tailorable thermal expansion hybrid structures

International Journal of Solids and Structures 46 (2009) 2372–2387

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International Journal of Solids and Structures

journal homepage: www.elsevier .com/ locate / i jsols t r

Tailorable thermal expansion hybrid structures

George Jefferson a,*, Triplicane A. Parthasarathy b, Ronald J. Kerans a

a Air Force Research Laboratory, Materials and Manufacturing Directorate, AFRL/RXLN, Wright–Patterson AFB, OH 45433, USAb UES, Inc., Dayton, OH 45432, USA

a r t i c l e i n f o

Article history:Received 7 January 2008Received in revised form 21 March 2008Available online 31 January 2009

Keywords:Tailorable thermal expansionComposite structuresBounds methodsThermoelasticityFinite element analysisParametric design

0020-7683/$ - see front matter Published by Elsevierdoi:10.1016/j.ijsolstr.2009.01.023

* Corresponding author. Tel.: +1 937 255 1307.E-mail address: [email protected] (G.

a b s t r a c t

A design concept is presented for a macro or microstructure that combines materials with differing ther-mal expansion to achieve an overall effective expansion that differs substantially from either of the con-stituents. Near-zero-CTE and isotropic negative expansion designs are achieved by creating compliantstructures where overall expansion is compensated by internal bending deformation. Such structureshave application where dimensional stability is required when subject to large thermal gradients, e.g.space mirrors. In this paper, we present closed form analytic expressions for prediction of the effectiveexpansion, and consequent internal stressing, of the structure, as well as several finite element simula-tions that demonstrate the design performance under non-uniform thermal load.

Published by Elsevier Ltd.

1. Introduction

In the design of most engineering structures which are subjectto significant temperature excursions or gradients, the thermalexpansion behavior of the structure is of key importance. The con-stituent material’s coefficient of thermal expansion (CTE) is thusone of the driving material properties considered in the engineer-ing materials selection and application process. The possibility ofcontrolling the thermal expansion of a sub-element will allowgreat design flexibility, and is the basis of this work. The mannerin which thermal expansion influences design varies with the spe-cific application. A few current applications of high technologicalinterest are outlined below, before presenting an exploration ofpossible solutions for tailored structures.

In applications, such as supports for space-based mirrors,dimensional stability under extreme variations in temperature aswell as steep gradients is a key consideration (Schuerch, 1972; Jac-quot et al., 1998). That is, ideally the structure should exhibit verylittle dimensional change when subjected to substantial changes intemperature that occur as the structure is exposed to changingradiation conditions. Another class of design challenge arises instructures subjected to space-varying thermal gradients, such asengine components subjected to hot combustion environment orhypersonic airframe surfaces subjected to aerothermal heating.The resulting gradient in thermal strain can result in design-limit-ing thermally induced stressing. In such cases, stresses may be re-duced by specifically varying the material’s CTE for compatibility

Ltd.

Jefferson).

with the thermal gradient. Additionally, structures which are spe-cifically designed to undergo significant thermally driven shapechange may be utilized as actuators or actively deforming aerostructures.

The ‘menu’ of intrinsic CTEs available with structural engineer-ing materials is, however, quite limited. In most cases availablematerials with the ideal expansivity for a given application are lessthan ideal for other reasons. For example, intrinsically low expan-sion glasses currently used for space mirror supports are inferior toceramic composites in terms of weight, fabricability and ultimatetemperature capability. As another example, ultra-high-tempera-ture ceramics (UHTCs) appear to be well suited to hypersonic lead-ing edge applications, however, their expansivity is too large forcompatibility with the much cooler support structure. It is there-fore desirable to design composite structures (macro- or micro-)with effective expansivities that are substantially different fromthat of the constituent materials. With these motivations, a com-posite material/structural design concept was explored to deter-mine the merits and limitations of such concepts. In particular,the possibility of devising structure that allow fabrication of struc-tures with tailored or designed-in CTE values is examined. Analyt-ical expressions are derived that will enable designing with thesestructures practical. The derivations are verified using finite ele-ment methods.

2. Background

Early attempts to design thermally stable structures formacro-scale applications resulted in only one-dimensional ther-mal stability and required pin-jointed designs (Schuerch, 1972).

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Nomenclature

L periodic structure characteristic lengthW internal angleR, C, Q position identifiers in periodic structureDT uniform temperature changeas, ah thermal expansion coefficientEs, Eh elastic modulusqs, qh mass densityAs, Ah cross section areaIh section moment of inertiats, th in plane thickness for extruded designz extrusion depthLh, Ls component lengthL�h curved-face arc lengthy(x) shape of curved facey0 amplitude of y(x)FC force in strut at point CFR; eF R reaction force in honeycomb at point R, normal and par-

allel to line R–CFRx, FRy forces resolved in directions R–R and Q–CN, M force and bending moment in curved faceU, Us, Uh complementary thermoelastic strain energy per unit

cell, total and component values

A1, A2, A3, B1, B2 constants appearing in energy expressionsuR displacement of point R relative to C in direction R–Cet effective thermal strain�a effective thermal expansionH normalized effective CTEr, rmax stress in curved face, max valuen thickness coordinate in curved faceeb effective strain due to biaxial applied stressingF̂ applied force per unit lengthMc unit cell massAc unit cell areaP stiffness indexamean, aratio mean and ratio of constituent CTEE, m constituent elastic modulus and Poisson ratiof, d area void area and solid area fractionsK;KU

HS 2D plane bulk modulus, effective and Hashin–Shtrik-man upper bound

Subscriptss, h constituent values, s – ‘strut’, h – ‘honeycomb’

G. Jefferson et al. / International Journal of Solids and Structures 46 (2009) 2372–2387 2373

More recently Steeves et al. (2007) have devised an isotropicpin-joint approach. Such linkage-based concepts are useful formacro-scale structures, however the complexity of fabricatingpin joints makes them less than ideal for consideration onmicrostructural scales. Non-jointed concepts, applicable to bothmacro- and micro-scale fabrication have been developed by Chenet al. (2001) who utilized a topology optimization scheme to de-velop a two-phase voided microstructure, and by Lakes (1996,2007) who devised a scheme for assembling bi-material stripsinto a structure with tailorable expansion. The model of Chenet al. has been demonstrated experimentally by Qi and Halloran(2004). Steeves et al. (2007) have also proposed ‘bonding’ thejoints solid in their pin-based model, however this results in sig-nificant stress concentrations in addition to reduced thermalexpansion performance.

All of these concepts, including the one presented in this paper,accommodate thermal expansion through the utilization of voidspace. Bounds for the stiffness of such voided, thermally tailored,structures are well established (Schapery, 1968; Rosen and Hashin,1970; Gibiansky and Torquato, 1997). The bounds prove that, gen-erally, the further the required deviation from the intrinsic constit-uent thermal expansion behavior, the lower the stiffness must be.Consequently some of the applications listed as ‘motivational’which are highly structural-integrity-driven will not likely be filledby any of these voided structures.

For completeness it should be noted that zero and otherwiselow/tailorable expansion materials/structures can be made with-out voids by use of intrinsically low or negative expansion materi-als such as ZrW2O8 and related ceramics (Korthuis et al., 1995;Mary et al., 1996; Evans et al., 1998) or Invar, alone or in a compos-ite form (Jacquot et al., 1998; Lommens et al., 2005). The voidedtailorable composites discussed in this paper will thus be primarilyof interest where these intrinsically low expansion materials arenot suitable for reasons such as fracture toughness, or limited tem-perature range of negative expansion (Steeves et al., 2007). SeeLakes (2007) and Steeves et al. (2007) for general order-of-magni-tude design space comparisons of intrinsic and voided-compositematerials.

3. Conceptual framework of tailorable design

The key idea examined in this paper relates to the possibility ofusing flexural rotation of components accommodating the overallexpansion of the structure. The idea is similar to that exhibitedin compounds that exhibit negative thermal expansion, e.g.ZrW2O8, where some of the units within the lattice rotate withincreasing temperature (Evans et al., 1998).

A straightforward way to introduce rotations in a fabricatedstructure is by bending of slender elements. In Lakes’ tailorableexpansion model, a lattice is assembled of (initially curved) bi-material strips. Where the two materials have a difference in ther-mal expansion coefficient, the individual strips flex. The change incurvature of an individual load-free strip is predicted by Timo-shenko’s (1925) analysis of a bi-metal thermostat strip, and thechange in length can hence be calculated (Lakes, 1996). Whenthese strips are assembled into a two- or three-dimensional latticethe overall expansion coefficient of the lattice will be exactly thesame as the relative lengthwise expansion of an individual strip– provided that no additional forcing occurs between individualmembers as a result of their relative deformation. Two infinitelyperiodic plane structures satisfying this requirement have beenidentified by Lakes (1996, 2007) which are square and hexagonalplane tessellations (The only other regular polygon plane tessella-tion, triangular, will necessarily have opposing rotations at ele-ment ends). Structures not satisfying the equal rotation conditionwill require pin jointing, or at least require an analysis that explic-itly accounts for the interaction.

In our proposed structure we will begin with the two Lakes-type bending element topographies, however induce bending bya secondary structural element inserted into each cell. An exampleof the structure is shown in Fig. 1. A continuous honeycomb-likestructure is fabricated of one material. A second material with dif-ferent CTE (larger, for sake of example) is inserted into each cell asshown. As the structure undergoes a temperature increase, the in-serts expand more-so than the honeycomb and thus put the facesinto bending. The shortening of the faces due to bending effectively

Fig. 2. Schematic representation of hexagonal thermally stable structure. Whiteand hatched areas represent thermoelastic materials with differing coefficient ofthermal expansion. The dark shaded area is a minimal repeating element. Labelsindicate conditions consequent of both symmetry and symmetric loading.

Fig. 1. Example geometry for a planar hexagonal grid thermally stable structure.

2374 G. Jefferson et al. / International Journal of Solids and Structures 46 (2009) 2372–2387

cancels the overall expansion so that, with appropriate specifica-tion of the materials and geometry, a zero, or other design target,effective thermal expansion results. The expansion (zero or other-wise) is notably isotropic in the plane of the ‘honeycomb’.

In this scheme it is possible to design a structure such that theinterface between the dissimilar materials is always in compres-sion. That is, while the inserts may be permanently bonded theycould simply be press-fit. Of course a press-fit arrangement wouldneed to be carefully designed to maintain compression throughoutits full range of thermal and mechanical loading. This feature is dis-tinctly different from the Lakes and Chen et al. topographies wherethe interface between the dissimilar materials must carry shearand or tensile stress, and so must be permanently bonded to eachother. Relaxing the bonding requirement potentially permits theuse of a wider variety of materials combinations as well as differ-ent fabrication schemes.

It is important to note that, while we have preserved the sym-metry necessary to ensure equal rotations (and thus no bendinginteraction) between the bending elements, there will be a netforce between each of the beams and thus the deformation analysismust explicitly account for these interactions. The analysis that fol-lows is thus that of the full periodic structure and not of an isolatedelement.

4. Analysis

A key to the usefulness of the design is the availability of anaccurate model for the behavior that allows the designer to (a)identify and (b) specify the relevant geometric parameters. In con-trast to models derived from numerical optimization methods(Bendsøe and Kikuchi, 1988; Sigmund and Torquato, 1997; Chenet al., 2001), the present model, is based on analytically tractableelements, and leads to a direct method of calculation of effectivethermal expansivity. This analytic result will be desirable bothfor rapid design adjustment as well as convenient for developmentof relatively simple fabrication routes. Note that the bi-materialstrip structure is also analytically tractable (Lakes, 1996).

The tailorable design is modeled by idealizing the structure asan infinite periodic array consisting of two-dimensional beam-likecomponents, and analyzing the response of a unit cell, as shown inFig. 2. The figure shows a hexagonal design in a stress-free (ambi-ent temperature) configuration. The centroid of each hexagonalcell is denoted ‘‘C”, the corners joining the curved faces are denoted‘‘R” and the center of each face is ‘‘Q”. The straight sections, Q–C,are of one thermoelastic material and the curved sections R–Q–Rare of a second material, which in general may have different ther-mal expansion as well as elastic properties. For simplicity of anal-ysis we will take the materials properties to be isotropic, althoughextension to anisotropic materials is straightforward so long as theanisotropy does not alter the symmetry of the structure.

The analysis begins with a characterization of the aspects of thedeformation that are a simple consequence of the periodic symme-

try. As the periodic structure undergoes a deformation resultingfrom a uniform temperature change, due to symmetry, the equilat-eral triangle indicated on the figure will remain an equilateral tri-angle. Clearly segments Q–C remain straight, that is they are puretensile elements which will be referred to as struts, with associatedquantities given a subscript s, i.e. the strut material CTE and elasticmodulus are denoted as and Es, respectively.

Point R remains at the centroid of the triangle, hence the angledenoted W is fixed with the value W = p/3. The R–Q–R segmentsform a continuous honeycomb-like grid (with curved faces), hencethe relevant material parameters will be denoted with subscript h.The specific shape of the curved section is a design variable to bespecified in order to achieve the desired thermal expansionbehavior.

Due to symmetry, the effective thermal expansion of the peri-odic structure will be isotropic, and can be calculated from the rel-ative motion of periodically equivalent points. For example thethermal strain may calculated as et ¼ DCC=CC, where the overbarindicates the length of the segment and D denotes a change inlength. Because R is the triangle centroid, we may also, more con-veniently, calculate the thermal strain as et ¼ DRC=RC. The strainet, so defined, is an overall, effective, or average thermal strain.All points R and C move as if embedded in a continuum whichundergoes a uniform plane dilatational strain et. The material at apoint R rotates, however, as the structure deforms, while the mate-rial at point C does not. All other points in the structure follow amore complex motion, and must be free to do so.

The related square-grid tailored CTE structure is shown sche-matically in Fig. 3. For the square case a nearly identical descrip-tion applies. In this case the dashed square identified on thefigure remains square with points R fixed at the centroid of thesquare and so on. Hence, the remainder of the analysis is the samefor both square and hexagonal configurations, except that in thesquare case W = p/4.

The minimum repeating element is indicated in grey on each ofFigs. 2 and 3, however for convenience of calculation we considerthe larger representative unit cell depicted in Fig. 4. Point C is takenas the origin, that is regarded as fixed. Points R are constrained toslide along the directions defined by W with displacement magni-tude uR as shown. Symmetry requires that there are no bending

Fig. 5. Schematic representation of curved beam.

Fig. 3. Schematic representation of a square-grid thermally stable structure. Labelsindicate conditions consequent of both symmetry and symmetric loading.

Fig. 4. Unit cell used for analysis, with periodic boundary condition shown.


moments at R or C and that the reaction force at C, FC, is directedalong QC and the reaction force at R is normal to the line RC. Thereaction at R is however assumed to have the normal component,FR, and a tangential component eF R, The tangential force eF R must bezero if the thermal expansion of the structure is unconstrained. It isretained here because we will ultimately determine uR by a virtualwork method.

Denoting the length CC as the fundamental unit length of thestructure, L (Figs. 2 and 3) the undeformed dimensions of the unitcell are:

RC ¼ L=ð2 sin WÞOR ¼ L=ð2 tan WÞOC ¼ L=2

ð1Þ

where O is the intersection point of straight line segments R–R andQ–C. The curve R–Q–R has an amplitude, y0, measured from R–O–R.The length of the strut portion of the cell is then Ls = L/2+y0. The se-cant half-length of the curved face OR is denoted Lh, so thatLs = Lh tanW + y0.

The shape of the curve, which is a design parameter, is given bythe even valued function y(x) with x the distance along O–R, whichmust satisfy y(Lh) = 0 and y(0) = y0. The form of y(x) may be se-lected, for example,

y ¼ y0 cosp2

xLh

ð2Þ

or

y ¼ y0 1� x=Lhð Þ2� �

ð3Þ

The remainder of this analysis does not assume a particular form,except as noted when presenting examples.

The forces, FR, FC and displacement uR are then determined fol-lowing a straightforward energy analysis. Since the strut is a sim-ple tensile element the thermoelastic complementary strainenergy is,

Us ¼12

F2CLs

EsAs� asLsDTFC ð4Þ

where As is the cross section area and DT is a uniform temperaturechange. See Tauchert (1974) for a formal treatment of energy meth-ods in thermoelasticity.

Fig. 5 shows a schematic of the curved element. The x–y coordi-nate components of the reaction forces are,

FRy ¼ FR cos W� eF R sin W

FRx ¼ FR sin Wþ eF R cos Wð5Þ

Noting for equilibrium, FRy = FC/2, the x-force is determined from Eq.(5) as,

FRx ¼12

FC tan Wþ eF R

�cos W ð6Þ

The deformation of the curved segment is calculated following theanalysis of a structural arch due to Timoshenko (1945). The approx-imations and assumptions inherent in this beam-theory calculationare discussed in detail by Timoshenko.

The bending moment, M and normal stress N in the curved seg-ment at a point located a distance x from the center are,

jMj ¼ FC

2Lh � jxjð Þ � FRxy ð7Þ

N ¼ FRx cos /þ FC

2sin / ð8Þ

where / is the angle between the curve and the line R–R . The nor-mal force expression is a small / approximation, and shear stressingis neglected following Timoshenko.

The complementary strain energy, Uh in the curved element is,

Uh ¼12

ZS

M2

EhIhþ N2

EhAhþ 2NahDT ds ð9Þ

where Ih is the moment of inertia of the beam section. The integralis evaluated over the arc length of the beam S. Combining Eqs. (6)–(9),

Uh¼L3

h

EhIh

eF 2R

cos2 WA2þ

FCeF R

cos WB1þ

14

B2eF 2

C

!þ 2Lh

cos WeF RþLsFC

� �ahDT

ð10Þ

where the strut length Ls has appeared by substitution for the quan-tity Lh tanW + y0. The constants B1 and B2 are,


B1 ¼ A3 þ tan WA2

B2 ¼ A1 þ 2 tan WA3 þ tan2 WA2ð11Þ

and the A are dimensionless constants, dependent only on the stiff-ness and shape of the curved segment,

A1 ¼1

2Lh

ZS

Ih

AhL2h

sin2 /þ ð1� x=LhÞ2 ds

A2 ¼1

2Lh

ZS

Ih

AhL2h

cos2 /þ ðy=LhÞ2 ds

A3 ¼1

2Lh

ZS

Ih

AhL2h

cos / sin /� 1� x=Lhð Þ y=Lhð Þds

ð12Þ

In deriving Eq. (10) we have evaluated the integrals,RS cos /ds ¼ 2Lh and

RS sin /ds ¼ 2y0, noting that ds = dx/cos /

and 1= cos / ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðdy=dxÞ2

q.

The integrals in Eq. (12) must be evaluated numerically, how-ever by selecting a form for y(x) and series expanding the trigono-metric functions to first order in y0, approximate closed formexpressions are obtained. For example, using Eq. (2) we obtain,

A1 ffi13þ 1

2Ih

AhL2h

p2

y0

Lh

� �2

A2 ffiIh

AhL2h

þ 12

y0

Lh

� �2

A3 ffiy0

Lh

Ih

AhL2h

� 2p

� �2 ! ð13Þ

Fig. 6 shows a comparison of the B constants based on the full inte-gration, Eq. (12), with the approximate form (13). The approximateexpressions are remarkably good for y0/Lh < 1, and for small valuesof Ih=AhL2

h. The full numerically integrated forms have been used forall examples except where noted, however.

The total energy is simply U = Us + Uh and the unknown forceand displacement are determined using Engesser’s theorem (Tauc-hert, 1974),

oUoFC¼ 0

oU

oeF R

¼ 2uR

ð14Þ

Fig. 6. Comparison of dimensionless constants, B1 and B2 based on

After differentiating, the tangential force eF R is set to 0 resulting in apair of linear equations in uR and FC.

Noting that the effective thermal expansion is

�a ¼ et=DT ¼ uR

RCDT¼ uR cos W

LhDTð15Þ

and using the solution to Eq. (14),

�a ¼ ah þ as � ahð ÞH ð16Þ

where H is the relative effective thermal expansion,

H ¼ B1 2EhAh

EsAs

Ih

AhL2h

þ B2

y0=Lh þ tan W

!,ð17Þ

The force in the strut resulting from a temperature change is,

FC

AhEh¼ Ih

AhL2h

2 as � ahð ÞB1

HDT ð18Þ

Usefully, Eqs. (16) and (17), along with the definitions of the A and Bconstants from Eqs. (11) and (12), show that the effective thermalexpansion, relative to the honeycomb material expansion, dependsonly on four non-dimensional quantities, that is

�aah¼ F

as

ah;

EsAs

EhAh;

Ih

AhL2h

;yðxÞLh

( )ð19Þ

If a form for y(x) is selected, i.e. (2) or (3) then the dependencyshown in Eq. (19) reduces to dependency on four constants.

It should be noted that the beam-theory analysis makes noassumption about the cross sectional shapes of the ‘beams’, how-ever in the special case of a simple geometric extrusion with outof plane depth z (see Fig. 1 for example),

Ih

AhL2h

¼ 112

th

Lh

� �2

andAs

Ah¼ ts

thð20Þ

where ts and th are the thicknesses of the strut and honeycomb ele-ments. The dependency in Eq. (19) hence reduces to,

�aah¼ F

as

ah;

Es

Eh

ts

th;

th

Lh;

y0

Lh

� ð21Þ

i.e. there is no dependence on extrusion depth.Further, it is clear that the ‘strut’ is a simple tension member, so

that the analysis is valid for non-slender strut geometries, such as

full integration, Eq. (12) or asymptotic approximation, Eq. (13).


shown in Fig. 1, so long as there is only a relatively small load pointat the connection with the bending element. In that case the quan-tity EsAs is replaced by a generalized force-deflection response ofthe ‘insert’ part of the structure.

Fig. 7 shows the effective expansion calculated from Eq. (16)using for example EsAs/EhAh = 1 and Ih=AhL2

h ¼ 1=200 (correspond-ing to an extruded geometry with th/Lh ffi 0.25), for various valuesof as/ah. With these parameters specified, the amplitude of the cur-vature, y0 (using form (2)), is varied to ‘adjust’ the effective expan-sion of the structure. The insets on Fig. 7 show schematically theperiodic structure for some selected values of y0. It can be seen thatfor large y0 the curvature will interfere with the next cell in thestructure, therefore y0/L must be held to roughly less than 1 (theexact limit depends on particular design features such as the beamcross section shape).

On Fig. 7, it should be noted that each curve is simply relatedto the dimensionless H function by Eq. (16) so that the normal-ized values are all equal to one where H = 0. One such H = 0point occurs for a small positive value of y0/Lh (on the order ofIh=AhL2

hÞ. Hence for y0 near zero the effective expansivity is equalto that of the continuous honeycomb material. For as > ah, ini-tially, increasing y0 reduces �a such that it is lower than eitherconstituent. For as < ah, and for small y0, �a is greater than eitherconstituent. In general the behavior for large and small as willbe opposite, due to the sign change in Eq. (16) so that for theremainder of this discussion we will consider only the as > ah casewith an understanding that generally opposite statements applyto the as < ah case.

The small y0 expansion behavior is as intended. The inducedbending shortens the cell faces resulting in reduced effective expan-sion. When as is sufficiently large (e.g. as = 4ah on the figure) theeffective expansion can be negative, while for smaller values (e.g.as = 2ah) the effective expansion can be reduced but not fully to zero.

However, for large y0, i.e. y0 > �0.5, there is an abrupt reversal inthe trend. In order to understand the behavior for large values of y0

we consider a pin-jointed form of the structure. By inserting a pinjoint at points ‘‘C” (i.e. a triple pin joint between the strut and theseparate halves of the curved face) the structure will be stress-freefor any thermal excursion and its expansivity is readily determinedkinematically to be,

Fig. 7. Effective expansion of a tailorable hexagonal grid structure for Ih=AhL2h ¼ 1

�aah¼ 1� as=ah � 1ð Þ y0

Lh

y0=Lh þ tan W1� y0=Lh tan W

ð22Þ

Note that this pin structure (unlike that of Steeves et al., 2007) willhave no structural integrity and is presented only to aid in under-standing the behavior of the solid joint design. Eq. (22) is alsoshown by dashed lines on Fig. 7. For small y0, Eq. (22) serves as agood approximation to the solid design, for sufficiently smallIh=AhL2

h . Indeed, Eq. (22) can be obtained from Eq. (16) by takingthe limit Ih=AhL2

h ! 0 and taking the honeycomb faces to be flat(y = y0(1 � |x/Lh|). However, the pin-joint structure behavior is sin-gular at y0/Lh = 1/tanW, which is the point where the link Q–R isnormal to the line R–C, (referring to Figs. 2 and 3). Near this config-uration, large motion at ‘‘R” is required to accommodate a small dis-placement at ‘‘Q”, and the required change of sign of the effectiveexpansion upon passing through the singular point can be under-stood from the kinematic requirements.

From Fig. 7, it can be seen that the solid structure behavior fol-lows the same general trend, except that in the neighborhood ofthe ’singularity’ the transition from positive to negative expansionis smooth and bounded. The trend reversal can further be under-stood by noting that the beam end force, FRx, given by Eq. (6) witheF R ¼ 0, is always of the same sign as the bending force FC, andworks against the desired bending-shortening, both by tending toflatten the beam, and by elastically stretching its arc length. Asy0 becomes large the extensional force becomes dominant andthe expansion reduction trend reverses.

Notably, by inspection, the minima occurs approximately wherethe end of the curved beam is normal to the R–C line, which for theexample shown in Fig. 7 occurs at,

y0

Lh¼ 2

p tan Wð23Þ

(The exact minima must be found numerically). The available tailor-ing for larger y0 does not appear to be of practical use. Additionally,as will be discussed in the ‘stiffness and bounding’ section, thestructural stiffness drops off sharply for large curvature. Thereforethe minima point should be regarded as the upper limit on y0 andas well as on the available range of expansion tailoring. Wherethe minima is negative, there will be a single �a ¼ 0 or zero expan-sion design value for y0 which we denote as y�0.

=200 and EsAs = EhAh, for selected constituent CTE ratios as a function of y0.


For completeness Fig. 7 extends to the range y0 < 0, that is a de-sign where the faces curve inward. The analysis here is valid forthat case, however less significant expansion tailoring is achievedin that regime and so it will not be considered further.

Fig. 8 shows the same results for a square configuration, i.e.W = p/4. The square is somewhat less effective in the sense thatgreater curvatures must be introduced to achieve a desired expan-sivity. For this reason the square configuration will not be dis-cussed in significant detail here, although the square geometrymay prove superior in some applications when fabrication issuesare taken into account.

As mentioned, the analysis does not depend on a specific choiceof the curved-face shape. Fig. 9 shows the difference between theforms (2) and (3) for as/ah = 2, Ih=AhL2

h ¼ 1=1200, and EsAs = EhAh

The effect of the specific shape is notably negligible in the regionof interest (small y0), as would be expected. For this reason the

Fig. 8. Effective expansion of a tailorable square-gri

Fig. 9. Comparison of trigonometric and quadratic curvature

remainder of the discussion will utilize Eq. (2), it is however animportant point that for practical purposes the curvature neednot follow a specific mathematical form.

5. Parametric studies

In order to explore the range of useful values of the material andgeometric parameters a numerical root finding algorithm was usedto find y�0 (the first �a ¼ 0 point). Fig. 10 shows y�0 for several valuesof Ih=AhL2

h as as/ah is varied (with EsAs = EhAh). In the limitIh=AhL2

h ! 0 a zero expansion structure can be designed with onlyan approximately 20% difference in intrinsic constituent expansiv-ity. Small values for the beam bending stiffness are undesirable asthey result in low overall stiffness of the structure. IncreasingIh=AhL2

h however, raises the minimum required CTE differential sothat an compromise must be reached in the design process. It must

d structure for Ih=AhL2h ¼ 1=200 and EsAs = EhAh.

forms with Ih=AhL2h ¼ 1=1200, EsAs = EhAh, and as/ah = 2.

Fig. 10. Availability of zero-CTE designs as a function of as/ah for selected Ih=AhL2h , with EhAh/EsAs = 1.


be noted that for designs where a reduced, but non-zero, effectiveexpansion is required greater flexibility in the choice of as/ah isavailable.

Finally, Fig. 11 shows the trend in y�0 with EhAh/Es As. It is mostuseful to note that EhAh/EsAs can be 0, that is the strut can be arbi-trarily stiff compared to the bending element. This is important asit permits use of more robust geometric forms such as shown inFig. 1 rather that the thin struts shown in Figs. 2 and 3.

6. Stressing

Thus far the analysis has focused on the kinematics of the defor-mation. Of course the internal stressing resulting from a tempera-ture excursion will additionally be a design-limiting issue. Prior toexamining the stress issue, it should first be noted that the actualpeak stress in the structure could (quite likely) be a result of the con-tact/interaction load between the strut and face elements. The anal-

Fig. 11. Availability of zero-CTE designs as a function

ysis presented in this section is based strictly on the beam theorymodel. Implicitly, the interaction force between the strut andcurved-face elements is evenly distributed over the cross section,that is any local concentration is ignored. In the numerical simula-tion section we will show that it is possible to design smooth connec-tion schemes that minimize such additional stressing.

The tangential stress in the curved beam, r is found by substi-tuting Eq. (6) into Eqs. (7) and (8), and using beam theory,

r ¼ NAhþMn

Ih

¼ FC

2Ahðtan W cos /þ sin /Þ þ AhL2

h

Ih

nLhð1� ðjxj � y tan WÞ=LhÞ

!ð24Þ

where n is the coordinate measured from the beam centroid. Thequantity AhL2

h=Ih is always large (no useful tailoring is achieved if

of relative elastic stiffness with Ih=AhL2h ¼ 1=1200.


it is small), and the second term is maximal at x = 0 for typical y(x)considered here. Hence the peak stresses (tensile and compressive)are,

rmax ¼NAhþMn

Ih¼ FC

2Ahtan Wþ AhL2

h

Ih

n�max

Lhð1� y0 tan W=LhÞ

!ð25Þ

where n�max represents the distance of the tensile and compressivesurfaces from the beam centroid.

The interaction force is found from Eq. (18) so that,

rmax

Eh¼ Ih

AhL2h

tan Wþ n�max

Lh1� y0

Lhtan W

� � !as � ahð ÞHDT

B1ð26Þ

In the special case of an extruded section, using (20),

n�max ¼ �th=2 ¼ �Lh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Ih=AhL2

h

qso that

rmax

Eh¼ Ih

AhL2h

tan W� 3Ih

AhL2h

!1=2

1� y0

Lhtan W

� �0@ 1A as � ahð ÞHDTB1

ð27Þ

Fig. 12 shows the peak bending stressing in an extruded design withEhAh/EsAs and as = 2ah as the beam bending stiffness is varied. As thebending parameter is varied, y0 is also adjusted to the value re-quired for zero expansion . For concreteness of example if the beamis fabricated of titanium with, ah � 8 � 10�6/�C, Eh � 100 GPa, and adesign allowable strength of 400 MPa, and the required tempera-ture excursion capability is 400 �C, then the dimensionless stressparameter, rmax/EhahDT, must be held to less than 1.25. Hence, fromthe figure, for this example, Ih=AhL2

h �< 0:003. For sake of complet-ing the example, aluminum would be a clear choice as a secondaryhigh expansion material with as � 2.5ah (Schuerch, 1972; Steeveset al., 2007). Note however that this is but one example, and thatexploring the full spectrum of potential materials combinations

Fig. 12. Bending stress for a zero expansion design. These are beam theory predicteconnection.

and related fabrication and applications issues are beyond the scopeof this paper.

7. Stiffness characterization and bounds

Clearly, a significant issue with an open structure consisting lar-gely of slender curved elements is the low structural rigidity whencompared with an efficient truss design. To evaluate the stiffnessissue we will calculate the uniaxial and biaxial stiffness of thestructure. Note that for a number of two- and three-dimensionaltruss structures formulas exist for such stiffness measures (Gibsonand Ashby, 1997; Lakes, 2007), however for the present structurewe will utilize the analysis methods that were used to calculatethe effective expansion, and hence account fully for the geometricparameters such as the initial curvature. The results will be com-pared with the pin-jointed structure (Steeves et al., 2007) andthe bi-material strip structure (Lakes, 1996), as well as withbounds for voided structures (Schapery, 1968; Rosen and Hashin,1970; Gibiansky and Torquato, 1997).

The biaxial stiffness of the structure is evaluated by reconsider-ing (14), now retaining eF R and with DT set to 0. Upon solving, thebiaxial strain is found as in (15) as,

eb ¼eF R

cos WL2

h

EhIhA22 �

B1Htan Wþ y0=Lh

� �ð28Þ

where H is from Eq. (17) and the corresponding force in the strut is,

FC ¼ �2eF RH

cos W tan Wþ y0=Lhð Þ ð29Þ

The force per unit length in the infinitely periodic structure is deter-mined by considering a cut along any line passing through points ‘C’and ‘Q’ (Figs. 2 and 3). Within each unit length L there are forces attwo points on that cut line, the normal force at the beam center, Ngiven by Eq. (8) with Eq. (6) and / = 0, and FC at point C. In the hex-agonal case, FC is oriented at angle W from the cut line, while in thesquare case the force is exactly normal to the cut line. The force perunit length, F̂, is then, for the hexagonal case,

d stresses and do not account for stress-concentration effects at the strut–beam


F̂ ¼ �2eF R

Lð30Þ

and for the square case,

F̂ ¼ �2eF R

L1þ H

1þ y0=Lhð Þ

� �ð31Þ

the biaxial stiffness Sb is calculated as Sb ¼ F̂=eb (using the samenotation as Steeves et al., 2007) so that on combining Eq. (28) withEq. (30) or (31) the arbitrary eF R cancels out.

The mass per unit area is determined by calculating the mass ofthe unit cell (Fig. 4), which is simply

Mc ¼ qsLsAs þ qhL�Ah ð32Þ

where qs, qh are the constituent mass densities and L* is the arclength of the curved segment,

L� ¼ 2Z Lh

0cos�1 /ðxÞdx � 2Lh ð33Þ

The approximation on Eq. (33) holds for small y0.The corresponding cell area, Ac is, for the hexagonal case, 1/3 of

the area of a hexagon with edge length 2Lh. For the square case thecell area is one-half the area of a square of edge length 2Lh. Bothcases result in the same expression,

Ac ¼ 2L2h tan w ð34Þ

finally (for consistency of comparison with their analysis) we calcu-late the dimensionless structural efficiency figure of merit used bySteeves et al. (2007),Y¼ F̂qhAc

ebMcEhð35Þ

Combining expressions (28)–(35), does not yield usefulsimplification.

The uni-axial stiffness index is defined similarly (Steeves et al.,2007), that is the ratio of the dimensionless axial force per unitlength in response to uniaxial strain to the area density. The uniax-ial stiffness for the present structure can not be determined di-rectly using the analysis presented in this paper because thesymmetry assumptions made in the present analytical context donot hold for uniaxial loading. Therefore we have developed a beamelement-based finite element model for this calculation, the detailsof which will be discussed in the numerical simulation section. Thefinite element model was also used to determine both bi- and uni-axial stiffness parameters for the bi-material strip and pin-jointmodels for comparison.

The possible range of thermal expansion coefficient for multi-phase composites is well known. Bounds were calculated by Scha-pery (1968) with proof given by Rosen and Hashin (1970). More re-cently Gibiansky and Torquato (1997) derived improved boundsbased on the Hashin–Shtrikman variational method (Hashin andShtrikman, 1963). The Gibiansky–Torquato bounds are believedto be best-possible bounds as they are nearly attained in somecases by both the Steeves et al. pinned structure and the Chenet al. topologically optimized structure. For the present analysiswe will utilize the Gibiansky–Torquato bounds, specialized to thecase of a three-phase material with one phase being void space.

For simplicity of comparison we will compare each of the mod-els and the two-dimensional bounds for the specific case where thetwo solid materials have equal elastic moduli and density (i.e. dif-fer only in thermal expansion). Further we will assume that thetwo solid constituents occur in equal proportion to each other.For this special case the formulae given by Gibiansky and Torquato(1997) reduce to the simple expression for the lower bound on theeffective thermal expansion coefficient,

�a amean P 1= � aratio � 1aratio þ 1

1� ð1þ mÞð1� f Þ2

� �1=2

KUHS=K � 1

� �1=2

ð36Þ

where amean is the arithmetic mean of the constituent thermalexpansion coefficients, aratio is the ratio of the high to low expansioncoefficients, m is the Poisson ratio of the (equal) constituents, f is thevoid area fraction, K is the effective biaxial compression modulusand KU

HS is the Hashin–Shtrikman upper bound on the bulk modulus,which is (for this special case), with E the constituent elasticmodulus,

KUHS ¼

E=22=ð1� f Þ � ð1þ mÞ ð37Þ

The effective plane bulk modulus is related to the biaxial stiffnessindex as,Y¼ 2K

Eð1� f Þ ð38Þ

Eqs. (36)–(38) are readily combined to yield an upper bound onQ

given f, m and the expansion coefficients. The result of this calcula-tion is show on Fig. 13 for area density d � 1 � f = 0.11 and d = 0.40.We have taken m = 0.2 and aratio = 4, for example.

Fig. 13 also shows the stiffness index for each of the three mod-els, the hexagonal forms of the present models and of Lakes’ bi-material strip model and the Steeves et al. pin-joint model. Foreach model we have adjusted appropriate model parameters to en-sure constant total solid area fraction and also a equal proportionof the two solid constituents. For example, for the hexagonalstrut-based model equal constituent area is fixed by setting thethickness ratio of the face and strut as,

th=ts ¼ Lh tan Wþ y0ð Þ=L� ð39Þ

with L* from Eq. (33). The solid area density is, from Eqs. (32) and(34),

d ¼ Lh tan Wþ y0ð Þts þ L�th

2L2h tan w

ð40Þ

Eqs. (39) and (40) uniquely determine th/Lh and ts/Lh given d,y0, Lh,and W. For example, for d = 0.11, y0 = 0, W = p/3 we obtain th/Lh = 0.1 and ts/Lh = 0.116, or Ih=AhL2

h ¼ 1=1200 and EhAh/Es

As = 0.866. The high-density example with d = 0.4 has the sameEhAh/EsAs and (for y0 ¼ 0Þ Ih=AhL2

h ¼ 1=100. The results on Fig. 13are generated by varying y0, adjusting the parameters, and then cal-culating both the effective thermal expansion coefficient and thebiaxial stiffness index. The procedure for the other two models issimilar and straightforward.

From Fig. 13 we can make a quantitative comparison of themodels. First we note that all of the models fall below the bound(as they must). The pin-joint model structure is the stiffest of thethree designs (over its useful range of expansion tailoring), essen-tially equaling the bound over much of the range (as was noted bySteeves et al., 2007). The bi-material strip structure shows by farthe greatest range of available tailoring, extending well off the neg-ative end of the plot (it is ultimately limited only by a self-interfer-ing geometric constraint). The present design shows the mostlimited range of tailoring, however it is somewhat stiffer thanthe bi-material strip structure in some regions, notably for a zeroexpansion, low-density structure. The range-limiting ‘knees’ onthe curves for the strut-based structure correspond to the extremaof figures such as Fig. 7, and, as noted, represent points of signifi-cantly diminishing stiffness.

Fig. 14 shows the same trends for uniaxial stiffness. Similarobservations apply except that the present model is the lowest stiff-ness of the three for the full range. The uniaxial stiffness of both

Fig. 14. Uniaxial stiffness index. The present design along with the bi-material strip and Steeves et al. pin-joint truss are compared for 11% and 40% areal densities.

Fig. 13. Biaxial stiffness index. The present design along with the bi-material strip and Steeves et al. pin-joint truss are compared with the Gibiansky–Torquato upper boundfor 11% and 40% areal densities. In all cases constituents appear in equal quantities and differ only in thermal expansion coefficient.


bending-dominated models notably falls off sharply at low areafractions compared with the pin-joint model. We should reiteratethat stiffness index is one criteria for comparison which should beconsidered along with materials selection and fabrication issues.

For completeness we should note that all of the models may be‘inverted’ to create high expansion structures. The resulting plotscorresponding to Figs. 13 and 14, extended to large values of �a,are exactly symmetric about �a ¼ amean (including the bounds).

8. Numerical simulation

To support the analytical predictions a number of numericalsimulations were performed. The simulations were designed to ad-dress a number of key issues. (1) Confirmation of the basic analysis

without making beam-theory assumptions. (2) Evaluation of thedesign performance using a finite rather than infinitely periodicgrid. (3) Evaluation of performance under symmetry-breaking con-ditions such as non-uniform temperature distributions and uniax-ial stiffness index calculation. (4) Consideration of largedeformations. (5) Consideration of stress-concentration effects,and of the effect of finite-size contact on the thermal expansionprediction.

To address these points the tailorable design concept was fur-ther explored using three separate numerical simulations. (1) a2D continuum model of the periodic unit cell, (2) a 3D solid contin-uum model, (3) a 3D beam element model. The unit cell model pro-vides the most accurate comparison with the analytical results,while the other models are useful for visualizing the geometry

Fig. 15. Periodic finite element model.


and demonstrating the behavior of bounded (i.e. not infinitely peri-odic) structures.

9. Periodic unit cell

An ABAQUS finite element simulation was preformed using theunit cell geometry as shown in Fig. 15. This is a true minimal unitcell, i.e. half of the geometry utilized for the analytic calculations.Reduced integration elements were used for best accuracy in thebending elements and a nonlinear geometry static equilibriumanalysis (NLGEOM) was performed. The materials were specifiedas linear elastic with equal Young’s modulus and Poisson’s ratio.The honeycomb expansivity was fixed at 2 ppm/�C while theexpansivity of the strut and the initial face curvature parametery0 were varied. The model was subjected to a uniform temperaturechange and the effective expansion determined from the displace-ment of the single node at R relative to the fixed center point C.

Fig. 16. Variation of effective thermal strain with intrinsic CTE ratio and cell face curva

The effective linear expansivity is derived from the displace-ment resulting from a 10 �C temperature change and is shown onFig. 16. The agreement with the analytical prediction is remarkablygood. Notably the agreement is nearly exact close to the importantfirst �a ¼ 0 point. For larger y0 there is some disagreement, which isnot unexpected due to the approximate nature of the Timoshenkobeam solution method. Nonetheless even for y0 � 1 here is a qual-itative agreement in the prediction of an amplified expansion (�agreater than either constituent).

The finite element solution, further, permits analysis of the non-linear nature of the expansion behavior due to large temperatureexcursions. Fig. 17 shows the thermal strain of a structure designedfor ‘‘0 CTE” at room temperature. The strain is nearly zero over 50–100 �C and becomes slightly negative for increasing temperature.This can be understood qualitatively as the bending due to thethermal strains effectively, incrementally, increases the value ofy0. Indeed it would be straightforward to iteratively account forthe progressive shape change following the analytical procedure,however that exercise has not been perused.

It follows then that a structure with zero effective CTE at someparticular target temperature could be designed by suitable adjust-ment of the initial y0 value.

Relying solely on the periodic unit cell calculation(s) leaves anumber of open issues. Any real structure will be bounded, hencesubject to edge effects. Also, realistically, temperature distributionsmay be far from uniform, so that it is important to verify that theessential behavior is not dependent on mathematically precisesymmetry. Indeed it is useful to check the correctness of the basicsymmetry assumptions which are shared by the analytic modeland the unit cell finite element model. Finally it is useful for con-ceptualization purposes to develop three-dimensional models toconsider some potentially useful deviations from the simple ex-truded two-dimensional geometry as well as to begin consideringfabrication issues.

10. 3D Continuum model

The 3D models developed here are of the 2D planar tailorablethermal expansion concept, but with possible out-of-plane varia-tion in cross section. Fig. 18 as well as Fig. 1, show examples of3D continuum finite element models. The 3D model is a complete

ture. Comparison of analytic model prediction with unit cell finite element model.

Fig. 17. Finite element prediction of nonlinear thermal strain over large temperature excursion. Solid lines show linear expansion of homogeneous constituents for reference.Dashed line connects numerically derived data points.

Fig. 18. Example of a thermally stable structure with a near-continuous surface.


discretization of a finite number of cells (i.e. 25 in Fig. 18), and isgenerated programmatically so that the geometric parameters(y0, th, etc.) can be readily specified. The 3D model proved to beessential for visualization purposes, as well as to explore a numberof non-planar variations on the concept.

As an example, in Fig. 18, the ‘strut’ parts are fabricated to besomewhat taller than the honeycomb and have faces that overlapthe top/bottom of the honeycomb so as to form a nearly continu-ous surface with only small expansion gaps. This type of design,of course, does not exactly follow the analytical predictions, how-ever with guidance from the predictions, tailored CTE designs canbe produced with some minimal iteration of parameters.

The 3D model demonstrates clearly that the concept doeswork well for the a finite number of cells. Whenever there arefree edges, the cells near the edges expand generally accordingto their constituent material expansivities, however it was found

that for a structure as small as 16 cells (4 � 4) all of the cellsaway from the edges show the expected tailored effective expan-sion behavior, to a good approximation. Edge cells locally expandpositively. The concept does require some extent of the gridstructure and will not work at all with, for example, a single unitcell. Of course for a real application a large number of cellsshould be used so that the edge effects can be neglected, butit is useful to know that a demonstration model could be de-vised with a fairly small number.

Fig. 19a shows contour plots of the bending (x-component)stress derived from the 3D continuum model using same-modulusmaterials perfectly bonded at the interface. The peak tension oc-curs, as predicted, at the center of the honeycomb faces and isslightly higher than the beam theory prediction (Eq. (27)) ofrmax = 1.115EaDT. The stress at the sharp corner on the compres-sive side of the beam is roughly twice the predicted value however.

Fig. 19. Stress contours showing typical interaction concentration. (a) Sharp cornerdesign shows the peak tension stress is only slightly higher than the simple beamprediction of rmax/EaD T = 1.115 while the compressive stress is more than doublepredicted value of 0.936 due to the sharp corner. (b) A fillet is one possible designmodification to address this issue. Here the peak stress is comparable to the beammodel value.

Fig. 21. Arbitrary non-uniform temperature distribution. Effective thermal strain ismeasured at the indicated nodes based on displacement relative to the indicatedreference node.


Fig. 19b shows an example of stress-concentration mitigation byfilleting the corner. In this example the peak stressing is broughtto be approximately equal to the beam theory prediction. Anothereffective approach is to round the strut end so the a Hertz-contactjoint is formed. It should be noted that these changes do impact theeffective thermal expansion behavior so that a final design proce-dure will require some adjustment of the curvature (y0), basedon simulation and/or experiment, to compensate for the effect ofconnection details such as fillet radius.

It must be noted that the normalized stress parameter rmax/EaDT is a critical design-limiting quantity. For example if a temper-ature excursion as large as 500 �C is to be tolerated then theparameter must be well less than one for most engineering mate-rials to survive, hence a rather thin-faced honeycomb will be re-quired. The ‘thicker’ end of the design space is however relevantfor applications where lower temperature excursions are expected,where highly elastic materials can be utilized, or in case whereonly small adjustments to the intrinsic expansivity is needed.

Fig. 20. Beam element modeling. Seventy-two-cell m

11. Beam element model

The utility of the continuum model for analysis of structureswith a large number of cells is limited by the scale of the compu-tation. The 25-cell model has, for example, more than 40,000 ele-ments. Extending this model to even larger numbers of cellswould require a prohibitive amount of computing time. Further,when the global behavior of large structures consisting of beam-like elements is examined, such a discrete solid model is not themost accurate computational approach in any case.

For these reasons a third finite element model was developed.Fig. 20 shows a beam element-based finite element model. Eachstrut is discretized as a single element, and each face by approxi-mately 6–10 elements. The ABAQUS 3D beam elements allow arbi-trary specification of the section properties (I, A) directly. It shouldbe noted, however, that the graphic representation of them, unfor-tunately, does not properly reflect their ‘thickness’ dimensions, i.e.the beam element method can properly model such geometricallycomplex designs such as shown in Fig. 18, but one must be awarethat the figures can give a misleading impression of a ‘wire frame’-like model.

The beam element model readily, and rapidly calculates thedeformation of solids with many hundreds of cells. The thermalexpansion of the structure resulting from a uniform temperaturechange again follows the analytic predictions. With this model

odel shown with approximately 1700 elements.


however, symmetry breaking loading can be examined, as forexample due to a non-uniform in-plane temperature distribution.Fig. 21 shows an arbitrary temperature distribution found simplyby applying different temperature boundaries to different partsof the model (the constituent thermal conductivity must be speci-fied and is taken to be uniform) With temperatures of 200 and400 �C applied along two edges, and the others insulated, the meantemperature, DT is found from the steady-state heat transfer solu-tion as approximately 280 �C.

With such a non-uniform temperature distribution one wouldexpect a structure to both expand and distort. In order to evaluatethe behavior of the thermally stable design we characterize thedistribution of thermal expansion by calculating displacement of‘stable’ (‘‘R” and ‘‘C’) nodes in the grid relative to a single centralreference node, as indicated on Fig. 21. The effective thermal strainis defined as et(r) = ur/r, with r the distance between the node andthe reference node and ur the magnitude of the relativedisplacement.

The analysis was performed for a structure designed for zeroexpansion under uniform temperature change. For comparisonthe analysis was repeated for the same structure with a homoge-neous material specified, that is the struts are the same materialas the honeycomb. Fig. 22 shows the distribution of thermalexpansion, normalized by the expected average expansion of ahomogeneous structure at the mean temperature change, and rel-ative to the central reference node. For clarity only the ‘‘C” nodeslying along the three lines of hexagonal symmetry are plotted,hence the distances from the reference node are all integer multi-ples of the cell spacing, however any of the ‘‘R” or ‘‘C” nodes in thestructure follow essentially the same behavior. The homogeneousstructure expands and distorts as expected, while the expansionof the stable structure is an order of magnitude lower and nearlyuniform. The furthest outlying points are edge effects, and it isnotable that even in this worst case the effective strain is quitesmall.

This is an important result, as the original calculations relied onthe symmetry of a uniform temperature change, it is valuable toknow that the essential character of the behavior is maintainedeven with a substantial asymmetry.

Fig. 22. Distortion due to in-plane thermal gradient depicted in Fig. 21. The homogenestable structure is an order of magnitude lower.

Finally, the beam element model was used to generate thenumerical stiffness index results shown on Figs. 13 and 14. Thiswas accomplished simply by applying the appropriate displace-ments to edge nodes and determining the total force response.

12. Closure

A composite material/structural design concept has beendevised to allow fabrication of structures with tailored ordesigned-in CTE values. The usefulness if the concept is signifi-cantly enhanced by an accurate analytic method for specifying therelevant materials and structural parameters to achieve a desiredexpansion characteristic. In closing brief mention will be made of anumber of unexplored issues, and potential new directions.

12.1. Fabrication and validation

At present the model has not been physically demonstrated.It is sufficiently general that it might be fabricated on a materialmicro-scale, in a manner similar to the Qi and Halloran (2004)co-extruded demonstration material, or it could be assembledfrom discrete components. It is key to note that the two constit-uent materials can be simply held together by compression andfriction, so that substantially different materials can be utilized(e.g. a metal honeycomb might be combined with a high expan-sion polymer).

12.2. Structural integrity and stability

The tailored CTE design achieves its expansion characteristicsby deliberately introducing compliant bending members. As suchthe overall stiffness of the structure is substantially sacrificed incomparison to a similar weight truss designed for stiffness ratherthat CTE. For this reason, the most promising applications appearto be where the structural loads are small, such as mirror plat-forms. Also, a potential pitfall, not yet explored in detail, is the factthat it is possible the internal stressing will produce an out-of-planebucking mode of deformation. Most likely bifurcation modes canbe suppressed though appropriate design adjustment, however

ous reference structure expands and distorts while the expansion of the thermally


this is an issue that will be most practically revealed experimen-tally rather than analytically.

12.3. Anisotropy and shape morphing

An obvious extension of the concept is to create structures that,instead of having a low/zero characteristic expansion, have designedin shape-change characteristic response to temperature change.Ongoing modeling efforts show promise of designing planar struc-tures with significantly anisotropic expansion, as well as structuresthat deform out-of-plane into desired, possibly complex, shapes.

Acknowledgments

The authors acknowledge the useful contributions and sugges-tions by Dr. W. Coblenz (DARPA/DSO), Dr. K. Buesking (MR&D) andDr. D.B. Marshall (RSC) throughout the course of this work. Thiswork was supported in part by DARPA and in part by USAF Con-tract No. FA8650-04-D-5233.

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