A study of negative Poisson's ratios in auxetic honeycombs based on a large deflection model

stretchedneycombs

inclinedave been

ratios ofct of the

ding. Allnd expand

r, negative

e specific

pyrolytic82) andratios.exampleslly, Lakesnegativerials that

aterials.ected to

European Journal of Mechanics A/Solids 23 (2004) 95–106

A study of negative Poisson’s ratios in auxetic honeycombsbased on a large deflection model

Hui Wan∗, Hideyuki Ohtaki, Shinya Kotosaka, Guoming Hu

Department of Mechanical Engineering, Faculty of Engineering, Saitama University, 255 Shimo-okubo, Saitama-shi,Saitama-kenn, 338-8570, Japan

Received 25 April 2003; accepted 22 October 2003

Abstract

Materials or structures that contract in the transverse direction under uniaxial compression, or expand laterally whenare called to have negative Poisson’s ratios. A theoretical approach to predict negative Poisson’s ratios of auxetic hohas been developed, which is based on the large deflection model. The equations of the deflection curves of themember of the re-entrant cell, strains and Poisson’s ratios of auxetic honeycombs in two orthogonal directions hderived. The deformed shapes of the inclined member of the re-entrant cell are calculated. The negative Poisson’sauxetic honeycombs are no longer a constant at large deformation. They vary significantly with the strain. The effegeometric parameters of the cell on the Poisson’s ratios is analyzed. 2003 Elsevier SAS. All rights reserved.

Keywords:Honeycomb; Poisson’s ratio; Deflection

1. Introduction

The Poisson’s ratio is defined as the negative ratio of the transverse strain and the axial strain in the direction of loacommon materials have positive Poisson’s ratios, i.e., the materials contract transversely under uniaxial extension, alaterally when compressed in one direction.

A negative Poisson’s ratio, therefore, has been treated as an abnormally elastic parameter for a long time. HowevePoisson’s ratios are theoretically permissible. For an isotropic material, the allowable range of Poisson’s ratio is from−1.0 to0.5, based on thermodynamic consideration of strain energy in the theory of elasticity. For anisotropic materials, theslimits do not apply.

Negative Poisson’s ratios have been observed in some anisotropic materials, such as crystal cadmium (Li, 1976) andgraphite (Garber, 1963). Materials with naturally elaborated construction, like cancellous bone (Williams and Lewis, 19rock with micro-cracks (Homand-Etienne and Houpert, 1989; Nur and Simmons, 1969), also have negative Poisson’s

Fabrication of man-made materials and structures, which exhibit negative Poisson’s ratios, has succeeded. Someare composite laminates, micro-porous polymers, two-dimensional honeycombs and three-dimensional foams. Specia(1987a, 1987b) had described a class of foams that constitutes perhaps the only known isotropic materials withPoisson’s ratios. Moreover, Rothenburg et al. (1991) proposed a general class of microstructure of isotropic mateleads to negative Poisson’s ratios.

All negative Poisson’s ratio materials and structures are termed by Evans et al. (1991) as auxetics or auxetic mAuxetic materials and their negative Poisson’s ratios have not been well understood. Materials of this sort are exp

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

0997-7538/$ – see front matter 2003 Elsevier SAS. All rights reserved.doi:10.1016/j.euromechsol.2003.10.006

96 H. Wan et al. / European Journal of Mechanics A/Solids 23 (2004) 95–106

have interesting mechanical properties, such as high energy absorption, fracture toughness, indentation resistance and enhancedang and

esign andresearch

whoted alongn (1985).oisson’schanical

roach foraluatingmalism’s ratio fors programcalculateTorquato

rren andased onestigated

eometricctures ora, 1991)This kindabsorptionformationhad beenof large

aper, wealculated

l

end.

ed

shear moduli, which may be useful in some applications. Here one can cite the works of Overaker et al. (1998) and WLakes (2002). They studied the application of auxetic material to medical anchors and cushions. Therefore recently, dfabrication of auxetic materials and analysis of their negative Poisson’s ratios have brought about great interest in thecommunity.

The first attempt to studyν, Poisson’s ratio of solid material, belongs to Poisson himself (Timoshenko, 1983),considered a material consisting of randomly packed smooth spherical particles interacting by forces that are directhe line connecting particle centers. Formal theoretical investigation of negative Poisson’s ratios was done by AlmgreWojciechowski and Branka (1989) investigated, both theoretically and by Monte Carlo simulation, the negative Pratio in a two-dimensional molecular system. Schajer and Robertson (1974), and Kolpakov (1985) studied the mebehavior of cellular structures and the averaged moduli of elastic gridworks. Warren and Kraynik (1987) made an appanalytically calculating effective elastic properties of polymeric foams. Wei (1992) presented a theoretical model for eveffective Poisson’s ratio of polymeric networks with special microstructure, which is a complete generalization of the forpresented by Warren et al. (1989). Gibson and Ashby (1997) had developed a model that successfully predicts Poissonmacroscopic honeycombs, assuming small deformation by flexure. Evans (1989, 1990) used a molecular mechanicwhich incorporates a standard valence force field to model the deformation of network microstructure and thus toPoisson’s ratio. Smith et al. (2000) developed a missing rib model to study auxetic behavior of honeycombs and foams.(2000) and Sigmund (2000) studied topology design of two-dimensional material with negative Poisson’s ratio. WaByskov (2002) studied the relationship of three-fold symmetry and mechanical isotropy of two-dimensional materials ba linear elastic micropolar model. Effects of structure geometric parameters on the negative Poisson’s ratio were invby Yang et al. (2003), using a finite element method.

Masters and Evans (1999) showed that materials that exhibit negative Poisson’s ratios do so as a result of gconfiguration of multiphase microstructures. Honeycombs are considered as a basic structure of these microstrugeometric units. An open cell in which the inclined members of the cell protrude inwards was described by Lakes (1987as re-entrant. Making the cell of a conventional hexagonal honeycomb re-entrant produces a negative Poisson’s ratio.of honeycomb is described as auxetic honeycomb here. When honeycombs are to be used as load bearing or energystructures, large deformation often occurs, and non-linear behavior becomes important. This suggests that a large demodel should be more appropriate. The in-plane buckling and non-linear compression of conventional honeycombsanalyzed by Zhang and Ashby (1992), and Zhu and Mills (2000), but these approaches either avoid the calculationdeflection itself, or divide the deformed member into segments and calculate the large deflection by iteration. In this plimit our analysis to the negative Poisson’s ratios of auxetic honeycombs, based on the large deflection model, and cthe large deflection directly by incomplete elliptic integrals.

2. Analysis of negative Poisson’s ratios of auxetic honeycombs by a large deflection model

2.1. Forces and moments on the cell members

For an auxetic honeycomb as shown in Fig. 1, assuming that re-entrant cells have uniform thickness, and thatt/L is small,the relative density given by simple geometry is:

R = ρ

ρS= t/L(H/L+ 2)

2cosλ(H/L − sinλ), (1)

whereR is relative density of the auxetic honeycomb;ρ is density of the auxetic honeycomb, kg/m3; ρs is density of the solidcell material, kg/m3; t is thickness of the cell member, m;L is length of the inclined cell member, m;H is length of the verticacell member, m; andλ is angle between the undeformed inclined member andX-axis, rad.

When the auxetic honeycomb is uni-axially loaded inX or Y direction as shown in Fig. 2, the re-entrant cell members bThe remote stressσX or σY produces forcePX or PY on inclined members of a unit cell, paralleling toX-axis orY -axis.PX andPY are given by Gibson and Ashby (1997):

PX = σX(H − Lsinλ)b, (2)

PY = σYLcosλb, (3)

wherePX , PY are forces applied on the inclined member due toσX andσY respectively, N;σX , σY are remote stresses applion the auxetic honeycomb inX-direction andY -direction respectively, Pa; andb is breadth of the cell member, m.

H. Wan et al. / European Journal of Mechanics A/Solids 23 (2004) 95–106 97

ormationded

Fig. 1. An undeformed auxetic honeycomb with re-entrant cells.

(a) (b)

Fig. 2. Cell deformation by inclined cell member bending. (a) Loaded inX-direction, (b) loaded inY -direction.

By equilibrium of the moment, giving the bending moments at the end of the inclined member as:

MX = PXLsinλ/2, (4)

MY = PYLcosλ/2, (5)

whereMX , MY are bending moments at the end of the inclined member due toσX andσY respectively, N·m.

2.2. Large deflection model

As can be seen from Fig. 2, there is zero bending moment at the mid-point of the inclined member. Then only the defshape of the half member will be analyzed. The moment distribution inOB is the same as if it were a cantilever beam, loa


etress.)

in

nde of therom

t

(a) (b)

Fig. 3. Large deflection model of the half inclined member. (a) Loaded inX-direction, (b) loaded inY -direction. (The solid line represents thdeformation caused by a remote compressive stress, while the dashed line represents the deformation caused by a remote tensile s

at the free end by forcePX acting in theX-direction (see Fig. 3(a)). A curvilinear co-ordinatesS, with origin O, is used todefine position in the bending member. The exact expression for the curvature ofOB is dθ/dS. Since the bending momentOB is equal to the flexural rigidity times the curvature, the differential equation of the deflection curve is:

ESIdθ

dS= −PXY, (6)

whereEs is Young’s modulus of the solid cell member material, Pa;I is second moment of inertia of the cell member aI = bt3/12, m4; θ is bending angle of a general point along a deformed member between the tangent of the shapinclined member and the loading direction, rad; andS is distance of a general point along the curvilinear co-ordinates forigin O, m. And the boundary conditions are:(

ESIdθ

dS

)s=0

= 0, (θ)s=0 = αX, (7)(ESI

dθ

dS

)s=L/2

= MX, (θ)s=L/2 = βX, (8)

whereαX andβX are bending angles between the tangent of the shape of the inclined member andX-axis loading direction aorigin O and endB respectively, rad.

Neglecting the change in length ofOB due to axial compression and differentiating (6) with respect toS and using therelation dY/dS = sinθ , we obtain:

ESId2θ

dS2= −PX sinθ. (9)

In solving (9), we begin by multiplying both sides bydθdS dS and integrating, so that:∫d2θ

dS2dθ

dSdS = −k2

X

∫sinθ

dθ

dSdS, (10)

wherek2X = PX/(ESI). Upon integrating, and using the boundary condition atO, leads to:(dθ

dS

)2= 2k2

X(cosθ − cosαX). (11)

It can be seen from Fig. 3(a) that dθ/dS is always negative, solving for dS gives:

dS = − dθ

kX√

2(cosθ − cosαX). (12)

The total length of the half member, after the limits of integration are interchanged, is:

L

2=

αX∫βX

dθ

kX√

2(cosθ − cosαX)= 1

2kX

αX∫βX

dθ√sin2(αX/2) − sin2(θ/2)

. (13)


Using the notationpX = sin(αX/2) and by introducing a new variableφ in such a manner that:

ure

lined

notedinclined

brizontal

sin(θ/2) = pX sinφ = sin(αX/2)sinφ (14)

it is seen from this relation that the limits of integration in theθ -coordinateθ = αX and θ = βX transform into that in theφ-coordinate, and thus areφ = π/2 and

δX = arcsin

(sin

βX

2

/sin

αX

2

). (15)

Differentiating (14) gives:

dθ = 2pX cosφ dφ√1−p2

X sin2φ

. (16)

Substituting (14) and (16) into (13), we obtain:

L = 2

kX

π/2∫δX

dφ√1− p2

Xsin2φ

= 2

kXF(αX), (17)

whereF(αX) is an incomplete elliptic integral of the first kind, and its value depends on the inclinationsαX andβX of thepointsO andB. The forcePX is related to this elliptic integral by:

PX = k2XESI = 4ESIF

2(αX)

L2. (18)

When the auxetic honeycomb is uniaxially loaded inY -direction (see Fig. 3(b)),PY can be obtained by the same procedof the solution forPX , and is expressed as:

PY = k2YESI = 4ESIF

2(αY )

L2, (19)

where

k2Y = PY

ESIand F(αY ) =

π/2∫δY

dφ√1− p2

Ysin2φ

.

In the above equation, the incomplete integralF(αY ) is related to two parameters:pY = sin(αY /2) and

δY = arcsin

(sin

βY

2

/sin

αY

2

)(20)

when the value ofF(αY ) is evaluated, andαY andβY are bending angles between the tangent of the shape of the incmember andY -axis loading direction at originO and endB respectively, rad.

For a given value ofαX or αY , PX or PY can be calculated through these incomplete elliptic integrals. It should bethat in the re-entrant cell (see Fig. 4), the direction of the forces in inclined members is reversed in contrast with that inmembers of a conventional hexagonal cell, while the forces in vertical members remain unchanged.

2.3. Strain and Poisson’s ratio for loading inX-direction

The horizontal projection dLXX of the element dS of the half memberOB is −dS cosθ when the auxetic honeycombears a remote tensile stress, or is dS cosθ when the auxetic honeycomb bears a remote compressive stress. So the hoprojected distanceLXX(θ) of a general point on the half member alongX-axis due toX-direction loading is:

LXX(θ) = 1

2kX

θ∫βX

±cosθ dθ√sin2(αX/2) − sin2(θ/2)

. (21)

Writing cosθ as 1− 2sin2(θ/2) and using (14) and (16), the above equation can be expressed in terms ofφ:

LXX(θ) = ±1

kX

ξX∫δX

1− 2p2X sin2φ√

1− p2X sin2φ

dφ = ±1

kX

[2

ξX∫δX

√1− p2

X sin2φ dφ −ξX∫

δX

1√1− p2

X sin2φ

dφ

], (22)


signated

th

r

Fig. 4. A re-entrant cell unit formed by triangular prisms.

whereξX = arcsin(sin θ2/sin αX

2 ) andβX � θ � αX.The first part of the right-hand side of this equation is an incomplete elliptic integral of the second kind, and can be de

asE(ξX), so the horizontal projected distance is:

LXX(θ) = ±1

kX

[2E(ξX) −F(ξX)

]. (23)

Letting the upper limit of the integration toπ/2, the total projected length of the half memberOB alongX-axis due toX-direction loading is:

LXX(αX) = ±1

kX

[2E(αX) − F(αX)

]. (24)

The strain is a measure of the change of the projected length of the deformed representative elementEABF relative to theprojected length of the undeformed one, where the representative elementEABF is shown in Fig. 4. As the horizontal lengof the undeformed inclining memberAB is Lcosλ (see Fig. 4), so the strain in the horizontal direction due toσX is:

εXX = 2LXX(αX)− Lcosλ

Lcosλ= ±2[2E(αX) −F(αX)]/kX − Lcosλ

Lcosλ. (25)

The vertical projected distanceLYX(θ) of a general point on the half member along theY -axis can be calculated in a similaway, and is expressed as:

LYX(θ) = 1

2kX

θ∫βX

sinθ dθ√sin2(αX/2) − sin2(θ/2)

. (26)

As sinθ = 2sin(θ/2)cos(θ/2) = 2sin(θ/2)(1− sin2(θ/2))1/2, and expressing the above equation in terms ofφ, we obtain:

LYX(θ) = 2pX

kX

ξX∫δX

sinφ dφ = 2pX

kX(cosδX − cosξX). (27)

Letting ξX = π/2, the total projected length of the half memberOB alongY -axis is

LYX(αX) = 2pX

kXcosδX. (28)

The vertical lengths of the undeformed inclined memberAB and the vertical half membersAE andBF areLsinλ andH

(see Fig. 4), respectively. The strain in the vertical direction depends on the total projected lengths ofAB, AE andBF alongY -axis. As vertical members do not deform in this large deflection model, the strain in the vertical direction due toσX is:

εYX = H − 2LYX(αX) − (H − Lsinλ)

H − Lsinλ= Lsinλ− 4pX cosδX/kX

H − Lsinλ. (29)


The Poisson’s ratio underX-direction loading is calculated as the negative transverse strain divided by the axial strain in the

member,

er

ngth

gative sign

ss

961).

loading direction:

νX = − εYX

εXX= −Lsinλ − 4pX cosδX/kX

H − Lsinλ

/±2[2E(αX)− F(αX)]/kX − Lcosλ

Lcosλ. (30)

2.4. Strain and Poisson’s ratio for loading inY -direction

With a similar approach of the above sub-section, we can calculate the projected distances of the deformed halfand the strains along horizontal and vertical directions, and therefore the Poisson’s ratio for loading inY -direction. The solutionfor the horizontal projected lengthLXY of the half member inX-axis for this loading direction is similar to solution ofLYX ,instead ofLXX, only with minor modification. The horizontal projected distanceLXY (θ) of a general point on the half memband the total horizontal projected lengthLXY (αY ) of the half memberOB alongX-axis due toY -direction loading are:

LXY (θ) = 1

2kY

θ∫βY

sinθ dθ√sin2(αY /2) − sin2(θ/2)

= 2pY

kY(cosδY − cosξY ), (31)

LXY (αY ) = 2pY

kYcosδY , (32)

whereξY = arcsin(sin θ2/sin αY

2 ) andβY � θ � αY .Similarly, the vertical projected distanceLYY (θ) of a general point on the half member and the total vertical projected le

LYY (αY ) of the half memberOB along theY -axis due toY -direction loading are:

LYY (θ) = 1

2kY

θ∫βY

±cosθ dθ√sin2(αY /2)− sin2(θ/2)

= ±1

kY

[2E(ξY )− F(ξY )

], (33)

LYY (αY ) = ±1

kY

[2E(αY )− F(αY )

], (34)

where the positive sign represents that the auxetic honeycomb undergoes a remote tensile stress, while the nerepresents that the auxetic honeycomb undergoes a remote compressive stress.

For loading inY -direction, the strains in the vertical direction and horizontal direction due toσY are:

εYY = H − 2LYY (αY )− (H − Lsinλ)

H − Lsinλ= Lsinλ∓ 2[2E(αY )− F(αY )]/kY

H − Lsinλ, (35)

εXY = 2LXY (αY ) − Lcosλ

Lcosλ= 4pY cosδY /kY − Lcosλ

Lcosλ. (36)

So the Poisson’s ratio for loading inY -direction is

νY = −εXY

εYY= −4pY cosδY /kY − Lcosλ

Lcosλ

/Lsinλ ∓ 2[2E(αY ) −F(αY )]/kYH − Lsinλ

. (37)

3. Results and discussion

3.1. Degeneration of the large deflection model

Referring back to Fig. 3(b), lettingλ = π/2, and considering that the half memberOB is under remote compressive streand that the deflection ofOB is very small, the compressive force predicted by (19) is

PY = π2EsI

4L2(38)

which is the value of the critical load for the elastic buckling of a vertical bar as described by Timoshenko and Gere (1Let λ = 0, and consider that the half memberOB undergoes aY -direction force at pointO. From (32), we have:

PY = ESI

(2pY cosδYLXY (αY )

)2. (39)


When the deflection ofOB is very small, we have:βY = π/2, αY = π/2 + γ , andLXY (αY ) = L/2. Through a set of

flexure.

eading

is time,a highercomb is

le

le

mathematical processing,γ is expressed as

γ = −PYL2

8ESI(40)

which is the expression of the bending angle at endO of a cantilever beam fixed at endB and loaded at free endO by forcePY ,predicted by the theory of small deformation of flexure.

Therefore, this large deflection model also contains the information of the elastic buckling and small deformation of

3.2. The shape of the deformed memberAB

Using (23) and (26), or (31) and (33), the co-ordinates of a general point on the deflection member alongX-axis orY -axiscan be calculated. The shape of the deformed member becomes aS-shape. Setting the mid-pointO as origin, the shapes of thdeflection curves for various values ofαX andαY are shown in Figs. 5 and 6, representing the compressive and tensile loin X-direction andY -direction respectively. In these figures, the cell’s geometric parameters considered are:H = 2L, λ = π/6.

Under compressive loading, the inclined members will touch each other, if the deformation is large enough. At ththe force and the moment distribution will change, which means that this large deflection model is not suitable fordeformation than the critical point of touch. The touch of the inclined members will not occur when the auxetic honeyunder tension.

(a) (b)

Fig. 5. Deformed shapes of an inclined member underX-direction loading withH/L = 2 andλ = π/6. (a) Compressive loading, (b) tensiloading.

(a) (b)

Fig. 6. Deformed shapes of an inclined member underY -direction loading withH/L = 2 andλ = π/6. (a) Compressive loading, (b) tensiloading.


3.3. Poisson’s ratio versus strain

smodel

ignificantly

2),

rmation.ycomb isesnegative as

ive toprojected

s ofeach other

dicted by

Poisson’s ratios forX-direction loading andY -direction loading are obtained from (30) and (37). The Poisson’s ratioνXandνY for loading inX-direction andY -direction are, as expected, independent of density, for this large deformationused here ignores axial compression.

Fig. 7 shows that Poisson’s ratios in the orthogonal directions versus strain for a case ofH = 2L andλ = π/6. It can be seenthat Poisson’s ratios are not constant, and have negative value generally. The magnitude of Poisson’s ratio decreases sat high strain when the auxetic honeycomb is compressed inX-direction loading or stretched inY -direction loading, whileincreases significantly at high strain when the auxetic honeycomb is stretched inX-direction loading or is compressed inY -direction loading. Poisson’s ratio at the limit value of small deformation is−1, the same result obtained by Gibson et al. (198and Masters and Evans (1999).

The variation of Poisson’s ratio versus strain is different for compressive loading and tensile loading at large defoFor X-direction loading, the magnitude of Poisson’s ratio decreases as strain increases when the auxetic honecompressed, and increases as strain increases when it is stretched. While forY -direction loading, Poisson’s ratio becomincreasingly negative as strain increases when the auxetic honeycomb is compressed, and becomes decreasinglystrain increases when it is stretched. Especially, when the auxetic honeycomb undergoes a tensile stress inY -direction orundergoes a compressive stress inX-direction at large deformation, the sign of Poisson’s ratio may change from negatpositive. This phenomenon can be explained from Figs. 5(a) and 6(b). As can be seen from Fig. 6(b), the horizontallength of memberAB increases asαY , the bending angle at originO, increases, whenαY changes fromπ/2 − λ to π/2, butwhenαY exceedsπ/2, it decreases asαY increases. While in Fig. 5(a), the vertical projected length of memberAB increasesasαX increases whenαX changes fromλ to π/2, but decreases asαX increases whenαX changes fromπ/2 toπ .

3.4. Effect of geometry on Poisson’s ratio

To investigate the effect of geometry of the re-entrant cell on the Poisson’s ratio,νX andνY at the limit of small deformationare tabulated in Tables 1 and 2 for the combinations ofH/L andλ. In reality, the negative Poison’s ratios in the parenthesethese tables are never achieved, because for geometric considerations, the inclined members of the cells will touchbefore deformation with these geometric parameters ofH/L andλ. νX andνY as they depends on the values ofλ andH/L arealso plotted in Figs. 8 and 9 for a given value ofλ and a varyingH/L, and for a givenH/L and a varyingλ, respectively.

As can be seen from Tables 1 and 2, negative Poisson’s ratios at the limit of small strain converge to the result presmall deformation of flexure:

Fig. 7. Poisson’s ratios vary with strain for a re-entrant cell with geometric parameters ofH/L = 2 andλ = π/6.

Table 1Poisson’s ratioνX at the limit value of small deformation

H/L 1.5 2 2.5 3 3.5 4

π/18 −4.211 −3.058 −2.40 −1.976 −1.679 −1.460π/9 −2.230 −1.557 −1.196 −0.971 −0.818 −0.706π/6 −1.5 −1 −0.75 −0.6 −0.5 −0.429

λ νX2π/9 −1.065 −0.673 −0.492 −0.387 −0.320 −0.2725π/18 (−0.735) −0.437 −0.311 −0.241 −0.197 −0.167π/3 (−0.455) −0.255 −0.177 −0.135 −0.110 −0.092


Table 2Poisson’s ratioν at the limit value of small deformation

onal

eters ofitude ofrection.omb. Annal

Y

H/L 1.5 2 2.5 3 3.5 4

π/18 −0.237 −0.327 −0.417 −0.506 −0.596 −0.685π/9 −0.449 −0.642 −0.836 −1.030 −1.223 −1.417π/6 −0.667 −1 −1.333 1.667 −2 −2.333

λ νY2π/9 −0.939 −1.487 −2.034 −2.582 −3.130 −3.6775π/18 (−1.361) −2.288 −3.215 −4.142 −5.069 −5.996π/3 (−2.196) −3.928 −5.660 −7.392 −9.124 −10.856

(a) (b)

Fig. 8. Effect ofH/L on Poisson’s ratios for a given value ofλ = π/6. (a)νX versusεXX , (b) νY versusεYY .

(a) (b)

Fig. 9. Effect ofλ on Poisson’s ratios for a given value ofH/L = 2. (a)νX versusεXX , (b) νY versusεYY .

νX = − cos2λ

(H/L − sinλ)sinλ, (41)

νY = − (H/L − sinλ)sinλ

cos2λ, (42)

νX and νY comply with the reciprocal relationνX ∗ νY = 1 for the same geometric parameters. In particular, orthogsymmetry of Poisson’s ratio occurs for some cells with their geometric parameters satisfying:

sinλH/L = 1 (43)

and Poisson’s ratio of this sort of auxetic honeycombs is minus one. A special case is the cell with geometric paramH/L = 2 andλ = π/6. In contrast with the conventional regular hexagonal honeycomb, they have the same magnPoisson’s ratio, but with different sign; and have the same inclining angle of the inclined member, but in different diThe relative density of this auxetic honeycomb calculated by (1) is larger than that of the conventional regular honeycauxetic honeycomb with geometric parameters ofH/L = 3 andλ = π/6 has the same relative density with the conventioregular honeycomb, but their magnitudes of Poisson’s ratios are different.


metricer thehe, dependspositive

astcal

structure,on’s ratiosion model,rstanding

tly withdel. Ateometrichey

etry of

Both

Fig. 10.νX ∗ νY versus strain for the case study with cell’s geometric parameters ofH/L = 2 andλ = π/6.

From Figs. 8 and 9, we know that Poisson’s ratio not only varies significantly with the strain in all cases of geoparameters, but also is influenced significantly by the geometric parameters of the cell. AsH/L becomes large, the magnitudof νX reduces while the magnitude ofνY increases generally, for a given value of strain. A similar tendency occurs foinfluence ofλ on νX andνY . The effect ofH/L on νY under tensile loading inY -direction becomes complicated when tdeformation is large enough. The sign of Poisson’s ratio changes to positive at a critical strain, which, on the other handon the value ofH/L. UnderX-direction compressive remote stress, the sign of Poisson’s ratio changes from negative toat a strain about 1.7.

In the effective intervals of tensileεX and compressiveεY , νX ∗ νY varied with the strain is showed in Fig. 10 by a lesquares fit. Geometric parameters of the re-entrant cell are:H/L = 2 andλ = π/6, for the case study. As can be seen, reciprorelation forνX andνY cannot hold when the deformation is large.

4. Conclusions

The non-linear behavior becomes important when honeycombs are to be used as load bearing or energy absorptionbecause large deformation often occurs. In this paper, we provide a theoretical approach for evaluating negative Poissof auxetic honeycombs. The prediction of negative Poisson’s ratios of these honeycombs is based on the large deflectwhich is also suitable to small flexure and elastic buckling. This large deflection model extends and refines our undeon negative Poisson’s ratios of auxetic honeycombs, giving a complete estimation of negative Poisson’s ratios.

It is found that Poisson’s ratios of auxetic honeycombs are not a constant at large deformation, vary significanthe strain, but converge to the results predicted by small deformation of flexure at the limit of small strain of this molarge deformation, the negative Poisson’s ratios are different in remote compressive loading and tensile loading. Gparameters of the re-entrant cell, i.e.,H/L andλ, have very significant effect on the magnitude of the Poisson’s ratio. Teven change the sign of the Poisson’s ratio from negative to positive, when this sort of honeycombs bears aY -direction tensileloading or aX-direction compressive load under the condition that the deformation is large enough. Orthogonal symmPoisson’s ratios occurs for some cell geometric parameters at small deformation, a re-entrant cell ofH/L = 2 andλ = π/6 is aspecial one. Reciprocal relation forνX andνY is conserved for all geometric parameters when the deformation is small.orthogonal symmetry and reciprocal relation of Poisson’s ratios cannot be maintained when the deformation is large.

Acknowledgements

The authors thank the anonymous referees very much for their helpful suggestions and useful comments.

References

Almgren, R.F., 1985. An isotropic three-dimensional structure with Poisson’s ratio equal to minus one. J. Elasticity 15, 427–430.Evans, K.E., 1989. Tensile network microstructures exhibiting negative Poisson’s ratios. J. Phys. D 22, 1870–1876.Evans, K.E., 1990. Tailoring the negative Poisson’s ratio. Chem. Ind. 20, 654–657.Evans, K.E., Nkansah, M.A., Hutchinson, I.J., Rogers, S.C., 1991. Molecular network design. Nature 353, 124–125.Garber, A.M., 1963. Pyrolytic materials for thermal protection systems. Aerospace Eng. 22, 126–137.Gibson, L.J., Ashby, M.F., 1997. Cellular Solids: Structure and Properties. Cambridge University Press, Cambridge, UK.


Gibson, L.J., Ashby, M.F., Schajer, G.S., Robertson, C.I., 1982. The mechanics of two-dimensional cellular materials. Proc. Roy. Soc. LondonSer. A 382, 25–42.

ech. Min.

olidi A 38,

xagonal

–472.rsity.

rib foam

atio foam

21, 779–

ls. Mech.

olids 37,

6–3233.iomech.

using the

Homand-Etienne, F., Houpert, R., 1989. Thermally induced microcracking in granites: characterization and analysis. Int. J. Rock MSci. & Geomech. Abstr. 26, 125–134.

Kolpakov, A.G., 1985. On the determination of the averaged moduli of elastic gridworks. Prikl. Mat. Mekh. 59, 969–977.Lakes, R.S., 1987a. Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040.Lakes, R.S., 1987b. Negative Poisson’s ratio materials. Science 238, 551.Lakes, R.S., 1991. Deformation mechanisms in negative Poisson’s ratio materials: Structure aspects. J. Mat. Sci. 26, 2287–2292.Li, Y., 1976. The anisotropic behavior of Poisson’s ratio, Young’s modulus, and shear modulus in hexagonal materials. Phys. Status S

171–175.Masters, I.G., Evans, K.E., 1999. Models for the elastic deformation of honeycombs. Compos. Struct. 35, 403–422.Nur, A., Simmons, G., 1969. The effect of saturation on velocity in low porosity rocks. Earth Planet. Sci. Lett. 7, 183–193.Overaker, D.W., Cuitiño, L.M., Langrana, N.A., 1998. Effects of morphology and orientation on the behavior of two-dimensional he

foams and application in a re-entrant foam anchor model. Mech. Mater. 29, 43–52.Rothenburg, L., Berlin, A.A., Bathurst, R.J., 1991. Microstructure of isotropic materials with negative Poisson’s ratio. Nature 354, 470Schajer, G.S., Robertson, C.I., 1974. Mechanical behavior of cellular structures. Project report, Pembroke college, Cambridge UniveSigmund, O., 2000. A new class of extremal composites. J. Mech. Phys. Solids 48, 397–428.Smith, C.W., Grima, J.N., Evans, K.E., 2000. A novel mechanism for generating auxetic behaviour in reticulated foams: missing

model. Acta Mater. 48, 4349–4356.Timoshenko, S.P., 1983. History of Strength of Materials. McGraw-Hill, New York.Timoshenko, S.P., Gere, J.M., 1961. Theory of Elastic Stability. McGraw-Hill, New York.Torquato, S., 2000. Modeling of physical properties of composite materials. Int. J. Solids Structures 37, 411–422.Wang, Y.-C., Lakes, R., 2002. Analytical parametric analysis of the contact problem of human buttocks and negative Poisson’s r

cushions. Int. J. Solids Structures 39, 4825–4838.Warren, W.E., Byskov, E., 2002. Three-fold symmetry restrictions on two-dimensional micropolar materials. Eur. J. Mech. A Solids

792.Warren, W.E., Kraynik, A.M., 1987. Foam mechanics: the linear elastic response of two-dimensional spatially periodic cellular materia

Mater. 6, 27–37.Warren, W.E., Kraynik, A.M., Stone, C.M., 1989. A constitutive model for two-dimensional non-linear elastic foams. J. Mech. Phys. S

717–733.Wei, G., 1992. Negative and conventional Poisson’s ratio of polymeric networks with special microstructures. J. Chem. Phys. 96, 322Williams, J.L., Lewis, J.L., 1982. Properties and an anisotropic model of cancellous bone from the proximal tibial epiphysis. J. B

Engrg. 104, 50–56.Wojciechowski, K.W., Branka, A.C., 1989. Negative Poisson ratio in a two-dimensional isotropic solid. Phys. Rev. A 40, 7222–7225.Yang, D.U., Lee, S., Huang, F.Y., 2003. Geometric effects on micropolar elastic honeycomb structure with negative Poisson’s ratio

finite element method. Finite Elements in Analysis and Design 39, 187–205.Zhang, J., Ashby, M.F., 1992. Buckling of honeycombs under in-plane biaxial stresses. Int. J. Mech. Sci. 34, 491–509.Zhu, H.X., Mills, N.J., 2000. The in-plane non-linear compression of regular honeycombs. Int. J. Solids Structures 37, 1931–1949.

A study of negative Poisson's ratios in auxetic honeycombs based on a large deflection model

Documents