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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