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Transcript of 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
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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
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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).
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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).
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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
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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.
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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$)v,,
E,
-cos24sin24
1 1
++_--
1 E2 (312
)I 83)
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M odels or t he el asti c deforma ti on of honeycombs
419
As can be seen from Figs 13 and 14, for the
flexure, hinging and stretching models, the
regular hexagon (h = I, 0 = 30) generates a
material which is truly isotropic in the plane.
For the re-entrant cases (h = 21, 0 = -30)
deforming by flexing and hinging (Figs 15 and
16) the honeycomb is clearly square symmetric,
not isotropic. For the stretching case (Fig. 17)
WKJ,
0 41
0 6
0 8
1 0
0.8,
0 6
0 4
0 2
lG,$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-
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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
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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.
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