Two-way reversible shape memory effects in a free-standing polymer composite

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Smart Mater. Struct. 20 (2011) 065010 (9pp) doi:10.1088/0964-1726/20/6/065010

Two-way reversible shape memory effectsin a free-standing polymer compositeKristofer K Westbrook1, Patrick T Mather2,3, Vikas Parakh2,Martin L Dunn1, Qi Ge1, Brendan M Lee4 and H Jerry Qi1,5

1 Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309, USA2 Department of Biomedical and Chemical Engineering, Syracuse University, Syracuse,NY 13244, USA3 Syracuse Biomaterials Institute, Syracuse University, Syracuse, NY 13244, USA4 Niwot High School, Niwot, CO 80503, USA

E-mail: qih@colorado.edu

Received 7 January 2011, in final form 8 April 2011Published 23 May 2011Online at stacks.iop.org/SMS/20/065010

AbstractShape memory polymers (SMPs) have attracted significant research efforts due to their ease inmanufacturing and highly tailorable thermomechanical properties. SMPs can be temporarilyprogrammed and fixed in a nonequilibrium shape and are capable of recovering the originalundeformed shape upon exposure to a stimulus, the most common being temperature. MostSMPs exhibit a one-way shape memory (1W-SM) effect since one programming step can onlyyield one shape memory cycle; an additional shape memory cycle requires an extraprogramming step. Recently, a novel SMP that demonstrates both 1W-SM and two-way shapememory (2W-SM) effects was demonstrated by one of the authors (Mather). However, toachieve two-way actuation this SMP relies on a constant externally applied load. In this paper,an SMP composite where a pre-stretched 2W-SMP is embedded in an elastomeric matrix isdeveloped. This composite demonstrates 2W-SM effects in response to changes in temperaturewithout the requirement of a constant external load. A transversal actuation of ∼10% ofactuator length is achieved. Cyclic tests show that the transversal actuation stabilizes after aninitial training cycle and shows no significant decreases after four cycles. A simple analyticmodel considering the programming stress and actuator dimensions is presented and shown toagree well with the transverse displacement of the actuator. The model also predicts that largeractuation can be achieved when larger pre-stretch of 2W-SMP is used. The scheme used for thispolymer composite can promote the design of new shape memory composites at micro- andnano-length scales to meet different application requirements.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Shape memory polymers (SMPs) are a class of environmen-tally activated materials that are capable of undergoing largedeformations which are recoverable upon application of anenvironmental stimulus. Out of all the environmental stimuli(light, humidity, electric fields, etc), temperature is the mostwidely studied. This is a consequence of the ease in ability totailor the thermomechanical properties of thermally activatedSMPs, such as the glass transition temperature (Tg) and the

5 Author to whom any correspondence should be addressed.

temperature range for the glass transition region (Lendleinet al 2004, Liu et al 2006, Mather et al 2009, Ortega et al2008). SMPs can be programmed and temporarily fixed in anonequilibrium shape and are capable of fully recovering theoriginal undeformed shape. A typical shape memory (SM)cycle in an amorphous network SMP is a two step process.First, the shape is programmed by deforming the material intoits required temporary shape at a temperature above Tg thencooling to a temperature below Tg to fix the shape. Second,the shape is recovered by increasing the temperature above Tg

under free or constrained recovery boundary conditions. Inthis case, for an amorphous network SMP, the SM behavior

0964-1726/11/065010+09$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA1

Smart Mater. Struct. 20 (2011) 065010 K K Westbrook et al

L L

Figure 1. Schematic showing the difference between the one-way and two-way shape memory effects where TH and TL are temperaturesgreater than and less than the material’s transition temperature. The top arrow represents the 1W-SM effect while the circularly orientedarrows represent the repeatable, thermally controlled 2W-SM effect occurring due to a constant load.

is the result of the glass transition (Lendlein et al 2004, Liuet al 2006, Qi et al 2008). Once the material has recovered,it requires an additional programming step for any furtherrecovery. Therefore, as shown schematically in figure 1 underfree recovery conditions, this behavior is referred to as a one-way shape memory (1W-SM) effect since additional recoverybehavior requires an additional programming step. There havebeen great efforts in modeling this behavior under both free andconstrained recovery conditions (Castro et al 2010, Chen andLagoudas 2008a, 2008b, Chen et al 2008, Lagoudas et al 2006,Liu et al 2006, Patoor et al 2006, Qi et al 2008, Tobushi et al1996, Westbrook et al 2010). In many applications, 1W-SMis sufficient. For example, the 1W-SM behavior for an SMPunder fully confined recovery conditions has been harnessed inthe field of self-healing materials where it has been shown thatmultiple crack closure or healing events can occur with onlyone programming step due to geometric confinement (Li andJohn 2008, Li and Nettles 2010, Xu and Li 2010).

The requirement of an additional programming step inSMPs exhibiting 1W-SM effects is a significant drawbackin some applications. Recent development of a novelsemicrystalline poly(cyclooctene) (PCO) based SMP by one ofthe authors (Mather) (Chung et al 2008) allows for an SMPwith the capability of exhibiting both 1W-SM and two-wayshape memory (2W-SM) effects. Here, both the 1W- and2W-SM effects are achieved via stretch induced crystallization(SIC) which can cause stress relaxation under a constant stretchwhen the temperature is lowered (Chung et al 2008, Gent1954, Westbrook et al 2010c). In the programming step,the SMP is initially stretched at a temperature above thetransition temperature (Tt) under a constant externally appliedtensile load, causing the chains to align in the direction ofthe applied strain. The SMP is then cooled below Tt whilemaintaining the external load. This leads to the formationof crystalline domains which are stress-free upon formationand result in an increase in material stretch to obtain an

equilibrium force balance in the material. Subsequentlyheating the material above Tt causes the sample to recoverits initially preloaded shape upon melting of the crystallinedomains. Under a constant load, these programming andrecovery steps are repeatable under thermal cycling as shownschematically in figure 1. This process is referred to as a2W-SM effect. Furthermore, the material can achieve a 1W-SM effect by removing the external load after programming,allowing full shape recovery with respect to the initiallyundeformed configuration. As with the amorphous SMP thatexhibits 1W-SM effects (1W-SMP) having a reprogrammingdrawback, the semicrystalline 2W-SMP has its own drawbackin which it requires a constant externally applied load in orderto achieve the 2W-SM effect.

In typical SMP applications, it is desirable to have theability to harness the 2W-SM effect without the requirementof a constant externally applied load. A very well-known application exhibiting these conditions of free-standingactuation in metals is the bimetallic strip which actuatesthrough the difference in the thermal expansion of the twometals. In order to achieve free-standing 2W-SM behavior,SMP composites must be developed. Using the mismatchin the change in thermal expansion and contraction betweenan epoxy-based resin and a fiber-reinforced epoxy-basedpolymer, Tamagawa (2010) introduced a polymeric laminatethat is capable of repeated thermal cycling (2W-SM behavior)without delamination and a maximum bending curvature of0.002 mm−1. Recently, Chen et al (2010, 2008) demonstrateda bilayer polymeric laminate composed of an elastic polymerand a crystallizable SM polyurethane capable of thermallyactivated 2W-SM effects in bending. Chen et al showedtwo repeatable cycles with some loss of actuation referred toas pre-training the actuator and showed dynamic mechanicalanalysis results of the laminate structure, suggesting that morethan five cycles can be repeated. The mechanism for the2W-SM behavior in the actuator by Chen et al is attributed

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Smart Mater. Struct. 20 (2011) 065010 K K Westbrook et al

to the difference in the temperature dependent elastic modulusbetween the two materials. In this paper, a method formanufacturing a polymer composite actuator (herein referredto as the actuator) and its characterization under free-standingthermal cycling are presented. The actuator was able todemonstrate more than four shape memory cycles withoutshowing much decay in its shape memory capability. Basedon the understanding of the SM mechanism for this material, asimple model was proposed and shown to compare favorablywith experimental results. The paper is arranged as follows. Insection 2, the materials used in this paper are introduced, thecharacterization of their thermally activated behavior is brieflypresented and the methods for fabricating and characterizingthe behavior of the actuator are discussed. In section 3, theresults from the actuator behavior experiments are shown. Insection 4, the behavior of the actuator is discussed and asimple model is introduced to quantify the actuator’s thermallyactivated deformation. Lastly, conclusions and future work arepresented.

2. Methods

2.1. Materials

2.1.1. 2W-SMP (Embedded material). Sheets of 2W-SMPswere synthesized following the detailed procedure by theauthors (Chung et al 2008). In particular, poly(cyclooctene)(Evonik-Degussa Corporation, Vestenamer 8012) with a transcontent of 80% and dicumyl peroxide (DCP) (>98% purity,Aldrich) were used as received. This polymer and crosslinkingreagent were blended intimately using a Randcastle singlescrew Microtruder (RCP 0625) with screw diameter 0.625inches and an effective L/D ratio of 24:1. The Microtrudertemperature set-points were 70 ◦C (feed zone), 75 ◦C (meltingzone), and 80 ◦C (metering zone). The 1 mm cylindricaldie temperature was set to 80 ◦C. The dried PCO pelletswere then fed into the feed chamber and mixed with 2 wt%DCP, based on the weight of PCO pellets to obtain the desirablecomposition in the final products. A rotation speed of 50 rpmwas chosen. Such a process is capable of melting the polymerand uniformly dispersing the peroxide, without inducing anydetectable cross-linking. The extrudate was collected as astrand from the microtruder and cut into small pellets. Thesepellets were re-extruded to ensure homogeneous mixing. Thefinal extrudate was cooled to room temperature. Next, theblended PCO/DCP mixture was molded into a crosslinked filmusing compression molding. In particular, the extrudate waspressed between two hot platens, with a square Teflon™ spacerinserted, preheated to 180 ◦C and then cured for 30 min undera load of 4448 N (about 1000 lb), which assured a good sealat the Teflon spacer. The press was then slowly cooled toroom temperature and the fully cured specimens isolated withuniformly white appearance. While samples featured DCPconcentrations varying from 1 to 2 wt% (based on the weightof PCO), only experimental results for PCO with 2 wt% ofDCP are presented and used in this paper. Results for the shapememory properties of PCO crosslinked with other DCP weightcontents can be found in (Chung et al 2008).

Figure 2. Two-way shape memory effect demonstrated by the2W-SMP with an applied stress of 700 kPa.

The 2W-SMP behavior was explored according to theprocedure described in recent works by the authors (Chunget al 2008, Westbrook et al 2010c). Here, a briefdescription of the experiment and the results showing the2W-SM effect are given. A specimen with dimensions of40 mm × 2.4 mm × 0.84 mm was sectioned from the fullycured PCO sheet and tested. The specimen is thermally cycledunder creep conditions with a constant nominal stress (hereinreferred to as programming stress) to measure the 2W-SMbehavior. The actuation strain Ract(T ) = L(T )/LH − 1 =λact − 1 (Chung et al 2008) is used to characterize the 2W-SM behavior where LH is the pre-stressed length of the sampleat TH = 70 ◦C, L is the length as the temperature varies andλact is the actuation stretch. Note, as previously reported Tg =−70 ◦C for the PCO material (Chung et al 2008). Figure 2shows the results of the experiments characterizing the 2W-SM effects. Briefly, two salient features can be seen from theseresults. First, the actuation strain at the end of the coolingstep and during the heating step reaches values of 24.8% and26.2%, respectively. Second, it is clear that the recovery of theactuation strain during heating shows a large hysteresis loopwith a temperature delay of approximately 17 ◦C. A more in-depth discussion on this behavior can be found in Chung et al(2008) and Westbrook et al (2010c) where the constant appliedstress dependence on the 2W-SM behaviors is also discussed.Furthermore, the initial Young’s modulus of the PCO–DCPmaterial is determined from the curve fitting Es = 1.4 MPaat 70 ◦C (Westbrook et al 2010c).

2.1.2. Matrix material. To achieve 2W-SM behavior inthe actuator presented in this paper, a polymeric materialwith a Tg below the actuator thermal cycle temperaturerange (effectively an elastomer) is required as the matrixmaterial. Here an acrylate-based polymer is used. The matrixmaterial was prepared following the procedure given in Ortegaet al (2008) where the effects of the polymer constituentcompositions on the thermomechanical properties of theacrylate-based SMP are studied. The prepolymer solution

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Smart Mater. Struct. 20 (2011) 065010 K K Westbrook et al

Figure 3. Schematic detailing the actuator fabrication process. Step 1: the PCO–DCP strip is programmed at TH and its shape is fixed at TL;step 2: the programmed strip is mounted inside the custom-manufactured aluminum mold; step 3: the matrix material is injected inside themold and photo-cured; step 4: the mold is thermally cured and the pre-sectioned actuator is removed; step 5: the actuator is sectioned to therequired geometry. The final dimensions of the actuator are referenced in table 1.

was prepared by the copolymerization of tert-butyl acrylate(tBA) (98% purity, Aldrich) with poly(ethylene glycol)dimethacrylate (PEGDMA), Mn = 750 (PEGDMA750)(Aldrich). Comonomer solutions were produced by mixing55 wt% PEGDMA750 with 45 wt% tBA. The photoinitiator2,2-dimethoxy-2-phenylacetophenone (99% Purity, Aldrich)was added to the comonomer solution at a concentration of0.2 wt% of the total comonomer weight and manually mixeduntil fully dissolved. For material characterization, such asdynamic mechanical analysis (DMA) and isothermal uniaxialtensile experiments, specimens were prepared by injecting thecomonomer solution between two Rain-X (SOPUS Products,Houston, TX) coated glass slides separated by glass spacers,secured by binder clips and exposed to UV light for a periodof time. Since the actuator testing range was between 15 and70 ◦C (discussed in section 2.2.2), the chemical compositionwas chosen from Ortega et al (2008) such that the Tg of thematrix material was below 0 ◦C. DMA experiments showedTg = −10 ◦C. Isothermal uniaxial tension experiments at70 ◦C were performed at a strain rate of 0.01% until fracture.Inspection of the initial slope of the resulting stress–strainbehavior (not shown) yielded an initial Young’s modulus,Em = 11.4 MPa. The remaining comonomer solution wasused in preparing the actuator discussed in section 2.2.

2.2. Polymer composite actuator

2.2.1. Actuator fabrication. A schematic showing thefabrication process for the actuator is shown in figure 3. Thefabrication process for the actuator was comprised of five steps.In step 1 the PCO–DCP strip was sectioned from the bulksheet to have dimensions of 40 mm × 2.4 mm × 0.84 mm andprogrammed following the steps discussed above. Specifically,the strip was stretched under a stress of 700 kPa at 70 ◦C andthen cooled to TL = 15 ◦C at a rate of 2 ◦C min−1. Theexternal load was then removed and the programmed stripwas used immediately in the actuator embedding step. Instep 2, the programmed PCO–DCP strip was inserted into themold in preparation for embedding. The mold was machinedwith slots on each end located off center in the thicknessdimension for this purpose. The enclosed, interior section ofthe mold was designed to be wider than the required actuatordimensions to allow for the removal of any edge defects thatmay be introduced during polymerization. The PCO–DCPstrip was secured in place to restrict it from recovering anyof its deformed length during the exothermic polymerizationof the chosen matrix material and the post-polymerizationthermal curing processes. In step 3 the mold was sealed by

Table 1. Material properties and dimensions (refer to figure 6) usedin the analytic model.

Description Parameter Value

ActuatorThickness h1 (mm) 2.76Width w1 (mm) 4.85Length L (mm) 46.73

Embedded PCO–DCP stripThickness h2 (mm) 0.81Width w2 (mm) 1.72Offset h B (mm) 0.63

Material propertiesMatrix material Em (MPa) 11.4PCO–DCP strip Es (MPa) 1.4

securing two Rain-X pretreated glass slides on the top andbottom using binder clips and the matrix material comonomersolution was then injected into the sealed mold using a syringe.After the mold was filled, it was placed under a UV lampfor a total of 4 min, turning over every minute to ensure thematrix material cured on both sides of the embedded strip. Instep 4 the sealed mold was then placed in an oven preheatedto 70 ◦C for 30 min to ensure the polymerization was completeand to remove any residual stresses introduced in the matrixmaterial during polymerization, and then was allowed to coolto room temperature. Afterward, the pre-sectioned actuatorwas removed from the mold. In step 5, the actuator wassectioned to the required actuator dimensions. The total widthof the actuator was cut to a length approximately three timesthe width of the programmed PCO–DCP strip giving overallactuator dimensions of 46.73 mm × 4.85 mm × 2.76 mm(reference table 1).

2.2.2. Actuator characterization experiments. The actuatorcharacterization experiments were performed using an MTSUniversal Materials Testing Frame (Model Insight 10)equipped with a customized Thermcraft thermal chamber(Model LBO) and a temperature controller (Model Euro 2404).The actuator was mounted inside the thermal chamber to thetop grip attached to the frame. A grid was placed behind thesample for tracking the actuator deformation via time-lapsedphotography. Once the actuator was mounted, the thermalchamber door was closed and a digital camera was set upoutside the thermal chamber. The chamber temperature wasdecreased to TL and 30 min was allowed to equilibrate beforestarting the test. The thermal history of one cycle is describedby the following. First, the temperature was held constant at TL

for 5 min and then heated to TH at a rate of 2 ◦C min−1. Once

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Smart Mater. Struct. 20 (2011) 065010 K K Westbrook et al

Figure 4. Actuator results for the first thermal cycle showing the free-standing 2W-SM effects. The top of the actuator is fixed while thebottom (actuator end denoted as point A) is free. The spacing of the square grid used to track the actuator deformation is 5 mm.

TH was reached, the temperature was held constant at TH for10 min and then cooled to TL at a rate of 2 ◦C min−1. Then, thetemperature was held constant at TL for an additional 5 min.Here, as before, TH = 70 ◦C and TL = 15 ◦C. The experimentconsisted of five thermal cycles and photos were taken at 5 sintervals.

3. Results

The results of the actuator characterization experiments areshown in figures 4 and 5. Figure 4 shows the actuatordeformation at different temperatures for the first cycle,referred to as the training cycle. During the heating step,the actuator begins to deform at a temperature of 40 ◦C andreaches the full actuation at approximately 60 ◦C. Duringcooling, the actuator starts to recover at 35 ◦C (not shown)and nearly reaches a fully recovered state at 20 ◦C. Smalldots on the side of the actuator were included to measure thedeformation behavior of the actuator during thermal cycling.Point A in figure 4 is used to track the transverse displacementof the actuator end. During the training cycle, the actuatorreaches an actuation of ∼6.00 mm but does not recover toits original position, losing ∼0.7 mm travel for the transversedisplacement.

Beyond this first training cycle, reversible bending actu-ation is observed with good repeatability among subsequentcycles. Figure 5(A) shows the transverse displacement ofpoint A for the remaining four thermal cycles. Recall fromfigure 2, the actuation strain of the PCO–DCP specimen

produced a large hysteresis loop between cooling and heatingwith a temperature span of approximately 17 ◦C. Similarbehavior is also observed for the actuator upon coolingand heating with the same 17 ◦C temperature span shownin figure 5(A). The major difference is that the actuatordemonstrates a temperature shift in this hysteretic behaviorcompared to the PCO–DCP strip. Specifically, for the coolingstep the PCO–DCP specimen by itself featured respective SICfusion and saturation temperatures of 38 and 32 ◦C whereas theactuator deformation presents a temperature delay of almost5 ◦C. Furthermore, this temperature delay is apparent inthe heating step between the actuator and the PCO–DCPspecimen where the PCO–DCP specimen displayed respectiveSIC melting and depletion temperatures of 50 and 55 ◦C.This can be attributed to the insulating properties of thematrix material in delaying the temperature response of theembedded 2W-SMP specimen and/or the constraint imposedby the matrix material on the recovery of the embedded 2W-SMP specimen. Considering that the heating/cooling rateswere slow in the present study (2 ◦C min−1) the latter appearsto be the more likely reason. Figure 5(B) shows the highlyrepeatable behavior of the actuator through the evolution ofthe maximum actuation during a thermal cycle (taken tobe the transverse displacement at 70 ◦C) for all cycles asit asymptotically reaches a saturation value ∼4.95 mm or∼10.6% of the actuator length. Although only a total of fiveactuation cycles were investigated, figure 5(B) shows that thereis no significant loss of actuation after the first training cycleand, moreover, the maximum actuation reaches a saturation

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Smart Mater. Struct. 20 (2011) 065010 K K Westbrook et al

Figure 5. Actuator response to thermal cycling. (A) Transverse displacement of the actuator end (point A in figure 4) for the four thermalcycles after training was performed and (B) the evolution of the maximum transverse displacement occurring at 70 ◦C (dashed line representsthe analytic solution discussed in section 4).

value which can be considered the working range of theactuator.

4. Modeling and discussion

To capture the bending deformation of the actuator, a simpleanalytic model is developed. The appendix describes thedetails of the mechanical model for the actuation of amultilayer beam and applies it to the special case of thepolymer composite actuator in this paper. For this actuator,the cross-section in its undeformed state is given in figure 6where the matrix material cross-section (unshaded) and theembedded PCO–DCP strip cross-section (shaded) are shownwith corresponding dimensions and the location of the neutralaxis for the intentional offset location of the embedded strip.The location of the mid-plane and the curvature κ can bedetermined by solving the linear equations:

Bκ + Nσ = 0 (1a)

Dκ + Mσ = 0 (1b)

where Mσ and Nσ are the resultant moment and force dueto the recovery stress in the 2W-SMP strip in the mid-plane,respectively, and B and D are constants determined from theactuator geometry, mechanical properties of PCO and matrixmaterials and the programming stress (see the appendix forcalculations). The curvature κ of the actuator during bendingis related to the actuator’s transverse end displacement d by

d = [1 − cos(Lκ)]/κ (2)

where L is the actuator length. The initial programming stressof the 2W-SMP strip (step 1 of the manufacturing process) was700 kPa, which led to a material stretch of 150%. During thecooling stage, the strip was further stretched by an additional27% (due to stretch induced crystallization), resulting in atotal 2W-SMP stretch of 177%. After the 2W-SMP strip was

Figure 6. Schematic showing the cross-section of the actuator withdimension labels (table 1) used in the analytical model.

embedded into the polymer matrix, the constraint of the matrixprevented the 2W-SMP strip from recovering to its originallength. Since the 2W-SMP has a lower modulus and a smallercross-sectional area than the matrix, the length of 2W-SMPcan be assumed to be nearly unchanged during the heating.Therefore, the total stretch of the 2W-SMP at high temperatureis 177%, leading to an actual actuator programming stressof 1.08 MPa. The actuator has a width w1 of 4.85 mm(the remaining dimensions are summarized in table 1). Theanalytic model predicts a transverse displacement of 7.00 mmfor point A at the actuator end which compares to 6 mm forthe end of the first cycle and an approximate saturation valueof 5.6 mm at the end of the fifth cycle for the experimentalresults (comparison shown in figure 5(B)). The discrepancybetween the model prediction and the experiment could be dueto two reasons. First, the interface between the 2W-SMP andthe matrix may not be perfect to transfer the load. This maycause some loss in actuation. Second, the model assumes thatthe length of the 2W-SMP strip does not change during heating,which could overestimate the programming stress, and thus

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Smart Mater. Struct. 20 (2011) 065010 K K Westbrook et al

Figure 7. Design space showing contours of the transversedisplacement of the actuator end normalized to the actuator lengthusing the analytical solution as a function of the programming stressand the actuator to 2W-SMP strip width ratio. The current actuatorresponse is represented by the bold black dot (w1/w2 = 2.82,σ = 700 kPa, d/L = 0.14).

over-predict the actuation displacement. As discussed later,a more detailed model should consider the thermomechanicalhistory of the 2W-SMP strip and its mechanical balance withthe matrix.

Given that the model is able to predict the maximumtransverse end displacement of the actuator at the hightemperature, the model is used to explore the actuatordesign space which is constrained by the mold geometry andprogramming stress. Here, the programming stress is variedfrom 500 to 900 kPa and the ratio of the total actuator width tothe embedded 2W-SMP strip width is varied from 2 to 4. Theresults are shown in figure 7 where the model prediction forthe actuator geometry used in this study is represented with abold black dot on the contour plot. The degree of actuation,expressed as normalized transverse displacement d/L, can beincreased either by reducing the amount of the matrix materialconstrained through a reduction in total actuator width or byincreasing the programming stress of the embedded strip. Thecontours can be used as a guide for the required deformationcharacteristics of an actuator in an application. It shouldbe noted that there are inherent drawbacks from a designperspective in decreasing the actuator width geometry (amountof constraining material) while increasing the programmingstress in the case of thermal cycling such as tear-out ofthe embedded specimen. Future work in this study willinclude experimentally investigating the design space (actuatordimensions and programming stress) using the same twomaterials introduced here along with presenting a temperaturedependent model for more accurately predicting the actuatorresponse.

Although the simple analytic model presented here isable to predict the isothermal maximum actuation behaviorfor the actuator presented in this work, to fully characterizethe behavior, such as the experimentally observed temperaturehysteresis and the SIC formation and melting characteristics

observed in both the actuator and 2W-SMP specimen itself,a more complicated model is needed. Taking advantageof a previous model (Westbrook et al 2010c) for the 2W-SM effect in the semicrystalline SMP used here, a modelto predict the temperature dependent actuation behavior isfeasible. Investigating the design space of the actuator asdescribed above and introducing this more complex model willallow for the adaptation of the actuator introduced here. Thiswork is currently carried out by the authors and will be reportedlater.

5. Conclusions

It is desirable to harness the 2W-SM effects in applicationswhere a free-standing actuator behavior is required withoutexternal constraints or stresses. In this paper, a method tofabricate a polymer composite actuator is presented. The free-standing 2W-SM effects of the actuator were realized throughembedding a 2W-SMP that exhibits 2W-SM behavior under aconstant external stress inside a polymer matrix. The actuatorwas characterized for multiple thermal cycles and showed thatafter an initial training cycle, the transverse displacement of theactuator end decreased slightly for each additional cycle andultimately approached a saturation value. A simple analyticmodel was presented to predict the maximum transversedisplacement relatively well. The model was then used toexplore the design space of the actuator where the actuatorgeometry and the programming stress of the embedded 2W-SMP specimen could be varied.

Acknowledgments

The authors gratefully acknowledge the support from anAFOSR grant (FA9550-09-1-0195) to PTM, HJQ, and MLD;a NSF career award to HJQ (CMMI-0645219); and a NSFgrant to PTM (DMR-0758631). The authors would also liketo acknowledge the initial work by Ryan J Latini.

Appendix

A.1. Multilayer theory

Figure A.1 shows a schematic of the geometry of a multilayerbeam consisting of N isotropic layers.

The layers are numbered i = 1, 2, . . . , N where 1 is thetop layer and N is the bottom layer. Each layer has a thicknessti where the total thickness of the structure is t . Each of thelayers consists of an isotropic material with a Young’s modulusEi and may have a constant internal stress σ̂i . It is assumedthat the stress state is one-dimensional, i.e. the only nonzeronormal stress component is σ = σX . The thickness t and widthb are both much less than the length L. The displacement uat any position z through the thickness of the layered beamis assumed to be described by the classical Euler–Bernoullihypothesis so that the displacement of any point in the layeredbeam is given by

u = uo − z∂wo

∂xw = wo (A.1)

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Smart Mater. Struct. 20 (2011) 065010 K K Westbrook et al

Figure A.1. Schematic showing an N-layer beam.

where uo and wo are the displacements of the geometric mid-plane at z = 0 in the x- and z-directions, respectively. Thenormal strain in this one-dimensional idealization is

ε(z) = ∂u

∂x= ∂uo

∂x− z

∂2wo

∂x2= εo + zκ (A.2)

where

εo = ∂uo

∂xκ = −∂2wo

∂x2. (A.3)

In equations (A.3), εo is the mid-plane strain and κ is the mid-plane curvature.

The stress–strain relations (Hooke’s law) for each layer are

σk(z) = Ek ε(z)︸︷︷︸

normal strain

+ σ̂k︸︷︷︸

internal stress

. (A.4)

Substituting equation (A.2) into equation (A.4) yields the stressdistribution through the thickness

σk(z) = Ek[εo + zκ] + σ̂k . (A.5)

Now the resultant force N and moment M in the mid-plane in the x-direction with width b in the y-direction isdefined as

N =∫ t/2

−t/2σk(z)b dz M =

∫ t/2

−t/2σk(z)bz dz. (A.6)

Substituting equation (A.5) into equations (A.6) and breakingthe integrals through the thickness into sums of integrals overeach layer yields

N =∫ t/2

−t/2σk(z)b dz =

(∫ t/2

−t/2Ekb dz

)

︸ ︷︷ ︸

A

εo

+(∫ t/2

−t/2Ekbz dz

)

︸ ︷︷ ︸

B

κ +∫ t/2

−t/2σ̂kb dz

︸ ︷︷ ︸

Nσ

M =∫ t/2

−t/2σk(z)bz dz =

(∫ t/2

−t/2Ekbz dz

)

︸ ︷︷ ︸

B

εo

+(∫ t/2

−t/2Ekbz2 dz

)

︸ ︷︷ ︸

D

κ +∫ t/2

−t/2σ̂kε(z)bz dz

︸ ︷︷ ︸

Mσ

(A.7a)

or

N = Aεo + Bκ + Nσ M = Bεo + Dκ + Mσ (A.7b)

where

A =∫ t/2

−t/2Ekb dz =

N∑

k=1

Ekb(zk − zk−1)

B =∫ t/2

−t/2Ekbz dz = 1

2

N∑

k=1

Ekb(z2k − z2

k−1)

D =∫ t/2

−t/2Ekbz2 dz = 1

3

N∑

k=1

Ekb(z3k − z3

k−1)

Nσ =∫ t/2

−t/2σ̂kb dz =

N∑

k=1

σ̂kb(zk − zk−1)

Mσ =∫ t/2

−t/2σ̂kbz dz = 1

2

N∑

k=1

σ̂kb(z2k − z2

k−1).

(A.8)

In the terminology of laminated composite materials, A,B , and D are the extensional, coupling, and bending constants,respectively, and are functions of the Young’s modulus ofeach layer and the arrangement of the layers through thethickness. Physically, A describes the connection betweenthe in-plane force and straining of the mid-plane, D describesthe connection between the moment and curvature and Bconnects the moment to the mid-plane strain and the force tothe curvature.

Equations (A.7b) can be written as either{

NM

}

=[

A BB D

]{

εo

κ

}

+{

Nσ

Mσ

}

(A.9a)

or{

N̄M̄

}

={

N − Nσ

M − Mσ

}

=[

A BB D

]{

εo

κ

}

. (A.9b)

A.2. Application to the polymer composite actuator

Figure 6 schematically shows the cross-section of the 2W-SMPactuator which consists of three isotropic layers. Layer 1 andlayer 3 with width w1 and Young’s modulus Em (Young’smodulus of the matrix) while layer 2 consists of two parts:in the middle there is the PCO–DCP strip with width w2

and Young’s modulus Es (Young’s modulus of the strip) andoutside of the PCO–DCP strip there is the matrix with widthw1 − w2 and Young’s modulus Em. There is a residual stressσ̂2 = S in the PCO–DCP strip of layer 2 that results from theprogramming of the PCO material and no residual stress in thematrix. Since no external force is applied εo = 0.

For this scenario, the resultant force and moment in themid-plane given in equation (7) can be computed as

N =∫ t/2

−t/2σk(z)(w1 − w2) dz +

∫ t/2

−t/2σk(z)w2 dz

= (B1 + B2)κ + (Nσ1 + Nσ

2 )

M =∫ t/2

−t/2σk(z)(w1 − w2)z dz +

∫ t/2

−t/2σk(z)w2z dz

= (D1 + D2)κ + (Mσ1 + Mσ

2 )

(A.10)

8

Smart Mater. Struct. 20 (2011) 065010 K K Westbrook et al

where

B1 =∫ t/2

−t/2Ek(w1 − w2)z dz = 1

2 Em

×3

∑

k=1

(w1 − w2)(z2k − z2

k−1)

D1 =∫ t/2

−t/2Ek(w1 − w2)z

2 dz = 13 Em

×3

∑

k=1

(w1 − w2)(z3k − z3

k−1)

Nσ1 =

∫ t/2

−t/2σ̂k(w1 − w2) dz = 0

Mσ1 =

∫ t/2

−t/2σ̂k(w1 − w2)z dz = 0

(A.11a)

and

B2 =∫ t/2

−t/2Ekw2z dz = 1

2

3∑

k=1

Ekw2(z2k − z2

k−1)

D2 =∫ t/2

−t/2Ekw2z2 dz = 1

3

N∑

k=1

Ekw2(z3k − z3

k−1)

Nσ2 =

∫ t/2

−t/2σ̂kw2 dz = Sw2(zk − zk−1)

Mσ2 =

∫ t/2

−t/2σ̂kw2z dz = 1

2 Sw2(z2k − z2

k−1).

(A.11b)

In equations (A.11b), Ek = Em when k = 1, 3 and Ek = Es

when k = 2. Defining A = A1 + A2, B = B1 + B2,D = D1 + D2, Nσ = Nσ

1 + Nσ2 , and Mσ = Mσ

1 + Mσ2 ,

equations (A.10) can be written as

N = Bκ + Nσ M = Dκ + Mσ . (A.12)

For the polymer composite actuator there are no externalapplied loads along the axial direction, therefore N = 0 andM = 0, further reducing equations (A.12) to

Bκ + Nσ = 0 Dκ + Mσ = 0. (A.13)

Using equations (A.13), the curvature κ and the locationof mid-plane z0 can be determined. Note the relationshipsbetween zi and the geometry of the actuator are z1 = z0 + hT ,z2 = z0 + hT + h2, and z3 = z0 + hT + h2 + hB .

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