extreme thermal expansion, piezoelectricity, and other...

JOURNAL OF APPLIED PHYSICS VOLUME 90, NUMBER 12 15 DECEMBER 2001

Extreme thermal expansion, piezoelectricity, and other coupled fieldproperties in composites with a negative stiffness phase

Y. C. Wang and R. S. Lakesa)

Department of Engineering Physics, Engineering Mechanics Program, University of Wisconsin–Madison,147 Engineering Research Building, 1500 Engineering Drive, Madison, Wisconsin 53706–1687

~Received 17 May 2001; accepted for publication 5 September 2001!

Particulate composites with negative stiffness inclusions in a viscoelastic matrix are shown to havehigher thermal expansion than that of either constituent and exceeding conventional bounds. It isalso shown theoretically that other extreme linear coupled field properties including piezoelectricityand pyroelectricity occur in layer- and fiber-type piezoelectric composites, due to negative inclusionstiffness effects. The causal mechanism is a greater deformation in and near the inclusions than thecomposite as a whole. A block of negative stiffness material is unstable, but negative stiffnessinclusions in a composite can be stabilized by the surrounding matrix and can give rise to extremeviscoelastic effects in lumped and distributed composites. In contrast to prior proposed compositeswith unbounded thermal expansion, neither the assumptions of void spaces nor slip interfaces arerequired in the present analysis. ©2001 American Institute of Physics.@DOI: 10.1063/1.1413947#

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I. INTRODUCTION

Microstructure in multiphase materials such as compites and biological tissues may be so complex that theoreprediction of aggregate physical properties from constituproperties becomes difficult or impossible. Consequentlyis useful to develop analytical bounds upon propertiesorder to constrain the range of properties which may bepected. Bounds1,2 have been developed for the thermal epansion coefficienta of composite materials of two solidphases in terms of constituent expansion coefficientsa1 anda2 . The upper bound is a rule of mixturesa5a1V1

1a2(12V1), in which V1 is the volume fraction of the firsphase. If the detailed microstructure of a compositeknown, the exact relations of overall mechanical or coupproperties and those of each constituent can be obtatheoretically. Even if the microstructure is unknown forparticular composite, the bound equations constrain attable behavior provided the assumptions of the boundtheorems are satisfied. Many bounding formulas are attable, that is, they correspond exactly to a known microstrture. In deriving the bounds on thermal expansion coecients, it was tacitly assumed that the two phasesperfectly bonded and with zero void content; moreover, teach phase is of positive stiffness with zero stored energequilibrium. We have shown that arbitrarily high thermal epansions can be achieved in composites with void fracti3

or in dense composites with interfaces which allow sli4

These composites contain rib elements of composite mistructure. Each rib element is a bilayer made of two bonlayers of differing thermal expansion coefficienta. Compos-ites with extremal thermal expansion coefficients have abeen presented based on topology optimization5 of structure

a!Author to whom correspondence should be addressed; [email protected]

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allowing void space. Inclusion of void space of appropriashape in a composite microstructure can also give riseunusual mechanical properties such as a negative Poissratio.6 Void space is not, however, a necessary conditionachieve negative Poisson’s ratio; a hierarchical laminate wdissimilar constituents also has this property.7 A commonaspect is nonaffine or heterogeneous deformation.8 Negativethermal expansion coefficients are known in certain oxsystems;9 they also occur in lattice composites if the highexpansion constituent is on the convex side of each bilaWe remark that molecular design of materials with modervalues of negative linear and volumetric thermal expanshas been conducted.10

It is the purpose of this article to explore the effectsnegative stiffness composite phases in achieving extrethermal expansion, piezoelectric, and pyroelectric coupfield properties. The possibility of extreme behavior in sucomposites was explored theoretically11 for viscoelastic sys-tems. High viscoelastic damping in composite cells wnegative stiffness has been demonstrated experimentally12 incompliant systems containing postbuckled tubes. High vcoelastic damping has also been observed in compositesnegative stiffness particulate inclusions.13 In this article, wefirst demonstrate the possibility to achieve extreme therexpansion of particulate composites via the exact relationthermal expansion coefficients and bulk moduli. Second,show theoretically that extreme piezoelectric constants,ezoelectric coefficients, and pyroelectric constants canbe observed in layer- and fiber-type piezoelectric composibased on available exact or approximate relations.

II. ANALYSIS

Thermal expansion in composites is intimately linkwith the bulk properties. The thermal expansion coefficieil:

8 © 2001 American Institute of Physics

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6459J. Appl. Phys., Vol. 90, No. 12, 15 December 2001 Y. Wang and R. Lakes

ac of a composite is related to the bulk propertiesfollows,14,15 provided the composite as well as each phasisotropic.

ac5a11a12a2

1

K22

1

K1

S 1

Kc21

K1D . ~1!

The composite thermal expansionac can be large if the composite bulk modulusKc is small. That is the case in thlattice structures considered in Refs. 3 and 4.ac can also belarge if bulk compliances of the constituents are out of pportion to their thermal expansion coefficients. Based onstatistically isotropic assumption, Eq.~1! was obtained as anexact solution by applying an eigenstress technique throthe uniform fields approach,16 described as follows. Thename for the uniform field was coined much later thanderivation done by Levin. First, separate the inclusions frthe matrix; let both of them be loaded by a hydrostatic strwith a uniform temperature change. Second, put those insions back into the matrix, constrain them to satisfy the trtion and displacement continuity conditions on the intfaces. Then, the expression is found for effective thermexpansion coefficients by the superposition of a hydroststress in the opposite direction on the outer boundary ofmatrix to cancel the one added artificially in the first step

As for the bulk modulus as input to Eq.~1!, consider theHashin–Shtrikman17 ~1963! formulas. The lower bounds fothe elastic shear modulusGL and bulk modulusKL of acomposite are:

GL5G21V1

1

G12G21

6~K212G2!V2

5~3K214G2!G2

, ~2!

KL5K21V1~K12K2!~3K214G2!

~3K214G2!13~K12K2!V2, ~3!

in which K1 andK2 , G1 andG2 , andV1 andV2 are the bulkmodulus, shear modulus, and volume fraction of phaseand 2, respectively. IfG1.G2 , thenGL represents the lowebound on the shear modulus. Interchanging the subscripand 2 results in the upper boundGU for the shear modulusThe bounds for isotropic composites are attainable. THashin–Shtrikman formula for the bulk modulus, Eq.~3!, isattained exactly by a coated sphere morphology. T‘‘lower’’ composite corresponds to the case of stiff sphecoated with a compliant layer. The shear modulus ofcoated sphere morphology approximates the corresponHashin–Shtrikman formula, Eq.~2!. Exact attainment of Eq~2!, however, is possible via a laminate morphology18 asshown by Milton.

We now allow one phase to havenegative stiffness.Negative stiffness entails a reversal of the usual directiorelationship between force and displacement in deformedjects. It does not violate any physical law, but an isolaobject with negative stiffness is unstable. Composite matals with negative stiffness inclusions can be stable provithe inclusion stiffness is not excessively negative. In elacomposites, the elastic moduli can become singular if


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phases of positive and negative stiffness are properlyanced. In viscoelastic composites11 with a negative stiffnessphase, an anomaly in stiffness is predicted, as well as a pin the mechanical damping. In that analysis, we appliedelastic–viscoelastic correspondence to the exact relation~1!and~3! to obtain corresponding relations for viscoelastic mdia. These formulas no longer represent bounds, howethey are exact solutions for particular microstructures.

If the bulk modulus of one phase is allowed to becomnegative, the thermal expansion coefficient can greatlyceed that of either phase, as shown in Fig. 1, which is baon Eq. ~1!, and the lower bound formula for effective bulmodulus, Eq.~3!. As inclusion stiffness is reduced, the composite thermal expansion attains its maximum magnitufirst, then the composite bulk modulus attains a minimuThe enhancement of thermal expansion becomes singulathe mechanical damping of the phases tends to zeroshown in Fig. 2. The singularity in thermal expansion occas the composite bulk modulus tends to zero. Therefore,present composites share with prior ones3–5 the characteristicthat extreme expansion is associated with bulk complianWe remark that singular thermal expansion can be achiewithout the composite bulk modulus passing through netive regions in which stabilization might be required. Speccally, in Fig. 2, as the inclusion bulk modulus is tunethrough progressively more negative values, the compothermal expansion becomes singular before the compobulk modulus.

As for the piezoelectric coupling problems, SkinneNewnham, and Cross19 proposed a set of rule-of-mixturetype equations to approximately estimate the effective pieelectric coefficients, as follows.

~d33c !series5

V1d33i «33

m 1~12V1!d33m«33

i

V1«33m 1~12V1!«33

i , ~4!

~d33c !parallel5

V1d33i S33

m 1~12V1!d33mS33

i

V1S33m 1~12V1!S33

i , ~5!

d31c 5V1d31

i 1~12V1!d31m , ~6!

FIG. 1. The thermal expansion coefficient~real part ofac! of a Hashin–Shtrikman composite assuming a matrix phase of mechanical dampingd50.05 in the bulk modulus with negative stiffness inclusions, as a funcof inclusion stiffness and volume fraction. Thermal expansion coefficientinclusion and matrix differ by a factor of two. The composite thermal epansion coefficient is normalized to that of phase 1.


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6460 J. Appl. Phys., Vol. 90, No. 12, 15 December 2001 Y. Wang and R. Lakes

where the symbolsdjk , « jk , andSjk are piezoelectric coefficients, permittivity constants, and elastic compliances,inverse of the elastic modulus (Cjk). The indexesj andk ofsecond rank tensor properties« jk are from 1 to 3, denotingthe axes of the coordinate system. The subscripts, seriesparallel, represent the type of connectivity, which corsponds to the Reuss and Voigt composites, respectivelgeometry. In the series case, 3-axis represents the axispendicular to the layers; the reverse applies in the paracase. The superscriptsi, m, and c represent the inclusionmatrix, and composite. The relationship between piezoetric constants (ejk) and piezoelectric coefficients (djk) isdjk5ejqSqk . The usual contracted index notation is usedthe higher order tensorial properties, such as elastic mo(Ci jkl ), elastic compliances (Si jkl ), and piezoelectric properties ~ei jk or di jk! in which only the last two indexes arcontracted, as used by Nye.20 The contraction means indexe11 becomeI 1, 22 becomeI 2, 33 becomeI 3, 23 become 4, 13become 5, 12 become 6. Each of the field quantities is msured when all others are fixed. A more rigorous derivatibased on the linear coupled field theory and uniform fitechnique, for this sort of two-phase layer-type piezoeleccomposites, consisting of transversely isotropic phasesgiven by Benveniste and Dvorak.21 The 17 independent material parameters for the overall orthotropic piezoeleccomposites are obtained exactly.

To simplify our theoretical analysis of this coupled fieproblem, we assume each of the piezoelectric transd~PZT! inclusion layers or rods is mechanically isotropic. Thassumption is justified since the degree of mechanical anropy of certain PZT materials, for example PZT-5A~Ref. 22!is not very pronounced. Moreover, most piezoelectric ceraics are rather stiff, therefore typical matrix materials canexpected to be much more compliant than piezoelectricclusions. In the vicinity of a phase transformation, a sindomain can go from positive to negative stiffness even if

FIG. 2. Thermal expansion properties of a Hashin–Shtrikman compoassuming a inclusion phase of mechanical damping tand50.05 or 0 andnegative stiffness. Thermal expansion is normalized to that of phase 1.volume fraction of the inclusion is 0.01.


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parent material is stiff away from the transition. Therefore,delineate phenomena, we allow the shear modulus ofinclusion to become negative. To aid the numerical calcution of the overall piezoelectric properties, we assumeelectrical and coupled properties of the inclusion to besame as those of PZT-5A material,24 which are25.4, 15.8,and 12.3~C/m2! for e31, e33, and e15, as well as 916«0 ,

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FIG. 3. ~a! Normalized piezoelectric coefficientd33 , in series,~b! normal-ized piezoelectric coefficientd33 , in parallel, and~c! normalized piezoelec-tric coefficientd31 of a layer-type piezoelectric composite versus inclusivolume fraction (V1) and normalized inclusion shear modulus (Gi /Gref),whereGref525 GPa. Calculated by Eqs.~4!–~6!.


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FIG. 4. ~a! Normalized piezoelectric coefficientd33 , ~b! normalized piezoelectric coefficientd32 , ~c! normalized piezoelectric coefficientd31 , and ~d!normalized piezoelectric coefficientd15 of a layer-type piezoelectric composite versus inclusion volume fraction (V1) and normalized inclusion shear modulu(Gi /Gref), whereGref525 GPa. Calculated by the equations, which are exact, in Ref. 21.

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830«0 for «11 and«33, where«0 represents the permittivityin a vacuum. For the isotropic matrix, there are no coupelectrical properties, but we assume its permittivity to4.43«0 . Figure 3 shows the existence of singularities, pdicted by Eqs.~4!–~6!, in this type of composites for theipiezoelectric coefficients when their piezoelectric phasesdergo a negative stiffness transition. Based on those etions obtained by Benveniste and Dvorak, the singularialso can be observed in the graphical results, Fig. 4, foroverall piezoelectric coefficientsd33, d32, d31, andd15, inwhich the index 3 indicates the axis perpendicular tolayers.

In the piezoelectric composites with 1–3 connectivicontrasted with the aforementioned piezoelectric composwhich are catalogued as those with 2–2 connectivity,piezoelectric phase is continuously self-connected in onemension and the matrix is connected in three dimensionsother words, they are fiber-type or transversely isotrocomposites. Throughout our analysis for fiber composithe 3 direction of the composite is along the directionwhich the fibers are aligned and poled. Benveniste23 pointedout, among the ten independent material constants foroverall properties, nine of them can be derived exactlysimple expressions in the model of composite cylinder


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semblage. Because of the independence of electric fieldrelation for effective plane strain bulk modulus, Eq.~7! isexact, based on the same argument used for the effecbulk modulus in the Hashin–Shtrikman bounds.

kc5km1V1

1

ki2km 112V1

k1GTm

, ~7!

where k is the plane strain bulk modulus, andGT is thetransverse shear modulus, differing fromGA , which is de-noted as the longitudinal shear modulus. Following the uversal connection for piezoelectric fiber-type compositfound by Schulgasser,24 we can obtain the following exacrelations.

C13c 5

~km2kc!C13i 2~ki2kc!C13

m

km2ki , ~8!

C33c 5V1C33

i 1V2C33m

2~C13

m 2C13c !~V1C13

i 1V2C13m 2C13

c !

km2kc , ~9!


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e33c 5V1e33

i 1~12V1!e33m

2~e31

m 2e31c !~V1C13

i 1~12V1!C13m 2C13

c !

km2kc , ~10!

e31c 5

e31i ~km2kc!2e31

m ~ki2kc!

km2ki , ~11!

«33c 5V1«33

i 1V2«33m 2

~e31m 2e31

c !~e31c 2V1e31

i 2V2e31m !

km2kc ,

~12!

FIG. 5. ~a! Normalized piezoelectric coefficiente33 , ~b! normalized piezo-electric coefficiente31 , and ~c! normalized permittivity constant«33 of a1–3 fiber-type piezoelectric composite versus inclusion volume frac(V1) and normalized inclusion shear modulus (Gi /Gref), where Gref

525 GPa. Calculated by Eqs.~10!–~12!, exact relations.


The rest of the exact relations fore15, GA , and«11, dem-onstrated by Milgrom and Shtrikman,25 are presented as follows.

L c5Lm@~12V1!L i1~11V1!Lm#21@~11V1!L i

1~12V1!Lm#, ~13!

where

L r5FGL e15

e15 2e11G r

, r 5 i ,m.

Figure 5 shows the singularities in piezoelectric costants,e31 and e33, and permittivity«33, predicted by theexact solutions, Eqs.~10!–~12!. For the transversely isotropic materials modeled by composite cylinder assemblageeffective transverse shear modulus can not be determexactly.23 However, deriving this effective property ispurely mechanical problem, since there is no coupling welectric field. Therefore, one can use, for example, geneized self-consistent method26 to calculate it approximatelyAlternatively, by the multiple-scattering technique, all of thten effective material properties can be computed by E~14!–~23!, plus the pyroelectric constantp3 by Eq. ~24!,27,28

presented next, in the sense of a first order approximatThe importance of these equations is that they represenoverall properties of the two-phase transversely isotropicezoelectric composites, whose microstructure is compriof infinitely long fibers. Mathematically, they are equivaleto the results of the self-consistent effective-mediutheory,29 after theV1 square terms are ignored in the effetive elastic moduli.

kc5k2~k11m2!1V1m2~k12k2!

k11m22V1~k12k2!~14!

mc5m2

k2~m11m2!12m1m21V1k2~m12m2!

k2~m11m2!12m1m22V1~m12m2!~k212m2!~15!

l c5C12

m ~k12kc!1C13i ~kc2k2!

k12k2~16!

nc5C11m 1V1~C33

i 2C11m !22V1

~C13i 2C12

m !2~k22kc!

~k11m2!~k12k2!~17!

pc5m2~m212C55

i !1V1m2~2C55i 2m2!

2~m212C55i !12V1~m222C55

i !~18!

e31c 5V1e31

i kc1m2

k11m2~19!

e33c 5V1e31

i 22V1e31i

C13i 2 l c

k11m2~20!

e15c 5

V1e15i «2~m212pc!

@~11V1!«21~12V1!«11i #~m212C55

i !~21!

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«11c 5«2

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i 22V1~e15c 2e15

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i !

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«33c 5«21V1~«33

i 2«2!22V1e31i

e31c 2e31

i

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p3c5V1S p3

i 2V2

V1

2b2e31c

k21m21

2b11i ~e31

i 2e31c !

k21m2D , ~24!

in which k/2, m/2, l, n, p are Hill’s30 elastic moduli under aconstant electric and temperature field. The single subscrepresent the inclusion, as 1, and the matrix, as 2. In contthe double subscripts represent second order tensors.symbolsp3 , b represent pyroelectricity and thermal strecoefficient. This pyroelectricity is the variation of electrdisplacement along the 3-axis due to a unit change of tperature. The relation ofb to the thermal expansion coefficients isb5Ca.

To investigate the anomaly in the pyroelectric constanwe adopt the CdS-epoxy piezoelectric composite,28 forwhich material parameters are20.074 and 1.32~C/m2! fore31 ande33, 1.17«0 for «33, and20.04 (1024 C2-m22-K21)for p3 . Also, the degree of mechanical anisotropy of tinclusion is low. The thermal properties of the inclusion amatrix are 6.5 and 4 (1026 K21) for a11 and a33 of theinclusion, and 60 (1026 K21) for the thermally isotropic ma-trix. It is noteworthy that, in all of our analysis, the absoluvalues of these material properties are not important beconly the normalized values are of interest to demonstrateanomalies. The standard normalization method used heto divide each of the overall composite properties by thaphase 1, correspondingly, except for the shear modulus oinclusion, which is normalized to that of phase 2, andoverall piezoelectric coefficients, which are normalizedthose evaluated atV151 andGi /Gref51. In our calculation,shear modulus is tuned through negative values while mtaining coupled field constants unchanged. Single dommay approximate such behavior but polydomain blocwould not. Here, since we are considering linear systemsinfluence of nonlinear behavior of piezoelectric materiasuch as ferroelectricity,31 upon the homogenization of composites is neglected. Figure 6 shows the singularity incomposite piezoelectric constantse31 and e33, as inclusionstiffness becomes negative. Figures 7 and 8 show that silarities also occur in the composite piezoelectric coefficiend33, d32, d31, and d15, and the permittivity«33, as theinclusion stiffness becomes negative. Figure 9 shows thefluence of inclusion negative stiffness on the compositeroelectric property. As with the other coupled fields, singlarities occur for proper values of inclusion stiffness. As wthe viscoelastic case,11 the singularities in the coupled fieldbecome peaks of finite magnitude when material dampproperties of either phase are considered.


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III. DISCUSSION

Although a giant magnitude of thermal expansion coficients can be also achieved by introducing the conceptvoid phase or slip interfaces,3,4 the present approach not onlargely increases the expansion of materials in magnitubut also improves the properties of stiffness and mechandamping simultaneously.11–13 In addition, the layer type ofthis kind of composites makes the actuator application, sas cantilever beam bimorph actuators,32 more feasible at adesignated temperature, without microdevice fabricatiThe achievement of enhanced composite stiffness by ntive stiffness inclusions, while supported experimentally13

raises questions of stability.33 For thermoelasticity or piezoelectricity, the sign is usually immaterial in transducersensor applications. Consequently, we expect it will be eato achieve overall stability in composites intended for etreme thermoelastic or piezoelectric coupling. Tuningnegative stiffness can be achieved by control of temperathrough a phase transition13 or by controlling prestrain upona postbuckled system.12 The negative stiffness approachdistinct from the use of structural resonance34 to achievehigh dielectric properties, since there are no inertial termsthe continuum elasticity equations used in the present w

Piezoelectric composite materials35 are of interest sincethe figure of merit for a variety of sensor and actuators c

FIG. 6. ~a! Normalized piezoelectric constante33 and~b! normalized piezo-electric constante31 of a 1–3 fiber-type piezoelectric composite versusclusion volume fraction (V1) and normalized inclusion shear modulu(Gi /Gref), whereGref525 GPa. Calculated by Eqs.~19! and ~20!.


tion


FIG. 7. ~a! Normalized piezoelectric longitudinal coefficientd33 , ~b! normalized piezoelectric transverse coefficientd32 , ~c! normalized piezoelectrictransverse coefficientd31 , and~d! normalized piezoelectric shear coefficientd15 of a 1–3 fiber-type piezoelectric composite versus inclusion volume frac(V1) and normalized inclusion shear modulus (Gi /Gref), whereGref525 GPa. Calculated by Eqs.~14!–~24!.

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be increased by using composites.36 In the present work, ithas been shown that piezoelectric and other coupledproperties can become unbounded, not just increased.

To extend our idea to predict other anomalies in otcoupled fields, phenomenologically, the effective modulitwo-phase composites with coupled multiple fields, such

FIG. 8. Normalized permittivity constante33 of a 1–3 fiber-type piezoelectric composite versus inclusion volume fraction (V1) and normalized inclu-sion shear modulus (Gi /Gref), whereGref525 GPa. Calculated by Eq.~23!.


ld

rfs

thermoelectric, magnetoelectric, and piezoelectric effecan be obtained exactly by a modulus matrix decomposiapproach.37 Therefore, the overall properties can be exaccalculated simply by summing a finite series, whose ter

FIG. 9. Normalized pyroelectric coefficientp3 of a 1–3 fiber-type piezo-electric composite versus inclusion volume fraction (V1) and normalizedinclusion shear modulus (Gi /Gref), whereGref525 GPa. Calculated by Eq~24!.


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are comprised of the phase moduli and volume fractionly. By allowing negative stiffness inclusions, we violathe common assumption of positive definiteness for modIn particular, it is possible to exceed commonly accepbounds. The consequences of negative stiffness inclusionmechanical systems have been explored theoretically11 andexperimentally.12,13Therefore, due to negative eigenvaluesthe modulus matrix as the coefficients of the terms infinite series, its sum, which represents the effective modmay lead to extremely small or large values of propertiwhereas the magnitude of the influence depends on the iaction of the degree of fields. The present theoretical analsupports this argument. In thermoelastic and piezoelecmaterials, elasticity is coupled with temperature and elecfield, respectively. Consequently, composites with negastiffness inclusions may find use in high performance senand actuators based on coupled fields.

IV. CONCLUSION

Extreme coupled field properties including thermal epansion, piezoelectricity, and pyroelectricity can occurcomposites in which one phase has negative stiffness. Inearly elastic materials, the properties become singular wpositive and negative stiffness are appropriately balancThe presence of time or frequency dependence in elastielectric properties, or their coupling will smooth out the sgularities to finite peaks. Following the same idea, it alsobe envisaged that similar anomalies can be found in ocoupled field problems.

ACKNOWLEDGMENT

The authors are grateful to the NSF for Grant No. CM9896284.

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-

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-

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