Enhancing buckling capacity of a rectangular plate under uniaxial compression by utilizing an auxetic material

Zhang Yongcun,Li Xiaobin,Liu Shutian

State Key Laboratory of Structural Analysis for Industrial Equipment,Department of Engineering Mechanics,Dalian University of Technology,Dalian 116024,China

Enhancing buckling capacity of a rectangular plate under uniaxial compression by utilizing an auxetic material

Zhang Yongcun*,Li Xiaobin,Liu Shutian

State Key Laboratory of Structural Analysis for Industrial Equipment,Department of Engineering Mechanics,Dalian University of Technology,Dalian 116024,China

Auxetic materials have previously been shown to enhance various performances due to its unusual property of becoming fatter when uniaxially stretched and thinner when uniaxially compressed(i.e.,the materials exhibit a negative Poisson’s ratio).The current study focuses on assessing the potential of an auxetic material to enhance the buckling capacity of a rectangular plate under uniaxial compression.The in-plane translational restraint along the unloaded edges that was of ten neglected in open literature is taken into consideration in our buckling model proposed in this study.The closed- form expressions for the critical buckling coefficient of the rectangle are provided and the predicted results agree well with those determined by the finite element method.Furthermore,the results indicate that the buckling performance of a rectangular plate under uniaxial compression can be significantly improved by replacing the traditional material that has a positive Poisson’s ratio with an auxetic material when there is in-plane translation restraint along the unloaded edges.

1.Introduction

The Poisson’s ratio defines the ratio between the transverse and axial strain in a load material.1Moreover,it is considered as an important material parameter that directly affects the mechanical properties of a structure.Most materials have Poisson’s ratio values that range between 0 and 0.5;however,some materials,known as auxetic materials,display a negative Poisson’s ratio.Auxetic materials behave contrary to what is expected.For example,when subjected to an axial tensile load,their transverse dimension increases.Furthermore,the counter-intuitive properties of auxetic materials lead to structures that also exhibit enhanced mechanical and other physical performances.Lakes2,3was the first to manufacture a novel foam structure with a negative Poisson’s ratio of -0.7,and presently,the main focuses of this field are discovering new auxetic materials and novel applications for them.

At the present,a large number of auxetic materials have been determined and manufactured.These materials encompass nearly all of the classes of materials,including polymers,composites,metals,and ceramics.For a detailed introduction,review literatures4–7can be examined.Furthermore,although natural auxetic materials exist,most auxetic materials are artificial.In addition,topology optimization can be employed to tailor special auxetic materials according to practical requirements,8,9and the results obtained by topology optimization can be directly manufactured by 3D printing technology(also called additive manufacturing).10,11Another popular topic in this field is the exploration of potential applications for autextic materials.The existing research results reveal that auxetic materials exhibit a higher resistance to indentation,shear resistance,12fracture resistance,13acoustic absorption,14damping,15energy absorption,16a wider band gap with lower frequency,17and so on.

Buckling is a common failure mode in the aerospace structure.How to improve the stability of structure becomes an attractive problem.Recently,the stability of auxetic materials has also received significant attention.One such instance was an investigation by Spadoni et al.18of the buckling behavior of a chiral cellular structure with a negative Poisson’s ratio under flat-wise compression.Additionally,the global buckling behavior of auxetic cellular tubes based on inverted hexagonal honeycombs has been discussed.19,20The results clearly indicate that the use of auxetic structures can significantly improve(or can result in a significant improvement on)the buckling behavior as compared to similar non-auxetic arrangements.Moreover,Lim21–23discussed the potential applications of the auxetic plate and shell,and the buckling behavior of rectangular and circular thick auxetic plates were investigated.From this investigation,a highly accurate shear correction factor in terms of a Poisson’s ratio from-1 to 0.5 was obtained.However,the effect of the unusual deformation mechanism of auxetic materials on buckling behavior remains unexplored.

The purpose of this paper is to enhance the buckling performance of a rectangular plate by replacing the traditional material with a positive Poisson’s ratio with an auxetic material.The structure of this paper is as follows:first,the mechanism of the enhanced buckling for the rectangular auxetic plate is provided in Section 2.Next,the critical buckling coefficient of the rectangular plate elastically restrained against in-plane translation under uniaxial compression is determined in Section 3.Section 4 introduces the results and discussion,and finally,Section 5 gives the conclusion.

2.Mechanism of enhanced buckling performance for rectangular auxetic plate

Fig.1 shows a simply supported rectangular thin plate of dimension a×b under biaxial compressive loads.The magnitude of the compressive load is N0at the edges x=0,a.Likewise,it is γN0at the edges y=0,b.For a rectangular isotropic plate,the buckling load under biaxial loading can be expressed as24

Fig.1 Plate subjected to uniform compression along x and y directions.

where β=a/b is the plate aspect ratio,E the elastic modulus of materials,ν the Poisson’s ratio,t the thickness of the plate,m the number of half waves along the x direction and y direction respectively.If the rotational constraint is full at the edges y=0,b,the buckling load under biaxial loading can be obtained through thefollowing expression

A simply support square thin plate(β=1)is selected as an example to demonstrate the mechanics of enhanced buckling performance for the rectangular auxetic plate.The critical buckling load subjected to a uniform compressive load N0on edges x=0 and a(i.e.when γ=0)can be calculated using Eq.(1)

If the same uniform compressive load(γ=1)is also applied along the y direction,the critical buckling load will decrease by 50%

In contrast,the critical buckling load will increase by 75%when a half uniform tensile load(γ=-0.5)is applied along the y direction

In other words,the tensile load along the x direction is beneficial for improving the critical buckling load for a square plate when the compressive load is applied in the y direction.

Due to the Poisson’s ratio effect,the plate becomes fatter when a material with a positive Poisson’s ratio is used and thinner when a material with a negative Poisson’s ratio is used.If the unloaded edges were to be subjected to elastic restraint against in-plane translation,then the induced equivalent load along the unloaded direction will be compressive for the material with a positive Poisson’s ratio and will be tensile for the plate with a negative Poisson’s ratio.Combined with the previous analysis,it can be predicted that an auxetic plate under uniaxial compression has a higher critical buckling load than one using a positive Poisson’s ratio material when the unloaded edges are subjected to the elastic restraint against in-plane translation.

It is well-known that in practice,the ideal free boundary conditions for simply-supported or clamped plate never occur,and there fore translational restraint exists.Furthermore,while the elastic stability of a rectangular plate with edges that are elastically restrained has been studied by many authors,25–28the influence of Poisson’s ratio on the buckling of elasticallyrestrained plates has not received attention yet.29

3.Buckling of rectangular plate elastically restrained against inplane translation

A simply supported rectangular thin plate elastically restrained against in-plane translation along the unloaded edges is subjected to uniaxial,uniform compressive load Nxas shown in Fig.2.The purpose of this section is to derive the analytical solution of the critical buckling load for the uniaxial compressive plate elastically restrained along the unloaded edges.

In order to obtain the critical buckling load,the induced equivalent load along the unloaded edges caused by the inplane translational restraint should befirstly determined.This is a classic plane stress problem and the strain in the ydirection can be written as

Moreover,the static equilibrium and deflection coordination condition at the elastic support should be satisfied:

where k is the elastic coefficient of support and Δx the de formation of elastic support.The induced load can be solved by combining Eqs.(6)–(8)

α is defined as the in-plane translational restraint coefficient.And the value range of it can be obtained by Eq.(10).When the elastic coefficient of support k is close to zero,the restraint coefficient α also approaches zero.On the other hand,when k tends towards infinity,α equals 1.Hence the restraint coefficient α varies from 0(free expansion condition)to 1(full restrained condition).

Based on the derivation above,the simply supported rectangular plate elastically restrained against in-plane translation that is subjected to uniaxial compressive load can be equivalent to one that is uniform compression along x and uniform tension along y.There fore,its buckling load can be given by substituting Eq.(9)into Eq.(1):

Fig.2 Uniaxial compressive plate elastically restrained along unloaded edges.

The minimum buckling load occurs at n=1 and it is given by

where K is the critical buckling coefficient,given in a generalized form as

When the unloaded edges are clamped,the critical buckling coefficient has been obtained by substituting Eq.(9)into Eq.(2):

4.Results and discussion

Thefinite element method using the commercial program NASTRAN 12.0 is constructed to verify the accuracy of the derived expressions for the buckling coefficient K of 13–14.In the finite element modeling,the geometry of the chosen plate is 100 mm×100 mm and the thickness is 1 mm.The shell element with four nodes is used to discretize it.The element size is 2 mm×2 mm and the total element number is 2500.The elastic modulus of material remains constant 70 GPa when the Poisson’s ratio changes.The Lanczos approach is selected to solve them.The results are shown in Fig.3.

The predicted results obtained by Eqs.(13)and(14)agree well with those obtained by the finite element analysis(FEA).Of particular importance,it can accurately capture the inflection point of the critical buckling coefficient K from one buckling mode to another.These results indicate that the analytical solutions are accurate and are an effective alternative to the computationally expensive FEA and there fore,the results are useful for later discussion.

The rest of this section focuses on the parametric effect on the critical buckling coefficient K.From Eqs.(13)and(14),it can be seen that the critical buckling coefficient K is dependent on the aspect ratio β,the Poisson’s ratio υ,and the in-plane translational restraint coefficient α for both the simplysupported boundary and the clamped boundary conditions.Moreover,it is important to note that K is unrelated to Poisson’s ratio when the restraint coefficient α equals zero.This is completely consistent with the previous analysis in Section 2.Obviously,the buckling coefficient K also is unrelated to the restraint coefficient α if the Poisson’s ratio of the material is zero.As an example,Fig.4 reveals the variation of the critical buckling coefficient of the square plate with the Poisson ratios for different in-plane translational restraint coefficients.The buckling coefficient K remains unchanged when the restraint coefficient α equals zero,and when Poisson’s ratio is zero,the critical buckling coefficient is the same for each of the various restraint coefficients.As a whole,if the in-plane translational restraint exists(α＞ 0),the critical buckling coefficient K gradually decreases as Poisson’s ratio increases.These results imply that the critical buckling coefficient K is a decreasing function of the material’s Poisson’s ratio,and hence the buckling performance of a rectangular plate under uniaxial compression can be enhanced by replacing the traditional material that has a positive Poisson’s ratio with an auxetic material.The existence of in-plane translational restraint is the necessary condition of this conclusion.

Fig.3 Comparison of predicted critical buckling coefficient for the case:β=1,α=1.

Figs.5 and 6 show the variations of the critical buckling coefficient of the square plate under uniaxial compression with in-plane translational restraint coefficients for various Poisson’s ratios.For both the simply-supported boundary and clamped boundary condition,the results demonstrate that the buckling coefficient K with respect to the restraint coefficient α is an increasing function when the material’s Poisson’s ratio is negative,but it is a decreasing function when the material’s Poisson’s ratio of is positive.This finding indicates that an auxetic material is beneficial for the buckling of a rectangular plate under uniaxial compression,and the traditional material with a positive Poisson’s ratio is harmful for the buckling when the in-plane translational restraint is greater than zero.The ideal situation is an in-plane translational restraint of 1 with a Poisson’s ratio that is as close to-1 as possible.In contrast,the worst situation is an in-plane translational restraint of 1 with a Poisson’s ratio that is the maximum value of 0.5.Compared to the following case:ν=0 or α =0,the maximum improvement for the buckling coefficient of the square plate under uniaxial compression is more than double for the simply-supported boundary condition and is nearly 1.5 times for the clamped boundary conditions.At the same time,the maximum reduction is approximately one-third for both of the two boundary conditions.

Fig.5 Variation of critical buckling coefficient with in-plane translational restraint coefficients for various negative Poisson’s ratios and β=1.

Fig.6 Variation of critical buckling coefficient with in-plane translational restraint coefficients for various positive Poisson’s ratios and β=1.

Fig.7 Variation of critical buckling coefficient with aspect ratios for various negative Poisson’s ratios and α =1.

Fig.8 Variation of critical buckling coefficient with aspect ratios for various positive Poisson’s ratios and α=1.

The aspect ratio of the rectangular plate influences the buckling coefficients and the buckling mode shape.Figs.7 and 8 display the variation of the critical buckling coefficient with the aspect ratios for various Poisson’s ratios and α =1.Thesefigures demonstrate that the critical buckling coefficient shows a relatively high variability when the aspect ratio is less than 1.5 and then tends to stability when the aspect ratio is greater than 1.5.Thus,the corresponding enhanced or weakened magnitude of the critical buckling coefficient K of the rectangular plate may be different for various aspect ratios compared to those utilizing a material with a Poisson’s ratio of zero.However,on the whole,the maximal improvement for the buckling coefficient of the rectangular plate under uniaxial compression is nearly double for the simply-support boundary condition and is nearly 1.5 times for the clamped boundary conditions.At the same time,the maximum reduction is approximately one-third for both of the two boundary conditions.This conclusion also implies that the aspect ratio of the rectangular plate has little effect on the enhanced buckling performance from utilizing the auxetic material.

5.Conclusion

The purpose of this paper is to assess the potential of an auxetic material to enhance the buckling capacity of a rectangular plate under uniaxial compression.In this study,the in-plane translational restraint along the unloaded edges that was of ten neglected in open literature is taken into consideration in our buckling model proposed in this study.The closed- form expression for the critical buckling coefficient of the rectangle is provided and its validity has been proven by thefinite element method.Some conclusions are drawn as follows:

The buckling performance of a rectangular plate under uniaxial compression can be enhanced by replacing a traditional material with a positive Poisson’s ratio with an auxetic material.Furthermore,the existence of in-plane translational restraint is the necessary condition of this conclusion.The critical buckling coefficient of the rectangle with respect to the in-plane translational restraint coefficient is an increasing function when the Poisson’s ratio of the material is negative,but it is a decreasing function when the Poisson’s ratio of the material is positive.Compared to those utilizing a material with a Poisson’s ratio of zero,the maximal improvement for the buckling coefficient of the rectangular plate under uniaxial compression can is nearly double for the simply-supported boundary condition and is nearly 1.5 times for the clamped boundary condition.At the same time,the maximum reduction is approximately one-third for both of the two boundary conditions.This study provides a solution for enhancing the buckling performance of the rectangular plate under uniaxial compression.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos.11572071,11332004,),the National Basic Research Program of China (No.2011CB610304),and the Program of Introducing Talents of Discipline to Universities(No.B14013).The financial supports are greatly appreciated.We would also like to thank the China Scholarship Council(No.201308210038).

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Zhang Yongcun is an associate prof essor and M.S.supervisor at Dalian University of Technology,China.He received the PH.D.degree from the same university in 2008.He current research interests are the multifunctional design of materials and structures oriented for thermal protection systems.

Liu Shutian is a prof essor and Ph.D.supervisor at Dalian University of Technology,China.He current research interests are material design,multi-scale analysis and design of composite materials,optimization,topology,sensors,machine design and band-gap materials.

23 June 2015;revised 14 March 2016;accepted 18 April 2016

Available online 22 June 2016

Auxetic material;

Buckling;

Elastically restraint;

Negative Poisson’s ratio;

Optimization

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E-mail addresses:yczhang@dlut.edu.cn(Y.Zhang),stliu@dlut.edu.cn(S.Liu).

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This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).