AIAA JOURNALVol. 34, No. 9, September 1996

Micromechanics of Composites with Shape MemoryAlloy Fibers in Uniform Thermal Fields

Victor Birman*University of Missouri-Rolla, St. Louis, Missouri 63121

Dimitris A. Saravanos^Ohio Aerospace Institute, Cleveland, Ohio 44142

andDate A. Hopkins*

NASA Lewis Research Center, Cleveland, Ohio 44135

Analytical procedures are developed for a composite system consisting of shape memory alloy fibers withinan elastic matrix subject to uniform temperature. A three-phase concentric cylinder model is developed for theanalysis of local stresses, which includes the fiber, the matrix, and the surrounding homogenized composite. Thesolution addresses the complexities induced by the nonlinear dependence of the in situ martensite fraction of thefibers on the local stresses and temperature and the local stresses developed from interactions between the fibersand matrix during the martensitic and reverse phase transformations. Results are presented for a Ni-44.8wt%Ti/epoxy composite. The applications illustrate the response of the composite in isothermal longitudinal loadingand unloading. The local stresses developed in the composite under various stages of the martensitic and reversephase transformations are also shown.

Introduction

SHAPE memory alloys (SMAs) have found increasing applica-tions as candidate smart materials and components of adaptive

structures.1 One potential application of shape memory alloys istheir use as fibrous sensors or actuators within a composite mate-rial system. The development of such fibrous shape memory (SM)composites will provide weight savings, directionality in actuationand sensing, and some control over the resultant SM effect. Sometechnical details of such applications were discussed by Schetkyand Wu,2 Schetky,3 and White et al.4 The use of shape memory al-loys as sensors may be related to the dependence of their propertieson the martensite fraction (for example, their electric resistance).5Conversely, the use of SMA fibers as distributed micro-actuators isalso very attractive because of the relatively high strains that can begenerated.

In summary, the SM effect is associated with a martensitic phasetransformation of the material from a parent phase.1 During suchmartensitic transformations, changes in crystallography result insubstantial strains, called transformation strains. Of particular inter-est are alloys that can recover their parent phase through a reversephase transformation accompanied by recovery strains. These areSMAs that can undergo thermoelastic or complete phase transfor-mations. In addition to the transformation strains, the phase trans-formations are accompanied by changes in the elastic, thermal, andelectric properties. The initiation of the martensitic or the reversetransformation depends on the free chemical energy in the material,which, in turn, depends on the temperature and stress state. Hence, ina SM composite, the actuation process may be achieved by forcing

Presented as Paper 95-1210 at the AIAA/ASME/AHS/ASC 36th Struc-tures, Structural Dynamics, and Materials Conference, New Orleans, LA,April 10-13,1995; received June 9,1995; revision received April 15,1996;accepted for publication May 19,1996. Copyright © 1996 by the AmericanInstitute of Aeronautics and Astronautics, Inc. No copyright is asserted inthe United States under Title 17, U.S. Code. The U.S. Government has aroyalty-free license to exercise all rights under the copyright claimed hereinfor Governmental purposes. All other rights are reserved by the copyrightowner.

* Professor, Engineering Education Center, 8001 Natural Bridge Road.f Senior Research Associate, 22800 Cedar Point Road. Member AIAA.* Acting Branch Chief, Structural Mechanics Branch, MS 23-3. Member

AIAA.

the phase transformation to occur in the fibers by changing eitherthe temperature or the stresses3 in the fibers.

Constitutive relations required to characterize the thermomechan-ical response of SMAs were outlined in the works of Baumgart etal.,6 Tanaka,7 Sato and Tanaka,8 and Brinson.9 Applications of theunidirectional shape memory effect can be found in Refs. 10-14.Liang and Rogers15 and Boyd and Lagoudas16 have provided for-mulations for the three-dimensional SM effect.

The analysis of composites incorporating continuous SMA fibersseems a rather open-ended research area involving solution of sev-eral problems including admissible micromechanics and the non-linear constitutive relations of SMAs. Indeed, the properties and theinduced strain of the SMA fibers depend on the fraction of marten-site, which, in turn, depends on the in situ temperature and effectivestress state. Because of the fiber-matrix interaction in the compos-ite, the developed in situ stresses will depend on phase transfor-mation strains, thermal strains, and the constituent properties. Thisinterdependence results in a nonlinear thermomechanical response(even by assuming thermoelastic austenite and martensite phases),which makes an iterative analysis seemingly necessary, although, asshown in the present paper, an analytical solution may be obtainedin some cases without iterations. Numerous micromechanical theo-ries may be used to formulate the present problem, yet limited workhas been reported in the area of SM composites. Three-dimensionalmicromechanics were reported by Boyd and Lagoudas16 using themicromechanics approach of Mori and Tanaka17 and by Sulivanand Buesking18 using primarily Hashin's19 vanishing fiber concen-tric cylinder assemblage method. The previous micromechanics arevalid approaches that may yield good approximations of the equiva-lent properties of composites20; however, the assumptions of averageconstituent stresses/strains may be an area of limitation, as they maynot yield the variation of stresses in the various constituents of thecomposite.

The present paper presents a combined micromechanical ap-proach that has the potential to provide better representations oflocal stresses in an SM composite. A three-phase concentric cylin-der model (SMA fiber, matrix, equivalent SM composite) is formu-lated. The third infinite cylinder of equivalent SM composite hasbeen introduced to account for the interactions with the surround-ing fibers/matrix. The properties of the composite medium enclosingthe SMA fibers are calculated by extending the multicell microme-chanics approach of Chamis.21 An analytical solution is obtained,

1905

1906 BIRMAN, SARAVANOS, AND HOPKINS

which yields the response of the SM composite in longitudinal load-ing and the corresponding fields of local stresses (microstresses).Results quantify the isothermal loading-unloading response of thecomposite under various temperature and initial stress conditions,the capability of the composite to undergo temperature-induced ac-tuation, and the resultant local stresses in the composite during aloading-unloading cycle with stress-induced phase transformationin the fibers.

Constitutive EquationsThe assumed constitutive equations of the SMA fibers, the ma-

trix, and the resultant SM composite are presented in this section.The fibers and the matrix are assumed to remain within the lin-ear elastic range, which is a reasonable assumption since plasticityseems to usually occur after the martensitic transformation. Both thefibers and matrix are assumed isotropic. However, the composite istransversely isotropic with a plane of symmetry transversely to thefibers.

FibersAssuming that both martensitic and parent phases are isotropic

and that there is no directionality in the alignment of the martensiteand austenite crystals, the SMA fibers may be assumed to remainisotropic, and their engineering properties are linear functions of themartensite fraction (rule of mixtures)14"16

(1)

where P represents the thermoelastic properties, subscript / in-dicates the fiber, £ is the current martensite volume fraction in thefiber, and superscripts M and A identify the martensite and austenite(parent) phases, respectively.

Considering that the austenite and martensite phases in an SMAare assumed to behave thermoelastically, the behavior of an SMAfiber during the martensite or reverse phase transformation may beeffectively described by Hooke's law in expanded form,16

fi = cfv(€fj ~ <*/; Ar - f , 7 = 1, . . . . 6 (2)

where afi is the stress vector, 6j is the elastic strain vector, AT isthe temperature variation, and C/7 and otj are the current elasticitytensor and the vector of thermal expansion coefficients, respectively.The term €*. is the transformation strain as a result of phase change.There is reasonably good understanding and experimental evidenceabout the uniaxial behavior of SM materials; that is, under uniaxialstress the rate of the transformation strain is directly proportional tothe rate of the martensite fraction,

€' = o>k (3)

where a) is the maximum transformation state in uniaxial isothermalloading. However, the behavior of SMAs under a three-dimensionalstress state seems to be less understood, particularly the effect thatindividual stress components may have on the reorientation of themartensite variants. The relationship between the transformationstrain and martensite fraction may have the general form15'16>22

(4)

Liang and Rogers15 and Boyd and Lagoudas16'22 have proposedvarious forms for the transformation tensor A/, which effectivelyassume that the transformation strain is also related to the devia-toric stress components. Although these are interesting approachesattempting to capture the interactions between the various stresscomponents and the corresponding transformation strain, the pro-posed forms of A, and underlying assumptions have been neithervalidated nor correlated with experimental data. In the present paper,the transformation tensor components were assumed proportionalto the maximum transformation strain &>/ being observed when thecorresponding uniaxial stress component was applied; i.e., A/ = &>/.Then the integration of Eq. (4) results in

(5)

where superscript 0 refers to an initial state. Equation (2) can beapplied at different time instances (as in the case of uniaxial phasetransformation10-14) to yield

afi - < =

(6)

In thermoelastic martensitic transformations, the martensite fractiondepends on the free chemical energy in the material, which, in turn,includes the thermal and strain energy components. This relationmay be represented by phenomenological equations7'8 that use theeffective stress to represent the three-dimensional stress state in thematerial.15'16 Accordingly, during the martensite transformation,

£ = 1 - exp [bM(TsM -T) + cMacf(]

and during the reverse transformation,

(7)

(8)

where 75M and 7/ are the martensite and austenite phase start tem-

peratures under reference stress-free conditions and oreff is the effec-tive stress defined as

The coefficients bM,cM,bA, and CA are given by

U).01 b*

bA =

M TS ~ LF

U).Q1

(9)

(10)cA =

where Tp and T* are the martensite and austenite finish temper-atures under stress-free conditions and dM and dA are constantsdefining the slopes of the martensitic and reverse transformationlines on the stress-temperature diagram, respectively. The charac-teristic temperatures depend also on the effective stress:

(11)

where the overbar indicates the start and finish temperatures of therespective phase transformation in the presence of stress. Equations(7) and (8) may be rearranged to provide the range of the effectivestress where stress-induced martensitic and reverse phase transfor-mations (at a prescribed temperature) will occur, respectively:

CMbJL(T^

(13)

Equations (12) and (13) are obtained in accordance with the met-allurgical approach, by identifying the values £=0.99 and 0.01with the completion of the martensite and reverse transformations,respectively.

MatrixThe matrix is assumed to be isotropic and elastic, such that

<rmi = Cmij(€mj-amj&T)9 i ,7 = l , . . . , 6 (14)

where AT is the difference between the current and stress-free tem-peratures and subscript ra indicates the matrix.

BIRMAN, SARAVANOS, AND HOPKINS 1907

- Matrix

Fiber

2, T

A: Matrix RegionB: Matrix and Fiber Region

a) Multicell model b) Concentric cylinders model

Fig. 1 Representative composite models.

CompositeThe equivalent constitutive equations for a composite medium

with SMA fibers undergoing a martensitic or reverse transformationcan be postulated to have the following form:

(15)

where subscript c indicates the homogenized composite and £ isthe martensite phase fraction in the fibers. The main difference withEq. (2) is that the resultant composite will exhibit orthotropy in theLT plane (see Fig. 1), including the components of the maximumtransformation strains. Equations (2), (6), and (15) remain validduring the martensitic and reverse transformation, and they reflectthe behavior of thermoelastic SMA materials stressed below theirsecond yield limit associated with the plastic slip of the martensiticphase.23 In all cases, the initial conditions corresponding to the par-ticular phase of the transformation should be considered; moreover,the continuity of stress-strain curves at the boundaries of adjacenttransformation phases is preserved. It is finally emphasized that al-though Eqs. (6) and (15) can describe both martensitic and reversetransformations, the corresponding stress-strain curves do not over-lap each other, because of the differences between the effective startand finish stresses and temperatures defining the onset and comple-tion of the martensite and reverse phase transformations [see Eqs.(7) and (8) or (12) and (13)].

MicromechanicsTwo different micromechanical approaches are presented. A con-

centric three-phase cylinder model is the primary micromechanicsapproach that is used to predict the response of an SM compositeand the corresponding microstresses. However, a priori knowledgeof the equivalent properties of the SM composite phase is requiredin the proposed concentric cylinder model. These equivalent proper-ties of the SM composite were calculated by extending the multicellmicromechanics method.21'24'25

Multicell ApproachGood approximations of the equivalent properties of the SM

composite may be obtained using the multicell micromechanicsapproach.21'24'25 According to the method, the representative cellis divided into a set of subregions (see Fig. la), each one contain-ing portions of a square SMA fiber and the matrix. It is furtherassumed that the strains/stresses in the various subregions of therepresentative cell are uniform, the Poisson's effect on transversedeformations is negligible, and the fibers are packed in a squarearray formation. The analysis employs a mechanics of materials ap-proach based on the equilibrium of stresses within each subregionand the compatibility of strains at the boundaries of each subregion.Thus, the equivalent constitutive relations of the SMA composite arederived in accordance with Eq. (15). The resultant expressions forthe elastic and thermal properties are similar to the ones reported forthermoelastic composites and are shown in the Appendix. Accord-ing to the constitutive relations for the fibers (2) and the composite(15), the phase transformation/recovery strain component has ananalogous form to the thermal strain component. Therefore, at a

given temperature and martensite fraction in the fibers, the equiv-alent maximum transformation strains of the composite o}ci maybe evaluated similarly to the thermal expansion coefficient of thecomposite (see the Appendix):

= k f c o f ( E f / E L )

(16)- (Em/Ef)] ET

where subscripts L and T indicate the longitudinal and transversedirection in the composite with respect to the fibers, kf is the fibervolume fraction, Ef and Em are the moduli of elasticity of the fiberand matrix, respectively, and EL and ET are the longitudinal andtransverse moduli of elasticity of the composite medium.

Concentric Cylinder ApproachConsidering the potential significance of local stresses (mi-

crostresses) in the response of the SMA fibers, a concentric cylindermodel can provide better representations of the local stresses, whichmay have an effect on the axial response of the SMA composite.The three-cylinder assembly model shown in Fig. Ib was formulatedcontaining a fiber, surrounded by a matrix and an infinite SM com-posite medium. The constitutive relations of the composite phaseare provided by Eq. (15), and equivalent properties are provided bythe previously described multicell micromechanics. The inclusionof the infinite composite cylinder is important, because it introducesthe interactions of the fiber and matrix with the surrounding fibers asa result of the mismatch in the respective elastic, thermal, and trans-formation strains. Thus, the consideration of a surrounding infinitecomposite instead of a surrounding infinite matrix extends the ap-plication of the approach beyond the low fiber volume ratio regime.

Considering the axisymmetry of the problem, and assuming thatthe displacement u in the direction of the fiber remains uniform inthe transverse r-9 plane, the nonzero strains are

w—,

dw— , ex = const (17)

where w is the displacement in the radial direction. The equations ofequilibrium within each material region of the concentric cylindermodel (fiber, matrix, and composite) are reduced to the followingone:

dr -rar - — u (18)

where subscripts jt, r, and 9 indicate longitudinal, radial, and tan-gential directions, respectively. Combination of Eq. (18) with con-stitutive equations results, in each region, in the following:

ar —- (cor -

€rtr — (Cr90i()

(19)

Considering the transverse isotropy of each phase (fibers, matrix,and composite), the uniform temperature (AT,. — 0), and combin-ing Eqs. (17-19), the following equation results:

Crr(r2w,rr j -w)- (Crr -

= 0 (20)

The preceding equation is further simplified by considering thetransverse isotropy of the transformation tensor and by assumingconstant martensite fraction in the fiber (£>r = 0). Under these as-sumptions, the admissible solution of Eq. (20) will have the form

wf

We

(21)

1908 BIRMAN, SARAVANOS, AND HOPKINS

where AJk are constants of integration and superscripts indicate the

constituent. The requirements of a bounded solution yield A\ =A £ = 0. The remaining constants of integration are related via theconditions of continuity in the displacements and stresses at thefiber-matrix and matrix-composite interfaces:

wm(rm) =(22)

where rf and rm are the radius of the fiber and the external matrixinterface (Fig. Ib), respectively.

The effective stress in the fibers [Eq. (9)], required for the deter-mination of the martensitic fraction [Eqs. (7) and (8)] and the stateof the composite, becomes

because the shear stresses are zero. The solution yields equal tangen-tial and radial strains in the fiber, that is, 60 = €r = A{ . Consideringthe isotropy in the transverse plane of the fiber, this also impliesthat the radial and tangential stresses in the fiber are equal and re-main constant along the radial direction. This simplifies further theexpression for the effective stress in the fibers:

= cf (24)

Computational ProcedureAs seen in the previous analytical formulation, the analysis of

both the properties and the local stresses in a fibrous SM compositemay require an incremental and iterative approach. This is becausethe properties of the fibers and, therefore, the average properties ofthe composite depend on the stresses via the martensite fraction. Inturn, the stresses are affected by the properties of the constituentmaterials. To further complicate the problem, the start and finishtemperatures of the martensitic and reverse transformation dependalso on the stresses.

However, the iterative process can be avoided, and the solutionmay be obtained directly. In the latter approach, the analysis beginsby specifying the value of the temperature. Rather than specifyingthe axial strain or stress, the effective stress in the fiber is specified.Then the martensite fraction in the fiber is calculated according toEq. (4) or (5). Once the martensitic function is defined, the prop-erties of the fibers and the equivalent properties of the surroundingcomposite are calculated. Then, Eqs. (22) and (24) are solved, andthe constants of integration A[, Af, AJ, and AC

2 and the axial strainex corresponding to the effective stress are found. The local strainsand stresses are calculated. A final integration over the radius yieldsthe average longitudinal stress in the composite,

1rrdr (25)

Application CasesThe application cases presented in the following paragraphs il-

lustrate the predicted stress-strain response of the composite un-der isothermal loading and unloading conditions, the capability ofthe composite to induce axial strains as a result of variations inthe temperature, and the developed local stress fields in the mate-rial during stress-induced martensitic and reverse transformationsin the fiber. The shape memory material considered in the exam-ples is Ni-44.8wt% Ti, whose properties are presented in Table 1as reported by Lei and Wu.26 An epoxy matrix was assumed withproperties shown in Table 1, and the fiber volume fraction was takenequal to 0.3.

Composite ResponseThe predicted loading and unloading cycle of the SM composite

at 40°C is shown in Fig. 2. It was assumed that local nonmechan-ical stresses (stresses arising from mismatch in the transformationstrains and the thermal strains between the fiber and the matrix)

Table 1 Properties of Ni-44.8wt% Ti fibers26

(annealing temperature 600° C)

Property Ni-44.8wt% Ti Epoxy

£A,GPa£M, GPaVaA, 10~6/°CaM, 10~6/°C"fM of*is , ^Tf ,°CJ"y , °C/ '

JA,'MPa/0C0)\ = 0)2 = 0)3 =0)4 = 0)5 = 0)6 =

30.013.00.3312.518.5235

2951

11.34.50.07

0

3.4475——0.3564.8————

——

————————

400

300

200

100

-100-0.02 0.02 0.04 0.06 0.08 0.1 0.12

Axial Strain

Fig. 2 Isothermal stress-strain cycle of 30% fiber volume ratio Ni-44.8wt% Ti/epoxy composite (40° C). Stress-free conditions correspondto austenite fibers.

are zero when the fibers are in the austenite phase. Practically, thisimplies that the SM composite was fabricated and matrix consoli-dation occurred at 40°C with the fibers in the austenite phase. Asseen in Fig. 2, the composite deforms linearly from point A to Bbecause the fibers remain in the austenite phase (£ =0). After pointC the effective stress in the fiber becomes sufficiently high for themartensite transformation to start. The tensile transformation strainin the fibers results also in elongation of the composite. As a resultof the transformation strain, local (compressive) longitudinal andradial stresses are also developed in the fiber, which start contribut-ing to the effective stress [Eq. (24)], thus resulting in the nonlinearsegment BC (0 < £ < 1). At point C the fibers are fully transformedto the martensite phase (£ = 1), and the subsequent linear sectionCD illustrates the behavior of the composite with elastic martensitefibers. The different slopes of sections AB and CD represent the dif-ferences in the moduli of the martensite and austenite phases of thefiber. The predicted unloading phase is also shown in section DEF.As the stress is reduced, the fibers remain in the martensitic phaseuntil the reverse transformation starts (point E). Further stress reduc-tions after point E reduce the martensite fraction in the fibers and therecovery strains, resulting in the nonlinear curve EF (1 > £ > 0).After point F the fibers are fully transformed to their parent phase(£ = 0) and the subsequent unloading follows the linear path FA.

The isothermal unloading of the previous SM composite system(reverse transformation of the SMA fibers) at four different temper-atures is shown in Fig. 3. As in the previous case, the composite isfree of local stresses when the fibers are austenite at 40°C. As seenin Fig. 3, at higher temperatures higher longitudinal stresses arerequired for starting and finishing the reverse transformation in thefibers. Figure 3 implicitly illustrates the potential of SM compositesto be used as strain or stress actuators by varying the temperatureof the composite, as well as quantifies the ranges of stress, strain,and temperature where such actuation is feasible. Within this range,

BIRMAN, SARAVANOS, AND HOPKINS 1909

400

25.

200

-200

-4000.05

Axial Strain0.1

Fig. 3 Isothermal unloading of 30% fiber volume ratio Ni-44.8wt%Ti/epoxy composite at various temperatures. Stress-free conditions cor-respond toaustenitefibers: ——,20°C; ——,40°C; -•—,60°C; and •-,80° C.

-0.04 -0.02 0Axial Strain

0.02

Fig. 4 Isothermal unloading of 30% fiber volume ratio Ni-44.8wt%Ti/epoxy composite at various temperatures. Stress-free conditions cor-respond to martensite fibers: —— , 20°C; —— , 40°C; — •-, 60°C; and

the composite has the potential to generate significant longitudinaldeformations or stresses. In all cases shown in Fig. 3, high recoverystrains on an order of 6-8% have been calculated. Outside this rangeof stresses, the SM composite cannot act as an actuator. However,as shown in the following case, it seems that this range may bemodified by changing the processing conditions.

Figure 4 also shows the isothermal unloading of the SM compos-ite; however, in this case fabrication and matrix consolidation wereassumed to take place when the fibers are in the martensitic phase at40°C (thermal residual stresses are zero at 40°C). By fabricating thecomposite with the fibers in martensite phase, different initial statesof local stresses are developed in the composite compared with theprevious case, which result in the drastically different unloadingbehavior of Fig. 4. The reverse transformation occurs at differentstress and strain ranges, and the recovery strains are smaller, on anorder of 4% or less. This illustrates that the range of actuation of SMcomposites may be modified by controlling the initial local stressesin the composite. The predicted nonlinear unloading curve is alsononmonotonic with respect to the stress. To qualitatively explainthis behavior, consider the stress-strain curve at 40° C, because thecomposite is free from thermal residual stresses at this temperature.First note that a tensile stress should be applied on the compositeto have martensite fibers, which correctly reflects the need to haveprestressed fibers, before and after the fabrication of the composite,for the martensite phase to exist at 40°C (T > Ty

M). As the axialstrain is reducing, the effective stress in the fibers becomes suf-ficiently small to initiate the reverse transformation. The onset ofreverse transformation in the fibers and the associated contractivetransformation strains result in substantial monotonic contractionof the composite. However, a strain mismatch is also initiated be-tween the fibers and the matrix when £ < 1 , which results in tensile

0.01 -

10 20 30 40Temperature, (degs 0)

Fig. 5 Temperature-induced axial strain actuation with of 30% fibervolume ratio Ni-44.8wt% Ti/epoxy composite with reverse phase trans-formation in the fibers. Composite remains under stress-free conditions.

radial and longitudinal stresses in the fiber. These stresses alter theeffective stress in the fiber [see Eqs. (23) and (24)], and to have con-tinuous reduction in the martensite fraction, the mechanical stresson the composite should be adjusted. It is believed that such an ad-justment is shown in Fig. 4, by the nonmonotonic stress variationduring the reverse transformation. Apparently, the evolution of ten-sile radial stresses in the fiber reduces the effective stress beyond apoint that makes it impossible to achieve certain ranges of marten-site fractions, unless the applied mechanical stress on the compositeis increased. In other words, Fig. 4 shows that to have continuousmonotonic decrease in the martensite fraction of the fibers, the axialstress needs to increase and decrease again. This type of behavioris expected to be observed in a strain controlled uniaxial test.

The predicted capability of the composite to induce axial strain,under zero axial stress conditions, by controlling the temperature inthe composite is shown in Fig. 5. The curve corresponds to a com-posite with stress-free austenite fibers. The curve begins at 20°Cwhen the fibers are in the martensitic phase. The reverse transfor-mation starts at 30°C and finishes at 55°C when the fibers havebeen fully transformed into austenite. It was predicted that a 25°Cvariation in temperature will result in a recovery strain of 6.3%.

Local StressesThe predicted local stress distributions in the composite are pre-

sented here. The calculation of the stresses in the fiber and the matrixis important to ensure the integrity of the composite and prevent mi-crocracking and fiber-matrix debonding. The longitudinal, radial,and hoop stresses are shown in Figs. 6-9 as functions of the normal-ized radius for various phases of the isothermal stress-strain cycleshown in Fig. 2. The normalized radius in the figures is defined asthe ratio of the local radius to the radius of the fiber (r/rf).

Figure 6 shows the stresses in the composite, when an aver-age longitudinal stress acx = -53.6 MPa is applied and the fibersstill remain in the austenitic phase. Figure 7 shows the stresses inthe composite when the fibers undergo martensitic transformation(£ = 0.83) under the application of a tensile stress (acx = 176 MPa).The applied tensile stress results in tensile longitudinal stresses inboth the fiber and the matrix and in a tensile hoop stress in thematrix. Figure 8 shows the local stress in the composite during un-loading with fully transformed fibers (£ = 1 and acx =288 MPa).In spite of the significant increase in the average applied stress, thereis little change in the magnitudes of the radial and hoop stresses,but the longitudinal stresses in the fibers and the matrix increasesignificantly. Figures 7 and 8 clearly show the significant contri-butions that the fiber transformation strains may have on the localstresses. The effect of these transformation-induced stresses on thecomposite response was discussed in the previous subsection.

Predicted stress distributions corresponding to the reverse phasetransformation in the fibers (£ = 0.40 and acx = 12 MPa) are shownin Fig. 9. A comparison indicates that the stresses tend to return to adistribution similar to the one shown in Fig. 6. Further reduction ofthe longitudinal stress to —53.6 MPa resulted in full recovery of theparent phase in the fibers (f = 0) and yielded a stress distribution

1910 BIRMAN, SARAVANOS, AND HOPKINS

OE+0

r -2E+7

• -4E+7

I| -6E+7

31 -8E+7

J -1E+8

0

_f!C*.R

-7E+6

ST -8E+6

| -9E+6}I* -1E+7

'

4 OC*T

1

0.5 1Normalize

-

-

Radial Stress•-

t

1.5 2d Radius

VV

V

** f c > >

„.-•"

t ,

4.OC1-O

3E+8

i

Ig 2.5E+8

g 2E+8

1 1.5E+8

€ 1E+8|

5E+7

_— _c

2E+8

J- 1E+8«•

-1E+8

-2E+8

-3E+8Q

-

-

_

i . i . i .> 0.5 1 1.5 2

Normalized Radius

V xx

"* -

Hoop Stress--

Radial Stress

~

,•'

r i i

0.5 1 1.5 2MMtmMirMH DM««I.M

0 0.5 1 1.5 2Normalized Radius

Fig. 6 Distribution of microstresses in Ni-44.8 wt% Ti/epoxy compositewith fibers in the parent (austenite) phase (crcx = — 53.6 MPa and £ = 0).Isothermal loading cycle at 40° C.

Fig. 8 Distribution of microstresses in Ni-44.8wt% Ti/epoxy compositewith fibers fully transformed to the martensite phase (crcx = 288 MPaand £ = 1). Isothermal loading cycle at 40° C.

2.5E+8

2E+8

! 1.5E+8

1E+8

5E+7

OE+00.5 1

Normalized Radius1.5

1E+8

I5E+7

OE+0

b-SE+7

-1E+8 -

0.5 1 1.5Normalized Radius

3t+8

2E+8

1E+8

ut*u

-1E+8

-2E+8

-3E+8fl

s

Hoop Stress

Radial Stress

_ _______ ! • ' '

0.5 1 1.5 2Normalized Radius

Fig. 7 Distribution of microstresses in Ni-44.8wt% Ti/epoxy compos-ite with fibers undergoing martensite phase transformation (crcx -176MPa and £ = 0.83). Isothermal loading cycle at 40°C.

1E+8

5E+7

OE+0

-5E+7

-1E+8

-1.5E+80.5 1 1.5

Normalized Radius

Fig. 9 Distribution of microstresses in Ni-44.8wt% Ti/epoxy compos-ite with fibers undergoing reverse transformation (crcx = 12 MPa and£ = 0.40). Isothermal loading cycle at 40°C.

BIRMAN, SARAVANOS, AND HOPKINS 1911

identical to the one shown in Fig. 6. This is not surprising, becausea reversible thermoelastic SM effect was assumed and at this pointboth loading and unloading curves coincide; hence, the correspond-ing internal stress states in the composite should be identical.

SummaryMicromechanics for the evaluation of the properties and analysis

of stress in unidirectional composite systems with continuous SMAfibers in an elastic matrix were presented. The multicell microme-chanical method was extended to calculate the average properties ofthe composite when the fibers undergo martensite or reverse phasetransformations. The axial response of the composite and the localstresses in the fiber and the matrix were determined using a concen-tric cylinder model, where the fiber and the surrounding matrix areembedded within an infinite composite medium.

Applications of the mechanics were presented for a Ni-44.8wt%Ti/epoxy composite system. The numerical results demonstratedthe longitudinal response of the composite under isothermal load-ing and unloading conditions. They also illustrated the capabilityof the composite to induce substantial longitudinal strains undervariation of temperature. The ranges of stress, strain, and tem-perature where martensite and reverse phase transformations oc-cur were also shown. It was predicted that substantial actuationstrains may be induced in composites of intermediate fiber vol-ume fraction (~0.3). It was also shown that the range of stressand strain where the composite may be used as actuator may bemodified by controlling the stress-free conditions in the composite.Finally, the distribution of microstresses was considered for an en-tire isothermal loading-unloading cycle of the composite materialwith stress-induced phase transformation. The characteristic dis-tribution of stresses remained unaffected, but their magnitude andsign changed drastically, depending on the martensite fraction ofthe fibers and the applied external load. In closing, this work rep-resents an effort towards understanding and modeling the complex,yet promising, active response of continuous SMA fiber composites.Additional analytical and experimental work should be conductedin the future.

Appendix: Thermoelastic PropertiesThe elastic properties of SM composites using the multicell

method21-25 are

E, = Efkf f Emkm

1 - J k f [ \ - (Gm/Gf)]

Ga

ET = Em\\-

GLT = '

- (Gm/Gf)]

VLT = vfkf + vmkm

VTT = (ET/2GTT) - 1

The coefficients of thermal expansion are

UT =

(Al)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

where Em,Gm,vm, and am are the normal and shear moduli of elas-ticity, the Poisson's ratio, and the coefficient of thermal expansionof the matrix, respectively; and v/ is the Poisson's ratio of the fiber,which is independent of the martensite fraction. Volume fractions

of the fiber and the matrix are denoted by kf and £m, respectively.Subscripts L and T identify the longitudinal and transverse direc-tions.

AcknowledgmentThis research was conducted during an American Society for

Engineering Education summer (1994) faculty fellowship of VictorBirman at the NASA Lewis Research Center.

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