Tension buckling and postbuckling of nanocomposite laminated plates with in-plane negative Poisson ’ s ratio

: Mechanical metamaterials with negative Poisson ’ s ratio (NPR) have emerged as a novel class of engineering material, and have attracted increasing attention in various engineering sectors. Most studies available on the buckling problem of laminated plates with positive or NPR are those under uniaxial compression. Here, we report that the buck-ling phenomenon may occur for auxetic nanocomposite laminated plates under uniaxial tension when the unloaded edges of the plates are immovable. Two types of nanocomposites are considered, including graphene/Cu and carbon nanotube/Cu composites. Governing equations of the auxetic nanocomposite laminated plates are formulated based on the framework of Reddy ’ s higher-order shear deformation theory. In modeling, the von Kármán nonlinear strain – displacement relationship, temperature-dependent material properties, thermal e ﬀ ects, and the plate – substrate interaction are considered. The explicit analytical solutions for postbuckling of auxetic nanocomposite laminated plates subjected to uniaxial tension are obtained for the ﬁ rst time by employing a two-step perturbation approach. Numerical investigations are performed for tension buckling and post-buckling behaviors of auxetic nanocomposite laminated rectangular plates with in-plane NPR rested on an elastic substrate under temperature environments.


Introduction
Buckling is an important problem for the thin plates subjected to uniaxial or biaxial compression [1].Under tensile loads, thin plates usually do not experience buckling.However, according to the literature survey, there are two special cases, in which buckling can occur under tensile loads, namely "tension buckling," for the thin plates/sheets.One case is if a plate contains a hole [2][3][4] or a crack [5][6][7], the compressive stresses arise locally near the hole or the crack under a uniaxial tensile load, where these compressive stresses may cause local buckling.Another case is wrinkling (i.e., local short-wavelength buckling), which is commonly observed in stretched-thin sheets [8][9][10][11][12] and single-layer graphene sheets [13][14][15][16] due to the variation of Poisson's effect.For nanocomposite structures, which are generally considered as the next-generation composite structures, although much research has been done on the buckling and postbuckling analyses of nanocomposite plates under uniaxial or biaxial compression, no attention is paid to the buckling of nanocomposite plates when they are subjected to a tensile load.The buckling of nanocomposite plates under tensile load represents a unique and crucial failure mode in the design of these structures, which has not been reported in the literature.To guide the design and optimization of nanocomposites for future engineering applications, it is of great significance to establish a scientific and theoretical tool to predict the critical tension buckling load, and hence, the factor of safety under this unique buckling failure mode.
Auxetic laminated composites with negative Poisson's ratio (NPR) are one class of mechanical metamaterials.With the development of nanotechnology and additive manufacturing technology [17,18], nanofillers such as graphene sheets [19] or carbon nanotubes (CNTs) [20] can be embedded in a single-crystal copper matrix to achieve auxetic nanocomposite laminates [21,22].Owing to their special properties and characteristics, auxetic nanocomposite materials have shown better performance in certain aspects than those of conventional materials and are expected to have a wide range of technological applications [23,24].
Recently, Shen et al. [25] combined the functionally graded (FG) concept with the auxetic concept to design the FG-GRMMC (graphene-reinforced metal-matrix composite) laminates.Considering the auxetic effect of GRMMCs, Shen and his co-authors [25,26] investigated the impact of in-plane NPR on the compressive postbuckling behaviors of FG-GRMMC laminated and sandwich plates subjected to uniaxial compression.Unlike fiber-reinforced composite (FRC), graphene-reinforced composite (GRC) and carbon nanotube-reinforced composite (CNTRC) laminated plates with positive Poisson's ratios [27][28][29] where the buckling loads and initial postbuckling load-deflection curves for the plate with unloaded edges that are movable (i.e., displacement is unconstrained in the inplane direction) are higher than those of the same plate with unloaded edges that are immovable (i.e., displacement is constrained in the direction perpendicular to the loaded edges), the auxetic GRMMC laminated plates showed the opposite behavior.Specifically, for auxetic laminated plates, the buckling loads and postbuckling load-deflection curves with unloaded edges that are movable in the inplane direction are lower than those of the same plate under immovable unloaded edges.This indicates that the unique lateral contraction of the auxetic laminates under uniaxial compressive load locally enhanced the buckling strength.Such an exceptional behavior leads us to believe that auxetic laminated plates with unloaded edges that are immovable may buckle under a uniaxial tensile load as compressive reaction force will be produced on unloaded edges to restrict the lateral expansion of the auxetic plates under uniaxial tension.This provides the motivation for the present investigation.
The present research is to investigate the buckling and postbuckling behavior of auxetic nanocomposite laminated plates with unloaded edges that are immovable and subjected to the uniaxial tensile load under thermal environmental conditions.In the current study, we chose two types of auxetic nanocomposite laminates.One is a graphene/Cu laminated plate with in-plane NPR and the other is a CNT/Cu laminated plate with in-plane NPR.The material properties of both graphene/Cu and CNT/Cu composites are temperature-dependent.The governing equations of the auxetic nanocomposite laminated plates are established based on the framework of Reddy's higher-order shear deformation theory (HSDT).In modeling, the von Kármán nonlinear strain-displacement relationship, the effect of temperature, the interaction between the plate and substrate, and the effect of the plate's initial geometric imperfection are also taken into consideration.The explicit analytical solutions for buckling and postbuckling of auxetic nanocomposite laminated plates subjected to uniaxial tension are obtained for the first time by employing a two-step perturbation approach.The impacts of the plate aspect ratio, the plate width-to-thickness ratio, temperature variation, and foundation stiffness on tension buckling and postbuckling behavior of auxetic nanocomposite laminated plates are discussed in the numerical investigation.the total thickness of the plate.The coordinate system (X, Y, Z) is located on the middle surface of the plate with its origin placed at one corner of the plate, where the X and Y axes are set along the length and the width directions, respectively, while the Z-axis is set along the thickness direction pointing downward.The plate is rested on an elastic substrate that is idealized as a Pasternak-type foundation model with two stiffnesses, where K ¯1 denotes the vertical spring stiffness and K ¯2 denotes the shearing layer stiffness.
The plate is exposed to elevated temperature and is subjected to uniaxial tensile load.Based on the framework of HSDT of Reddy [30] and that coupled with the von Kármán nonlinear strain-displacement relationships, the governing equations for the postbuckling of the nanocomposite laminated plate with in-plane NPR are given by Shen [1] where W ¯is the plate displacement in the Z direction, W ¯⁎ is the initial geometric imperfection, F ¯is the stress function defined by , and Ψ ¯x andΨ ¯y are two rotations with respect to the Y and X axes.L ˜ij ( ) represent the linear operators [1], and L ˜( ) representing the nonlinear operator involving geometric nonlinearity in the von Kármán sense is expressed by Meanwhile, the interaction of the plate foundation defined by along with the thermal effect is included in equations (1a)-(1d).The thermal forces, the thermal moments, and the higher-order thermal moments (N ¯T , M ¯T , P ¯T ) due to elevated temperature are given by In equation (3a), T Δ = T − T 0 denotes the temperature increase with reference to temperature T 0 , and where α 11 and α 22 are the thermal expansion coefficients in the longitudinal and transverse directions of the kth ply, respectively, and Q ¯ij represent the transformed elastic con- stants, defined by where Besides the governing equations (1a)-(1d), for the boundaryvalue problem, we require the boundary conditions for the auxetic laminated plate.In order to derive the theoretical solution, the four edges of the auxetic laminated plate are assumed to be SSSS (i.e., simply supported on all four edges).Specifically, on the X = 0 and X = a edges, and on the Y = 0 and Y = b edges, where M ¯x and M ¯y are the bending moments, and P ¯x and P ¯y are the higher-order moments, as given in the study of Reddy [30].
Tension buckling and postbuckling of nanocomposite laminated plates  3 The tensile loads are applied on the X = 0 and X = a edges and the two loaded edges are freely movable (i.e., the displacement can move in the X-direction), while the other two unloaded edges are immovable (i.e., the displacement is constrained in the Y-direction).The in-plane boundary condition on the Y = 0 and Y = b edges is where V ¯is the displacement of the plate in the Y direction, and equation ( 7) can be expressed in an average sense as Although the governing equations (1a)-(1d) have the same forms for the compressive buckling and tension buckling problems, unlike the compressive buckling problem [25,26], in the current study, the equilibrium of force in the X-direction is expressed as where σ x denotes the average tensile stress at X = 0 and X = a edges.
In equation (8b), the reduced stiffness coefficients are contained in the reduced stiffness matrices, including where the plate stiffness coefficients A ij , B ij , etc., are expressed as .
3 Solution procedure In order to solve the buckling and postbuckling problems of nanocomposite structures analytically, the Ritz and Galerkin methods are usually employed [31][32][33][34][35][36][37][38].The accuracy of applying the Ritz and Galerkin methods depends strongly on the chosen modal shape functions.Shen [39] developed a two-step perturbation approach that gives explicit analytical expressions of all the variables in the large deflection region.The advantage of this method is that it is unnecessary to guess the form of the modal shape function, which can be obtained step by step, and such solutions satisfy both the governing equations and the boundary conditions accurately in the asymptotic sense.This approach has been applied to successfully solve various plate nonlinear boundary-value problems by other research teams [40][41][42][43][44][45][46][47][48][49][50].To employ this approach for solving the postbuckling problem of auxetic laminated plates with in-plane NPR, the governing equations (1a)-(1d) are first converted into the non-dimensional forms as x y T 11 12 13 14 14 16 where the non-dimensional operators L ij ( ) and L( ) are given in the study of Shen [1].Note that the operators L 15 ( ), L 25 ( ), L 35 ( ), and L 45 ( ) vanish due to the uniform temperature field.The dimensionless parameters are defined by where k 1 and k 2 are only utilized for numerical examples in Section 4, E 0 is the reference value of Young's modulus for the single-crystal copper matrix at T = 300 K, and is set to be E 0 = 101.14GPa [22].A x T , D x T , F x T , etc., are given by Accordingly, the plate boundary conditions on the x = 0 and x = π edges can be nondimensionalized as and those on the y = 0 and y = π edges can be nondimensionalized as And, hence, the in-plane boundary condition for the plate becomes where γ 230 and γ 526 are defined in the study of Shen [1].
For the auxetic laminated plate subjected to uniaxial tension, equation (9) becomes The initial geometric imperfection of the auxetic laminated plate in the dimensionless form is given by where ε is the small perturbation parameter and μ is the imperfection parameter.By employing the two-step perturbation approach, the asymptotic solutions of equations (12a)-(12d) along with the boundary conditions (15a) and (15b) are obtained as follows: Note that to restrict the lateral expansion of the auxetic laminated plate subjected to uniaxial tensile loads, compressive stresses are produced in two unloaded edges that are immovable, so that It is worth noting that solution ( 22) is different from that obtained for the compressive postbuckling problem, as reported in the study of Shen et al. [25].Thereafter, by substituting equation (22) into equation (17), one has Similarly, we have Tension buckling and postbuckling of nanocomposite laminated plates  5 From perturbation procedure, we can obtain the and substituting W, Ψ x , Ψ y , and F into equation ( 16) yields From equations ( 25) and ( 26), we obtain where (A ε 11 1 ( ) ) is treated as the second perturbation parameter and is related to the non-dimensional plate deflection W m .From equation (19), one has In the current study, buckling is caused by compressed stress on the unloaded edges.By minimizing the compressive stress in equation ( 27) with respect to m and n, the buckling load of a perfect plate can readily be numerically obtained by setting = 0 (or W h ¯/ ⁎ = 0), while taking W m = 0 (or W h ¯/ = 0).The corresponding buckling mode (m, n) is obtained simultaneously, which determines the number of half-waves in the X and Y directions, respectively.
After the buckling mode (m, n) and buckling loads are determined, the postbuckling tensile load-deflection curve can be obtained as All symbols used in equations ( 27)-( 29) are explained in Appendix.

Numerical results
In this section, we will present numerical studies for tension buckling and postbuckling behavior of perfect and imperfect auxetic nanocomposite laminated plates with in-plane NPR that are rested on elastic foundations.Two types of nanocomposite materials are selected for each ply of the laminated plate.One is the graphene/Cu composite with the graphene weight fraction w G = 3%, while another is the CNT/Cu composite with the CNT weight fraction w CNT = 3%.The material properties of the two composites are both temperature-dependent and are taken from the molecular dynamics simulation results of Fan et al. [21] and Zhang et al. [22], respectively, which are listed in Tables 1  and 2, respectively.
In the current research, symmetric (0/90/0) S and antisymmetric (10/−10) 3T laminated rectangular plates are considered.The thickness of each ply is identical and the total thickness of the plate is h = 1.2 mm.The plate aspect ratios are selected as a/b = 2, 3, 4, and 5. To avoid cases where stresses increase beyond the elastic range, the plate widthto-thickness ratio is set as b/h = 50, 100, and 200.The in-plane effective Poisson's ratio (EPR) ν e 12 of these nanocomposite laminated plates are determined by employing the method as reported in the study of Shen et al. [51], and listed in Table 3. From Table 3, it can be seen that the absolute value of NPR of (0/90/0) S CNT/Cu laminated plate is the largest, whereas that of the (10/−10) 3T graphene/Cu laminated plate is the smallest at all three temperature levels considered.
The buckling load is of practical concern of the nanocomposite laminated plates and, therefore, we need to determine the buckling tensile load and the corresponding buckling mode for the auxetic nanocomposite laminated plates first.Since the tension buckling of the auxetic laminated plates is investigated for the first time, no experimental data are currently available for model validation.
In order to validate the accuracy and reliability of the present solution method, the finite element analysis is performed and the results are depicted in Figure 2. The buckling load associated with the buckling mode agrees well in each case, which clearly shows the validity and accuracy of the current solution for the tension buckling analysis of auxetic laminated plates.Buckling tensile loads P cr (kN) of the (0/90/0) S and (10/−10) 3T graphene/Cu laminated plates and (0/90/0) S CNT/Cu laminated plates, with different values of plate aspect ratios (a/b = 2, 3, and 4) rested on the elastic foundations under temperature conditions are presented in Tables 4-6.The thermal environments are set as T = 300, 500, and 700 K. Two foundation models with (k 1 , k 2 ) = (10, 0) for the Winkler foundation and (k 1 , k 2 ) = (10, 1) for the Pasternak foundation are considered.Owing to the NPR effect, the applied tensile load acting in the longitudinal direction produces a compressive force and, hence, compressive stress, when the transverse displacement is constrained.Buckling occurs when the compressive stress reaches a critical value.As can be observed, the buckling tensile loads for the (10/−10) 3T graphene/Cu laminated plates are larger than those of the (0/90/0) S graphene/Cu laminated plates, whereas the buckling tensile loads of the (0/90/0) S CNT/Cu laminated plates are lower than those of the (0/90/0) S graphene/Cu laminated plates, even if the (0/90/ 0) S CNT/Cu laminated plate is relatively thicker (i.e., lower b/h) than the (0/90/0) S graphene/Cu laminated plate.This is because the (10/−10) 3T graphene/Cu laminated plate has the lowest  Tension buckling and postbuckling of nanocomposite laminated plates  7 negative EPR, while the (0/90/0) S CNT/Cu laminated plate has the highest negative EPR among the three.In other words, a higher negative EPR will exacerbate the compressive stress caused by the reaction force, and hence, make the composites more prone to buckle.We also observe that the buckling mode (m, n) changes from (1, 3) to (1, 2) for the (0/90/0) S CNT/Cu and graphene/Cu laminated plates, whereas the buckling mode (m, n) changes from (1,4) to (1,2) for the (10/−10) 3T graphene/Cu laminated plates when the plate aspect ratio a/b changes from 2 to 4. The changes in the buckling mode can also be observed when the temperature increases from 300 to 700 K.These simulation results indicate that changing the aspect ratio of the plates and applying the temperature condition significantly affect the distribution of the compressive stress along the unloaded edges, thereby resulting in shifts in the buckling mode.Additionally, similar to cases of compressive buckling [25], the buckling tensile loads decrease when the temperature increases and increase when the foundation stiffnesses are increased.
Figure 3 shows the effect of plate aspect ratio a/b ( = 3, 4 and 5) on the postbuckling behavior of (0/90/0) S and (±10) 3T graphene/Cu and (0/90/0) S CNT/Cu laminated plates under tension at T = 300 K.The depicted curves in Figure 3 indicate that increasing the plate aspect ratio significantly reduces the critical tension buckling load as well as the peak of the postbuckling load-deflection curves.These simulation results suggest that in the postbuckling range, the increase of the amplitude of the plate deflection will counteract the lateral expansion displacement of the auxetic plate.The stress on the unloaded edges changes from compressive stress to tensile stress when the amplitude of the plate deflection reaches a certain value, which leads the tensile load-deflection curve to switch from rising to falling at the peak point.The postbuckling curves of tensile load vs deflection for the imperfect auxetic laminated plates are also plotted in Figure 3, where W h ¯/ ⁎ = 0.1 (or 0.05) denotes the non-dimensional maximum initial geometric imperfection of the auxetic laminated plates.
The effect of plate width-to-thickness ratio on the postbuckling behavior of the three configurations of auxetic laminated plates with a/b = 3 under tension at 300 K is illustrated in Figure 4. Three cases of b/h = 50, 100, and 200 are chosen for the (0/90/0) S CNT/Cu and graphene/Cu laminated plates, while b/h is set as 100, 150, and 200 for the (10/−10) 3T graphene/Cu laminated plates.Such choices of the b/h ratios for the two types of laminates have ensured that the compressive or tensile stresses stay within the    The effect of temperature change on the postbuckling behavior of three configurations of auxetic laminated plates with a/b = 3 is depicted in Figure 5. Three thermal environmental conditions, T = 300, 500, and 700 K, are considered.The temperature effect is included in the simulations by using the temperature-dependent material properties of both graphene/Cu and CNT/Cu composites.As the temperature increases, the elastic moduli reduce and the strength degrades for nanocomposites.As a result, the postbuckling tensile load-deflection curves are decreased as the temperature increases.Note that the initial geometric imperfection is set as W h ¯/ ⁎ = 0.05 in this example.Our results show that the (10/−10) 3T graphene/Cu laminated plate has no buckling mode change.Additionally, the postbuckling tensile load-deflection curve becomes extremely low at T = 700 K compared to the case at T = 300 K.
Figure 6 illustrates the effect of foundation stiffnesses on the postbuckling behavior of three configurations of auxetic laminated plates with a/b = 3 rested on elastic foundations at T = 300 K.In this example, two foundation models with (k 1 , k 2 ) = (50, 0) for the Winkler foundation and (k 1 , k 2 ) = (50, 5) for the Pasternak foundation are considered.Similar to conventional observations in laminated plates [25], the postbuckling curves of tensile load vs deflection increase with higher foundation stiffnesses.This implies that the tensile buckling could be mitigated by using a stiffer foundation as a substrate.Furthermore, no change in the buckling mode is observed when the auxetic laminated plate is supported by either the Winkler foundation or the Pasternak foundation.

Conclusion
Tension buckling and postbuckling phenomena of auxetic nanocomposite laminated plates with immovable unloaded edges are presented.The material properties of the two nanocomposites considered in this study, CNT/Cu and graphene/Cu composites, are both dependent on temperature.Based on the two-step perturbation method, the explicit analytical solutions are obtained for the first time.The numerical studies have been presented for three configurations of laminated plates, including (0/90/0) S and (10/−10) 3T graphene/Cu laminated plates and (0/90/0) S CNT/Cu laminated plates, with or without an elastic substrate under temperature environments.The buckling tensile loads and associated buckling modes have been verified by FE simulation.The results presented explore for the first time the important issue of auxetic nanocomposite laminated plates, and thus are greatly useful in the engineering design of the novel nanocomposite structures.The findings are summarized as follows: 1) Due to the impact of in-plane NPR, the buckling tensile loads exist for the auxetic laminated rectangular plates with unloaded edges that are "immovable" under uniaxial tension.The tensile loads on "movable" ends lead to compressive stress on "immovable" ends, which is exactly the origin of tension buckling and postbuckling.2) Unlike the traditional compressive postbuckling case where the postbuckling load-deflection curve rises slowly, in the tension postbuckling case, the postbuckling tensile load-deflection curve rises rapidly, in particular for cases when the temperature variation is under consideration.3) In the postbuckling range, the increase of the amplitude of the plate deflection will counteract the lateral expansion displacement of the auxetic plate.The stress on the unloaded edges may shift from compressive stress to tensile stress when the amplitude of the plate deflection reaches a certain value.In most cases, a peak point exists on the postbuckling tensile load-deflection curves.

(A2)
Tension buckling and postbuckling of nanocomposite laminated plates  13 and S ¯T in equations (1c) and (1d) are given by