Models for Elastic Deformation of Honeycombs

8/9/2019 Models for Elastic Deformation of Honeycombs

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ELSEVIER

Composirr Structures 35 (1996) 403-422

0 1997 Elsevier Science Ltd

Printed in Great Britain . All rights reserved

0263~8223/96/ 15.00

Pll:SO263-8223(96)00054-2

Models for the elastic deformation of

honeycombs

I. G. Masters K. E. Evans

School

fEngineet ing, Uni versit y f Exeter; Nor th Park Road, Exet er EX4 4QE UK

A theoretical model has been developed for predicting the elastic constants

of honeycombs based on the deformation of the honeycomb cells by

hexme, stretching and hinging. This is an extension of earlier work based

on flexure alone. The model has been used to derive expressions for the

tensile moduli, shear moduli and Poissons ratios. Examples are given of

structures with a negative Poissons ratio. It is shown how the properties

can be tailored by varying the relative magnitudes of the force constants for

the different deformation mechanisms. Off-axis elastic constants are also

calculated and it is shown how the moduli and Poissons ratios vary with

applied loading direction. Depending on the geometry of the honeycomb

the properties may be isotropic (for regular hexagons) or extremely

anisotropic. Again, the degree of anisotropy is also affected by the relative

magnitude of the force constants for the three deformation mechanisms. 0

1997 Elsevier Science Ltd. All rights reserved.

INTRODUCTION

Honeycomb core materials are widely used in

the manufacture of stiff, lightweight sandwich

panels mainly for use in aircraft. Commercial

honeycombs are most commonly based on a

hexagonal cell shape which is simple to produce

and ideal for the manufacture of flat sandwich

panels. A disadvantage of the hexagonal cell

honeycomb is that if it is bent out-of-plane it

produces an anticlastic or saddle-shaped curva-

ture due to the effective in-plane Poissons ratio

being positive. With such a honeycomb doubly

curved structures, e.g. radomes can only be pro-

duced by forcing a sheet of honeycomb into the

desired shape, causing local crushing of the

cells, or by machining a block to the required

profile which is expensive. However, if the

effective Poissons ratio is made negative by

altering the cell shape the domed or synclastic

curvatures can be achieved naturally.

The value of the in-plane Poissons ratio is

determined by the cell geometry alone whereas

the stiffness in bending of the sheet of honey-

comb is related to the mechanism by which the

individual cells deform, which in turn, is deter-

mined by the material properties of the cell wall

material.*

Honeycombs can be envisaged to deform

when loaded in the plane by flexing and stretch-

ing of the cell walls and by hinging at the cell

wall junctions. Several workers have formulated

mathematical models based on one or two of

these mechanisms for specific geometries. The

in-plane moduli of the hexagonal cell honey-

comb has been successfully modelled by

assuming that the cell walls flex like beams.2%4

Using simple mechanics to calculate the deflec-

tions in each beam the strains induced in an

individual cell, and hence the whole network

can be determined; enabling expressions for the

moduli and Poissons ratios to be written for the

condition of uniaxial loading. This simple model

has been shown to give good agreement with

experimental results for both metal and silicon

rubber honeycombs.2*4

The flexure model was

extended4 to include stretching and shear

deflections but these refinements were found to

provide negligible improvement to the model.

Only in the particular case of a hexagonal cell

honeycomb subjected to biaxial loading was the

contribution of stretching considered significant.

403


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404

I. G.

Masters, K. E. Evans

A related approach has been proposed for

predicting the in-plane properties of graphite by

assuming that deformation of the cellular net-

work occurred by stretching of the atomic

bonds and changes in bond angle, i.e. hinging.

Other workers have used the flexure model to

determine the Youngs moduli and Poissons

ratio of theoretical molecular structures, and

compared the results with those obtained by

molecular modelling and finite element analysis.

The simple flexure model was found to consist-

ently overestimate the values of E and 1

predicted by molecular modelling. This implies

that although flexure might be the dominant

mechanism there must be contributions from

stretching, and or hinging. In molecular net-

works the stretching of molecular chains tends

to increase the longitudinal deformation at the

expense of transverse thus reducing the Pois-

sons ratio. In cardboard honeycombs hinging

has been shown to be the dominant mechan-

ism,77x the low forces required to operate the

hinges in

these materials giving rise to

extremely flexible honeycombs.

Stretching and hinging mechanisms have

been combined to develop a model for predict-

ing the Poissons ratio of both hexagonal and

re-entrant cell three-dimensional (3D) struc-

tures. which in a generalized form can

describe the Poissons ratio of polymer mole-

cules. For the latter types of structure the cell

orientations are random unlike periodic honey-

combs and hence produce very different results.

Flexing and stretching have also been com-

bined*

to describe the elasticity of rigid,

disordered 3D networks.

These mathematical models show that for

regular hexagonal-celled honeycomb structures

Poissons ratios in excess of +1 can easily be

achieved as a result of the open structure. The

re-entrant cell shape identified by various work-

ers

4,0,3.4 however, is shown to have a negative

Poissons ratio, which may also be much less

than -1.

In this paper all three mechanisms of flexing,

hinging and stretching are combined. New

expressions for shear moduli as well as Youngs

moduli and Poissons ratio are derived to

explore the off-axis properties of the honey-

combs

using the

axis

transformation

equations.

Polar plots of properties are

obtained that enable us to determine the com-

bination of material properties that lead to an

auxetic honeycomb which is nearly isotropic, as

opposed to the highly anisotropic behaviour

seen in honeycombs deforming by one mechan-

ism alone.

MODELS

To aid comparison of the models each can be

written in terms of a force constant Kj which

also facilitates combining the three mechanisms

to generate a general model.

Force constants

The elastic constants of a two-dimensional (2D)

honeycomb can be described by considering the

displacement of the single cell, from which the

honeycomb is produced by translational repeti-

tion, under appropriate loading conditions.

The force constants relate the displacement

of the cell walls of a honeycomb to the applied

force which causes it. For all three mechanisms

the force constant is defined by the general

relationship

F = K;6

(1

where

F

is the applied force,

K,

is the force

constant and 6 is the displacement. The force

constant contains details of the mechanical

properties of the cell wall material and the net-

work structure itself. For example, in a

molecular network

Kj

can be related directly to

the atomic force constants. The conventional

case of macroscopic honeycombs of cell wall

lengths I and h, thickness

t

and depth b (see Fig.

1) is considered here and it is assumed that the

elastic constants of the material forming the cell

walls are known; E,s being the Youngs modulus

and G, being the shear modulus. Explicit rela-

tionships between

Kj

and the properties of the

cell wall can therefore be derived for each of

the deformation mechanisms of flexure, stretch-

ing and hinging.

Flexure force constant Kf

A cell wall of length 1 deforming by flexing can

be likened to a cantilever beam loaded and

guided at one end and fixed at the other. The

deflection of the guided end due to flexingI is

given by


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Models for the elastic deformation of honeycombs

405

(a) 2

t

h + 1 sin0

0))

2

t

/t + sin -8)

*

w

I

Fig. 1. Cell geometry and coordinate system used for (a)

hexagonal and (b) re-entrant cells.

Ml2

a= -

12E.J

(2)

where M is the applied moment=t;l, I is the

second moment of area of the cell wall=bt/12.

Therefore

b=z_

E,Tbt

Comparing this with eqn (1) gives

K = E.&t

.f

1

(3)

(4)

Stretching force constant KS

The extension of a cell wall, length I, due to the

axial force

F

acting along it, is

Fl

6= -

bt Es

Comparing this with eqn (1) gives

btE,

= -

1

(5)

Hinging force constant Kh

Finally, for a cell deforming by hinging, we

assume that the cell wall is rigid along its length

and deflection occurs at the junction with other

cell walls by a change of angle AO. Hence for a

wall of length

1

(5=lsinAOzlA(I

(7)

if A0 is small. Substituting into the general for-

mula (eqn (1)) gives

F = KJAO

(8)

The actual mechanism by which hinging occurs

can be envisaged as one of two processes;

global shear or local bending.

In global shear the relationship of

Kh

to

material parameters is obtained by assuming

that hinging occurs by shearing of the material

at the cell wall junction (Fig. 2). Using the

standard definition of shear modulus we can say

F

Gs= -

btAO

where Gs is the

shear modulus of the cell wall

(9)

material and

F

is the force applied perpendicu-

lar to the beam. Comparing with eqn (8) gives

G,bt

K,, = -

1

(10)

The shear mechanism is important when con-

sidering small-celled foams and molecular

networks but is unrealistic for the macro-net-

works,

like honeycombs, where hinging is

Fig. 2.

Schematic diagram of hinging due to shearing of

the cell wall material.


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I. G. Masters K

E. Evans

06

/

Fig. 3.

Schematic diagram of hinging due to local bending

of the cell wall.

obviously a local effect. To model the latter

type of behaviour we can imagine the hinge to

be a short, curved beam, of axial length q, in the

vicinity of the cell node (Fig. 3).

Assuming that a curved beam behaves in

accordance with simple bending theory16 then

the change in angle II/ to I+Vdue to an applied

moment M is given by

i.e.

(11)

(12)

But as can be seen from Fig. 3, II/ = 90 - 0 and

rl/ = 90-O--A( ). Hence we can say

m

Aij=Ao= -

E.J

(13)

where 19 is the cell angle. If q


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M odel s or t he el asti c deformati on of honeycombs

407

where El, E,,

v12, v21

and G,, are the elastic

constants for the honeycomb. This model has

been shown to be very successful for modelling

a great variety of conventional honeycombs and

reticulated foams.4 If 8 is made negative (Fig.

1) then the Poissons ratio of the cell becomes

negative in value; a condition known as aux-

etic.

The Youngs moduli and Poissons ratio

expressions for the flexure model comply with

the reciprocal relation E,v,, = E2~,2 as required

for a symmetric stiffness matrix.

Stretching

This model assumes that the cell walls are only

able to deform by stretching along their axes

with no change in angle. This is akin to a set of

connected shock absorbers.

Consider a hexagonal cell (Fig. 4(a)) sub-

jected to a tensile load g2 in the 2-direction.

The load acting on the unit cell due to the

applied stress c2 is w = b(Z sin 8 +h)o, and the

component P of w acting along the cell wall of

length 1 s

P = bcr,(Z

sinO+h)cos 8

(22)

But

P = K .,S,

where K.:, is the force constant for

stretching, therefore

-mm__

4

\

,I

\

\

On+:

0

\

,

\

\

I

\

\

8

-____

(b)

Fig. 4. Hexagonal cell deforming by stretching of the cell

walls due to tensile load applied in (a) direction 2 and (b)

direction 1. Forces acting on the walls length 1 are shown

on the right.

6

I

=

b

rr2cos O(I sinO+h)

KS

(23)

The strain 6, in the 2-direction caused by the

extension 6, is

b c2

cos 0 (h/l+sin 0)

c2 =

K,

(24)

Therefore the modulus in direction 2 is

E, =

KS

b cos O(h/l+sin 0)

(25)

The strain in the l-direction due to the exten-

sion 6, is

bo, cos0

sin0

c, =

K

The Poissons ratio is therefore

v2,=

-

siz ;,l

(26)

(27)

By considering the forces acting on the cell

edge when loaded in direction 1 (Fig. 4(b))

similar equations can be derived. However, in

this orientation it should be noted that the cell

walls of length h also extend. The force constant

(Kf)

for these walls is

E,bt

K -

h

Comparison with eqn (6) enables us to write K t

in terms of K, the force constant for a wall

length 1, .e.

and thus the strain in direction 1 is given by

2ba,h cos0

= IK,(h/l+sinO)

+

(bo,f cos0

sinO)sinO

lKJhll+sin0


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408

I . G. Mast ers, K . E. Evans

bo, cosU(2h/l+sin*U)

I:, =

&(h/f+sin 0)

(28)

and the strain in the 2-direction is

bo,cosO sin0

c2 =

KS

(29)

The modulus in the l-direction is therefore

E, =

K,(h/l+sin 0)

b cos 0(2h/l+sin*U)

(30)

and the Poissons ratio is

-sin O(h/l+sin 0)

v,* =

2h/1+sin20

(31)

Note that when 0 is positive, i.e. the cell is

hexagonal in shape, both v,* and v2, are nega-

tive in value. Substituting (-0) into these

equations produces the equivalent expressions

for the re-entrant cell. It should also be noted

that in the re-entrant case the direction of the

forces in the walls of length

1

is reversed whilst

the forces in the other walls remain unchanged.

Youngs moduli and Poissons ratio expres-

sions have been derived4 for a hexagonal cell

honeycomb subjected to a biaxial stress deform-

ing by both flexure and stretching of the cell

walls. It can be shown that for the conditions of

a uniaxial applied stress and deformation only

occurring by stretching that these equations

reduce to eqns (25) (30) (27) and (31) respec-

tively.

The in-plane shear modulus can be obtained

by considering the shear stresses acting on the

cell node shown in Fig. 5. The shear stress 5

acting on the unit cell is given by the expression

where F, and F, are the forces acting in the l-

and 2-directions, respectively. Rearranging this

equation enables F, to be written in terms of

F2, i.e.

F, =

F,(h +Z sin 0)

21 cos 0

(32)

If we let F2 = F then the component P* of F,

acting along member AC is given by

t/2

\i

t/2

1

\t-

Fig. 5. In-plane shear deformation at a cell node due to

stretching of the cell walls.

F, cos 6, F cos0

p* = =

2

2

(33)

and the component P* * of F, acting along AC

is

P** = F, sin6 =

F(h +I

sin B)sin 0

21 cos 0

(34)

The total force P acting along AC is therefore

P=P**+P*

(35)

F cos6

P=

+

F(h +I

sin 0) sin 0

2

21 cos 0

(36)

The point A moves an amount 6,c to A due to

the force P. This extension is obtained from the

expression for the stretching constant K, (eqn

(6))

P

6

- -

AC K,y

(37)

Substituting for P gives

F

6

- -

AC - 26,

cos o+

(h+l sinO)sin0

1 case

1

(38)

The horizontal deflection L& due to the exten-

sion 6A,

is given by


7/20

M odels or the elasti c deformati on

of

honeycombs

409

6, =

GA&OS

0

F

6,= -

2Ks

cos2 0 +

(h +I sin 0) sin 0

1

1

(39)

The vertical deflection 6, across the unit cell is

6, = 2bAV sin 0

(40)

The factor of two arises because the member

CB shortens as much as AC extends. Therefore

F sinU

ti, = ___ cosU+

K

i

fh+l sinU)sinU

1 os

1

(41)

If the unit cell is considered to deform by

simple shear then, as can be seen from Fig. 6,

the shear strain 7 is given by

A, = ;,+f =

4

62

i

+

21 cos 0 h+l sinU

F (h+l

sinU)sinU

= - cosU+

KS

LCOSU

I

x (cosU) + (sin 0)

h+l

sinU 1 cos U

1

(42)

(43)

The shear modulus G is the ratio of the shear

stress and the shear strain, i.e.

G =

z/l

(44)

Therefore

G,*= K,

1 cosU(h+l sinU)

h (1 cos*U+(h+l sinU)sinU)*

1 45)

Pure shear

Simple shear

Fig. 6. Schematic diagram showing the assumed relation-

ship between pure shear and simple shear.

0

t

HINGE

P

s

1 4t

?

8

,

:

,I

;I3

I

I

0

\

A6

Fig. 7. Hexagonal cell deforming by hinging due to a

compressive stress applied in direction 2 also showing the

forces acting on a wall length 1.

Hinging

The hinging model relies on the cell walls being

stiff in both the axial and transverse directions.

Elastic hinges at the joints enable the cell to

deform when a load is applied and restore the

cell to its original shape when the load is

removed. The cell deforms by changes in the

cell angle alone. Consider a hexagonal cell as

shown in Fig. 7. If we assume that the material

from which the cell is manufactured has a force

constant K,, which determines the deflection 8,

caused when a load is applied to the cell wall

(Fig. 7) we can say that

P=K,J

(46)

where P is the applied load and 6 is the deflec-

tion. If the cell is subjected to a compressive o2

in direction 2, then the forces acting on the cell

edge of length 1 are given by

P = cr2(h +I

sin U)b sin U

(47)

where h is the honeycomb thickness. Substitut-

ing for P gives

n =

o& l sin U(h/l+sin U)

K,,

(48)

The strain in direction 2 is therefore given by

-o,b

sin2U(h/l+sinU)

+ =

K,, cos U

(4%


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410

I . G. Masters. K. E. Evans

and the modulus in direction 2 by

E =

K,z os 0

b sin*O(h/l+sin 0)

(50)

The strain in direction 1 is

g2b sin 0 cos 0

c, =

K/1

(51)

and hence the Poissons ratio in the 2-direction

is

--El

12,= -

62

cos*

= (h/l+sinO)sin 0

(52)

If the honeycomb is compressed in direction 1

then by a similar method we can determine that

c,b cos0

sin0

c2 =

Kh

(53)

-o,b cos0

Cl =

Kh h/l+sin 0)

(54)

E _ K,(h/l+sin 0)

I-

b cos 0

(55)

sin O(h/l+sin 0)

12 =

cos2 0

(56)

Comparing this result with the expression for

v2, we can see that

1

I2 =

21

Again by substituting (- 0) into these equations

we obtain expressions to describe the behaviour

of a re-entrant cell.

As in the stretching model the in-plane shear

modulus is obtained by considering the effects

of the shear stresses z acting aIong the sides of

Fig. 8. In-plane shear deformation at a cell node due to

hinging.

the unit cell in both the 1 and 2 directions (Fig.

8). The cell walls are rigid so that no flexing

or

stretching occurs. All the movement occurs due

to the hinging at point C.

The point A is subjected to a force F, acting

in direction 1 and a force F2 acting in direction

2. As previously shown if

F,=F

then

F1

F(h+l

sin0)

21 cos 0

(57)

The component P* of F2 acting perpendicular

to the member AC is given by

F sin0

p*= -

2

(58)

The component

P**

of F, acting perpendicular

to AC is given by

P+* =

-F cos8 = -

F(h +I sin 0)

21

(59)

Note that this expression is negative because

P

* causes counter-clockwise rotation of the


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M odels or t he el asti c deforma ti on of honeycombs

411

member AC. The total force

P

acting perpen-

dicular to the member AC is therefore

P = P+P*

P= -

[

sin&

(h+l

sin0)

t

1

P-5

(60)

Theorce P rotates AC through an angle A8

such that the point A is deflected an amount

6

AC*

From eqn (1) we know that

P

6

--

AC-

I

(61)

The displacement in direction 2 due to 6,, is

Psin8

6*,=6,,sinQ= -

&

(62)

and the horizontal displacement of point D due

to the force

F,

is

F F

((j**=_=_

2 K; K/ ,C

(63)

where C is a constant, enabling Ki, the force

constant for a cell wall of length h, to be written

in terms of Kh, the force constant for a wall

length

1.

For the shear model (eqn (10))

C=l/h

and for bending model (eqn (16)) C=(l/h)2. The

total deflection in direction 2 is therefore

PsinO

F F

6,=-

-Ch sirit)+

+-=-

K/Z K4 &I

2Cl

1

and the total displacement in direction 1 is

A general model

By summing the deflections in directions

(64)

2P cos 0

Fh cos 0

6, =

K, =- K,,l

(65)

If we assume that the unit cell shown in Fig. 8

deforms by simple shear then the shear strain y

is given by

1= *

I

62

21cosUh+lsinO

F 21-Ch

sin0

h

21K,

C(h +I

sin0) -7

1

66)

The negative sign in eqn (65) can be ignored

since we are only interested in the length 6,.

The remote shear stress z is given by

F

z=

21b cos 0

Hence the shear modulus is

F

G*=

21K,

X-

21b COSU F

X

Cl(h +1 sin8)

Ch(h+lsin8)+1(21-Ch

sin0)

1

(67)

(68)

K/l

G,2=-

b cosU

X

Cl(h +f

sin0)

Ch(h+Isin0)+1(21-Ch sin0)

1

(69)

1 and 2, we can combine the three models to obtain a

. _

general expression. For example, if we consider a honeycomb loaded in direction 1 then the strains

in direction 1 arising from deformation by stretching and hinging are given by eqns (28) and (54).

The strain in direction 1 caused by flexure of the cell walls has been shown34 to be


10/20

412

I. G. Masters, K. E. Evans

o,h14 osu

I= 12E,,I(h+f sin(I)

where I is the second moment of area of the cell wall. Rewriting this in terms of the force constant

for flexure F$ (eqn (4)) gives

I _

o,hcos3u

>

K,(h/f + sin 0)

The total strain in direction 1, obtained by summing eqns (28) (56) and (72) is thus

(70)

(71)

The modulus in direction 1 is then given by

hence

1

cos2U + (2h/I+sin*h)

(72)

-+-

K,,

KY

1

The strain in direction 2, due to flexing, arising from the applied stress o-, is given by the expression

6, bl -3 os 0

sin 0

Cz=

12E,,I

(73)

Writing in terms of the force constant this becomes

o,bcosOsinU

C2=

K,

Summing this expression with (29) and (53) g

ives the total strain in direction 2

I;pta=o,b cos(,I sin0

1 1 1

------+y

Kf K,,

s

1

(74)

(75)

Dividing this expression by (71) gives the Poissons ratio


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M odel s o r t he el asti c deformati on of honeycombs

Using a similar method the following general expressions can be obtained for E2 and r2,

1

E,=

sin2 0 sin 0

cos 0

b (h /I + sin 0)

Kfcos 0 + K ,, cos 0

+-

IY,

1

1 1

1

-sinO.cosO K~F-K

L

.f I1

s

1

v

1=

sin* 0 sin 0

cos 0

(h/Z+sinO)

Kr cos 0 + K /, cos 0

+-

K,s

1

Gibsons expression for the shear strain written in terms of the flexure force constant is

Fh2(f +2h)

r =

2K,Z(h +I sin0)

Summing this with expressions (45) and (69) the general shear modulus expression is obtained

1

G,2=

bh(l+2h)cosO

1

Ch2+212

h cos 0

Kb2(h +I sin0 1 L

-

K,,

Cl(h +I sin 0) 1

1

b(f

cosZO+(h+ZsinO)sinO

cos 0 sin 0

+

KS

(h

+fsinO) cos0

413

(76)

(77)

(78)

(79)

(80)

If, for example, we put

K,=K,y= rx,

the above equations reduce to those for the flexure model.

DISCUSSION

The force constants

The properties of the force constants Kf KS and

Kh are compared in Fig. 9 having evaluated the

eqns (4) (6) (10) and (16) for the conditions

E,s=l=b=l, G,YzE,J3=1/3, q=Z/lO

The shear modulus of the bulk cell wall

material G,

z EJ3 is a general assumption for

an elastic material and using this value we can

evaluate the global shear model for

K/*

(eqn

(10)). In reality to evaluate this shear model for

K,, w e

need to know G/,, the shear modulus of

the material in the hinge, which is not neces-

sarily the same as G, and is likely to be

considerably lower due to the material being

damaged. Using the local shear model, Kh (eqn

(16)) is obtained directly from

E,s.

1/10 may at

first appear to be rather large as an estimate for

the effective length

(q)

of the hinge, but for the

0*21-mm thick card honeycombs used in the

experiment8 the folds were typically 1 mm in

width and 1 was a constant length of 10 mm. It

is apparent from the three models discussed

here that if the force constant K , is high in value


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I . G. Masters, K. E. E vans

I . , 1 , . , I , ,

0

0.2 0.4 0.6 0.8 I

t/1

Fig. 9.

Plot of K, versus t/l showing the relative behaviour

of the flexing, stretching and hinging force constants. The

constant with the lowest value determines the dominant

mechanism.

then the contribution of that particular mechan-

ism to the overall deformation will be small.

For the honeycombs used in these tests the

value of t/l is in the range 0.01 to 0.02 so that K,

(eqn (6)) is large compared to Kf (eqn (4)) and

Kh (eqns (10) and (16)) and stretching can be

ignored (Fig. 9). Stretching only becomes a sig-

nificant mechanism when

t/l >

1 which is

obviously unrealistic for an open honeycomb

structure, or unless KS is decoupled from Kh and

Z$ as in molecular structures. Kf (eqn (4)) has

the lowest value and thus explains why the flex-

ure model has been so successful. However, as

already stated the properties of the material

operating as the hinge can be significantly dif-

ferent from those of the bulk cell wall material.

If they are lower, as might be expected, due to

local damage from the folding of the card then

Kh 0 and h = 21, 0 < 0 enabling a conventional

hexagonal cell honeycomb to be compared, on

the same plot, with an auxetic honeycomb con-

taining cells of approximately the same area.

The density of the auxetic honeycomb is higher

than the conventional structure. Figures lo-12

show the elastic properties E, v and G vary with

the cell angle 0 for each model. In each case

E,Y=

I = b =

1,

t = 0.1.

The respective force con-

stants are &

= O-001, Kh = 0.03 (shear) and

KS = 0.1. The discontinuities at 0 = 0 occur

because of the change in the value of h.

The E versus 0 plots (Figs 10 and 11) for the

flexure and hinging models are of identical

shape as expected from comparing eqns (17)

and (55) for E,, and (18) and (50) for

E2.

The

difference in numerical values is determined by

0.06

E

0.04

0.02

0.00

-90

-60

-30 0

30 60 90

0 (Deg)

8.0

6.0

4.0

2.0

v

-90 -60

-30 0 30 60

0 (Deg)

0.010 - ---~._--..~ __._ ,

_I_

0.008

0.006

G

12

o.ooo2

0.004

0.002

-90

-60

-30 0 30 60

0 (De13

Fig. 10.

Plot of

E, v

and G versus 0 for the flexure model

E=l=b=l, t=O+l,

K.=O.OOl,

h=2

for 0O).


13/20

Model s or t he el asti c deforma ti on of honeycombs

415

the respective values of Kj. Square symmetry is

obtained when 0 =

-30 for the re-entrant cells

and when 8 = +30 for the hexagonal cells.

Higher modulus values are obtained for the re-

entrant than the corresponding hexagonal cells

of positive angle. It must be remembered how-

ever that in the case of honeycombs this

0.8

0.6

E

0.4

n.n

._

_.

_i__I~

-90

-60

-30 0 30 r--90

8.0

6.0

4.0

V

-6.0. 12

1

-60 -30

0

30 60

90

0 (Deg)

0.00 1

-90 -60

-30 0

30 60 90

0 (Deg)

increase in modulus is achieved at the expense

of increased weight due to the higher density of

the re-entrant cell honeycomb.

The Poissons ratios for the flexure and hing-

ing models are identical as can be seen from

eqns (19) and (56) (20) and (52). The re-

E,

0.8

0.6

0.4

0.2

0 a

l-O--._

.~

0.8

v2,

0.6

.

\

.

0.4

. .

0.2 -v,*

V 12 0.0

-0.2

-0.4

-0.6

-0.8

_,.o_-b ~l.m-..i-___i mmml__ -

-90 -60

-30 0

30 60 90

8 (De&

0.40

o.ot

1.00

0.80

0.60

0

-60

-30 0 30 60 90

fl (Deg)

Fig. 11.

Plot of E, v and G versus 6 for the hinging model

(E=l=b=1,t=~1,K,,=0-03,h=2forOO).

model (E = 1= b = 1,

t =

O-1, K- = 0.1, h = 2 for O-CO and

h = 1 for $>O).


14/20

416

1. G. Masters, K. E. Evans

entrant cells have a negative Poissons ratio and

square symmetry is again apparent at +30.

Unlike the Youngs modulus and Poissons

ratio the expressions for shear modulus in the

flexure and hinging models are not identical.

This is because in the Gibson model the shear

displacement is entirely due to the flexing of the

walls length

h

and rotation of the cell wall junc-

tion. The walls of length

1

do not bend and their

relative positions do not change. In the hinging

model each cell wall deflects under the applied

1.0

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

1.0

Fig. 13. Polar plots for a regular hexagon (h = I= 1,

Fig. 14. Polar plots for a regular hexagon (A = I= 1,

0 = 30) deforming by flexure or hinging showing the vari-

f1 = 30) deforming by stretching showing the variation of

ation of elastic properties with orientation in the

elastic properties with orientation in the I,Zplane.

E, v

1,2-plane.

E 1

and G are isotropic.

and G are isotropic.

shear force so we need to introduce the con-

stant C so the hinging constant (Ki

I)

for a wall

length h can be written in terms K,, for a wall

length 1. However for the case of a regular

hexagon (0=30, h/1=1) eqns (21) and (68) both

reduce to

1.0

0.8

0.6

0.4

0.2 ,

WKJ, 0.0

0

2 ,

0 . 4 / .

0.6

0.8

1.0

1.0

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

1.0

( -ve values) 0.1

I

,

0.2


15/20

M odels or the elasti c deformati on of hontycomhs

417

This incidently is the same result as that

obtained from the Gillis graphite model when

only hinging occurs, i.e. K, = 30. In both the

flexure and hinging models the shear modulus is

lowered by adopting the re-entrant cell shape.

The value of G is particularly low and insensi-

tive to cell angle in the range 0 > 0 > - 60.

For the stretching model (Fig. 12) the mod-

ulus E, is reduced for the re-entrant cell but E,

remains higher than that obtained from the

/

1.0

1

I

0.5

[EKJ, 0.0 a ; i,(AiL : +-;j$

0.5

.,i,:

:

1.0

\

..I., ;

1.5

2.0. L

id

1~

2.5

Fig. 15.

Polar plots for a re-entrant cell (h = I= 2,

U = -30) deforming by flexure showing the variation of

elastic properties with orientation in the 1,2-plane. Note

that v is negative in value along the principle axes but

positive between them.

corresponding hexagonal cells when - 60 < 0

< -90. Square symmetry occurs at 0 and

+30. This occurs because the walls of length h

have no effect when the cell is loaded in direc-

tion 2.

The significant feature of the stretching

model is that the Poissons ratio is negative

(11 = -l/3) for th

e regular hexagonal cell and

positive (V = +3/17) for the re-entrant cell (h/

2 0

1.5

1.0

WK,l+ i::

0.5

1.0

1.5

2.0

2.5

2.5

2.0

1.5

1.0

N;, ,K,I+ 1::

0.5

1.0

1.5

2.0

2.5

-ve

1.01

u

Fig. 16. Polar plots for a re-entrant cell (/I =l= 2,

0 = -30) deforming by hinging (-

bending, ...

shear) showing the variation of elastic properties with

orientation in the 1,Zplane. Note that 11 s negative in

value along the principle axes but positive between them.


16/20

418

I. G. Masters, K. E. Evans

I= 2, 0 = -30). Again square symmetry is

achieved when

h/l =

1, 0 = 30 and

hll

= 2,

0 = o, and unlike the other models, it is only

for these specific conditions that the relation-

ship

1

v

12=-

V2I

1.6,

1.2

0.8

0.4

[EKJ, 0.0

0.4

0.8

1.2

1.6(

5 0

4.0

3.0

2.0

1.0

W, *KS],0.0

t

1.0

2.0

3.0

4.0

5.0

1.2

0.9 ,

0.6

0.6

0.9

1.2

Fig 17 Polar plots for a re-entrant cell (h = I= 2,

0 = -30) deforming by stretching showing the variation

of elastic properties with orientation in the 1,2-plane.

Note that v is positive in value along the principle axes

.

but negative between them.

Table 1 Cell geometries necessary for square symmetry in

the flexure stretching and hinging models

hi1

ell

Flexure/

angle 0

hinging

Stretching

-30 2.00 3.00

-40

1.56

3.29

-50

1.31

353

-60

1.16

3.73

-70

1.06

3.88

applies. The reciprocal relation v,& = v,,E,

however, holds true for all cell geometries. The

shear modulus for the stretching model is

increased when the cell is re-entrant although

the maximum at -30 is difficult to explain.

Off-axis properties-near isotropic honeycombs

To determine the effect of the load orientation

(4) in the 1,2-plane the elastic properties were

calculated using the transformation equations15

derived for an orthotropic material. Assuming

that a thin sheet of honeycomb behaves as if

under plane stress conditions, and that the com-

pliances s12 and s2, are equal, then the

transformation equations are simplified to

[

1

z

[

1

=z

cos4

=-+cos2~ sin24

4 E,

[&-?I

sin4 4

+ ~52

=2cos2~ sin24

@

1

cos44 +sin44

+

712

(81)

(84

b121+=Kp

[

(cos4~+sin4$)v,,

E,

-cos24sin24

1 1

++_--

1 E2 (312

)I 83)


17/20

M odels or t he el asti c deforma ti on of honeycombs

419

As can be seen from Figs 13 and 14, for the

flexure, hinging and stretching models, the

regular hexagon (h = I, 0 = 30) generates a

material which is truly isotropic in the plane.

For the re-entrant cases (h = 21, 0 = -30)

deforming by flexing and hinging (Figs 15 and

16) the honeycomb is clearly square symmetric,

not isotropic. For the stretching case (Fig. 17)

WKJ,

0 41

0 6

0 8

1 0

0.8,

0 6

0 4

0 2

lG,$J,o.ol

0 2

0 4

0 6

0 8

0 8

0 6

0 4

0 2

Iyl, 0.0

0.2

0 4

0 6

0 8

Fig 18 Polar plots for square symmetric, re-entrant cells

(of the geometries listed in Table 1) deforming by stretch-

ing showing the variation of elastic properties with

orientation in the 1,Zplane. Near isotropy is achieved at

8 = -6O, h/l = 3.73. (- e = -5O, h/l = 3.53. - - -

0 = -6O, h/l = 3.73. 0 = -7O, h/l = 3.88.)

the honeycomb is clearly not isotropic or square

symmetric.

For each mechanism the maximum shear

modulus occurs when E is at minimum, i.e.

when 4 = 45 for flexure (Fig. 15(b)) and hing-

ing (Fig. 16(b)) and when 4 = O, 90 for

stretching (Fig. 17(b)).

For flexure and hinging the Poissons ratio

(Figs 15(c) and 16(c))

is interesting in that it

only remains negative when the direction of the

load lies close to the principal axes. At 45 and

adjacent angles the Poissons ratio becomes

positive. This effect can be clearly seen when a

square specimen of material is stretched diago-

nally. Similarly if a square section of

honeycomb is bent out-of-plane across the diag-

onals little or no secondary curvature is

observed in the orthogonal direction. For

stretching the Poissons ratio is positive along

the principal axes becoming negative at 4 = 45

(Fig. 17(c)).

To investigate the possibility of obtaining a

re-entrant cell honeycomb which is more truly

isotropic E, G and v are plotted against the

orientation $ for the combinations of 0 and

h/f

listed in Table 1. These values are simply

obtained by equating the v12 and v2, equations

for the appropriate model, which is the condi-

tion for square symmetry in the 1 and 2

directions.

For the hinging and flexure models E, G and

v remain highly anisotropic for all conditions.

However, for stretching we notice the symmetri-

cal shape of the E plot (Fig. 18(a)) as a result

of using geometries which ensure square sym-

metry (Table 1). As 0 becomes increasingly

negative the peak value of E at 4 = -45

reduces while the minimum values at 4 = 0 and

4 = -90 increase until approximate isotropy is

achieved at 0 = - 60. At 0 = -70 the struc-

ture is no longer isotropic and has returned to a

square symmetry although the maximum values

now occur at 4 = 0 and 90 with the minimum

at 4 = 45.

The values of G (Fig. 18(b)) are reduced as 0

becomes more negative again achieving

approximate isotropy at at 0 = -60.

Similarly the Poissons ratio (Fig. 18(c))

reduce as 0 becomes more negative until

approximate isotropy is achieved at 0 = - 60

and v = +0*3.

It is apparent then that near isotropy can only

be achieved in a re-entrant cell deforming by a

single mechanism if that mechanism is stretch-


18/20

420

I. G.

M asters, K. E. Evans

Table 2. Values of

K,,IIC,

or re-entrant cells in hinging/stretching model obtained from eqn (84). K,,/& must be positive so

negative values can be ignored

rzir =

1 2 3

4

&.)

(R(L)

( 2,)

( fk,)

(&

( &,

0 0~0000 ~

1 oooo 0.3333 0.1250

- 10

-

0.1745 -0.7041

-

I$470 0.3685

0.0778

~ 20 -0.3491

-

0.4903 -

0.3793

-

0.0558

- 0.1690

- 30

-

0.5236

-

0.3333

0~0000 3&15

- 0.8333

-40 ~ 0.6981 -0.2174

0.2762

-2.5716

-50 - 0.8727 -0.1325

05709 4.1815

-7.7155

- 60

-

1.0472 -0.0718

I.0415 6.1608

- 27.2583

- 70 ~- 1.2217 ~ 0.03 1 1

2.2267 12.1016

- 1462914

-80 - 1.3963 - 0.0077

8.4072 44.8926

- 2405.6680

- 90 - I.5708 0~0000

73 x

%

ing. Is it possible to achieve isotropy if a

combination of deformation mechanisms are

allowed to operate ? It has already been shown

that for the flexing and hinging models E,, E,,

I,~ and vzI are identical functions of & and &,

respectively. It is therefore unlikely that if these

mechanisms were combined that isotropy would

result since the deflections caused by each

mechanism will be additive. We will therefore

consider the combination of stretching and

hinging although as we are working in terms of

Kj we could have equally well chosen flexure.

Taking the expressions for E and v from the

general model equations (72) (76) (77) and

(80) (assuming local hinging, i.e. C = l/4 in eqn

(SO)) and letting K~-+x we obtain a stretching/

hinging model. For square symmetry or isotropy

we know that 1~~= r2, and that E, =E, from

which we obtain the relationship

&=

(h/Z+sin0)2sin20-cos40 .K

cos20[(2h/l + sin()) -(h/l + sin O)2]

(84)

enabling K,, to be written in terms of K,. Evalu-

ating this expression for various values of 0 and

h/f we obtain Table 2. Since K,/K, must be posi-

tive we can ignore the negative values. These

then are the parameters we expect to produce

square symmetry or isotropy. Substituting these

values into the stretching/hinging model equa-

tions and letting KS = 1 we obtain the values of

E and I? listed in Table 3. As expected v,? =

v2,, E,b/K, = E,b/K,. Substituting these values

listed in Table 3 into the transformation equa-

tions we can generate polar plots like that

shown in Fig. 19(a-c). These show the plots for

E, 11 nd G, respectively for the following para-

meters:

0 = - 40, h/l = 2, E ,/K, = O-2710,

E,/K, = 0.2710, 1= -0.3497 and G/K,, = 0.0167.

As can be seen these plots are highly aniso-

tropic although square symmetry along the

principle axes. The other values listed in Table

3 produce similar degrees of anisotropy.

CONCLUSIONS

Three mechanisms can be identified by which

honeycombs can deform, namely flexure, hing-

ing and stretching. Using simple mechanics each

mechanism can be expressed mathematically in

Table 3. Elastic properties for re-entrant, square symmetry cells in a hinging/stretching model calculated using positive

values of K,& listed

in Table 2. (Calculations assume hinging

occurs by local bending i.e. C = l/4 in eqn (69))

0

(rad.)

~ 0.5236

-0.5236

0.5236

- 0.698 I

-0.6981

- 0.698 1

- 0.8727

~ 0.8727

- 0.8727

-

I.0472

-

I.0472

- I.0472

hi1

2

3

2

3

4

2

3

4

2

3

4

c x;;,, cR k

(&

I

1

2

(X%

0~0000

0~0000 0~0000

-

1 oo

~

140

0~0000

-

oG333

0.5799

OG99 0:238 02238

- (I:;47 1

0.2762

0.2710 0.2710

-

0.3497 - 0.3497 0.0406

3.8815

0.4688 0.4688 0.1714 0.1714

0526 1

-2.5716

0.5354 0.5354 0.3662 0.3662

- 0.8549

0.5709

0.3615 0.3615 -0.1338 -0.1338 0.0884

4.1815

05198 0.5198 0.1948

0.1948

0.4267

-7.7155

0.5896 0.5896 0.3279

0.3279

0.6986

1.0415

0.4545 0.4545 0.0078 0.0078 0.1660

6.1068 0.6285 0.6285 0.2276 0.2276 0.3283

-

272583

0.7171 0.7171 0.3219 0.3219 0.2678


19/20

M odels for the elasti c deformat ion

of

oneycombs

421

terms of the properties of the cell wall material

and the cell geometry. By writing in terms of

force constants the three mechanisms can be

combined to produce a general model.

Each model can be used to predict the elastic

properties of both hexagonal and re-entrant

cells. Regular hexagonal cells are truly isotropic

in the 1,2-plane for all three deformation mech-

anisms. Re-entrant cells are highly anisotropic;

0.3

0.2

0.1

W,/KJ, 0.0

0.1

0.2

0.3

0.3

0.2

0.1

[G, ,/q, 0.0

0.1

0.2

0.3

0.5

0.4

0.3

0.2

0.1

Id, 0.0

0.1

0.2

0.3

0.4

0.5

Fig. 19.

Polar plots for square symmetric, r-e-entrant cell

(0 = -4o,

h/l =

2) deforming by hinging and stretching.

E, v and G are not isotropic.

even when square symmetric the off-axis

properties vary considerably.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the finan-

cial support of the Engineering and Physical

Sciences Research Council of the UK. K. E.

Evans currently holds an EPSRC Advanced

Fellowship.

REFERENCES

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2.

3.

4.

5.

6.

7.

8.

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16.

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Models for Elastic Deformation of Honeycombs

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