 1/3, 1 and 3, respectively. The sandwich specimens with different front-to-back thickness ratios but the same the core density difference of 0.25 g/cm3 were numerically examined under the three different TNT masses (i.e. 50, 60 and 70g) and a constant SOD of 100 mm. Fig. 24 compares the maximum back facesheet deflection vs. TNT charge; and energy absorption vs. TNT mass, respectively. It is found that the sandwich with the thinnest front facesheet (0.5 mm) exhibited highest blast resistance than the other specimens when the TNT charge mass is at a low level (50 or 60 g). However, the blast resistance of sandwich panel with the front-to-back thickness ratio of 1.0 is superior to the others under the high TNT charge (70 g), as shown in Fig. 24 (a). Over the whole range of TNT mass considered, the sandwich spec­ imen with front facesheet thickness of 0.5 mm absorbed more energy than the other specimens as shown in Fig. 24 (b). The energy absorption capacity decreased as the front facesheet thickness increased. In order to facilitate comparison, the sandwich specimens with the front facesheet thickness of 0.5, 1.0, and 1.5 mm were marked as Case 1, Case 2 and Case 3 here. From Fig. 24 (b), it can be found that the front facesheet and foam core absorbed more blast energy in Case 1 than in the other cases (Cases 2 and 3). Nevertheless, the difference of the energy absorption of back facesheet was small of all these three cases. As the thickness of back

Fig. 23. Comparison of attenuation ratio of contact force in sandwich speci­ mens UD, TG-HML, and TG-LMH.

4.5. Effect of the facesheet thickness The conventional front and back facesheets are usually of the same properties including material and thickness. Compared with the con­ ventional design, the non-uniform facesheets could offer a large design 12

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the graded core. The thickness of each core layer t1 ​ t2 , t3 and the density of each core layer ρ1 , ρ2 , ρ3 were considered to be the design variables of core configuration. The thickness ratio of the front-to-back facesheets is equivalent to the allocation of thicknesses tF ; tB of front and back facesheets, which were taken as the other two design variables. The TNT mass of 60 g and SOD of 100 mm, and graded core in thickness direction were utilized for the design optimization. In this study, single and multiple objective optimizations were respectively carried out to seek the optimal parameters for the graded core sandwich panels aiming to enhance the blast resistance. For the single-objective optimization, the design criterion was defined to mini­ mize the peak deflection under the condition of equal mass. To facilitate comparison, the mass and height of the graded foam core were remained to be the same as those of uniform foam core (i.e uniform density of 0.45 g/cm3 and core thickness of 30 mm), where the thickness of front and back facesheets was kept to be 1.0 mm for the uniform sandwich. Besides, the total height of sandwich panel was kept to be 32 mm, spe­ cifically tF þ tB ¼ 2 mm and t1 þ t2 þ t3 ¼ 30 mm .This implies that there

were only five independent design variables, i.e. x ¼ ðtF ; t1 ; t2 ; ρ1; ρ2 ÞT for the single objective optimization, which is thus expressed mathemati­ cally as: 8 min : δ ðt ; t ; t ; ρ ; ρ Þ 8 max F 1 2 1 2 > > > > 0:5 mm � tF ; tB � 1:5 mm > > > > 6:0 mm � t ; t ; t � 12:0 mm > > > < > 1 2 3 > < 0:25g�cm3 � ρ ; ρ ; ρ � 0:65g�cm3 1 2 3 s:t: : > > > > tF þ tB ¼ 2:0mm > > > > > > t þ t þ t ¼ 30:0mm > 1 > > 2 3 : � : ρ1 t1 þ ρ2 t2 þ ρ3 t3 ¼ 30:0 mm � 0:45g cm3

(7)

where tF and tB represent the thicknesses of front and back facesheets, respectively. According to the above constraints, it can be seen that the total thickness of facesheets was kept at 2.0 mm. The graded core can be determined by the density and core layer thicknesses. The multiobjective optimization for the graded sandwich panel aimed at minimizing the structural mass and minimizing the peak deflection. In this case, there were no constraints on total thickness of core layers and total weight; the thicknesses of foam core and facesheets were purposely varied in a relatively larger space in comparison with the single objective optimization problem. Therefore, there are 8 indepen­

Fig. 24. The maximum deflection and energy absorption of sandwich panels with different thickness ratios of front-to-back facesheets under different TNT charge masses (tF : front facesheet thickness, tB : back facesheet thickness): (a) the maximum back facesheet deflection vs. TNT charge; (b) the energy ab­ sorption vs. TNT charge.

dent design variables, i.e. x ¼ ðtF ; tB ; t1 ; t2 ; t3 ; ρ1 ; ρ2 ; ρ3; ÞT . The optimiza­ tion problem is defined mathematically as:

facesheet in Case 1 (1.5 mm) was the thickest, the deformation degree of the back facesheet was smaller than the other cases. Consequently, the blast resistance with a thinner front facesheet (Case 1) outperformed the conventional even facesheets. In other words, the blast resistance of sandwich panel can be effectively enhanced by decreasing the thickness ratio of front to back facesheets when subjected to low explosive mass.

min8: m and δmax < 0:5 mm � tF ; tB � 2:5 mm s:t: 6:0 mm : � � t1 ; t2 ; t3 � 15:0 mm � : 0:25g cm3 � ρ1 ; ρ2 ; ρ3 � 0:65g cm3

(8)

The dynamic behavior of blast-loaded sandwich structure is highly nonlinear, and it is very costly to explore all the response information of structural parameters through numerical simulations. Surrogate models provide an effective way to characterize the relationship between the design variables and functional responses [32,44]. Kriging model was selected here for both the single and multiobjective optimization attributable to its characteristics in modeling highly nonlinear problems [45–47]. The flowchart of the optimization procedure is depicted in Fig. 25. First, the optimization problem is defined. Second, some sample points are generated by using Design of Experiment (DOE) method which is set to be the design variables to calculate the response values through finite element analysis. Third, the Kriging models are constructed based upon the sample responses and then the accuracies are validated. Finally, optimization algorithms, such as NSGA-II, are implemented to obtain the optimum results.

5. Design optimization for graded foam core sandwich panels 5.1. Optimization problems According to the above parametric study, it has been known that both the foam core gradient and thickness ratio of front-to-back face­ sheets greatly affect the blast resistance of sandwich panels. In the above section, the thickness of each core layer was kept constant (10 mm) and only limited number of front-to-back facesheet ratios was investigated. The question is what the optimal design would be if taking both the thickness ratio of front-to-back facesheets and the thickness of each core layer into consideration? Thus, the density and thickness of each core layer as well as the front-to-back facesheet ratio were selected as the design variables in the optimization problem here. Generally speaking, the computing cost of optimization increases significantly as the number of design variables increases [42,43]. For this reason, three layers of different density foams are still adopted for 13

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Fig. 25. Flowchart of the optimization by using Kriging model.

5.2. Single-objective optimization results

Table 1 Design points and relevant numerical results on these points.

Central composite design (CCD) method was employed here to generate sample points. For these five design variables

Points

T

​ x ¼ ðtF ; t1 ; t2 ; ρ1; ρ2 Þ , a total number of 43 sample points were ob­ tained. The sample points and the maximum back facesheet deflection were listed in Table 1. The accuracy of the Kriging model was usually assessed by using R-square (R2 ), maximum relative error (emax ) and average relative error (eavg ) [45]. In the optimization process, the Kriging approximate model of δmax ðtF ; t1 ; t2 ; ρ1; ρ2 Þ was constructed using these 43 sample points and

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

validated by the additional 10 points. The results showed that R2 was close to 1 and emax and eavg were smaller than 4%, indicating that the constructed Kriging model is of sufficient accuracy. The optimization problem was solved using NSGA-II and the opti­ T

mum point was found to be x*S ¼ ð0:86; 9:25; 10:25; 0:64; 0:47Þ . As per the constraints, the optimum design of the sandwich structure was tF ¼ 0:86 mm tB ¼ 1:14 m t1 ¼ 9:25 mm, t2 ¼ 10:25 mm, t3 ¼ 10:50 mm and ρ1 ¼ 0:64g=cm3 , ρ2 ¼ 0:47g=cm3 , ​ ρ3 ¼ 0:27g=cm3 .It can be seen that the optimum design has a descending gradient of layer density across thickness direction and a thicker back facesheet, which agreed well with the above discussion in Sections 4.4 and 4.5. However, the thickness of core layer increased from top layer (t1 ) to bottom layer (t3 ). The predicted value of maximum back facesheet deflection was 15.98 mm. To compare the surrogate based optimal result, the uniform foam core sandwich specimen (marked as initial baseline design) and the optimum one were examined by the FE analysis respectively. The time histories of back facesheet deflection were compared in Fig. 26. The maximum back facesheet deflection was 15.53 mm, which is only 2.82% lower than the predicted optimum value. This indicates that the Kriging model was sufficiently accurate to provide reliable predictions on the responses. In comparison with the initial design, the optimal design decreased 24.58% maximum back facesheet deflection (from 20.59 mm to 15.53 mm). Therefore, it can be concluded that an optimal configu­ ration of core layers (density and thickness) as well as facesheet thick­ ness could significantly enhance the blast resistance of sandwich structures. 5.3. Multiobjective optimization results The multiobjective optimization was conducted as above depicted in Fig. 25, in which 85 sample points were first selected from the space of 8 independent design variables as per the Latin hypercube sampling 14

Design variables

δmax (mm) 3

3

tF (mm)

t1 (mm)

t2 (mm)

ρ1 (g/cm )

ρ2 (g/cm )

0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.75 1.25 0.50 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

7.5 7.5 10.5 10.5 7.5 7.5 10.5 10.5 7.5 7.5 10.5 10.5 7.5 7.5 10.5 10.5 7.5 7.5 10.5 10.5 7.5 7.5 10.5 10.5 7.5 7.5 10.5 10.5 7.5 7.5 10.5 10.5 9 9 6 12 9 9 9 9 9 9 9

7.5 7.5 7.5 7.5 10.5 10.5 10.5 10.5 7.5 7.5 7.5 7.5 10.5 10.5 10.5 10.5 7.5 7.5 7.5 7.5 10.5 10.5 10.5 10.5 7.5 7.5 7.5 7.5 10.5 10.5 10.5 10.5 9 9 9 9 6 12 9 9 9 9 9

0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.45 0.45 0.45 0.45 0.45 0.45 0.25 0.65 0.45 0.45 0.45

0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.45 0.25 0.65 0.45

20.31 18.81 20.52 20.18 20.75 20.80 20.36 19.18 18.51 20.28 17.97 20.62 18.10 20.23 18.04 20.19 20.20 20.36 20.68 18.96 20.39 20.31 20.89 19.69 18.86 20.94 17.77 17.93 17.88 19.33 16.77 17.41 19.31 21.22 20.78 20.64 20.62 20.81 19.52 16.25 16.75 16.85 20.55

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16.5 mm; hence, it seems that the Kriging models are fairly suitable for the design optimization within this deflection range. The sandwich panels with a uniform foam core and identical face­ sheet are commonly used in the real applications [15]. Such uniform sandwich structure was considered to be the baseline design for com­ parison here. To be consistent with the above study, the density and thickness of uniform foam core were also selected to be 0.45 g/cm3 and 30 mm here. Two different facesheet thicknesses (1.5 mm and 2.0 mm) were considered as two baseline designs here, namely baseline Design 1 (1.5 mm) and baseline Design 2 (2.0 mm), respectively. The peak deflections of these two uniform sandwiches were plotted together with the optimal designs in the Pareto space as shown in Fig. 28 (a) and (b). It can be seen that the structural mass of optimal designs reduced by 24.25% and 11.83% respectively when compared with Design 1 and Design 2 at the same deflection. This means that the optimal designs exhibit significant potential to reduce weight without compromising the blast resistance. While for the similar mass, the peak deflection of the optimized designs decreased by 17.68% and 19.49% compared with Design 1 and Design 2. Moreover, the optimized peak deflection in Pareto space was 15.58 mm at a similar mass of the baseline design (uniform sandwich with the facesheet thicknesses of 1.0 mm), which agreed well with the optimal results (15.98 mm) obtained from the single-objective optimization. These findings provided us with some guidelines for design of sandwich structure against blast loading.

Fig. 26. Comparison of the deflection-time curves for the initial baseline design (uniform) and single objective optimal design.

method (LHS). The responses of these sample points were obtained through numerical simulations. 75 sample points were used to construct the Kriging approximate model, and the other 10 points to validate the model accuracy. As R2 was close to 1 and emax and eavg were smaller than 5.5%, it means that the accuracy of Kriging approximate model was sufficient. The optimal results were obtained by using the NSGA-II optimization algorithm, as displayed in Fig. 27. Note that these 85 sample design points were all used as the initial population and the algorithm was considered to converge when the maximum number of generations reached 200. Each point of the “Pareto front” represents an optimal design in different situations. However, the peak deflection increased as decrease in structural mass, exhibiting a strong conflicting relationship between the target design goals of minimum mass and minimum peak deflection. Five optimal designs were selected from the “Pareto front” in Fig. 27, and were further examined by using the FE analysis again; the detail results are presented in Table 2. As can be seen from Table 2, the de­ flections obtained from the finite element analysis (FEA) exhibited relatively larger values in comparison with the surrogate-based opti­ mization results (seen in Fig. 27 (b)). When the deflection was large (26.40 mm), the optimized result was much smaller than the FEA result. Nevertheless, the surrogate-based optimization results well agreed with the FE simulation results for the deflection ranged from 10.5 mm to

6. Conclusions This study explored dynamic behavior of the sandwich panels with homogeneous (uniform) and graded foam cores under blast loading. The numerical model was established and validated against the dedicated experimental test results. The deformation modes and dynamic re­ sponses of the sandwich panels were analyzed. The effects of some key parameters, such as blast intensity, stand-off distance, foam core gradient and facesheet thickness on the blast resistance and energy ab­ sorption of sandwich panels were investigated in detail. Finally, the single- and multi-objective optimization was conducted respectively to search for the optimal designs of graded core sandwich panels against blast loading. Based upon the time histories of velocity and deflection of the facesheets, the dynamic responses under blast loading could be classi­ fied into three stages, namely (1) central core compression, (2) localized indent at the center with global deformation and (3) free vibration. In this course, most of the blast energy is absorbed by the foam core and its percentage counts for more than 80% for the applied loading intensity or stand-off distance (SOD). The absorbed energy proportion of foam core with respect to the total energy increased with the increase of blast loading, but decreased with increasing SOD. The peak deflections of

Fig. 27. Multiobjective optimization results: (a) sample design points and Pareto front; (b) comparison of the FE analysis and Pareto solution. 15

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Table 2 Comparison between the Kriging and FEA results for the optimal design points. No.

1 2 3 4 5

Facesheet

Foam core layer

Kriging model

FEA result

tF =tB

t1 =t2 =t3

ρ1 =ρ2 =ρ3

m

δmax

m

δmax

2.50/2.38 2.08/2.50 1.57/2.36 0.73/2.16 0.66/1.35

9.98/6.00/15.00 7.83/6.06/6.06 6.93/6.52/6.45 6.00/6.12/6.35 6.00/600/6.10

0.52/0.65/0.25 0.38/0.48/0.25 0.31/0.27/0.26 0.29/0.28/0.25 0.29/0.26/0.26

4828.73 3984.91 3321.52 2461.90 1809.97

8.07 10.56 12.95 16.36 19.32

4696.39 3988.10 3345.75 2547.81 1894.79

9.26 11.22 13.34 17.39 26.40

Fig. 28. Comparison of the FE analysis of initial (uniform core) designs with Pareto solution: (a) baseline Design 1; and (b) baseline Design 2.

Discovery project (DP190103752), National Natural Science Foundation of China (51575172, 11602161), and the Open Fund of State Key Lab­ oratory of Advanced Design and Manufacturing for Vehicle Body (31615008).

facesheets are almost linearly proportional to the TNT mass. The core gradient configuration is of great influence on the energy absorption and blast resistance of sandwich panels. It was found that decreasing core density across the thickness can effectively reduce the transmitted stress to the back facesheet, thus providing higher blastresistance than the other core configurations. The optimal gradient in the thickness direction is superior to that in the lateral direction. Moreover, the blast resistance can be greatly enhanced by enlarging the density difference of core layer especially under medium or high level blast intensity. Keeping the total thickness of front and back facesheets to be a constant, a relatively thick back facesheet exhibited advantage in enhancing blast resistance under relative low blast intensity. However, the conventional even facesheet sandwich performed better than uneven front facesheet configurations at a high blast intensity. Kriging surrogate models and NSGA-II algorithm were utilized to the design optimization of graded core sandwich structures for enhancing blast resistance. For the single-objective optimization, the optimum design exhibited a descending gradient of core layer density in the thickness direction and a thicker back facesheet, which significantly decreased the peak deflection of back facesheet compared with the initial (uniform core and even facesheet) design. While for the multiobjective optimization, which had a larger design space, the optimal designs obtained from the Pareto solution were more efficient in light­ weight and blast resistance.

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Declaration of competing interest The authors declared that they have no conflicts of interest to this work. Acknowledgements The project was supported by Australian Research Council (ARC) Discovery Early Career Researcher Award (DE160101633) and 16

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