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1 Copyright © 2011 by ASME
Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference
IDETC/CIE 2011 August 29 – August 31, 2011, Washington, DC, USA
DRAFT: DETC2011-48790
SHEAR COMPLIANT HEXAGONAL CELLULAR SOLIDS WITH A SHAPE MEMORY ALLOY
Jaehyung Ju Research Associate
Department of Mechanical Engineering Clemson University
Clemson, South Carolina 29634-0921 jju@clemson.edu
Joshua D. Summers Associate Professor
Department of Mechanical Engineering Clemson University
Clemson, South Carolina 29634-0921 jsummer@clemson.edu
ABSTRACT In this study, hexagonal honeycombs with a shape memory
alloy (SMA) are explored for super-compliant meso-structural
design. A nitianol (NiTi) SMA based shear compliant
hexagonal cellular materials are introduced and their elastic
properties in shear are investigated. The constitutive relation of
SMA and Cellular Materials Theory (CMT) are used to develop
analytical constitutive equations of SMA honeycombs under
isothermal shear loading. A fixed volume based SMA
honeycombs are designed with a target shear modulus, (GA*)12,
of 10MPa and minimum uni-axial moduli (E11* and E22
*) of
10MPa. About 27 to 70% of elastic shear strains are obtained
with NiTi SMA honeycombs when they are designed with a
G12*of 10MPa.
Keywords: Shape Memory Alloys, Honeycombs, Auxetic,
Cellular Solids, Meso-Structure Design, Compliant Design
1 INTRODUCTION
Nitanol (NiTi) shape memory alloys (SMA) have been
paid much attention associated with the two extraordinary
properties - shape memory effect and superelasticity (often
called pseudoelasticity) for actuator applications [1]. For light
weight and low stiffness purposes, porous SMAs have also
been investigated with a sintering manufacturing technique [2-
4]. The porous SMAs with a low modulus have been suggested
for artificial bones to minimize the modulus mismatch between
human cancellous bones (~3GPa) and conventional metallic
implant (~110GPa) [4].
With an invent of a new metallurgical bonding method
called the reactive eutectic brazing using niobium, an ultra-
superelastic property appears to be used in many practical
applications with SMA honeycombs [5, 6]. Several researchers
have modeled and prototyped hexagonal [7-10] and chiral [11,
12] honeycombs with SMAs to investigate in-plane uni-axial
elastic properties of SMA honeycombs.
Cellular materials provide unique elastic properties - low
modulus, high strength, high strain or combination of those
properties depending on meso-structures [13, 14]. Some
researchers turn their interest from simple light weight design
with cellular solids to compliant structural design for aircraft
morphing applications [15, 16]. The cellular material design
group of Clemson University (CU) has been working on
compliant cellular structures with metals and polymers for a
high elastic shear stain at a given gross modulus [17-22].
Motivated by the SMA‟s isothermal superelasticity, we explore
shear compliant properties of SMA honeycombs while
considering geometric constraints of cellular materials.
In this study, constitutive relations of SMA honeycombs
are developed by decomposing austenite and martensite
material properties of a NiTi SMA and by using cellular
materials theory (CMT) [22-25]. Under a given gross volume
of cellular structures, effective properties of SMA honeycombs
are obtained from the developed constitutive equations. For a
target shear modulus, (GA*)12 of 10MPa and minimum uni-axial
moduli, E11* and E22
* of 10MPa, shear strains of SMA
honeycombs are obtained and compared with metallic cellular
solids. About a 70% maximum shear strain is obtained with
SMA auxetic honeycombs, which are 4.2 and 5 times higher
than those of Ti-alloy and Al-alloy auxetic honeycombs,
respectively.
2 MODEL DEVELOPMENT
NiTi SMA compliant hexagonal structures are investigated
by combining CMT with the NiTi SMA‟s constitutive equation.
Effective properties of NiTi SMA honeycombs are obtained
from a linear elastic model of NiTi SMA.
2 Copyright © 2011 by ASME
2.1. In-Plane Effective Properties of NiTi SMA Hexagonal Honeycombs
A uni-axial stress-strain behavior of SMA under the
isothermal deformation is often simplified as a two step linear
elastic model which is given by
A M AE E E E (1)
where, EA and EM denote the moduli at the austenite and
martensite phases, respectively. ξ is the martensite phase
fraction. For ξ=1, which is the maximum elastic point, the
constitutive relations are expressed as
,
,
forfor
A A
A A M A A
E
E E
(2)
where, εA is the starting strain of the martensite phase.
The constitutive relation of a NiTi SMA is plotted in
Figure 1 which is similar to the conventional elasto-plastic
behavior [5,6,8]. Elastic behaviors of an aluminum alloy (Al-
7075-T6) and a titanium alloy (Ti-6Al-4V) are plotted as well
for comparison of our previous compliant cellular design with
the current SMA honeycombs, which will be discussed in
Section 3.2
The behavior is initially affected by the austenite
microstructure and followed by the martensite one. The
maximum tensile strain recovery of a NiTi SMA is in the range
of 5~8% in a low cyclic loading and 2.5~3.2% for a high cyclic
one [5,6,8]. In the current study, a yield strain, εys of 3.1% is
used as the elastic limit of a NiTi SMA considering a high
cyclic loading.
Figure 1. Tensile stress-strain relation of NiTi SMA [5,6,8], Al-
7075-T6[26-28], and Ti-6Al-4V [26, 29]
Cell geometries with conventional and auxetic hexagonal
honeycombs are shown with the honeycomb layer height (H)
and the honeycomb layer length (L) in Figure 2. The critical
geometric parameters include the cell angle (θ), the vertical cell
length (h), the inclined cell length (l), and the wall thickness (t).
Figure 2. Unit Cell Geometry for (a) Conventional and (b) Auxetic
Hexagonal Honeycombs
The effective moduli of NiTi SMA honeycombs of a two
step linear model can be modified from CMT [23-25] and are
given by
3
*
11 2
cos
sin sinA A M A
tE E E E
hll
(3)
3
*
322
sin
cosA A M A
ht l
E E E El
(4)
3
*
212
sin
1 2 cosA A M A
ht l
G E E El h h
l l
(5)
where subscripts A and M denote „austenite‟ and „martensite‟,
respectively. The superscript „*‟ denote the effective property.
Therefore, (EA*)11 and (EM
*)11 are the effective tensile moduli in
the x1 direction at the austenite and martensite phases,
respectively. Similarly, (EA*)22 and (EM
*)22 are the effective
tensile moduli in the x2 direction at the austenite and martensite
phases, respectively. (GA*)12 and (GM
*)12 are the effective shear
moduli at the austenite and martensite phases, respectively.
The primary initial failure mode of metallic hexagonal
honeycombs under tensile and shear loading is known to be the
plastic collapsing of cell walls [21-23]. CMT assumes that
honeycombs start collapsing plastically when the bending
moment in the cell walls reaches the fully plastic moment and
provides a yield point of a honeycomb as a function of a
material‟s strength. The in-plane effective plastic collapse
strengths of NiTi SMA hexagonal honeycombs are modified
from CMT and are given by:
0
100
200
300
400
500
600
700
800
900
1000
0% 1% 2% 3% 4%
σ(M
Pa)
ε
NiTi SMA
Al-7075-T6
Ti-6Al-4V
3 Copyright © 2011 by ASME
2
*
112 sin sin
pl A A M ys A
tl
E Eh
l
(6)
2
*
222 2cospl A A M ys A
tl
E E
(7)
2
*
12 4 cospl A A M ys A
tl
E Eh
l
(8)
where σys is the yield strength of the solid wall. The subscript
„pl‟ denotes „plastic‟. (σpl*)11, (σpl*)22, and (τpl*)12 are the
effective yield strengths in the x1, the x2 and the shear
directions, respectively.
The corresponding in-plane elastic strains of NiTi SMA
hexagonal honeycombs are given by
2
*
* *11
11 112 sin sin
M ys AA Apl
A M
tEE l
hE El
(9)
2
*
2* *22
22 222cos
M ys AA Apl
A M
tEE l
E E
(10)
2
*
* *12
12 124 cos
M ys AA Apl
A M
tEE lhG G
l
(11)
where (εpl*)11, (εpl
*)22, and (τpl*)12 are the elastic strains of NiTi
SMA honeycombs in the x1, the x2 and the shear directions,
respectively.
The relative density, often used for the cellular material
design, is only a function of cellular geometry is given by [6]:
* 21
2 cos sins
ht l
hll
(12)
It should be noted that the analytical expressions for the
effective properties and maximum effective strains are
restricted to be used in the linear elastic range.
Table 1. Properties of a NiTi SMA[5,6,8]
Base Material
ρs
[kg/m3]
Es [GPa]
EM [GPa] νs
(σys)A
[MPa]
(σys)M
[MPa]
NiTi SMA 6400 [4] 75 11 0.3 262 309
2.2. Design of Honeycombs for a Given Target Shear Modulus and Constraints
When designing cellular meso-structures considering load
carrying capability, cell geometry should be designed based on
target effective properties, e.g., effective moduli. We first set a
design target; a shear modulus of an elastomer, e.g., G12* of
10MPa.
The layer height, H and the layer length, L of honeycombs
is chosen to be 50mm in the x1 and x2 direction as shown in
Figure 2. This aligns with other design considerations of the
structure that are outside the scope of this paper. For a given H
and L, the cell angle, θ, the vertical cell length, h, and the
inclined cell length, l are defined as
4 cos
Ll
N (13)
sin2
Hh l
M (14)
where M and N are the number of unit cells in the vertical and
horizontal directions (the x1 and x2 directions in Figure 2). In
the present study, the integer M and N are varied from 1 to 3.
Honeycomb geometries with M=N=3 are shown in Figure 3 as
exemplary honeycomb structures.
(a) θ=15o
(b) θ=30o
(c) θ=45o
(d) θ=60o
(e) θ=0o
(f) θ= -15o
(g) θ= -30o
(h) θ= -45o
(i) θ= -60o
Figure 3. Honeycomb geometries with the same gross area design
There are additional constraints for the honeycomb design.
The simple beam theory, which is used in CMT, is valid for t/l
or t/h <1/4 [23-25]. Related to the manufacturing limitation, a
4 Copyright © 2011 by ASME
minimum cell wall thickness should be set. In this study, the
minimum cell wall thickness of a NiTi SMA is set to be 0.1mm
considering the wire Electrical Discharge Machining (EDM)
technique. Cell angles also have a limitation to avoid collision
of the vertical cell walls with the adjacent inclined cell walls;
we set the range of cell angle as 70 70o o . To avoid
elastic nonlinear deformation such as elastic buckling, the
relative density should be high, which means the cell wall
thickness should be high. For example, Thompson et al. used
the minimum criteria of the relative density of 0.07 when they
set a point, which may be used in this study [30]. The goal of
the shear compliant honeycomb design is to maximize the shear
yield strain, (γpl*)12 of hexagonal honeycomb meso-structures at
a given target (GA*)12. As an additional in-plane stiffness
requirement, minimum uni-axial tensile moduli are designed to
be higher than 10MPa; (EA*)11>10MPa and (EA
*)22>10MPa.
Those properties are related to our specific product requirement
which is outside scope of this paper.
When multiple geometric options are available to satisfy
the goals, the ones that minimize the relative density are
selected for a light weight design. The detailed formulation of
the shear compliant honeycomb design is shown in Figure 4.
Given:
Target property: effective initial shear modulus,
(GA*)12(=10MPa)
Property of constituent material: NiTi SMA - Es, νs and
ys
Geometry: Layer height (H) and layer length( L)
(=50mm) and the number of vertical and horizontal
cells, M, N (=1, 2 and 3)
Find:
Honeycomb geometry of *
120.6pl : t, h, l, and θ
Satisfy:
Constraints:
0.1t mm (manufacturing constraint)
2 sinh l (negative cell angle constraint)
Relative density must be greater than 0.07 to avoid
buckling: *0.07 s
0.25t l
0.25t h
70 70o o
Requirements:
(EA*)11>10MPa, (EA
*)22>10MPa
Goals:
Maximize the effective shear yield strength, *
12pl
Maximize the effective shear yield strain, *
12pl
Minimize:
Relative density of honeycombs
Figure 4. Word formulation of the shear compliant NiTi SMA
honeycomb design
3 RESULTS AND DISCUSSION
In this section, resultant cellular geometries with NiTi
SMA are presented when honeycombs are designed with the
constraints and requirements of Figure 4. Corresponding ultra-
superelastic property in shear of NiTi SMA honeycombs is also
presented.
3.1. Cellular Geometries of NiTi SMA Hexagonal Honeycombs
Figure 5 shows the cell wall thicknesses, t of honeycombs
as a function of the number of vertical and horizontal cell
stacks, which are noted to be M and N, respectively. The t
varies from 0.5mm to 2.5mm depending on cell geometries. At
a lower cell angle, θ, a higher t is needed to have the target
shear modulus, (GA*)12 (=10MPa). On the other hand, a lower t
is required at a higher θ to maintain the target shear modulus
(Figure 5).
The 1×N stack shows many options in honeycomb design
due to the lower limitation with re-entrant cellular geometires;
e.g., satisfying a geometric condition of h>2l|sinθ |as shown in
Figure 5 (a). On the other hand, design with the 2× N and the
3× N stacks shows a limitation with negative cell angles related
to the geometric constraint as shown in Figures 5(b) and 5(c).
For example, honeycombs with -45o or a lower cell angle for
the 2x1 design and honeycombs with -30o or a lower cell angle
for the 1x1 design are not available because they do not satisfy
the geometric constraint, h>2l|sinθ|.
For the 1×N stack, a higher cell wall thickness (1.08 to
2.57mm) is required to keep the effective moduli;
(GA*)12=10MPa, (EA
*)11>10MPa, and (EA*)22>10MPa. For the
2×N and 3×N stacks, a relatively lower cell wall thickness
(0.77 to 1.82mm for 2×N and 0.71 to 0.35mm for 3×N) are
required to maintain the (GA*)12.
NiTi SMA honeycombs with a higher cell angle (30o or a
higher cell angle for the 1×1 stack, 15o or a higher cell angle for
the 2×1 stack, and 0o or a higher cell angle for the 3×1 stack)
are not available because they do not satisfy the uni-axial in-
plane modulus requirements, (EA*)11>10MPa and
(EA*)22>10MPa.
5 Copyright © 2011 by ASME
Figure 5. Cell wall thickness, t of NiTi SMA honeycombs for a
target shear modulus, (GA*)12 of 10MPa
Figure 6 shows the vertical cell length, h of NiTi SMA
honeycombs as a function of number of M and N. The h varies
from 6.4 to 46.7mm depending on cell geometries. The h shows
a similar trend as t for varying cell angle, θ; a higher h at a
lower θ and a lower h at a higher θ (Figure 6).
For the 1×N stack, a higher h (17.5 to 46.7mm) is required
to keep the effective moduli; (GA*)12=10MPa, (EA
*)11>10MPa
and (EA*)22>10MPa. For the 2×N and 3×N stacks, a lower h
(9.0 to 23.4mm for the 2×N and 6.4 to 14.6mm for the 3×N) are
required to maintain the effective moduli.
Figure 7 shows inclined cell length, l of honeycombs as a
function of the M and N. The l varies from 4.6 to 25.0mm
depending on cell geometries. At a lower cell angle, θ, a higher
l is needed to have the target shear modulus, (GA*)12 (=10MPa).
The l decreases with an increasing θ and slightly increases at a
0o or a higher θ.
For the 1×N stack, a higher l (4.2 to 25.0mm) is required to
keep the effective moduli; G12*=10MPa, E11
*>10MPa, and
E22*>10MPa. For the 2×N and 3×N geometries, a lower h (4.2
to 16.3mm for the 2×N and 4.2 to 13.8mm for the 3×N) are
required to maintain the effective moduli. It should be noted
that the lowest l is found at a θ of 0o with the N×3 stack (Figure
7).
0
0.5
1
1.5
2
2.5
3
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
cell
wal
l th
ickn
ess
, t(m
m)
cell angle, θ (degree)
1X1
1X2
1X3
(a)
0
0.5
1
1.5
2
2.5
3
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
cell
wal
l th
ickn
ess
, t(m
m)
cell angle, θ (degree)
2X1
2X2
2X3
(b)
0
0.5
1
1.5
2
2.5
3
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
cell
wal
l th
ickn
ess
, t(m
m)
cell angle, θ (degree)
3X1
3X2
3X3
(c)
6 Copyright © 2011 by ASME
Figure 6. Vertical cell length, h of NiTi SMA honeycombs for a
target shear modulus of 10MPa
Figure 7. Inclined cell length, l of NiTi SMA honeycombs for a
target shear modulus of 10MPa
0
5
10
15
20
25
30
35
40
45
50
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
vert
ical
ce
ll le
ngt
h ,
h(m
m)
cell angle, θ (degree)
1X1
1X2
1X3
0
5
10
15
20
25
30
35
40
45
50
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
vert
ical
ce
ll le
ngt
h ,
h (
mm
)
cell angle, θ (degree)
2X1
2X2
2X3
0
5
10
15
20
25
30
35
40
45
50
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
vert
ical
ce
ll le
ngt
h ,
h (
mm
)
cell angle, θ (degree)
3X1
3X2
3X3
0
5
10
15
20
25
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
incl
ine
d c
ell
len
gth
, l(
mm
)
cell angle, θ (degree)
1X1
1X2
1X3
0
5
10
15
20
25
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
incl
ine
d c
ell
len
gth
, l(
mm
)
cell angle, θ (degree)
2X1
2X2
2X3
0
5
10
15
20
25
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
incl
ine
d c
ell
len
gth
, l (
mm
)
cell angle, θ (degree)
3X1
3X2
3X3
7 Copyright © 2011 by ASME
3.2. Superelasticity of NiTi SMA Hexagonal Honeycombs under In-Plane Shear
Combining compliant design of honeycombs with the
superelasticity of a NiTi SMA allows a NiTi SMA honeycomb
super-compliant. Figure 8 shows elastic constitutive relations
of NiTi SMA honeycombs (θ=-60, -35, -10, and 15o with the
1x1 stack design) when a linear elastic behavior of honeycombs
is assumed. SMA‟s two phase (austenite and martensite in
Figure 1) is also noticed in the SMA honeycomb behavior as
shown in Figure 9. The constitutive relation may not be
accurate enough to describe the nonlinear stress-strain behavior
of honeycombs associated with the local cell wall micro-
rotation. According to our previous study on metallic
honeycombs, the CMT based constitutive relations of
honeycombs in shear overestimate an elastic strain in shear by
about 20% [19, 20]. In spite of the overestimation of elastic
limit of honeycombs, CMT is known to be accurate to
qualitatively rank the shear flexibility of honeycombs [19, 20].
A finite element based numerical analysis is planned for our
near future study.
Figure 8. Elastic stress-strain behaviors of honeycombs in shear
(1X1 design)
Figure 9 shows elastic limits of SMA honeycombs in shear for
all available geometries while meeting the design requirements
in Figure 4. As can be seen in Figure 8, high elastic limits in
shear are achieved with NiTi SMA honeycombs; shear strains
of 27.4% to 70.8% depending on cell geometries. A yield strain
of about 70% is obtained at a highly negative cell angle. The
re-entrant cell geometry is known to have a high shear flexible
property [21, 22]. NiTi SMA honeycombs with a high cell
angle even show high shear strains up to 38.8% due to the NiTi
SMA‟s superelasticity. The maximum shear strain (70.8%) of a
NiTi SMA honeycomb is about 4.2 and 5 times higher than
those of Ti-alloy and Al-alloy honeycombs, respectively [20].
Figure 9. Yield strains of SMA honeycombs
Michailidis et al. reported a uni-axial compressible elastic strain
of 41.7% with a NiTi SMA honeycomb in their finite element
based analysis when the honeycomb has a uni-axial modulus of
about 4MPa [8]. The present study on the shear flexibility of
SMA honeycombs also show a possibility of SMA
honeycombs‟ use for the superelastic applications. Numerical
validation with finite element analysis and experimental
validation are scheduled for our future works.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.0 0.2 0.4 0.6 0.8
she
ar s
tre
ss, τ
* (M
Pa)
shear strain, γ*
theta=-60
theta=-35
theta=-10
thata=15
0%
10%
20%
30%
40%
50%
60%
70%
80%
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
yie
ld s
trai
ns
of
ho
ne
yco
mb
s, (γ*
)pl
cell angle, θ (degree)
1X1
1X2
1X3
0%
10%
20%
30%
40%
50%
60%
70%
80%
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
yie
ld s
trai
ns
of
ho
ne
yco
mb
s, (γ*
)pl
cell angle, θ (degree)
2X1
2X2
2X3
0%
10%
20%
30%
40%
50%
60%
70%
80%
-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70
yie
ld s
trai
ns
of
ho
ne
yco
mb
s, (γ*
)pl
cell angle, θ (degree)
3X1
3X2
3X3
8 Copyright © 2011 by ASME
4 CONCLUDING REMARKS
In this study, we introduced NiTi SMA honeycomb design for a
ultra-superelastic compliant application in shear. Under a target
shear modulus of 10MPa and minimum uni-axial moduli of
10MPa, shear compliant NiTi SMA honeycombs were found
under a fixed volume. The major findings in this study are as
follows:
About 27 to 70% of elastic shear strains are obtained
with NiTi SMA honeycombs when they are designed
with a (GA*)12 of 10MPa;
Two phases (austenite and martensite) are identified in
the constitutive relations of NiTi SMA honeycombs;
The 1xN stack design of honeycomb is preferable in
terms of covering broad range of cell angles to satisfy
the requirements of minimum uni-axial moduli - (EA*)11
>10MPa and (EA*)22>10MPa;
Finite element based numerical validation and physical tests for
experimental validation are planned for our future work.
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