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Research Papers

# Buckling and Vibration of Circular Auxetic PlatesPUBLIC ACCESS

[+] Author and Article Information
Teik-Cheng Lim

School of Science and Technology,
SIM University,
S599491, Singapore
e-mail: alan_tc_lim@yahoo.com

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received December 18, 2013; final manuscript received January 27, 2014; published online February 19, 2014. Assoc. Editor: Irene Beyerlein.

J. Eng. Mater. Technol 136(2), 021007 (Feb 19, 2014) (6 pages) Paper No: MATS-13-1235; doi: 10.1115/1.4026617 History: Received December 18, 2013; Revised January 27, 2014

## Abstract

This paper evaluates the elastic stability and vibration characteristics of circular plates made from auxetic materials. By solving the general solutions for buckling and vibration of circular plates under various boundary conditions, the critical buckling load factors and fundamental frequencies of circular plates, within the scope of the first axisymmetric modes, were obtained for the entire range of Poisson's ratio for isotropic solids, i.e., from −1 to 0.5. Results for elastic stability reveal that as the Poisson's ratio of the plate becomes more negative, the critical bucking load gradually reduces. In the case of vibration, the decrease in Poisson's ratio not only decreases the fundamental frequency, but the decrease becomes very rapid as the Poisson's ratio approaches its lower limit. For both buckling and vibration, the plate's Poisson's ratio has no effect if the edge is fully clamped. The results obtained herein suggest that auxetic materials can be employed for attaining static and dynamic properties which are not common in plates made from conventional materials. Based on the exact results, empirical models were generated for design purposes so that both the critical buckling load factors and the frequency parameters can be conveniently obtained without calculating the Bessel functions.

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## Introduction

Auxetic solids are materials that possess negative Poisson's ratio. Such materials expand transversely when stretched axially, and contract transversely when axially compressed. Auxetic materials have been initially observed in ferromagnetic films [1], face centered cubic crystals [2], hard cyclic hexamers [3,4], and foams [5-8]. Due to their counter-intuitive properties, auxetic materials have been investigated as smart materials for potential applications including cushion materials [9], stents [10,11], pressure vessels [12], sensors [13], morphing airfoils [14,15], smart folding structures [16], smart metamaterials [17], aeroengine fan blades [18], and vibration dampers [19], to name a few. Arising from their unique properties, the mechanical performance of auxetic solids has been investigated [20-33], including their elastic stabilities [34-36]. Additionally, investigations in the dynamic behavior of auxetic solids and structures have also been performed [37-45]. Although a comprehensive set of available results for the frequencies and mode shapes of free vibration of plates have been provided [46], this information are based on limited range of Poisson's ratio, typically between 0.25 and 0.333. Similarly, established information on the buckling behavior of circular plates is presently confined to conventional Poisson's ratio, especially at $v=0.3$. This paper establishes information on the buckling load and fundamental frequencies of circular plates with Poisson's ratio in the range $-1≤v≤0.5$, so as to allow a comparison to be made between conventional and auxetic circular plates. This paper is therefore divided into two parts, i.e., (a) buckling and (b) vibration of circular auxetic plates. These two seemingly unrelated modes of deformation are consolidated herein on the basis of their overlapping similarities in analysis, particularly in the use of Bessel functions for extracting the critical bucking load and the natural frequencies of circular plates.

## Analysis

###### Preliminaries.

It is expedient to use Bessel functions when dealing with the elastic stability and vibration frequencies of circular plates. To pave a way for understanding the effect of material auxeticity on circular plates, recall that the Bessel function of the first kind of order $n$ is defined as Display Formula

(1)$Jn(x)=∑m=0∞(-1)mm!(m+n)!(x2)2m+n$

while the modified Bessel function of the first kind of order $n$ is defined as Display Formula

(2)$In(x)=∑m=0∞1m!(m+n)!(x2)2m+n$

for an argument $x$. The Bessel functions of the first kind of orders zero and one are used for determining the critical buckling load, while both the Bessel function and its modified form are used for determining the frequency parameters. At orders $n=0$ and $n=1$, it will be shown later that sufficient accuracy can be obtained by using the first eight terms of the Bessel functions, i.e., Display Formula

(3)$J0(x)=1-(x2)2+14(x2)4-136(x2)6+1576(x2)8 -114400(x2)10+1518400(x2)12-125401600(x2)14$
Display Formula
(4)$J1(x)=x2-12(x2)3+112(x2)5-1144(x2)7+12880(x2)9 -186400(x2)11+13628800(x2)13-1203212800(x2)15$
Display Formula
(5)$I0(x)=1+(x2)2+14(x2)4+136(x2)6+1576(x2)8 +114400(x2)10+1518400(x2)12+125401600(x2)14$
Display Formula
(6)$I1(x)=x2+12(x2)3+112(x2)5+1144(x2)7+12880(x2)9 +186400(x2)11+13628800(x2)13+1203212800(x2)15$

Physically, $n$ represents the number of nodal diameters. The flexural rigidity of the circular plate of radius $R$ is described as Display Formula

(7)$D=Eh312(1-v2)$

where $h$, $E$, and $v$ refer to the plate's thickness, Young's modulus and Poisson's ratio, respectively.

###### Effect of Negative Poisson's Ratio on Critical Buckling Load.

Figure 1(a) depicts a circular plate under the action of in-plane radial load $N$ acting at the plate rim toward the plate center, while Fig. 1(b) shows the various boundary conditions considered herein. The general solution for obtaining the critical buckling load, $Ncr$, of a circular plate with rotational stiffness at the rim, as shown in Fig. 1(b) (bottom) is given as [47,48]

where Display Formula

(9)$α=ND$

(10)$β=kRD$

in which $k$ is the rotational stiffness. The smallest root for Eq. (8), i.e., $αR$, gives the critical buckling load, which is normally expressed in the form Display Formula

(11)$Ncr=(αR)2DR2$

The term $(αR)2$ is also known as the critical buckling load factor, $N¯$, as defined by Display Formula

(12)$N¯=NcrR2D=(αR)2$

By incorporating a rotational stiffness, the general solution includes cases of fully clamped edge ($β→∞$) and simply supported (SS) edge without rotational stiffness ($β=0$). To date all computed data of $N¯$ are based on positive Poisson's ratio, normally within the range $1/4≤v≤1/3$, with the value corresponding to $v=0.3$ being the most widely computed. An exception is in the case of clamped edge, which gives a constant value of $N¯$ independent from the Poisson's ratio. Since the Poisson's ratio for isotropic solids falls within the range $-1≤v≤0.5$, a set of $N¯$ values have been computed to include the auxetic region. As a way for verification, the results of $N¯$ were initially computed at $v=0.3$ for comparison with the corresponding values by Reddy [49], as furnished in Table 1.

Having verified against the critical buckling load factor by Reddy [49] for $v=0.3$ using the first eight terms of the Bessel functions as described in Eqs. (3) and (4), the numerical results of $N¯$ for the entire range of Poisson's ratio were similarly computed and furnished in Table 2. Figure 2 shows the critical buckling load factor plotted against the Poisson's ratio for various extent of rotational restraint.

Although both the rotational restraint and Poisson's ratio exert equal effect in influencing the critical buckling load factor, as obviously described in Eq. (8), the fact that the range of Poisson's ratio for isotropic solids being very much limited ($-1≤v≤0.5$) in comparison to the range of rotational restraint ($0≤β≤∞$) means that in practice the Poisson's ratio plays a secondary role to the rotational restraint. Here we observe that circular plates that possess negative Poisson's ratio have lower elastic stability due to their lower buckling load, especially for simply supported boundary condition. On the other hand, it was recently shown that simply supported plates that are laterally loaded with uniform load and central pint load encounter the least bending stress when the plate material is auxetic at $v=-1/3$ and $v=-1$, respectively [50]. The advantage and disadvantage in the use of auxetic circular plates for withstanding combined lateral load and axisymmetric buckling load, respectively, must be considered in the mechanical design of circular plates made from auxetic materials. For the convenience of design engineers, the critical buckling load factors that were obtained herein by the exact method were surface fitted to give an empirical model Display Formula

(13)$N¯=n¯0+n¯1v-n¯2v2$

where Display Formula

(14)${n¯0n¯1n¯2}=[5.784212.2889.2057-0.56763.45020.89683.33732.98021.77142.86830.35561.14540.96590.42260.5476] ×{+(tan-1(β))4-(tan-1(β))3+(tan-1(β))2-(tan-1(β))1+(tan-1(β))0}$

Comparison between the exact and empirical models for the critical buckling load factors is furnished in Fig. 3. The good approximation by the empirical model suggests its possible use in the design consideration of circular auxetic plates that are subjected to combined lateral and axisymmetric buckling loads.

###### Effect of Negative Poisson's Ratio on Fundamental Frequencies.

The frequency equations for circular plates are [49,49,51-52] Display Formula

(15)$Jn+1(λ)Jn(λ)+In+1(λ)In(λ)=0$

for clamped edge Display Formula

(16)$Jn+1(λ)Jn(λ)+In+1(λ)In(λ)=2λ1-v$

for simply supported edge, and Display Formula

(17)$λ2Jn(λ)+(1-v)[λJn'(λ)-n2Jn(λ)]λ2In(λ)-(1-v)[λIn'(λ)-n2In(λ)]=λ3Jn'(λ)+(1-v)n2[λJn'(λ)-Jn(λ)]λ3In'(λ)-(1-v)n2[λIn'(λ)-In(λ)]$

(18)$Jn'(λ)=nλJn(λ)-Jn+1(λ)$

(19)$In'(λ)=nλIn(λ)+In+1(λ)$

The eigenvalue $λ$ is the frequency parameter given by Display Formula

(20)$λ2=R2ωρhD$

where $ρ$ is the density of the plate material while $ω$ is the circular frequency of the plate vibration. The effect of negative Poisson's ratio is herein investigated for the lowest frequency that gives axisymmetric deformation mode. In the case of fully clamped and simply supported edges, $n=0$ since $n$ refers to the number of nodal diameters. For circular plates with free edge, the fundamental frequency takes place at $n=2$ without nodal circles. For consistency in comparing axisymmetric mode shapes, the choice of $n=0$ is made for circular plates with free edge to represent the lowest frequency with axisymmetric deformation. Hence substitution of $n=0$ into Eqs. (15)–(17) leads to Display Formula

(21)$J1(λ)J0(λ)+I1(λ)I0(λ)=0$

for clamped plate Display Formula

(22)$J1(λ)J0(λ)+I1(λ)I0(λ)-2(λ1-v)=0$

for simply supported plate, and Display Formula

(23)$J0(λ)J1(λ)+I0(λ)I1(λ)-2(1-vλ)=0$

for free plate. As a means of verification, Eqs. (21)–(23) were used for obtaining values of $λ2$ within $0.25≤v≤0.333$ for comparing with earlier computed results [52-59]. The numerical scheme for solving Eqs. (21)–(23) involves, for each value of Poisson's ratio, a value of $λ$ being initially assigned zero and gradually incremented by 1 until such as extent when there is a change in the sign, in which the increment is then lowered to 0.1. The procedure continues such that the increment is decreased by an order each time a change in sign is encountered. The numerical search stops once the magnitude of Eqs. (21)–(23) attains an order of $10-9$ or lower, and the corresponding value of $λ$ is accepted. Having demonstrated that the use of the first eight terms of the Bessel functions, as described in Eqs. (3)–(6), enable Eqs. (21)–(23) to agree well with past results shown in Table 3, especially with the most recent and reliable results [53,54], further computation of $λ2$ values was performed for the full scale of Poisson's ratio $-1≤v≤0.5$. The numerical and schematic results are furnished in Table 4 and Fig. 4Fig. 4

Plots of $λ2$ versus Poisson's ratio of circular plates for clamped edge (C), free edge (F), and SS edge

, respectively.

The fact that $λ2$ is constant for fully clamped circular plates is obvious since Eq. (15) is independent of the plate's Poisson's ratio. Almost all available data of $λ2$ parameter for simply supported and free boundary conditions are based on $v=0.3$. Considering the fact that the Poisson's ratio for most materials fall within the range $0.2, the use of $λ2$ at $v=0.3$ means that the percentage error for $λ2$ is less than $±3%$ for $0.2≤v≤0.4$. Even if the $λ2$ parameter used is based on $v=0.25$, the percentage error is less than $±10%$ for the entire range of conventional isotropic materials ($0≤v≤0.5$). These approximations are not valid for auxetic plates due to the rapid drop in the fundamental frequency as the Poisson's ratio approaches the lower limit. This means that, when dealing with auxetic plates with nonclamped boundary conditions, the $λ2$ parameter must be calculated for the corresponding Poisson's ratio. Perusal to Fig. 4 further implies that the natural vibration frequencies of plates can be effectively controlled by selecting appropriate Poisson's ratio. Reference to Fig. 4 also indicates that while the fundamental frequency of the free circular plate is closer to that of the clamped plate at highly positive Poisson's ratio, the former is closer to the fundamental frequency of simply supported plate at highly negative Poisson's ratio. Specifically, the fundamental frequency of the free plate is closer to the clamped plate and simply supported plate when $v≥(-0.37)$ and $v<(-0.37)$, respectively.

As with Eqs. (13) and (14) in which surface-fitting was performed to obtain an empirical model for the critical buckling load factor, curve-fitting was performed for obtaining an empirical model Display Formula

(24)$λ2=a0+a1v+a2v2+a3v3$

where the coefficients $ai(i=0,1,2,3)$ are listed in Table 5. The applicability of the empirical model for $λ2$ is attested in Fig. 5Fig. 5

Comparison between exact (circles) and empirical (curves) results of $λ2$ versus Poisson's ratio of simply supported and free circular plates

, which indicates very good agreement with the exact model, thereby suggesting its usefulness for the convenience of the design engineer. The case of fully clamped edge is not shown in Fig. 5 for obvious reason, but whose sole coefficient is nevertheless listed in Table 5 for completeness' sake.

## Conclusions

The critical buckling load factors and fundamental vibration frequencies for the first axisymmetric mode of circular plates have been calculated and tabulated herein, with special emphasis on the effect of negative Poisson's ratio of the mechanical response. The critical buckling load and the $λ2$ parameter are independent of the plate's Poisson's ratio only when the edge is fully clamped. In other boundary conditions, both the buckling load and fundamental frequency reduce as the plate's Poisson's ratio becomes more negative. This is especially so for the vibration frequency in which the $λ2$ parameter for nonclamped boundary conditions reduces sharply as the Poisson's ratio becomes highly negative. The generated results suggest the Poisson's ratio of a plate can be selected as a way to obtain certain desired mechanical properties, such as to suppress vibration frequency of a plate. Empirical modeling has been performed in order to obtain a set of critical buckling load factors $N¯$ and the $λ2$ parameters in the form of design equation so that these parameters can be calculated without the need to compute the corresponding Bessel functions.

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## References

Popereka, M. Y. A., and Balagurov, V. G., 1969, “Ferromagnetic Films Having a Negative Poisson Ratio,” Fizika Tverdogo Tela, 11(12), pp. 3507–3513.
Milstein, F., and Huang, K., 1979, “Existence of a Negative Poisson Ratio in FCC Crystals,” Phys. Rev. B, 19(4), pp. 2030–2033.
Wojciechowski, K. W., 1989, “Two-Dimensional Isotropic System With a Negative Poisson Ratio,” Phys. Lett. A, 137(1&2), pp. 60–64.
Wojciechowski, K. W., and Branka, A. C., 1989, “Negative Poisson's Ratio in Isotropic Solids,” Phys. Rev. A, 40(12), pp. 7222–7225. [PubMed]
Lakes, R., 1987, “Foam Structures With Negative Poisson's Ratio,” Science, 235(4792), pp. 1038–1040. [PubMed]
Lakes, R., 1987, “Negative Poisson's Ratio Materials,” Science, 238(4826), pp. 551. [PubMed]
CaddockB. D., and Evans, K. E., 1989, “Microporous Materials With Negative Poisson's Ratios. I. Microstructure and Mechanical Properties,” J. Phys. D: Appl. Phys., 22(12), pp.1877–1882.
Evans, K. E., and Caddock, B. D., 1989, “Microporous Materials With Negative Poisson's Ratios. II. Mechanisms and Interpretation,” J. Phys. D: Appl. Phys., 22(12), pp. 1883–1887.
Wang, Y. C., and Lakes, R. S., 2002, “Analytical Parametric Analysis of the Contact Problem of Human Buttocks and Negative Poisson's Ratio Foam Cushions,” Int. J. Solids Struct., 39(18), pp. 4825–4838.
Dolla, W. J. S., Fricke, B. A., and Becker, B. R., 2007, “Structural and Drug Diffusion Models of Conventional and Auxetic Drug-Eluting Stents,” ASME J. Med. Dev., 1(1), pp. 47–55.
Tan, T. W., Douglas, G. R., Bond, T., and Phani, A. S., 2011, “Compliance and Longitudinal Strain of Cardiovascular Stents: Influence of Cell Geometry,” ASME J. Med. Dev., 5(4), p. 041002.
Ieşan, D., 2011, “Pressure Vessel Problem for Chiral Elastic Tubes,” Int. J. Eng. Sci., 49(5), pp. 411–419.
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## Figures

Fig. 1

Schematics for buckling of a circular plate: (a) top and side views and (b) boundary conditions

Fig. 2

A family of critical buckling load factor plotted against the Poisson's ratio of circular plates for various rotational restraints

Fig. 3

Critical buckling load factor versus Poisson's ratio of circular plates for various rotational restraints: exact solution (circles) and empirical model (curves)

Fig. 4

Plots of λ2 versus Poisson's ratio of circular plates for clamped edge (C), free edge (F), and SS edge

Fig. 5

Comparison between exact (circles) and empirical (curves) results of λ2 versus Poisson's ratio of simply supported and free circular plates

## Tables

Table 1 Verification of computed critical buckling load factor at $v=0.3$ [49]
Table 2 List of computed critical buckling load factor for the entire range of rotational restraint and Poisson's ratio for isotropic solids
Table 3 Verification of $λ2$ values with earlier investigations on conventional plates
Table 4 Values of $λ2$ in auxetic and conventional regions
Table 5 Coefficients for empirical model of $λ2$

## Errata

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