Full-scale finite element modeling and nonlinear bending analysis of sandwich plates with functionally graded auxetic 3D lattice core

Abstract

This paper investigates the nonlinear bending behavior of sandwich plates with functionally graded auxetic 3D lattice core. First and foremost, an auxetic 3D lattice metamaterial with negative effective Poisson’s ratio (EPR) is designed and examined via theoretical and finite element methods with experimental verifications using specimens fabricated by 3D printing. Furthermore, three functionally graded configurations of the auxetic 3D lattice core through the plate thickness direction are proposed and compared with the uniform distribution case. Full-scale finite element modeling and nonlinear thermal-mechanical analysis are performed for the sandwich plates, with the temperature-dependent material properties of both core and face sheets taken into account. Numerical results revealed that the auxetic core can remarkably reduce the lateral deflections, with comparison to their non-auxetic counterpart with positive EPR. Parametric studies are further carried out to demonstrate the effects of functionally graded configurations, temperature rises, facesheet-to-core thickness ratios, boundary conditions, and strut radii on the nonlinear bending load-deflection curves, along with EPR-deflection curves in the large deflection region.

Keywords

Auxetic metamaterial 3D lattice 3D printing functionally graded sandwich plates nonlinear bending

Introduction

Metamaterials [2] are designed and made to possess macro-scale properties that are not found in naturally occurring materials [3], and this novelty is not gained from the constituent materials, but their microstructural design, which can achieve benefits that go beyond the limitation of conventional materials. Recently, auxetic [4] metamaterials have received a great deal of attention in virtue of their novel property of negative Poisson’s ratio (NPR) [5]. In 1987, the NPR material of thermoplastic open cell foam was manufactured by Lakes [6], which demonstrated that auxetic effects can be induced through microstructural design, e.g. making cell shape inverted [7]. In the existing literature, a variety of such metamaterials have been proposed, including re-entrant, chiral, as well as rotating rigid structures. However, most of them merely have two-dimensional (2D) microstructures.

Three-dimensional (3D) auxetic metamaterials are more suitable in many applications, and advanced manufacturing techniques, like additive manufacturing, have opened new opportunities to fabricate such metamaterials with complex microstructures. Using selective laser melting (SLM) method, Li et al. [8] built an NPR TiNi-based shape memory alloy (SMA) structure. Using finite element analysis (FEA) method, Xiong et al. [9] carried out a quantitative optimization of a modified re-entrant NPR structure whose overhanging struts were replaced with inclined ones to avoid the support structures, which were generally required in the SLM process. Chen and Fu [10] proposed a 3D lattice metamaterial by extending the existing 2D enhanced auxetic model to a 3D one and carried out theoretical and numerical analyses to gain a deeper understanding of the elastic behavior of the new 3D structure and its dependence on the geometric parameters.

From another point of view, sandwich construction is playing an increasingly important role in structures because of its exceptionally high-flexural stiffness-to-weight ratio compared to monocoque and other architectures [11]. A typical sandwich plate is composed of a lightweight core, such as foams [12–14] and honeycombs [15–17], to which two relatively thin, dense, high-strength, and high-stiffness face sheets or skins are adhered. The auxetic metamaterials have exhibited a great many of engineering advantages, such as increased shear resistance [6], indentation resistance [18,19], energy absorption [20–22], crashworthiness [23,24], and buckling capacity [25–27]. As a result, they are potentially the ideal core of sandwich structures.

Although many research works have been carried out on the bending and vibration responses of sandwich plates with 2D auxetic metamaterial core [15–17], few studies have been conducted on the sandwich plates with 3D auxetic lattice metamaterial core. Using the selective electron beam melting technique, Novak et al. [28] fabricated the auxetic cellular structures 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 can be utilized to model lattice cores, instead of massive solid elements.

As a new generation of composite material, functionally graded material (FGM) [30] has microstructural details that vary in a spatially pattern and thus possess outstanding designability [31]. There are many new kinds of FGM, such as carbon nano-tube reinforced composite (CNTRC) material [32,33], graphene-reinforced composite material [34,35] and nano-scaled FGM [36–39]. Functionally graded auxetic metamaterials have gradual variations of cell size, shape, or strut radius over a prescribed volume [40], and the gradient configuration may lead to continuous distributions of effective stiffness and Poisson’s ratio. Two classes of gradient cellular layout were compared by Boldrin et al. [41], which has continuously varying internal cell angle or gradient cell wall aspect ratio. Furthermore, the nonlinear bending, thermal postbuckling, nonlinear vibration, and dynamic response of sandwich beams with functionally graded NPR honeycomb core were investigated by Li et al. [42–45]. In their studies, the EPR-deflection curves of the sandwich beams were obtained for the first time. More recently, Li et al. [27] investigated the in-plane effective Poisson’s ratio (EPR) of auxetic 3D lattice metamaterials and the postbuckling behavior of sandwich plates with such core, as well as their nonlinear vibration [45] and nonlinear dynamic response [46] behaviors. However, to the best of authors’ knowledge, the out-of-plane EPR of such metamaterials and nonlinear bending of sandwich structures with functionally graded auxetic 3D lattice core are remain to be explored.

In the present work, an auxetic 3D lattice metamaterial with NPR is firstly designed, modeled, analyzed and verified by experiments. The nonlinear bending behavior of sandwich plates with functionally graded (FG) auxetic 3D lattice core is then investigated. Three FG configurations are considered, in which the strut incline angles of microstructures vary along the sandwich plate thickness direction. The same sandwich plate with uniform distribution (UD) of auxetic 3D lattice core is also considered for comparison purpose. Moreover, the effect of thermal environmental conditions is considered, and the constituent material properties of both auxetic core and face sheets are taken to be temperature-dependent. Under transverse uniform pressure, large deflections of the plate will occur when the pressure is sufficiently large. In such a case, the shape of both sandwich plates and microstructures of the 3D lattice core, along with their effective stiffnesses, must be changed accordingly. Consequently, full-scale nonlinear FE modeling and analysis are conducted to investigate the nonlinear bending behavior of sandwich plates with FG–NPR 3D lattice core.

NPR of the auxetic 3D lattice metamaterial

A 3D lattice microstructure with NPR is expanded from the classical 2D re-entrant honeycombs, as illustrated in Figure 1. As an obvious advantage, the 3D lattice metamaterial possesses lower relative density compared with the 2D honeycombs. Furthermore, when the 3D lattice microstructure is compressed in one direction, both the other two directions will shrink due to NPR.

Figure 1.

The microstructures of auxetic metamaterials: (a) 2D honeycomb; (b) 3D lattice; (c) subjected forces of oblique strut AC at endpoint A; (d) subjected forces of oblique strut AC at endpoint C including virtual load F.

The geometric features of the 3D lattice unit cell may be characterized by five parameters: the horizontal strut length a₀, the oblique strut length b₀, the link strut length c₀, the strut incline angle θ, and the strut radius r, as shown in Figure 1(b), in which in-plane x₁- and x₂-directions are parallel to the horizontal struts, whereas the out-of-plane x₃-direction is determined according to the right-handed coordinate system.

An obvious precondition of the microstructural design is that the struts should not contact or even intersect with each other, which requires

a_{0} - 2 r + 2 b_{0} \sin θ > 0 \Rightarrow \sin θ > - (a_{0} - 2 r) / 2 b_{0}

(1)

The overall dimensions of a unit cell could be expressed as

{\begin{matrix} L_{u} = a_{0} + c_{0} + 2 b_{0} \sin θ \\ H_{u} = 2 \sqrt{{(b_{0} \cos θ)}^{2} - a_{0}^{2} / 4} \end{matrix}

(2)

where L_u and H_u are the overall in-plane and out-of-plane dimensions, respectively. If the value of L_u, H_u, and a₀ is given, there is only one independent parameter left. In the present study, the incline angle θ is selected as the design parameter, while b₀ and c₀ will be determined accordingly.

Moreover, the relative density could be expressed as

\frac{ρ^{*}}{ρ_{c}} = \frac{m^{*} / V^{*}}{ρ_{c}} = \frac{π r^{2} (4 a_{0} + 16 b_{0} + 2 c_{o})}{H_{u} \cdot L_{u}^{2}} \propto r^{2}

(3)

in which ρ* and ρ_c are, respectively, the mass density of 3D lattice metamaterials and its constituent material, m* is the mass in the volume V* of one microstructure.

Effective Poisson’s ratio

Assuming that the microstructure illustrated in Figure 1(b) is linearly elastic and in equilibrium state under the action of compressive forces along x₃-direction, the effective strain ε₃ apparently comes from the relative displacement of two layers of horizontal struts, and the strain ε₁ (or ε₂) along the in-plane direction can be calculated from the relative displacement of link struts. Then, the EPR can be evaluated. To this end and according to the Castigliano’s second theorem, the total strain energy is established and used to determine the deflections.

Considering the symmetry about the plane of horizontal struts, there is only axial deformation within them. Assume the axial force in a horizon strut AB is 2T, then the displacement of its end point A would be

Δ = \frac{2 T (a_{0} / 2)}{E S} = \frac{T a_{0}}{E S}

(4)

where E is the elastic modulus of constituent material and S is the cross-section area of struts.

For the special microstructure with the condition of $a_{0} = H_{u}$ , the angle between strut AC and T is exactly equal to that between AC and P, i.e. α. The internal forces in strut AC are shown in Figure 1(c), and

{\begin{array}{l} \begin{array}{l} N = P \cos α + T \cos α \\ Q^{2} = {(P \sin α)}^{2} + {(T \sin α)}^{2} - 2 (P \sin α) (T \sin α) \cos (π - β) \end{array} \\ M^{2} = \frac{Q^{2} b_{0}^{2}}{4} {(1 - \frac{2 x}{b_{0}})}^{2} \end{array}

(5)

where

\sin α = \frac{\sqrt{{(b_{0} \sin θ)}^{2} + {(a_{0} / 2)}^{2}}}{b_{0}}

\cos β = - \frac{\cos^{2} α}{\sin^{2} α}

Therefore, the total strain energy could be expressed as

U_{T} = U_{N} + U_{Q} + U_{M}

(6)

where

U_{N} = \int_{b_{0}} \frac{N^{2}}{2 E S} d x

U_{Q} = \int_{b_{0}} \frac{10}{9} \frac{Q^{2}}{2 G S} d x

, and

U_{M} = \int_{b_{0}} \frac{M^{2}}{2 E I} d x

are, respectively, the strain energy induced by axial force N, transverse shear force Q, and bending moment M;

G = \frac{E}{2 (1 + ν_{c})}

is the shear modulus of constituent material with isotropic property and ν_c is the corresponding Poisson’s ratio; I is the cross section inertia moment and, for circular sections,

I = \frac{r^{2}}{4} S

The force T could then be determined by the axial displacement of point A using the following equation

- Δ = \frac{\partial U_{T}}{\partial T} = δ_{A}^{N} + δ_{A}^{Q} + δ_{A}^{M}

(7)

where

\begin{array}{l} Δ = \frac{T a_{0} r^{2}}{4 E I}, δ_{A}^{N} = \frac{b_{0} r^{2}}{8 E I} \frac{\partial (N^{2})}{\partial T}, \\ δ_{A}^{Q} = \frac{5 b_{0} r^{2} (1 + ν_{c})}{18 E I} \frac{\partial (Q^{2})}{\partial T}, δ_{A}^{M} = \frac{b_{0}^{3}}{24 E I} \frac{\partial (Q^{2})}{\partial T} \end{array}

(8a)

And

\frac{\partial (N^{2})}{\partial T} = 2 T \cos^{2} α + 2 P \cos^{2} α

(8b)

\frac{\partial (Q^{2})}{\partial T} = 2 T \sin^{2} α - 2 P \cos^{2} α

(8c)

The relationship between T and P can be written as

T = P \cdot \frac{(\frac{b_{0}^{3}}{24} + \frac{5 b_{0} r^{2} (1 + ν_{c})}{18}) \cos^{2} α - \frac{b_{0} r^{2}}{8} \cos^{2} α}{(\frac{b_{0}^{3}}{24} + \frac{5 b_{0} r^{2} (1 + ν_{c})}{18}) \sin^{2} α + \frac{b_{0} r^{2}}{8} \cos^{2} α + \frac{a_{0} r^{2}}{8}}

(9)

Subsequently, we can obtain the relative displacement along x₃-direction, which is twice that of the point A

\begin{array}{l} δ_{3} = - 2 \cdot (δ_{3}^{N} + δ_{3}^{Q} + δ_{3}^{M}) \\ = - 2 \cdot [\frac{b_{0} r^{2}}{8 E I} \frac{\partial (N^{2})}{\partial P} + (\frac{b_{0}^{3}}{24 E I} + \frac{5 b_{0} r^{2} (1 + ν_{c})}{18 E I}) \frac{\partial (Q^{2})}{\partial P}] \end{array}

(10)

where

\frac{\partial (N^{2})}{\partial P} = 2 P \cos 2 α + 2 T \cos^{2} α

(11a)

\frac{\partial (Q^{2})}{\partial P} = 2 P \sin^{2} α - 2 T \cos^{2} α

(11b)

To calculate the displacement along in-plane direction, two pairs of opposite forces are then applied as virtual additional loads. As illustrated in Figure 1(d), the internal forces in the incline strut AC are

{\begin{array}{l} \begin{array}{l} N = P \cos α + T \cos α + F \sin θ \\ \begin{array}{l} Q^{2} = {F - \sin θ [F \sin θ + (T + P) \cos α]}^{2} \\ + {- T + a_{0} / 2 b_{0} [F \sin θ + (T + P) \cos α]}^{2} \\ + {- P + H_{u} / 2 b_{0} [F \sin θ + (T + P) \cos α]}^{2} \end{array} \end{array} \\ M^{2} = \frac{Q^{2} b_{0}^{2}}{4} {(1 - \frac{2 x}{b_{0}})}^{2} \end{array}

(12)

Thus, the relative displacement along in-plane direction could be expressed as twice that of point C as

δ_{1} = 2 \cdot (Δ + δ_{1}^{N} + δ_{1}^{Q} + δ_{1}^{M})

(13)

where

δ_{1}^{N} = {\frac{b_{0} r^{2}}{8 E I} \frac{\partial (N^{2})}{\partial F} |}_{F = 0}

(14a)

δ_{1}^{Q} = \frac{5 b_{0} r^{2} (1 + ν_{c})}{18 E I} {\frac{\partial (Q^{2})}{\partial F} |}_{F = 0}

(14b)

δ_{1}^{M} = \frac{b_{0}^{3}}{24 E I} {\frac{\partial (Q^{2})}{\partial F} |}_{F = 0}

(14c)

and

{\frac{\partial (N^{2})}{\partial F} |}_{F = 0} = 2 \sin θ (P \cos α + T \cos α)

(15a)

\begin{array}{l} {\frac{\partial (Q^{2})}{\partial F} |}_{F = 0} = - 2 (\cos^{2} θ) [(T + P) \cos α \sin θ] \\ + 2 [(\sin θ) \frac{a_{0}}{2 b_{0}}] [- T + (T + P) \cos α \frac{a_{0}}{2 b_{0}}] \\ + 2 [(\sin θ) \frac{H_{u}}{2 b_{0}}] [- P + (T + P) \cos α \frac{H_{u}}{2 b_{0}}] \end{array}

(15b)

Finally, the EPR is established as

ν_{31}^{e f f} = - \frac{ε_{1}}{ε_{3}} = \frac{(Δ + δ_{1}^{N} + δ_{1}^{Q} + δ_{1}^{M}) / L_{u}}{(δ_{3}^{N} + δ_{3}^{Q} + δ_{3}^{M}) / H_{u}}

(16)

It is worth mentioning that all the displacements in this equation contain a factor of EI, so the value of $ν_{31}^{e f f}$ should be independent of it. However, the Poisson’s ratio of constituent material will play a part in EPR of auxetic 3D lattice metamaterials.

Finite element simulation

Before experiments, finite element (FE) models are created to predict the EPR of the lattice metamaterial. To clarify the influence of incline angles, models are designed to have different values of θ as −15°, −20°, and −25°. The specimen consists of $4 \times 1 \times 6$ unit cells, and two cover plates to be compressed upon when the specimens are tested. The detailed dimensions are as follows: L_u = 12 mm, H_u = 8 mm, a₀ = 8 mm, and r = 0.5 mm.

The unit cells are modeled by Timoshenko beam elements B31 that allow for transverse shear deformation, and two cover plates are modeled by SC8R elements. As shown in Figure 2(a), in which the beam profiles are rendered, the boundary conditions are considered to be that the bottom layer of nodes is constrained by all freedoms while the nodes lying on the top layer are constrained except for the freedom along the loading direction.

Figure 2.

The FE model and test setup of EPR of the auxetic 3D lattice metamaterials.

For the auxetic 3D lattice metamaterials, we can trace and calculate the average displacement in each boundary of a specified effective region (marked by the red lines) and obtain the EPR $ν_{31}^{e f f}$ defined as

ν_{31}^{e f f} = - \frac{({\bar{U 1}}_{R} - {\bar{U 1}}_{L}) / L_{E R}}{({\bar{U 3}}_{B} - {\bar{U 3}}_{T}) / H_{E R}}

(17)

where the

{\bar{U 1}}_{R}

and

{\bar{U 1}}_{L}

denote the displacement along X-direction of the right and left line, while the

{\bar{U 3}}_{B}

and

{\bar{U 3}}_{T}

are, respectively, the displacement along Z-direction of the bottom and top line;

L_{E R}

and

H_{E R}

is the overall length and height of the effective region. Within this section, as illustrated in Figure 2(a), a region that consists of

2 \times 1 \times 2

unit cells in the central area is expected, and consequently,

L_{E R}

= 2L_u and

H_{E R}

= 2H_u.

Experimental verification

The specimens are manufactured using 3D printing technology, as shown in Figure 2(b). The Poisson’s ratio of the PA 12 material is v = 0.37, as determined in our laboratory according to the ASTM D638-14 standard test method.

The test system includes a test machine with its control and data acquisition software installed in a computer, a camera with its images lively displayed on a monitor, as shown in Figure 2(c). For the purpose of keeping boundary and loading conditions consistent with the previous FE models, the specimens are rested in the supporting plate with the bottom facesheet prevented from moving and compressed by the uniaxial loading plate in the normal direction of both plates. The images of the specimens before and after deformation are processed using the Image Processing Toolbox of MATLAB (R2015a) to determine distances between selected pixels located on the effective boundaries, as illustrated in Figure 2(a), in the images.

Experimental results of EPR are obtained and compared with the theoretical results of equation (16) and FEM results of equation (17), as listed in Table 1, from which good agreements are observed. Furthermore, we can draw conclusion that the magnitude of EPR is decreased as the incline angles increase.

Table 1.

The EPR of the auxetic 3D lattice metamaterials with different incline angles.

Incline angle	Theoretical results	FE results	Experimental results
−15°	−1.0869	−1.0647	−1.0386
−20°	−0.9737	−0.9567	−0.9371
−25°	−0.8442	−0.8334	−0.8206

Further discussion

According to equation (16), the NPR values of different strut incline angles and slenderness ratios are illustrated in Figure 3. The theoretical results revealed that the curves of different strut slenderness ratios possess significant differences, which will further enlarge for smaller magnitude of incline angles. Second, magnitudes of NPR values are decreased as the strut slenderness ratios increase, indicating a higher curve in the Figure 3. In conclusion, both strut incline angles and slenderness ratios will have significant effects on NPR of the auxetic 3 D lattice metamaterials.

Figure 3.

The EPR of auxetic 3D lattice metamaterials with different strut incline angles and slenderness ratios.

Furthermore, within the range of [−30, −5], the variation trends for cases of $r / b_{0} \geq 0.05$ are not monotonously decreasing, but have a point of minimum. While as the magnitude of incline angles decrease, the magnitude of EPR for the case of very small slenderness ratio becomes very large. Nevertheless, such cases might not exist in practice.

Full-scale FE modeling and nonlinear analysis of sandwich plates

Consider a sandwich plate with length a, width b, and thickness $h = h_{c} + 2 h_{f}$ , in which the thickness of facesheets and core layer is, respectively, h_f and h_c. As illustrated in Figure 4(a), a right-handed coordinate system (X, Y, Z) with its origin located at the center of mid-surface is established, in which X- and Y-directions are, respectively, along the length and width of sandwich plates, while the out-of-plane Z-direction points to the bottom facesheet. Two facesheets are assumed to be made from isotropic metal, and the 3D lattice core is assumed to have six microstructures in the plate thickness direction. The section bending stiffness of sandwich plates $(= \int E (Z) Z^{2} d Z)$ is influenced by the position (Z) and effective stiffness (E) of each layer of materials and will then control the bending response.

Figure 4.

(a) Construction of a sandwich plate; (b) the substructure of the auxetic core (effective region for the EPR calculation); (c) the substructure of a non-auxetic core.

Specifically, three types of functionally graded configurations are considered. For the type V, i.e. [( $θ_{1}$ )₂/( $θ_{2}$ )₂/( $θ_{3}$ )₂] (assume $| θ_{1} |$ < $| θ_{2} |$ < $| θ_{3} |$ ), referred to as FG-V, where $θ_{i}$ (i = 1,2,3) is the strut incline angle. For type O, a mid-plane symmetric-graded distribution of microstructural strut incline angles is achieved, i.e. [ $θ_{3}$ / $θ_{2}$ / $θ_{1}$ ]_S, and for type X, the microstructures are assumed to have strut incline angles of [ $θ_{1}$ / $θ_{2}$ / $θ_{3}$ ]_S, referred to as FG-O and FG-X, respectively. A sandwich plate with uniform distributed (UD) auxetic 3D lattice core is also considered for comparison, of which all the strut incline angles are equal to $θ_{2}$ .

Full-scale FE modeling

The sandwich plates would have bending deformation when subjected to transverse uniform pressure, and large deflection will occur when the applied load is sufficiently large. In such a case, the shape of the sandwich plate and, hence, its stiffness changes as it deforms. Therefore, the geometric nonlinearity is necessary to be taken into account to achieve reasonable simulations.

For the intension of more precisely obtaining the nonlinear bending behaviors, 3D full-scale FE modeling is conducted, which indicates that all the structural details are modeled, rather than using the effective homogenization models. The main reason is that, in the large deflection region, the microstructures of auxetic 3D lattice core may deform to a different shape, and the effective mechanical properties will be changed accordingly. Therefore, the effective homogenization models are suitable for linear problems only.

For the nonlinear bending problem of functionally graded sandwich plates with auxetic 3D lattice core, we need to determine the load–deflection curves, along with the EPR-deflection curves. Moreover, in the structural analysis, the expected region to calculate the EPR is set to be the all the six unit cells in the thickness direction, with 2 × 2 unit cells in the XY-plane. It is worth noting that equation (17), directly modified from the definition of the Poisson’s ratio, is applicable for both linear and large deformation problems.

To perform the NPR effect of 3D lattice metamaterials in bending, the auxetic core is arranged to have the same out-of-plane direction with the sandwich plates, and the correspondence relationship between their two coordinate systems can be expressed as: $x_{1} \Leftrightarrow X; x_{2} \Leftrightarrow Y; x_{3} \Leftrightarrow Z$ . In such a layout, when the sandwich plates are subjected to transverse loads, the microstructures will response in a bi-directional NPR way. The construction of sandwich plates is illustrated in Figure 4(a).

The full-scale nonlinear finite element modeling is conducted using FEA software ABAQUS. The thickness of core is assumed to be h_c = 24 mm with H_u = 4 mm, and the facesheet-to-core thickness ratio might be h_f/h_c = 1/100, 2/100, 3/100, and 5/100. The width-to-thickness ratio of sandwich plates is taken to be b/h = 25, while the aspect ratio is fixed as a/b = 1, i.e. only square plates are considered. The number of 3D lattice unit cells along the width direction is then N = b/L_u, in which the length of a microstructure is L_u = 6 mm. For the UD microstructure configuration, the strut incline angle is taken to be −20°. While for the functionally graded cores, the unit-cell strut incline angles are arranged from the top one to the bottom one through the Z-direction of plate, as listed in Table 2.

Table 2.

Configuration codes of the auxetic 3 D lattice core.

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]

The facesheets are meshed with one layer of continuum shell element SC8R, which can allow for changes in the thickness. For the unit cells of 3D lattice cores, Timoshenko beam element B31 is adopted, as in the analysis of auxetic 3D lattice metamaterials. To achieve a satisfactory compromise between costs of time and memory and accuracy of results, a convergence analysis was carried out, and finally the mesh size for facesheets is 1 mm, while the a₀, b₀, and c₀ struts are meshed using one, two, and four elements, respectively. For the sake of simplicity, the face sheets and core are assumed to be perfectly connected, and the meshes are carefully arranged so as to share nodes along their interfaces. The detailed overall geometric parameters and meshes of sandwich plates with different facesheet-to-core thickness ratios are summarized in Table 3.

Table 3.

Overall geometric parameters and meshes of sandwich plates.

h_f /h_c	N = b/L_u	Elements	Nodes	Model scale
1/100	51	187,272 SC8R + 676,260 B31	835,384	Quarter
2/100	52	194,688 SC8R + 703,040 B31	868,404	Quarter
3/100	53	202,248 SC8R + 730,340 B31	902,064	Quarter
5/100	55	217,800 SC8R + 786,500 B31	971,304	Quarter

The loading is assumed to be uniform pressure and applied upon the top surface ( $Z = - (0.5 h_{c} + h_{f})$ ) of the sandwich plates. It is worth noting that such a loading is symmetric about X = 0 and Y = 0 plane.

Three types of boundary conditions are considered, i.e. CCCC, CSCS, and SSSS, 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 at the plate edge are constrained, while for “S,” the boundary edges are constrained with only one rotation DoF is allowed. It is notable that three boundary conditions are also symmetric about X = 0 and Y = 0 plane, so only a quarter of sandwich plates needs to be modeled, as illustrated in Figure 4. In such cases, symmetric constraint conditions are applied upon nodes located on the symmetric planes. These three boundary conditions can be expressed as

C : U 1 = U 2 = U 3 = U R 1 = U R 2 = U R 3 = 0

(18a)

S : {\begin{array}{l} U 3_{U P R} + U 3_{L W R} = 0 \\ \begin{array}{l} U 1_{U P R} + U 1_{L W R} = 0 (for X = a / 2) \\ U 2_{U P R} + U 2_{L W R} = 0 (for Y = b / 2) \end{array} \end{array}

(18b)

S Y M M : {\begin{array}{l} U 1 = U R 2 = U R 3 = 0 (for X = 0) \\ U 2 = U R 1 = U R 3 = 0 (for Y = 0) \end{array}

(18c)

where U1, U2, and U3 are translational displacements along the X-, Y-, and Z-directions; UR1, UR2, and UR3 are rotational displacements about the X-, Y-, and Z-directions, respectively; and the subscripts UPR and LWR refers to nodes symmetrically located on the upper and lower side of mid-surface. Moreover, as shown in Figure 4, the region to calculate the EPR will now reduce to only one microstructure in the in-plane direction, with two symmetric conditions. Consequently, we have

L_{E R}

= L_u and

H_{E R}

= 6H_u.

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 if a thermal expansion coefficient is given for the material. Accordingly, a sequentially coupled thermal-mechanical analysis is performed. Specifically, a predefined field is created to define the initial temperature field as T₀ = 300 K. Sequentially, a new analysis step is created and inserted to modify the predefined field as expected, after which mechanical analysis is performed. Based on the displacement field that is induced by the previous temperature field changes, mechanical loads of transverse pressure loading are then applied upon the sandwich plates.

Moreover, in different thermal environments, significant changes in mechanical properties of the constituent materials are to be expected. For accurate prediction of the nonlinear bending of the sandwich plates, it is essential to take this temperature-dependency into consideration. Thus, the Young’s modulus E and thermal expansion coefficient α are assumed to be temperature-dependent, while the Poisson’s ratio ν remains constant. The face sheets are made from aluminum alloy. As for the 3D lattice core, the Ti-6Al-4V material is adopted, and the expressions of the mechanical properties are defined in Li et al. [42]. In the present FE simulations, three thermal environments are used, i.e. T = 300, 325, and 350 K, and the properties for the constituent materials of both facesheets and 3D lattice core under these thermal environments are listed in Table 4.

Table 4.

Temperature-dependent properties of facesheets and 3D lattice core.

Temperature	300 K	325 K	350 K
E_f (MPa)	68,750	67,835	66,921
ν_f	0.33	0.33	0.33
α_f (/℃)	2.31E-05	2.35E-05	2.39E-05
E_c (MPa)	105,698	104,293	102,888
ν_c	0.29	0.29	0.29
α_c (/K)	6.9415E-06	6.6946E-06	6.4179E-06

Comparison studies

Before parametric studies, it is essential to validate the present FE method. As mentioned earlier, there are no existing studies covering the nonlinear bending of sandwich plates with NPR cores. Two reduced examples of nonlinear bending problems are re-calculated and compared. By default, the SC8R element is used to mesh the plates, and the seed density is determined by convergency analysis.

Example 1

Consider a CNTRC plate with different boundary conditions, which has an aspect ratio of a/b = 1 and width-to-thickness ratio of b/h = 100. The effective material properties of the CNTRC have been determined by Shen [47], and the matrix is poly{(mphenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene) vinylene]}, referred to as PmPV, with E_m = 2100 MPa when T = 300 K (room temperature). For the CNTRC lamina with V_CN = 0.17 and resting at T = 300 K, E₁₁ = 144,771.38 MPa, E₂₂ = 3493.88 MPa, ν₁₂ = 0.3120, G₁₂ = 1303.66 MPa, and G₂₃ = G₁₃ = G₁₂. The load–deflection curves of the CNTRC plate with SSSS and CCCC boundary conditions and subjected to a uniform pressure are presented in Figure 5, in which we can observe good agreements between present results and those obtained by Zhang et al. [48] using the element-free IMLS-Ritz method.

Figure 5.

The load–deflection curves of CNTRC plates with different boundary conditions and subjected to a uniform pressure.

Example 2

Consider a simply supported sandwich plate with FGM facesheets, which has an aspect ratio of a/b = 1, width-to-thickness ratio of b/h = 20, and core-to-facesheet thickness ratio of h_core/h_F = 8. The effective Young’s modulus E_F of the FGM facesheets may be expressed as E_F = E_cV_c + E_mV_m, where V_c and V_m are the ceramic and metal volume fractions and V_c + V_m = 1. Furthermore, the plate is mid-plane symmetric, and for the lower facesheet, $V_{m} = {[(Z - h / 2) / (h_{c o r e} / 2 - h / 2)]}^{N}$ , where N is the volume fraction index. The material properties of the ceramic and metal, which is also the used as core material, are E_c = 168,062.96 MPa and E_m = 105,698.20 MPa at T = 300 K (from Reddy and Chin [49]). Moreover, Poisson’s ratio is 0.29 for both constituent materials. The load–deflection curves of the FG sandwich plate with N = 2 and subjected to a uniform pressure are presented in Figure 6, from which good agreement can be observed between results herein and those of Wang and Shen [50] using a two-step perturbation approach.

Figure 6.

Comparison of nonlinear bending load–deflection curves of the sandwich plate with FGM face sheets subjected to a uniform pressure.

Parametric studies and discussion

In the following sections, parametric studies are carried out to demonstrate the novelty of NPR core, the effect of FG configurations, temperature rises, facesheet-to-core thickness ratios, boundary conditions, and strut radii on the nonlinear bending behavior and EPR variation of sandwich plates, in which EPR variation refers to the effective region of $1 \times 1 \times 6$ 3D lattice microstructures located in the center of plates, as illustrated in the Figure 4(b). Moreover, the non-dimensional amplitude is defined by W_mid/h, where W_mid is the displacement along Z-direction of the node located at the center of top face.

The novelty of auxetic cores

To illuminate the novelty of auxetic cores, non-auxetic cores with positive EPR are modeled and compared. The only difference between two cores is the inclined angle, i.e. a negative one for the auxetic core with negative EPR, while a positive one for the non-auxetic core with positive EPR, and the microstructures of them are exhibited in Figure 4(b) and (c), respectively. The remaining dimensions are the same: a₀ = 4 mm, b₀ = 3 mm, c₀ = 2 mm, and the same pressure loading of p = 0.01 MPa is applied upon the sandwich plates with a = b = 540 mm and h_f = 0.5 mm. The resulting deflections are listed in Table 5, from which we conclude that the deflections of the sandwich plates with non-auxetic core are distinctly larger than that of their counterpart with auxetic core, and the differences enlarge with the thickness of struts increasing.

Table 5.

Comparisons of transverse deflection between sandwich plates with auxetic and non-auxetic 3D lattice cores.

3D lattice cores	r = 0.15 mm	r = 0.20 mm	r = 0.25 mm	r = 0.30 mm
Auxetic	2.9290	1.1400	0.5563	0.3357
Non-auxetic	3.2050	1.3190	0.6626	0.4058
Differences^a	9.42%	15.70%	19.11%	20.88%

Difference = $(\frac{W_{N o n - a u x e t i c}}{W_{A u x e t i c}} - 1) \times 100 %$ .

The effects of FG configurations

The effects of functionally graded configurations on the nonlinear bending behavior and EPR variation of sandwich plates are shown in Figure 7(a) and (b), respectively. Three types of FG configurations, namely, FG-V, FG-O, and FG-X, are considered and compared with the UD case. All FG and UD sandwich plates have the same facesheet-to-core thickness ratio of h_f/h_c = 1/50, strut radius r = 0.15 mm, and are resting at thermal environment of T = 300 K. The boundary conditions are assumed to be CCCC. As shown in Figure 7(a), among the four configurations, the FG-X plate processes the best flexural property, which indicates that its deflection is smallest when subjected to the same pressure loading. Moreover, it is observed that the load–deflection curves are nonlinear because of the slopes gradually decrease as the increase of applied loads. From Figure 7(b), the EPR-deflection curves of sandwich plates with symmetric FG-O and FG-X configurations have the same variation trend with the UD plate, while the behavior of FG-V plate is quite different, which is due to the asymmetry of its 3D lattice core. Moreover, it is observed that all the EPRs will change significantly when W_mid/h < 0.3, whereas the changes become very limited in the large deflection region. Only FG-X and UD sandwich plates are included and compared in the following studies.

Figure 7.

Effects of FG configurations on the nonlinear bending behavior and EPR variation of sandwich plates with FG–NPR 3D lattice cores: (a) nonlinear bending load–deflection curves; (b) EPR-deflection curves.

The effects of temperature changes

Figure 8 demonstrates the effects of temperature changes on the nonlinear bending behavior and EPR variation of sandwich plates with facesheet-to-core thickness ratio of h_f/h_c = 1/50. The plates have CCCC boundary conditions. The load–deflection curves of FG-X and UD plates are higher with increasing temperature, indicating that the stiffness decreases with temperature rise. However, as shown in Figure 8(a), they all share the same variation trend. On the other hand, the FG-X plates denoted by curves with hollow symbols have better flexural property than UD plates denoted by curves with solid symbols, and this is consistent with the results of the previous sub-section. From Figure 8(b), it is observed that when the higher temperature environments are considered, the variation of EPR-deflection curves is significantly different from that of the sandwich plate at T = 300 K. The explanation is that the temperature changes will bring about initial displacement fields. Therefore, we can conclude that thermal environments will greatly affect the load–deflection curves and in particular have prominent effect on the EPR of the effective region.

Figure 8.

Effects of thermal environmental conditions on the nonlinear bending behavior and EPR variation of sandwich plates with FG–NPR 3D lattice cores: (a) nonlinear bending load–deflection curves; (b) EPR-deflection curves.

The effects of facesheet-to-core thickness ratios

The effects of facesheet-to-core thickness ratios (h_f/h_c) on the nonlinear bending behavior and EPR variation of sandwich plates are shown in Figure 9. The plates have the same boundary conditions of CCCC, and the thermal environment is assumed as T = 300 K. It can be seen in Figure 9(a) that the deflection decreases with increasing facesheet thickness. Moreover, as shown in Figure 9(b), the EPR curves are higher for thinner facesheets, meaning smaller magnitudes of EPR. Consequently, the conclusion can be drawn that facesheet-to-core thickness ratios not only have obvious effect on the flexural properties but also will significantly affect the EPR variation.

Figure 9.

Effects of facesheet-to-core thickness ratios on the nonlinear bending behavior and EPR variation of sandwich plates with FG–NPR 3D lattice cores: (a) nonlinear bending load–deflection curves; (b) EPR-deflection curves.

The effects of boundary conditions

Figure 10 presents the effects of boundary conditions on the nonlinear bending behavior and EPR variation of sandwich plates at T = 300 K. The plates have facesheet-to-core thickness ratio of h_f/h_c = 1/20 and strut radius r = 0.275 mm. As shown in Figure 10(a), the CCCC plate has the lowest bending load–deflection curve, while the SSSS plate has the highest ones. However, the differences between different boundary conditions are not quite significant, indicating that the deformation of such sandwich plates is shear dominant, not bending dominant. From Figure 10(b), the EPR-deflection curves all show the same trend, and the difference merely exists in small deflection region. Thus, we come to a conclusion that boundary conditions have an only limited effect on the bending behavior of sandwich plates with auxetic 3D lattice core.

Figure 10.

Effects of boundary conditions on the nonlinear bending behavior and EPR variation of sandwich plates with FG–NPR 3D lattice cores: (a) nonlinear bending load–deflection curves; (b) EPR-deflection curves.

The effects of strut radii

Figure 11 is dedicated to study the effects of strut radius r on the nonlinear bending behavior and EPR variation of sandwich plates with FG-NPR 3D lattice cores. The sandwich plate is subjected to a UD pressure at T = 300 K. The boundary condition is assumed to be CCCC. As the strut radius decreases, the load–deflection curves are lowered, which indicate a higher bending stiffness. Moreover, the EPR-deflection curve of the sandwich plates with bigger r is much lower than that of the plate with thinner structs, as shown in Figure 11(b). This indicates that the strut radius r will significantly affect the EPR variation of the effective region.

Figure 11.

Effects of strut radii on the nonlinear bending behavior and EPR variation of sandwich plates with FG–NPR 3D lattice cores: (a) nonlinear bending load–deflection curves; (b) EPR-deflection curves.

Concluding remarks

Modeling and analysis of the nonlinear bending behavior of sandwich plates with FG auxetic 3D lattice core has been presented. An auxetic 3D lattice metamaterial expanded from the classical 2D re-entrant honeycomb was designed and studied using theoretical, FE, and experimental methods. It was found that both strut incline angles and slenderness ratios have significant effects on EPR of the auxetic 3D lattice metamaterials. Numerical studies were carried out for FG-V, FG-O, FG-X, and UD sandwich plates by using full-scale nonlinear finite element simulations, with consideration of temperature-dependent material properties of both face sheets and core. Finite element results revealed that, compared with the sandwich plates with non-auxetic 3D lattice core, the transverse deflection of those with auxetic core is distinctly smaller. Furthermore, the EPR-deflection curves of sandwich plates were obtained for the first time. Numerical results also indicated that the FG configurations, temperature changes, facesheet-to-core thickness ratios, and strut radii can significantly influence the EPR-deflection curves, while boundary conditions only have small effects. In addition, when the plate deflection is sufficiently large, all the EPR-deflection curves will enter to a stable region. It is expected that the present results could shed light on the EPR of auxetic 3D lattice metamaterials and the nonlinear bending responses of sandwich plates with FG-NPR 3D lattice core and could be conducive to further investigations.

Footnotes

Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The support for this work, provided by the National Natural Science Foundation of China under Grant 51779138, is gratefully acknowledged.

ORCID iD

Hai Wang

References

Kolken

HMA

Zadpoor

. Auxetic mechanical metamaterials. RSC Adv 2017; 7: 5111–5129.

Kshetrimayum

. A brief intro to metamaterials. IEEE Potential 2004; 23: 44–46.

Evans

Nkansah Hutchinson

et al. Molecular network design. Nature 1991; 353: 124–125.

Shen

H-S

Reddy

. Large amplitude vibration of FG-CNTRC laminated cylindrical shells with negative Poisson’s ratio. Comput Methods Appl Mech Eng 2020; 360: 112727.

Lakes

. Foam structures with a negative Poisson’s ratio. Science 1987; 235: 1038–1040.

Gibson

Ashby

. Cellular solids: structure and properties. Cambridge, UK: Cambridge University Press, 1997.

Hassanin

Attallah

et al. The development of TiNi-based negative Poisson’s ratio structure using selective laser melting. Acta Mater 2016; 105: 75–83.

Xiong

Chen

et al. Structural optimization of re-entrant negative Poisson’s ratio structure fabricated by selective laser melting. Mater Des 2017; 120: 307–316.

Chen

. A novel three-dimensional auxetic lattice metamaterial with enhanced stiffness. Smart Mater Struct 2017; 26: 105029.

10.

Vinson

. The behavior of sandwich structures of isotropic and composite materials. Lancaster: Technomic Publishing Company, 1999.

11.

Wang

Chen

et al. Experimental and numerical study of the elastic properties of PMI foams. J Mater Sci 2010; 45: 2688–2695.

12.

Wang

Waas

Wang

. Experimental and numerical study on low-velocity impact behavior of foam-core sandwich panels. Compos Struct 2013; 96: 298–311.

13.

Choi

Lakes

. Analysis of elastic modulus of conventional foams and of re-entrant foam materials with a negative Poisson’s ratio. Int J Mech Sci 1995; 37: 51–59.

14.

Wang

. Bending behavior of sandwich composite structures with tunable 3D-printed core materials. Compos Struct 2017; 175: 46–57.

15.

Duc

Cong

. Nonlinear dynamic response and vibration of sandwich composite plates with negative Poisson’s ratio in auxetic honeycombs. J Sandw Struct Mater 2018; 20: 692–717.

16.

Cong

Khanh

Khoa

et al. New approach to investigate nonlinear dynamic response of sandwich auxetic double curves shallow shells using TSDT. Compos Struct 2018; 185: 455–465.

17.

Lakes

Elms

. Indentability of conventional and negative Poisson’s ratio foams. J Compos Mater 1993; 27: 1193–1202.

18.

Evans

Alderson

. Auxetic materials: functional materials and structures from lateral thinking!. Adv Mater 2000; 12: 617–628.

19.

Evans

. Auxetic polymers: a new range of materials. Endeavour 1991; 15: 170–174.

20.

Scarpa

Yates

Ciffo

et al. Dynamic crushing of auxetic open-cell polyurethane foam. J Mech Eng Sci 2002; 216: 1153–1156.

21.

Liu

Wang

Luo

et al. In-plane dynamic crushing of re-entrant auxetic cellular structure. Mater Des 2016; 100: 84–91.

22.

Hou

Liu

Zhang

et al. How does negative Poisson’s ratio of foam filler affect crashworthiness?. Mater Des 2015; 82: 247–259.

23.

Mohsenizadeh

Alipour

Shokri

et al. Crashworthiness assessment of auxetic foam-filled tube under quasi-static axial loading. Mater Des 2015; 88: 258–268.

24.

Zhang

Liu

. Enhancing buckling capacity of a rectangular plate under uniaxial compression by utilizing an auxetic material. Chinese J Aeronaut 2016; 29: 945–951.

25.

Asemi

Shariyat

. Three-dimensional biaxial post-buckling analysis of heterogeneous auxetic rectangular plates on elastic foundations by new criteria. Comput Method Appl Mech Eng 2016; 302: 1–26.

26.

Shen

H-S

Wang

. Postbuckling behavior of sandwich plates with functionally graded auxetic 3D lattice core. Compos Struct 2020; 237: 111894.

27.

Novak

Starčevič

Vesenjak

et al. Blast response study of the sandwich composite panels with 3D chiral auxetic core. Compos Struct 2019; 210: 167–178.

28.

Imbalzano

Tran

Ngo

et al. Three-dimensional modelling of auxetic sandwich panels for localised impact resistance. J Sandwich Struct Mater 2015; 19: 291–316.

29.

Shen

H-S

. Functionally graded materials nonlinear analysis of plates and shells. Boca Raton, FL: CRC Press, 2009. [Database][Mismatch]

30.

Shen

H-S

. Modeling and analysis of functionally graded carbon nanotube reinforced composite structures: a review. Adv Mech 2016; 46: 478–505.

31.

Wang

Z-X

Shen

H-S

. Nonlinear vibration and bending of sandwich plates with nanotube-reinforced composite face sheets. Composites B 2012; 43: 411–421.

32.

Fan

Wang

. Nonlinear bending and postbuckling analysis of matrix cracked hybrid laminated plates containing carbon nanotube reinforced composite layers in thermal environments. Composites B 2016; 86: 1–16.

33.

Song

Yang

Kitipornchai

. Bending and buckling analyses of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Composites B 2018; 134: 106–113.

34.

Zhao

Feng

Dong

et al. Geometrically nonlinear bending of functionally graded nanocomposite trapezoidal plates reinforced with graphene platelets (GPLs). Int J Mech Mat Des. Epub ahead of print 29 January 2019. DOI: 10.1007/s10999-019-09442-4.

35.

Lü

Chen

Lim

. Elastic mechanical behavior of nano-scaled FGM films incorporating surface energies. Compos Sci Technol 2009; 69: 1124–1130.

36.

Lü

Lim

Chen

. Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. Int J Solid Struct 2009; 46: 1176–1185.

37.

Barretta

Fabbrocino

Luciano

et al. Closed-form solutions in stress-driven two-phase integral elasticity for bending of functionally graded nano-beams. Physica E 2018; 97: 13–30.

38.

Barretta

Faghidian

de Sciarra

. Stress-driven nonlocal integral elasticity for axisymmetric nano-plates. Int J Eng Sci 2019; 136: 38–52.

39.

Hou

Tai

Lira

et al. The bending and failure of sandwich structures with auxetic gradient cellular cores. Compos Part A 2013; 49: 119–131.

40.

Boldrin

Hummel

Scarpa

. Dynamic behavior of auxetic gradient composite hexagonal honeycombs. Compos Struct 2016; 149: 114–124.

41.

Shen

H-S

Wang

. Nonlinear bending of sandwich beams with functionally graded negative Poisson’s ratio honeycomb core. Compos Struct 2019; 212: 317–325.

42.

Shen

H-S

Wang

. Thermal postbuckling of sandwich beams with functionally graded negative Poisson’s ratio honeycomb core. Int J Mech Sci 2019; 152: 289–297.

43.

Shen

H-S

Wang

. Nonlinear vibration of sandwich beams with functionally graded negative Poisson’s ratio honeycomb core. Int J Str Stab Dyn 2019; 19: 1950034.

44.

Shen

H-S

Wang

. Nonlinear dynamic response of sandwich beams with functionally graded negative Poisson’s ratio honeycomb core. Eur Phys J Plus 2019; 134: 79.

45.

Li C, Shen H-S, Wang H, et al. Large amplitude vibration of sandwich plates with functionally graded auxetic 3D lattice core. Int J Mech Sci 2020; 174: 105472.

46.

Li C, Shen H-S, Wang H. Nonlinear dynamic response of sandwich plates with functionally graded auxetic 3D lattice core. Nonlinear Dyn. Epub ahead of print 25 May 2020. DOI: 10.1007/s11071-020-05686-4.

47.

Shen

H-S

. Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite cylindrical shells. Composites B 2012; 43: 1030–1038.

48.

Zhang

Song

Liew

. Nonlinear bending analysis of FG-CNT reinforced composite thick plates resting on Pasternak foundations using the element-free IMLS-Ritz method. Compos Struct 2015; 128: 165–175.

49.

Reddy

Chin

. Thermoelastical analysis of functionally graded cylinders and plates. J Therm Stress 1998; 21: 593–626.

50.

Wang

Z-X

Shen

H-S

. Nonlinear analysis of sandwich plates with FGM face sheets resting on elastic foundations. Compos Struct 2011; 93: 2521–2532.