Large amplitude vibration of sandwich plates with functionally graded auxetic 3D lattice core

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Large amplitude vibration of sandwich plates with functionally graded auxetic 3D lattice core Chong Li , Hui-Shen Shen , Hai Wang , Zhefeng Yu PII: DOI: Reference:

S0020-7403(19)34309-7 https://doi.org/10.1016/j.ijmecsci.2020.105472 MS 105472

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

12 November 2019 16 January 2020 22 January 2020

Please cite this article as: Chong Li , Hui-Shen Shen , Hai Wang , Zhefeng Yu , Large amplitude vibration of sandwich plates with functionally graded auxetic 3D lattice core, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105472

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Highlights 

Functionally graded auxetic 3D lattice metamaterials are designed for the first time.



Numerical and experimental investigates on the fundamental frequencies.



Full-scale modelling and nonlinear vibration analysis for sandwich plates.



FG configurations have distinct effect on the linear & nonlinear vibration behaviors,

1

Large amplitude vibration of sandwich plates with functionally graded auxetic 3D lattice core Chong Li, Hui-Shen Shen, Hai Wang*, Zhefeng Yu

School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

Abstract Full-scale modelling and nonlinear FEA are presented for large amplitude vibration of sandwich plates with functionally graded (FG) auxetic 3D lattice core. For the first time, auxetic 3D lattice metamaterials with FG configurations along the out-of-plane direction are designed, of which the fundamental vibration frequencies are analyzed and verified by experiments using 3D printed specimens. Both results suggested that the effects of FG configurations and strut incline angles are significant, and the FG-X specimen possesses the highest fundamental frequency. Subsequently, by means of full-scale nonlinear FE simulations, the large amplitude vibration characteristics are investigated for the sandwich plates, in which the novel construction of auxetic 3D lattice core with three FG configurations along the thickness direction is proposed. And the constituent material properties are taken to be temperature-dependent. Results revealed that FG configurations have distinct effect on the natural frequencies, nonlinear-to-linear frequency ratios of sandwich plates, along with EPR-amplitude curves, which will become stable when the vibration amplitude is sufficiently large. Keywords: Auxetic metamaterial; 3D Lattice; Functionally graded; 3D printing; Sandwich plates; Large amplitude vibration.

* Corresponding author. E-mail address: [email protected] (H. Wang). 2

1.

Introduction Metamaterials [1] are invented to possess macro-scale properties that are not found in

naturally occurring materials [2], and the properties are gained from their micro-structural design, rather than the constituent materials. In recent years, auxetic [3] metamaterials have attracted extensive attentions owing to their novel property of negative Poisson‟s ratio (NPR). In 1987, an auxetic foam material was manufactured by Lakes [4], who then indicated that auxetic effects could be induced through micro-structural design, e.g. making cell shape inverted [5]. Afterwards, a variety of such metamaterials have been proposed, including re-entrant [6], chiral [7], and other new varieties such as [8]. Nevertheless, most of them are actually two-dimensional (2D) metamaterials. Three-dimensional (3D) auxetic metamaterials are more suitable in many applications, and the developments in additive manufacturing techniques have enabled the fabrication of such metamaterials with complex microstructures. Li et al. [9] used Selective Laser Melting (SLM) method to build a Negative Poisson's Ratio (NPR) TiNi-based Shape Memory Alloy (SMA) structure. Xiong et al. [10] carried out a quantitative optimization of a modified re-entrant negative Poisson's ratio (NPR) structure whose overhanging struts were replaced with inclined ones to avoid the support structures which were generally required in the selective laser melting (SLM) process using finite element method (FEM). Chen et al. [11] proposes a three-dimensional lattice metamaterial by extending the existing 2D enhanced auxetic model to a 3D one. And theoretical and numerical analyses were carried out to gain a deeper understanding of the elastic behavior of the new 3D structure and its dependence on the geometric parameters. The auxetic metamaterials have exhibited a great many of engineering advantages, such as increased shear resistance [4], indentation resistance [12,13], energy absorption [14-16], crashworthiness [17, 18]. Thus, they are potentially the ideal core of sandwich structures. A typical sandwich plate is composed of a lightweight core, such as foams [19-21] and honeycombs [6, 22-27] etc., to which two relatively thin, dense, high-strength and high-stiffness facesheets are adhered. Lots of research works have been carried out on the bending and vibration behaviors of sandwich plates with 2D auxetic metamaterial core [6,

3

22-27]. However, only a few works have been made on the sandwich plates with 3D auxetic lattice metamaterial core. Novak et al. [28] fabricated the auxetic cellular structures using the Selective Electron Beam Melting (SEBM) technique and showed that using the designed auxetic cellular cores can improve the dynamic response of sandwich structures. Imbalzano et al. [29] proposed sandwich panels with auxetic lattice cores confined between metallic facets for localized impact resistance applications, and indicated that beam elements, instead of massive solid elements, can be used to model lattice cores. As a new generation of composite materials, functionally graded material (FGM) [30] has microstructural details that vary in a spatially pattern, and thus have outstanding designability [31]. There are many kinds of FGM, such as CNTRC material [32, 33], and GRC material [34, 35]. Functionally graded auxetic metamaterials, including 3D lattices and 2D honeycombs, possess gradual variations of cell size, shape or thickness, and the gradient configuration can give rise to the continuous distribution of mechanical properties [36], such as effective stiffness and Poisson‟s ratio. Ma et al. [37] developed a functionally graded NPR metamaterial for blast protection, in which the microstructures are varying along the in-plane direction to place stiffer ones in the central region. Their results indicated that optimal distribution can further improve the protection. Jin et al. [38] numerically investigated the dynamic responses and blast resistance of the honeycomb sandwich structures under blast loading, in which the 2D re-entrant honeycombs have varying cell wall thicknesses. By means of tests and full-scale finite element simulations, Boldrin et al. [39] compared two gradient cellular layouts of 2D re-entrant honeycombs: gradient internal cell angles and cell wall aspect ratios. Furthermore, the nonlinear bending, thermal postbuckling, nonlinear vibration and dynamic response of sandwich beams with functionally graded negative Poisson‟s ratio honeycomb core were investigated by Li et al. [40-43], in which the auxetic core is still 2D re-entrant honeycombs. However, to the best of authors‟ knowledge, no existing researches have been carried out on sandwich structures with auxetic 3D lattice core possessing functionally graded configurations along the plate thickness direction, which can achieve the FG distribution of effective properties of the auxetic core, and the section stiffness of sandwich plates will then be influenced.

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In the present study, we first examine fundamental vibration frequencies of the designed auxetic 3D lattice metamaterials with functionally graded configurations, by FE simulations and experiments. The proposed FG auxetic 3D lattices would have a wide range of versatility in their application, including for anomalous elastic wave polarization [44], as acoustic metamaterials for controllable band gaps [45, 46] and as core of sandwich structures, which are increasingly used in aircrafts, spacecrafts, deep submergence vehicles and pressure vessels, etc. These structures, however, are often subjected to severe operational conditions [47], in which the transverse deflections are no longer small compared to the plate thickness. Under these circumstances, geometric nonlinearity must be included in the vibration analysis, and nonlinear vibration frequencies will change as the increment of amplitudes. focused on the large amplitude vibration behavior of sandwich plates with three functionally graded auxetic 3D lattice core, in which the strut incline angles of microstructures vary along the plate thickness direction. The uniform distributed (UD) core is also taken into account for comparison. With thermal effects are further considered, full-scale modelling and nonlinear thermal-mechanical analysis are performed, and the material properties of both auxetic 3D lattice core and facesheets are taken to be temperature-dependent.

2.

Vibration behavior of the FG auxetic 3D lattice metamaterials The microstructure of a 3D lattice metamaterial with negative Poisson‟s ratio, expanded

from the classical 2D re-entrant honeycombs, is illustrated in Fig. 1. Five parameters are needed to characterize the microstructure, and apparently we can use the horizon strut length a0, the incline strut length b0, the connecting strut length c0, the strut incline angle θ, and the strut radius r, as shown in Fig. 1(b), in which in-plane x1- and x2- directions of the right handed local coordinate system are parallel to the horizon struts, while the out-of-plane x3-direction is then determined accordingly. The overall dimensions of a microstructure can be expressed as

 Lu  a0  c0  2b0 sin   2 , 2  H u  2 (b0 cos )   a0 2 

(1)

where Lu and H u are respectively the overall in-plane and out-of-plane dimensions.

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Moreover, the incline angle θ is defined to be negative when it is clockwise. An obvious precondition of the micro-structural design is that, the struts should not contact or even intersect with each other, which requires:

a0  2r  2b0 sin   0  sin     a0  2r  2b0 .

(2)

The main advantage of this 3D microstructure is that, all six effective Poisson‟s ratios (EPRs) are negative. While for re-entrant honeycombs compressed along x3-direction, only x1-direstion will generate strain like in the plain strain condition, because of the fact that

 32eff  0 . As a preliminary study on the EPR of auxetic 3D lattice metamaterials, we firstly calculate the variations of Lu and H u :

 Lu  2b0 cos   2b02 cos sin   H  u  (b0 cos ) 2  (a0 2) 2 

(3)

eff Then the  31 is

 31eff  

 Lu Lu H 1   u 3  H u H u Lu

(b0 cos )2  (a0 2)2 b0 sin 



H u 2 2Lu b0 sin 

(4)

eff eff Apparently, the microstructure will have negative  31 (and  32 ) when   0 .

On the other hand, the 3D lattice metamaterial possesses lower relative density compared with the 2D re-entrant honeycombs. For the same volume of

6  6  4 (mm) ,

t  2r  0.2mm , l0 sin   b0 sin   1mm and a0=4mm, in which the parameters t, l0, h0, α of

honeycombs are illustrated in Fig. 1(a), the relative density of them are evaluated as: for 2D re-entrant honeycombs, * honeycomb t  3h0  4l0    0.1745 ; c H u  Lu

(5a)

while for 3D lattice, * honeycomb  r 2 8a0  16b0  2c0  lattice   0.0192  11%  . 2 c H u  Lu c *

Therefore, the 3D lattice metamaterials do have the lightweight advantage. 2.1. Functionally graded configurations 6

(5b)

Unlike the in-plane FG configurations in [38], the microstructures in present study are varying along the x3-direction, i.e. out-of-plane direction. Note that the strut radius r can be independent, and if the values of Lu , H u and a0 are given, there will be only one independent parameter left. In the current study, the incline angle θ is chosen as the design parameter, while b0 and c0 will be determined accordingly. As a qualitative analysis, the effective stiffness of 3D lattice microstructures will decrease with the magnitude of incline angle θ enlarge, and would vanish when    2 . Of course, this is impossible because of the restriction of Eq. (2). Therefrom the structural section bending stiffness will depend on the FG configurations, which will then affect the vibration frequencies. Three types of functionally graded (FG) configurations are considered, indicating that the strut incline angle of microstructures is varying along the out-of-plane direction. For FG-V configuration, the strut incline angles are

[( 1 )2/(  2 )2/(  3 )2], in which:  i ( i =1,2,3) is

negative and 1 <  2 < 3 ; the subscript „2‟ indicates two microstructures having the same inline angle. The FG-X and FG-O configurations possess incline angles of [ 1 /  2 /  3 ]S and [  3 /  2 / 1 ]S, in which the subscript „S‟ denotes symmetric about the mid-plane of sandwich plates. Fig. 1(c) and (d) show the FE model and 3D printed specimen with the FG-X configuration. In addition, the auxetic 3D lattice metamaterial with uniform distributed (UD) microstructures is also taken into account for comparison, of which all the strut incline angles are equal to  2 . 2.2. Specimen design and manufacture In the current research, the specimens consist of 4 1 6 microstructures, and the detailed length dimensions are as follows: Lu=12mm, Hu=8mm, a0=8mm, r=0.5mm. For the UD configuration, three values of the strut incline angles are considered and compared, namely, -15°, -20° and -25°. For functionally graded auxetic 3D lattice metamaterials with FG-X and FG-O configurations, the strut incline angles of six microstructures along the x3direction are listed in Table 1. Besides, two cover plates are fixed at the top and bottom to 7

apply boundary conditions and excitations. The specimens, as illustrated in Fig. 1(d), are manufactured using additive manufacture technology, and the Jet Fusion 4200 3D printing solution is utilized in consideration of its high quality and low price. The elastic modulus and mass density of the PA-12 material are taken to be 1800MPa and 1.01g/cm3, while the Poisson‟s ratio is v =0.37, as determined in our lab under the ASTM 638-14 standard test method. 2.3. Results and discussion As a prediction of experimental results, a set of finite element models are created and analyzed using the FEA software ABAQUS, and boundary conditions are considered to be clamped at the bottom facesheet. Numerical results, solved by a linear perturbation analysis step, revealed that the fundamental vibration mode is bending, as shown in Fig. 1(c). On the other hand, the test system includes a laser displacement transducer (LK-H050) with its controller (LK-G5001) connected to the dynamic signal acquisition & analysis system, and a computer to display the analysis panel and conduct post-processing. Consistent with the boundary conditions of FE models, the bottom surface of specimens is adhered to a heavy steel plate, and are excited along x2-direction on the top facesheet, of which the displacements are then measured by the laser transducer. Experimental results of fundamental frequencies are compared with FE predicted results in Table 2 and Table 3, from which good agreements are observed, and the differences can be attributed to the beam assumption for the struts of auxetic 3D lattice microstructures in the FE models. In 3D printed specimens, the struts with smaller incline angles will have a small “grown-together” region around their connecting end, which will make the microstructure stiffer. Therefore, the differences reduce with an increase in the incline angle. From Table 2, we conclude that as the increase of magnitudes of the incline angles, the fundamental frequency will decrease. As has been noted in Section 2.1, the specimen consists of microstructures with the incline angle of -15°, has higher bending stiffness than that of [(-25°)6] specimen. On the other hand, the relative density will decrease as the incline angle increase, because the decrease of b0 and c0 in Eq. (3b). In conclusion, the natural frequencies will increase as the decrease in incline angle, because of the increment in specific stiffness.

8

Furthermore, Table 3 suggested that the functionally graded configuration can obviously affect the vibration behavior, and the FG-X specimen has higher natural frequency. As shown in Table 1, the FG-X configuration has microstructures with the incline angle of -15° in the outer layer, which possess higher effective stiffness than those having incline angle of -25° and located around the midplane. Therefore, the effective stiffness of FG-X configuration is higher than that of UD and FG-O configurations. Considering their equal mass, the FG-X specimen has larger specific stiffness and thus higher natural frequencies.

3.

Full-scale nonlinear FE simulation of sandwich plates Consider a sandwich plate with total thickness of h  hc  2h f , in which hc and h f are

respectively the thickness of core layer and facesheet, and having width-to-thickness ratio b/h, length-to-width ratio a/b. Let (X, Y, Z) be the right-handed global coordinate system with its origin located at the center of mid-surface, and X- and Y-direction are respectively along the length and width of sandwich plates, while Z-direction points to the bottom facesheet along thickness direction. As illustrated in Fig. 2, the auxetic 3D lattice core is arranged to have the same out-of-plane direction with the sandwich plates, and the correspondence relationship between the local and global coordinate systems is x1  X ; x2  Y ; x3  Z . 3.1. Full-scale finite element modelling In the instances of large amplitude vibration, the shape of the sandwich plate and, hence, its stiffness changes as it deforms at different amplitudes. Therefore, the geometric nonlinearity is necessary to be taken into account to achieve reasonable simulations. The large amplitude deformation of sandwich plates will bring about local large deformation of the auxetic 3D lattice core. As a result, the shape of microstructures, as shown in Fig. 2(b), along with their effective properties will change accordingly. With regard to this, in the large amplitude vibration problems, we will obtain and present not only the nonlinear-to-linear frequency (NLFR) ratios, but also the EPR-amplitude curves. On the other hand, for the intension of more precisely obtaining the large amplitude vibration behaviors, 3D full scale finite element modelling is conducted, which indicates that

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all the structural details are modeled, instead of using the effective homogenization models. The main concern is that, the effective mechanical properties of the microstructures are quite sensitive to their shapes, which will change significantly in the large amplitude region. In contrast, the determination of effective properties often based on the assumptions of linear elastic and small deformation. Therefore, theoretical investigations using effective homogenization models are inadequate for geometric nonlinear problems, and suitable for linear problems only. The thickness of core layer is hc =24mm with Hu=4mm, and the facesheet-to-core thickness ratio is selected as h f hc =1/20, 1/50 and 1/100. The width to thickness ratio of sandwich plates is taken to be b/h =25, and length-to-width ratio is fixed as a/b=1. The number of 3D lattice microstructures along the in-plane direction is then N=b/Lu, in which the overall length of a microstructure is taken to be Lu=6mm. Moreover, the strut incline angles of microstructures with different configurations has been presented in Table 1. For the sake of brevity, the core and facesheets are assumed to be perfectly adhered and their meshes are carefully arranged so as to share nodes along the interfaces. Each facesheet is meshed with one layer of continuum shell element SC8R, while for the microstructures of auxetic 3D lattice core, the Timoshenko beam element B31 is adopted, which allow for transverse shear deformation. To achieve a satisfactory compromise between accuracy and costs of time and memory, a convergence analysis was carried out, and finally the mesh size for facesheets is 1mm, while the a0, b0 and c0 struts are meshed using one, two and four elements, respectively. Finally, detailed overall geometric parameters and meshes of the sandwich plates with different parameters are summarized in Table 4. Three types of boundary conditions, i.e. CCCC, CSCS and SSSS, are considered, where “C” signifies the fully clamped boundary condition, “S” signifies the simply supported boundary condition with in-plane immovable conditions, and “CSCS” indicates that “C” for X=±a/2 and “S” for Y=±b/2. For “C”, all the degrees of freedom (DoFs) of the nodes located at the plate edges are constrained, while for “S”, the boundary edges are constrained with only one rotation DoF is allowed. In large amplitude vibration problems, we only consider the vibration mode (1, 1), which is symmetric about both X=0 and Y=0 plane, thus 10

only a quarter of sandwich plates need to be modelled, as shown in Fig. 2(a). In this case, symmetric constraint conditions are applied upon the symmetric planes. Finally, these three boundary conditions can be expressed as:

C : U1  U 2  U 3  UR1  UR2  UR3  0 ,

(6a)

U 3UPR  U 3LWR  0   S :  U 1UPR  U 1LWR  0 ( for X   a 2) , U 2  U 2  0 ( for Y   b 2) LWR  UPR

(6b)

U1  UR 2  UR3  0 ( for X  0) SYMM :  , U 2  UR1  UR3  0 ( for Y  0)

(6c)

in which, U1, U2 and U3 are translational DoFs along the X-, Y- and Z-directions; UR1, UR2 and UR3 are rotational DoFs about the X-, Y- and Z-directions; and the subscripts UPR and LWR refers to nodes symmetrically located on the upper and lower side of mid-surface. 3.2. Solution procedures and temperature-dependent material properties The sandwich plates are further considered in different thermal environments, and the difference between the applied and initial temperatures will cause thermal strain on the condition that the thermal expansion coefficient is given for the material. Accordingly, a sequentially coupled thermal-mechanical analysis is carried out. To begin with, a predefined field is created, in which the initial temperature field is defined as T0=300K. Sequentially, the temperature field is modified as expected in a new analysis step, after which mechanical analysis is then performed, as shown in the flowchart presented in Fig. 3. For the nonlinear vibration problem, a large amplitude displacement field is firstly applied upon the sandwich plates, as shown in Fig. 2(a), which will be then deactivated to allow free vibrations. The nonlinear frequency ωNL can be obtained by tracking the motion of the sandwich plate. Then the nonlinear-to-linear frequency ratio (NLFR) ωNL/ωL is determined accordingly, and the natural frequency ωL is solved as an eigenvalue problem, which can be expressed as

- L 2 M  K  vij = 0 , ij  

(7)

where M and K are respectively the mass and stiffness matrix, (ωL) ij and vij are respectively the natural frequency and corresponding vibration mode, in which the subscript ij refers to the 11

mode shape of (i, j), and i, j is the half wave numbers in the X- and Y- direction. As mentioned earlier, only the mode (1, 1) is considered to calculate the NLFR, thus by default the ωL refers to (ωL)11. Moreover, in different thermal environments, the mechanical properties of the constituent materials are expected to have significant change. For accurate prediction of the large amplitude

vibration

of

the

sandwich

plates,

it

is

essential

to

ensure

this

temperature-dependency are taken into account. Considering the realistic materials used in aircraft industry and possible 3D printing techniques for metal specimens (including SLM and EBM), the auxetic 3D lattice core and facesheets are respectively made from Ti-6Al-4V and aluminum alloy, and the expressions for their mechanical properties are detailly described in [42]. In the present FE simulations, three thermal environments are used, i.e. T=300, 325 and 350 K. The material properties for the constituent materials of both auxetic 3D lattice core and facesheets under these thermal environments are summarized and listed in Table 5.

4.

Numerical examples and discussion The numerical results presented herein were obtained using the implicit solver

ABAQUS/Standard, which solves nonlinear problems iteratively using the Newton‟s method with the Nlgeom option is switched on, and the natural frequency problem is solved by a linear perturbation analysis step. 4.1. Comparison studies As introduced earlier, no existing researches have been carried on the large amplitude vibration behaviors of sandwich plates with auxetic 3D lattice core, three reduced problems are utilized to verify present FEM, including one natural frequency and two large amplitude vibration problems. The SC8R elements are employed to mesh the plates, and seed density is determined by convergency analysis. Example 1. Consider a square thick plate with clamped boundary conditions, which has a width-to-thickness ratio of b/h=10. The material property is isotropic with a Poisson‟s ratio of ν=0.3. The comparison results are listed in Table 6, in which the natural frequencies are

12

non-dimensional and defined by =a  E . From the Table 6, in can be concluded that present results using 10 SC8R elements in the thickness direction agree well with that of Lim et.al [48] using a higher-order plate theory with 17 terms in the admissible shape functions. Example 2. Consider a square [0/90/core/90/0] sandwich plate with simply supported boundary conditions, which has a width-to-thickness ratio of b/h=40 and core-to-facesheet thickness ratio of hc h f =10. The material properties are: for facesheets, EL=139000MPa, ET = 9860 MPa, GLT= GTT = 5240 MPa,  LT = 0.3 and ρ = 1590 kg/m3; for core, E = 90 MPa, G = 32 MPa, ν=0.45 and ρ = 170.6 kg/m3. From Fig. 4, in can be concluded that present results using 10 SC8R elements in the thickness direction agree perfectly well with the finite element results of Ganapathi et.al [49] based on the HSDT theory with 7 DoFs per node. Example 3. Consider a square [0/90/core/90/0] sandwich plate with different boundary conditions, which has a width-to-thickness ratio of b/h=20 and core-to-facesheet thickness ratio of hc h f =8. The material properties are the same as used in Example 2. The nonlinear-to-linear frequency ratios (ωNL/ωL) of the sandwich plate with SSSS and CCCC boundary conditions are listed and compared in Table 7, from which it can be concluded that present results agree well, but are slightly lower than that of Madhukar and Singha [50] using normal deformation theory. 4.2. Parametric studies In the following subsections, parametric studies are carried out to demonstrate the effect of FG configurations, temperature rises, boundary conditions, strut radii and facesheet-to-core thickness ratios ( h f hc ) on the nonlinear-to-linear frequency ratio (NLFR) and EPR variation of sandwich plates. The non-dimensional amplitude is defined as Wmid /h, in which h is the total thickness and Wmid is the displacement along Z-direction of the node located at the center of top surface. Considering the symmetry condition, the effective region for calculating EPRs is set to be 1×1×6 microstructures in the central region of the sandwich plate, as shown in Fig. 2(a) and (c). The definition is expressed as 13

 31eff  

U 1 LER (U 3B  U 3T ) H ER

(8)

in which, LER and H ER are respectively the overall length and height of the effective region, and satisfy LER  Lu , H ER  6 H u . It is notably that Eq. (8), directly modified from the primal definition of the Poisson‟s ratio, is applicable for both linear and large amplitude vibration problems. 4.2.1. The linear vibration frequencies As a basis to determine the NLFRs, we firstly extract the first four linear vibration frequencies of sandwich plates with different geometrical parameters under environmental conditions, and results are summarized in Tables 8, 9 and 10. From Table 8, we can conclude that as the temperature rises, the natural frequencies will decrease. As expected, the natural frequencies of CSCS plates lie between those of the CCCC plates and SSSS plates, as shown in Table 9. It is notable that, for the CSCS cases, the natural frequencies of mode (1, 2) and (2, 1) is different, and the latter one is higher. As presented in Table 10, the natural frequencies will increase significantly with the increment in strut radii. Moreover, the sandwich plates in Tables 8, 9 and 10 have facesheet thickness of h f hc =1/50, 1/20, and 1/100. It is observed that as the increase of thickness ratio, the natural frequencies will decrease. Furthermore, among the four different configurations in all cases, the FG-X plate has the highest fundamental natural frequencies. As has been noted in Section 2.3, the FG-X configuration possess stiffer microstructures located in outer layer, while microstructures with larger compliance are arranged to be around the mid-plane. Therefore, the section bending stiffness of the FG-X configurations is larger than the others, which can be expressed as

 E Z Z

2

dZ , and Z has been defined in the beginning of Section 3. Considering their equal

mass and based on Eq. (7), the fundamental frequency will become higher as the increase of structural specific stiffness ( Κ Μ ), which can further explain the above four conclusions. 4.2.2. The effects of FG configurations The effects of functionally graded (FG) configurations on the NLFR and EPR variation of

14

sandwich plates are respectively demonstrated in Fig. 5(a) and (b). Four different configurations, namely FG-V, FG-O, FG-X and UD, are considered, in which the UD configuration is taken as the basis for comparison. The dimensions are: width-to-thickness ratio b/h=25, plate aspect ratio a/b=1, strut radius r=0.200mm and facesheet-to-core thickness ratio h f hc =1/50. The sandwich plates are clamped at four edges and rested in the temperature filed of T=300K. It can be observed from Fig. 5(a) that, among the NLFR curves of four configurations considered, the FG-X one is the lowest, which is contrary to the natural frequency problem. In other words, as the increase of structural specific stiffness, the nonlinear-to-linear frequency ratios will become smaller. As presented in Fig. 5(b), EPR-amplitude curves of two symmetric configurations have the similar variation trend with the UD case, while that of the FG-V plate is quite different. For FG-O, FG-X and UD plates, the EPR curves possess significant changes when Wmid /h <0.5. Moreover, all the curves will gradually become stable in the large amplitude region. The physical explanations are as follows. Under in-plane immovable boundary conditions, when the structure is subjected to a large amplitude deformation, its mid-plane will extend to a curved surface, and the in-plane strain ε1 is then positive. Because of the negative Poisson‟s eff ratio  31 , the resulting out-of-plane strain ε3 will also be positive. The change of ε3 is the eff outcome of ε1 (as well as negative  31 ), and then will delay. However, as the accumulate of

strains, in the large amplitude region their ratios become stable. In the following analysis, only FG-X and UD plates are further considered. 4.2.3. The effects of temperature rise Fig. 6 presents the effects of thermal environments on the NLFR and EPR variation of sandwich plates with strut radius of r=0.200mm, facesheet-to-core thickness ratio of h f hc =1/50, and boundary conditions of CCCC. As illustrated in Fig. 6(a), as the elevation of environmental temperature, the NLFR-amplitude curves will become higher, implying the increase of the nonlinear frequency ratios. The physical explanation is that, the material softening in high temperature environments will induce stiffness decrease, and obviously the specific stiffness becomes smaller. 15

From Fig. 6(b), it is observed that when the thermal effects are taken into consideration, the EPR-amplitude curves become distinctly different from those of T=300K. This phenomenon is on account of the fact that the initial displacements are caused by temperature changes. Therefore, conclusions can be drawn that thermal environments have significant effects on the large amplitude vibration behaviors of sandwich plates. Furthermore, the NLFR-amplitude curves of FG-X plates denoted by hollow symbols are lower than that of UD plates with solid symbols. On the other hand, the FG-X plates have higher ERP variation curves than UD plates. Both results further demonstrate the distinct effect of FG configurations. 4.2.4. The effects of boundary conditions Fig. 7 illuminates the effects of boundary conditions on the NLFR and EPR variation of sandwich plates with a large facesheet-to-core thickness ratio of h f hc =1/20 and are rested at T=300K. As shown in Fig. 7(a), the CCCC plate has the lowest NLFR-amplitude curve, while that of SSSS plate is the highest, and the differences are owing to the effects of boundary conditions on the structural stiffness. Fig. 7(b) revealed that that the EPR-amplitude curves of sandwich plates with different boundary conditions all possess the similar variation trend, but limited distinctions. It is concluded that boundary conditions can distinctly affect the NLFRs, but only have small effects on the EPR-amplitude curves. However, it is observed that in large amplitude region, all the EPR curves of FG-X marked by hollow symbols, are higher than that of UD by solid symbols. Besides, the NLFR curves of FG-X plates are lower than that of UD. 4.2.5. The effects of strut radii The effects of strut radii r on the NLFR and EPR variation of sandwich plates with FG-NPR 3D lattice core are illustrated in Fig. 8. The sandwich plate has a small facesheet-to-core thickness ratio of h f hc =1/100 and are rested at the temperature filed of T=300K. The boundary conditions are assumed to be CCCC. Results presented in Fig. 8(a) indicated that, the NLFR-amplitude curves will become lower as the strut radius increases, implying that the specific stiffness is becoming larger. Moreover, the EPR-amplitude curves

16

of the sandwich plate with bigger strut radius is lower than that of the plate with thinner structs, as shown in Fig. 8(b). Therefore, the strut radii will significantly affect the NLFR-amplitude curves, and EPR variation of the effective region. Moreover, the effect of FG configurations is also distinct: FG-X plates possess smaller NLFRs while higher EPR-amplitude curves. 4.2.6. The effects of facesheet-to-core thickness ratios Finally, the effects of facesheet-to-core thickness ratios ( h f hc ) on the NLFR and EPR variation of sandwich plates are summarized in Fig. 9. The plates are fully clamped and are rested at T=300K. As presented in Fig. 9(a), as the increase of facesheet-to-core thickness ratio, the NLFR-amplitude curves are getting higher, indicating that the specific stiffness is decreasing. Therefore, the effects of strut radii and facesheet thickness are quite the opposite. From Fig. 9(b), it can be concluded that the EPR-amplitude curves of sandwich plates with different facesheet-to-core thickness ratios all share the same trend, and the EPRs of plates with thinner facesheets are obviously smaller. As a result, facesheet-to-core thickness ratios have significant effect on both the NLFR-amplitude and EPR-amplitude curves. Consistent with previous conclusions, the FG-X plates have lower NLFR-amplitude curves, while higher EPR variation curves than that of UD plates.

5. Concluding remarks For the first time, the fundamental frequencies of auxetic 3D lattice metamaterials with functionally graded configurations are investigated numerically and experimentally. Moreover, the large amplitude vibration analysis of sandwich plates with FG-NPR 3D lattice core in different thermal environments has been presented. And major conclusions include: 

The effects of FG configurations and strut incline angles on the fundamental frequencies of auxetic 3D lattice metamaterials are significant, and the FG-X specimen possesses the highest fundamental frequency.



The FG configurations of the auxetic 3D lattice core have distinct effect on the linear and nonlinear vibration behavior of sandwich plates.



As the increase of structural specific stiffness, the fundamental frequency will 17

increase while the NLFRs of sandwich plates will decrease. 

The EPR-amplitude curves of sandwich plates are obtained, which will become stable when the vibration amplitude is sufficiently large.

It is expected that the present results could shed light on the vibration behavior of functionally graded auxetic 3D lattice metamaterials and sandwich plates with such core, and would be conducive to further investigations.

Acknowledgments The support for this work, provided by the National Natural Science Foundation of China under Grant 51779138 is gratefully acknowledged. Author Statement Chong Li: Conceptualization, Methodology, Investigation, Software, Validation, Formal analysis, Data curation, Visualization, Writing Original draft, Writing - Review & Editing; H-S Shen: Conceptualization, Writing - Review & Editing, Funding acquisition; Hai Wang: Resources, Supervision; Zhefeng Yu: Investigation, Resources. Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Mater 2018; 20: 692-717. [24] Cong PH, Khanh ND, Khoa ND, Duc ND. New approach to investigate nonlinear dynamic response of sandwich auxetic double curves shallow shells using TSDT. Compos Struct 2018; 185: 455-465. [25] Duc ND, Kim SE, Cong PH, Anh NT, Khoa ND. Dynamic response and vibration of composite double curved shallow shells with negative Poisson‟s ratio in auxetic honeycombs core layer on elastic foundations subjected to blast and damping loads. Int J Mech Sci 2017; 133: 504-512. [26] Hajmohammad MH, Nouri AH, Zarei MS, Kolahchi R. A new numerical approach and visco-refined zigzag theory for blast analysis of auxetic honeycomb plates integrated by multiphase nanocomposite facesheets in hygrothermal environment. Eng Comput 2019; 35: 1141-1157. [27] Hajmohammad MH, Kolahchi R, Zarei MS, Nouri AH. Dynamic response of auxetic honeycomb plates integrated with agglomerated CNT-reinforced face sheets subjected to blast load based on visco-sinusoidal theory. Int J Mech Sci 2019; 153-154: 391-401. [28] Novak N, Starčevič L, Vesenjak M, Ren Z. Blast response study of the sandwich composite panels with 3D chiral auxetic core. Compos Struct 2019; 210: 167-178. [29] Imbalzano G, Tran P, Ngo TD, Lee PV. Three-dimensional modelling of auxetic sandwich panels for localised impact resistance. J Sandw Struct Mater 2015; 19: 291-316. [30] Shen H-S. Functionally graded materials nonlinear analysis of plates and shells. First ed., CRC Press, Raton, 2009. [31] Shen H-S. Modeling and analysis of functionally graded carbon nanotube reinforced composite structures: A review. Adv Mech 2016; 46: 478-505. [32] Wang Z-X and Shen H-S. Nonlinear vibration and bending of sandwich plates with nanotube-reinforced composite face sheets. Composites B 2012; 43: 411-421. [33] Fan Y and Wang H. Thermal postbuckling and vibration of postbuckled matrix cracked hybrid laminated plates containing carbon nanotube reinforced composite layers on elastic foundation. Compos Struct 2016; 157: 386-397. [34] Yang J, Chen D, Kitipornchai S. Buckling and free vibration analyses of functionally graded graphene reinforced porous nanocomposite plates based on Chebyshev-Ritz method. Compos Struct 2018; 193: 281-294. [35] Gao K, Gao W, Chen D, Yang J. Nonlinear free vibration of functionally graded graphene platelets reinforced porous nanocomposite plates resting on elastic foundation. Compos Struct 2018; 204: 831-846. [36] Hou Y, Tai YH, Lira C, et al. The bending and failure of sandwich structures with auxetic gradient cellular cores. Compos Part A 2013; 49: 119-131. 20

[37] Ma ZD, Bian H, Sun C, et al. Functionally-graded NPR (negative Poisson‟s ratio) material for a blast-protective deflector, in: Proceedings of the 2009 Ground Vehicle Systems Engineering and Technology Symposium (GVSETS), Michigan, 17-19 August, 2010. p. 1-12. [38] Jin XB, Wang ZH, Ning JG, Xiao GS, Liu EQ, Shu XF. Dynamic response of sandwich structures with graded auxetic honeycomb cores under blast loading. Compos. B Eng. 106; 2016: 206-217. [39] Boldrin L, Hummel S, Scarpa F. Dynamic behavior of auxetic gradient composite hexagonal honeycombs. Compos Struct 2016; 149: 114-124. [40] Li C, Shen H-S, Wang H. Nonlinear bending of sandwich beams with functionally graded negative Poisson‟s ratio honeycomb core. Compos Struct 2019; 212: 317-325. [41] Li C, Shen H-S, Wang H. Thermal postbuckling of sandwich beams with functionally graded negative Poisson‟s ratio honeycomb core. Int J Mech Sci 2019; 152: 289-297. [42] Li C, Shen H-S, Wang H. Nonlinear vibration of sandwich beams with functionally graded negative Poisson's ratio honeycomb core. Int J Struct Stab Dyna 2019; 19: 1950034. [43] Li C, Shen H-S, Wang H. Nonlinear dynamic response of sandwich beams with functionally graded negative Poisson‟s ratio honeycomb core. Eur Phys J Plus 2019; 134: 79. [44] Patil GU, Shedge AB, Matlack KH. 3D auxetic lattice materials for anomalous elastic wave polarization. Applied Physics Letters 2019; 115: 091902. [45] Krödel S, Delpero T, Bergamini A, et al. 3D Auxetic Microlattices with Independently Controllable Acoustic Band Gaps and Quasi-Static Elastic Moduli. Advanced Engineering Materials 2013; 15: 9999. [46] Chen M, Xu WS, Liu Y, et al. Band gap and double-negative properties of a star-structured sonic metamaterial. Applied Acoustics 2018; 139: 235-242. [47] Vinson JR. The Behavior of Sandwich Structures of Isotropic and Composite Materials. Technomic Publishing Company, Lancaster, 1999. [48] Lim CW, Liew KM, Kitipornchai S. Numerical aspects for free vibration of thick plates Part I: Formulation and verification. Comput Methods Appl M 1998; 156: 15-29. [49] Ganapathi M, Patel BP, Makhecha DP. Nonlinear dynamic analysis of thick composite/sandwich laminates using an accurate higher-order theory. Compos Part B 2004; 35: 345-355. [50] Madhukar S and Singha MK. Geometrically nonlinear finite element analysis of sandwich plates using normal deformation theory. Compos Struct 2013; 97: 84-90.

21

Table 1 Configuration codes of the auxetic 3D lattice metamaterials.

Configuration

Code

FG-V

[(-15)2/(-20)2/(-25) 2]

FG-O

[-25/-20/-15]S

FG-X

[-15/-20/-25]S

UD

[(-20)6]

22

Table 2 The fundamental natural frequencies of auxetic 3D lattice metamaterials with different strut incline angles.



FEA prediction (Hz)

Experiment (Hz)

-15°

114.84

120.28

-20°

88.980

91.186

-25°

70.944

71.381

23

Table 3 The fundamental natural frequencies of auxetic 3D lattice metamaterials with different functionally graded configurations.

Configurations

FEA prediction (Hz)

Experiment (Hz)

UD

88.980

91.186

FG-O

81.651

83.264

FG-X

92.320

94.702

24

Table 4 Overall geometric parameters and meshes of sandwich plates with a=b=25h and different facesheet-to-core thicknesses ratios.

Problem

Natural frequency

Nonlinear vibration

h f hc

N=b/Lu

Elements

Model-scale

1/100

102

749088 SC8R+2705040 B31

Whole

1/50

104

778752 SC8R+2812160 B31

Whole

1/20

110

871200 SC8R+3146000 B31

Whole

1/100

51

187272 SC8R+676260 B31

Quarter

1/50

52

194688 SC8R+703040 B31

Quarter

1/20

55

217800 SC8R+786500 B31

Quarter

25

Table 5 Material properties of the auxetic 3D lattice core and face sheets.

Temperature

300K

325K

350K

Ec (MPa)

105698.2

104293.0

102887.9

νc

0.29

0.29

0.29

6.94150E-06

6.69461E-06

6.41790E-06

ρc (kg/m )

4429

4429

4429

Ef (MPa)

68749.5

67835.2

66921.0

νf

0.33

0.33

0.33

2.31134E-05

2.35274E-05

2.39414E-05

2707

2707

2707

αc (/K) 3

αf (/℃) ρf (kg/m ) 3

26

Table 6 Comparison of natural frequencies for a square thick plate (ν=0.3, b/h=10) with clamped boundary conditions.

Sources

XSYMM-YSYMM

XASYMM-YSYMM & XSYMM-YASYMM

XASYMM-YASYMM

27

C.W. Lim et.al [48]

Present

0.98636

0.99754

3.1200

3.09999

3.1485

3.13351

1.8848

1.88972

3.7269

3.73059

3.7765

3.77877

2.6457

2.63177

4.4395

4.45314

5.1393

5.03918

Table 7 Comparison of the nonlinear frequency ratios (ωNL/ωL) for a symmetric [0/90/core/90/0] sandwich plate with b/h = 20 and hc h f = 8.

SSSS

CCCC

Wmax /h Madhukar and Singha [50]

Present

Madhukar and Singha [50]

Present

0.2

1.02657

1.01962

1.01822

1.01401

0.4

1.10352

1.08221

1.07059

1.05805

0.6

1.22408

1.18795

1.15064

1.12913

0.8

1.38006

1.33472

1.25143

1.22237

1.0

1.56534

1.51059

1.36799

1.32524

28

Table 8 The natural frequencies (Hz) of sandwich plates in different thermal environments. (CCCC, r=0.200mm, h f hc =1/50)

Temperature

Configuration

(ωL)11

(ωL)12=(ωL)21

(ωL)22

UD

157.81

251.79

323.33

FG-V

159.02

253.86

325.97

FG-O

156.89

250.14

321.24

FG-X

163.27

260.89

335.20

UD

127.12

238.32

296.62

FG-V

128.58

240.46

299.46

FG-O

126.06

236.60

294.31

FG-X

133.69

247.75

309.41

UD

84.658

223.47

265.17

FG-V

86.754

225.71

268.35

FG-O

83.230

221.63

262.47

FG-X

94.066

233.37

279.47

T = 300K

T = 325K

T = 350K

29

Table 9 The natural frequencies (Hz) of sandwich plates with different boundary conditions. (T=300K, r=0.200mm, h f hc =1/20)

Boundary condition

Configuration

(ωL)11

(ωL)12

(ωL)21

(ωL)22

UD

135.50

216.16

216.16

276.39

FG-V

136.62

218.00

218.00

278.72

FG-O

134.60

214.61

214.61

274.45

FG-X

140.67

224.52

224.52

287.10

UD

128.67

207.45

213.25

269.34

FG-V

129.82

209.65

215.17

271.98

FG-O

127.93

206.28

211.79

267.69

FG-X

133.25

215.18

221.39

279.56

UD

121.60

204.23

204.23

262.45

FG-V

122.81

206.52

206.52

265.42

FG-O

121.05

203.16

203.16

261.11

FG-X

125.53

211.70

211.70

272.17

CCCC

CSCS

SSSS

30

Table 10 The natural frequencies (Hz) of sandwich plates with different strut radii. (T=300K, CCCC,

h f hc =1/100)

Strut radius

Configuration

(ωL)11

(ωL)12=(ωL)21

(ωL)22

UD

125.10

199.09

254.26

FG-V

126.31

201.14

256.81

FG-O

124.20

197.49

252.27

FG-X

130.04

207.24

264.70

UD

144.47

230.68

296.51

FG-V

145.53

232.51

298.82

FG-O

144.07

229.88

295.53

FG-X

148.77

237.91

305.94

UD

159.84

256.83

332.30

FG-V

160.67

258.32

334.19

FG-O

160.14

257.20

332.83

FG-X

163.04

262.41

339.68

r=0.150mm

r=0.175mm

r=0.200mm

31

Fig. 1. The auxetic metamaterials ： (a) 2D honeycomb microstructure; (b) 3D lattice microstructure; (b) FE predicted first-order vibration mode of the FG-X specimen; (d) 3D printed specimen with FG-X configuration.

32

Fig. 2. (a) Construction of full-scale FE model and predefined displacement field for large amplitude vibrations; (b) local deformation of the auxetic 3D lattice core; (c) Effective Region for calculating effective Poisson‟s ratios.

33

Fig. 3. Flowchart of linear and nonlinear vibration analysis process with consideration of thermal effects.

34

NL/L

1.07 1.06

[0/90/core/90/0] sandwich plate a/b=1, b/h=40 hc / hf=10

1.05

SSSS

1.04 1.03 1.02 1.01

Ganapathi et. al. [49] Present

1.00 0.99 0.0

0.1

0.2

0.3

0.4

Wmid / h

Fig. 4. Comparison of NLFR-amplitude curve for a square [0/90/core/90/0] sandwich plate with simply supported BCs.

35

1.6

b/h=25, a=b r=0.200mm hf /hc=1/50

NL /L

CCCC T=300K

UD FG-V FG-O FG-X

1.4

1.2

1.0 0.00

0.25

0.50

0.75

1.00

Wmid / h

(a)

0

eff 31

-2

-4 b/h=25, a=b r=0.200mm hf /hc=1/50

-6

CCCC T=300K

-8 0.00

UD FG-V FG-O FG-X

0.25

0.50

0.75

1.00

Wmid / h

(b) Fig. 5. The effects of FG configurations on the nonlinear-to-linear frequency ratio (NLFR) and EPR variation of sandwich plates with FG-NPR 3D lattice core: (a) NLFR-amplitude curves; (b) EPR-amplitude curves.

36

2.4 2.2

b/h=25, a=b r=0.200mm hf /hc=1/50 CCCC

NL /L

2.0 1.8

1: T=300K 2: T=325K 3: T=350K

UD&1 FG-X&1 UD&2 FG-X&2 UD&3 FG-X&3

1.6 1.4 1.2 1.0 0.00

0.25

0.50

0.75

1.00

Wmid / h

(a)

0

eff 31

-2

-4 b/h=25, a=b r=0.200mm hf /hc=1/50

-6

CCCC 1: T=300K 2: T=325K 3: T=350K

-8

-10 0.00

UD&1 FG-X&1 UD&2 FG-X&2 UD&3 FG-X&3

0.25

0.50

0.75

1.00

Wmid / h

(b) Fig. 6. The effects of temperature rise on the nonlinear-to-linear frequency ratio (NLFR) and EPR variation of sandwich plates with FG-NPR 3D lattice core: (a) NLFR-amplitude curves; (b) EPR-amplitude curves.

37

2.25

2.00

b/h=25, a=b r=0.200mm hf /hc=1/20 T=300K

NL/L

1.75

1: CCCC 2: CSCS 3: SSSS

FG-X&1 UD&1 FG-X&2 UD&2 FG-X&3 UD&3

1.50

1.25

1.00 0.00

0.25

0.50

0.75

1.00

Wmid / h

(a)

0

 eff 31

-10

b/h=25, a=b r=0.200mm hf /hc=1/20

-20

T=300K -30

-40 0.00

1: CCCC 2: CSCS 3: SSSS 0.25

0.50

FG-X&1 UD&1 FG-X&2 UD&2 FG-X&3 UD&3 0.75

1.00

Wmid / h

(b) Fig. 7. The effects of boundary conditions on the nonlinear-to-linear frequency ratio (NLFR) and EPR variation of sandwich plates with FG-NPR 3D lattice core: (a) NLFR-amplitude curves; (b) EPR-amplitude curves.

38

1.8

b/h=25, a=b hf /hc=1/100 CCCC T=300K

NL/L

1.6

1: r=0.150mm 2: r=0.175mm 3: r=0.200mm

UD&1 FG-X&1 UD&2 FG-X&2 UD&3 FG-X&3

1.4

1.2

1.0 0.00

0.25

0.50

0.75

1.00

Wmid / h

(a)

0 -1

UD&1 FG-X&1 UD&2 FG-X&2 UD&3 FG-X&3

b/h=25, a=b hf /hc=1/100 CCCC T=300K

 eff 31

-2

1: r=0.150mm 2: r=0.175mm 3: r=0.200mm

-3 -4 -5 -6 0.00

0.25

0.50

0.75

1.00

Wmid / h

(b) Fig. 8. The effects of strut radii on the nonlinear-to-linear frequency ratio (NLFR) and EPR variation of sandwich plates with FG-NPR 3D lattice core: (a) NLFR-amplitude curves; (b) EPR-amplitude curves.

39

2.25

2.00

NL/L

1.75

b/h=25, a=b r=0.200mm CCCC T=300K 1: hf /hc=1/20 2: hf /hc=1/50

FG-X&1 UD&1 FG-X&2 UD&2 FG-X&3 UD&3

3: hf /hc=1/100 1.50

1.25

1.00 0.00

0.25

0.50

0.75

1.00

Wmid / h

(a)

0

 eff 31

-10 b/h=25, a=b r=0.200mm CCCC T=300K

-20

FG-X&1 UD&1 FG-X&2 UD&2 FG-X&3 UD&3

1: hf /hc=1/20

-30

2: hf /hc=1/50 3: hf /hc=1/100 -40 0.00

0.25

0.50

0.75

1.00

Wmid / h

(b) Fig. 9. The effects of facesheet-to-core thickness ratio on the nonlinear-to-linear frequency ratio (NLFR) and EPR variation of sandwich plates with FG-NPR 3D lattice core: (a) NLFR-amplitude curves; (b) EPR-amplitude curves.

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Graphical abstract

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