Thermomechanical behavior of a two-way shape memory composite actuator

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Smart Mater. Struct. 22 (2013) 055009 (11pp) doi:10.1088/0964-1726/22/5/055009

Thermomechanical behavior of a two-wayshape memory composite actuator

Qi Ge1, Kristofer K Westbrook1, Patrick T Mather2,3, Martin L Dunn1

and H Jerry Qi1

1 Department of Mechanical Engineering, University of Colorado, Boulder, CO, USA2 Syracuse Biomaterials Institute, Syracuse University, Syracuse, NY, USA3 Department of Biomedical and Chemical Engineering, Syracuse University, Syracuse, NY, USA

E-mail: qih@colorado.edu

Received 4 January 2013, in final form 4 March 2013Published 3 April 2013Online at stacks.iop.org/SMS/22/055009

AbstractShape memory polymers (SMPs) are a class of smart materials that can fix a temporary shapeand recover to their permanent (original) shape in response to an environmental stimulus suchas heat, electricity, or irradiation, among others. Most SMPs developed in the past can onlydemonstrate the so-called one-way shape memory effect; i.e., one programming step can onlyyield one shape memory cycle. Recently, one of the authors (Mather) developed a SMP thatexhibits both one-way shape memory (1W-SM) and two-way shape memory (2W-SM) effects(with the assistance of an external load). This SMP was further used to develop a free-standingcomposite actuator with a nonlinear reversible actuation under thermal cycling. In this paper, atheoretical model for the PCO SMP based composite actuator was developed to investigate itsthermomechanical behavior and the mechanisms for the observed phenomena during theactuation cycles, and to provide insight into how to improve the design.

(Some figures may appear in colour only in the online journal)

1. Introduction

Shape memory polymers (SMPs) are a class of emergingsmart materials that feature both a temporary shape and apermanent shape with an ability to switch between the twoon the application of an environmental stimulus, such astemperature (Ames et al 2009, Anand et al 2009, Castro et al2010, Di Marzio and Yang 1997, Diani et al 2012, 2006,Ge et al 2012a, 2012b, Lendlein and Kelch 2002, 2005,Lendlein and Langer 2002, Liu et al 2007, Nguyen et al 2008,O’Connell and McKenna 1999, Qi et al 2008, Srivastava et al2010a, 2010b, Westbrook et al 2011, 2010, Williams et al1955, Xie 2010, Xie et al 2009), light (Jiang et al 2006,Koerner et al 2004, Lendlein et al 2005, Li et al 2003, Longet al 2009, Ryu et al 2012, Scott et al 2006, 2005), moisture(Huang et al 2005), electricity (Leng et al 2008, Yu et al2011a, 2011b) or magnetic field (Mohr et al 2006). SMPscan be further categorized as one-way SMPs (1W-SMPs)and two-way SMPs (2W-SMPs), based on whether theactuation is reversible or not. In the first kind, the transition

from the temporary shape to the permanent shape cannotbe reversed by simply reversing the stimulus. In order toachieve the temporary shape again after recovery, a newprogramming step is necessary. In the second kind, thetransition between the temporary shape and the permanentshape is reversible. Both the one-way shape memory(1W-SM) effect and two-way shape memory (2W-SM)effect have found applications. However, from a materialsprocessing and fabrication point of view, it is relatively moredifficult to achieve the 2W-SM effect. Several attempts weremade in the past to achieve a free-standing 2W-SM effect bydeveloping a SMP composite such as a polymeric laminatedeveloped by Tamagawa (2010); a bilayer polymeric laminatewas introduced by Chen et al (2010, 2008). Recently, Chungreported a SMP, poly(cyclooctene) (PCO), which is capableof exhibiting both 1W- and 2W-SM effects (Chung et al2008). However, in order to achieve the 2W-SM effect, aconstant load (such as a dead weight) has to be applied toassist the reversible actuation. In typical SMP applications,it might not be desirable to apply a constant external load toachieve the 2W-SM effect. To overcome this disadvantage, we

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Smart Mater. Struct. 22 (2013) 055009 Q Ge et al

developed a PCO based composite actuator that demonstrateda nonlinear reversible actuation under the thermal stimulus(Westbrook et al 2011). This composite was fabricated byembedding a PCO strip in its pre-stretched temporary shapeinto an elastomeric matrix. Upon a heating and cooling cycle,the composite actuator could be actuated from a straight shapeto a bent shape, and then be returned to its straight shape.Figure 1(A) shows the actuation during a heating–coolingcycle and figure 1(B) shows the transverse displacement of theend of the composite actuator as a function of temperature. Infigure 1(B), there are three key features of the free-standing2W-SM effect: (1) the actuation is a highly nonlinear functionof temperature with a large hysteresis loop. (2) The trainingcycle of heating–cooling leads to an actuation that is differentfrom the other cycles. After the training cycle, the amountof actuation achieved is slightly reduced. (3) During heating,the actuator initially and unusually moves in a directionopposite to the total net actuation direction. As the heatingcontinues, the actuator reverses this initial direction andmoves dramatically along the actuation direction.

The goal of this paper is to theoretically investigate thethermomechanical characteristics of this PCO SMP compositeactuator and the mechanisms for the observed phenomenaduring the actuation cycles, and to provide insight intohow to improve the design. The paper is arranged in thefollowing manner. In section 2, materials and fabrication forthe actuator are briefly introduced and the thermomechanicalbehavior for the PCO SMP is presented. In section 3, theexisting thermomechanical constitutive model for the PCOSMP is reviewed. A new model for the actuator is developedin section 4 by integrating the constitutive models for thePCO SMP into a laminate composite theory. In section 5,predictions from the analytical model are presented and thecharacterization of the actuator is analyzed. A parametricstudy examining the effects of width ratio, modulus of thematrix and programming stress is presented.

2. Materials and experiments

2.1. Materials

The programmed PCO SMP serves as a strip embeddedinto the matrix. The PCO SMP was synthesized by us-ing poly(cyclooctene) (PCO) (Evonik-Degussa Corporation,Vestenamer 8012 with a trans-content of 80%), crosslinkedwith 2 wt% dicumyl peroxide (DCP) (>98% purity, Aldrich).Details about the preparation of the sample were discussed byChung et al (2008) and Westbrook et al (2011, 2010).

Figure 2(A) shows the 1W-SM effect for this crosslinkedPCO: the sample was stretched by an external load (700 kPa)at TH (here TH = 70 ◦C). Then, the temperature decreasedfrom TH to TL (here TL = 15 ◦C), while the external load wasmaintained. After unloading at TL, PCO fixed its temporaryshape. Finally, the sample recovered to the permanent shapeas it was heated back from TL to TH (Chung et al 2008,Westbrook et al 2010). The 2W-SM effect can be achievedessentially by following the same steps as those in the 1W-SMeffect, except that the external load is maintained throughout.

Figure 1. An actuation during a heating–cooling cycle: (A) theactuation shows the free-standing 2W-SM effect; (B) transversedisplacement of the actuator end for five thermal cycles (Westbrooket al 2011).

The actuation strain Ract(T), which was used to characterizethe 2W-SM effect, is defined as:

Ract(T) =L(T)− L (TH)

L (TH), (1)

where L(T) is the current length as temperature varies andL (TH) is the length of the sample at TH, figure 2(B) showsRact(T) versus temperature plots for the 2W-SM behaviorof PCO under three different external loads (500, 600 and700 kPa). During cooling, at temperatures above ∼35 ◦C, theactuation strain increased almost linearly with a relativelysmall slope. At temperatures between ∼35 and ∼30 ◦C, theactuation strain increased dramatically during crystallization.At a temperature ∼30 ◦C, the actuation strain saturated to∼25% for the 700 kPa case. The magnitude of the actuationstrain depends strongly on the external load, i.e., a higherexternal load results in a higher actuation strain. Upon heating,at temperatures below ∼50 ◦C, the actuation strain increasedslightly. At temperatures between ∼50 and ∼55 ◦C, theactuation strain decreased dramatically due to melting of the

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Smart Mater. Struct. 22 (2013) 055009 Q Ge et al

Figure 2. PCO exhibits both 1W- and 2W-SM behaviors: (A) stress–temperature–strain plot for 1W-SM behavior under an external load700 kPa. (B) Actuation strain–temperature plot for 2W-SM behavior under three different external loads (Chung et al 2008, Westbrook et al2010). Copyright 2010, ASME.

crystalline phase. Finally, at temperatures above ∼55 ◦C, theactuation strain decreased with increasing temperature witha relatively small slope similar to that of the cooling firstregion’s slope (Chung et al 2008, Westbrook et al 2010).

An acrylate-based polymer was chosen as the matrix forthe actuator. It was synthesized by 55 wt% poly(ethyleneglycol) dimethacrylate (PEGDMA), Mn = 750 (PEGDMA750, Aldrich) and 45 wt% tert-butyl acrylate (tBA) (98%purity, Aldrich) and 2,2-dimethoxy-2-phenylacetophenone(99% Purity, Aldrich) was added as the photoinitiator.The glass transition temperature of this polymer is Tg =

−10 ◦C, lower than the temperature range of the actuatorheating–cooling cycle (15–70 ◦C). Details were introduced byOrtega et al (2008).

2.2. Actuator fabrication and characterization

The polymer composite actuator was fabricated by using PCOand the PEGDMA/tBA elastomer. The details of fabricationhave been provided in Westbrook et al (2011). Briefly, thePCO strip was first programmed into a stretched shape under astress of 700 kPa at 70 ◦C, followed by cooling to TL = 15 ◦C,then unloading. The programmed PCO strip was immediatelyplaced into an aluminum mold. After sealing the mold withtwo glass slides, the PEGDMA/tBA solution was injectedinto the mold using a syringe. After photo-polymerizationand curing, the composite material was removed from themold and sectioned into the required actuator dimensions.Specifically, for the actuator tested in this paper, the lengthwas 46.73 mm, the width ratio between the actuator andthe PCO SMP strip was 2.82 (4.85 mm/1.72 mm), and thethickness ratio between them was 3.41 (2.76 mm/0.81 mm).

The actuator characterization experiments followed fiveheating–cooling cycles with each cycle as follows: first, thetemperature was held constant at TL = 15 ◦C for 5 min andthen heated to TH = 70 ◦C at a rate of 2 ◦C min−1. Once

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. Lastly,the temperature was held constant at TL for an additional5 min. The results of the characterization experiments werepresented in the introduction (figure 1).

3. Constitutive models for PCO SMP andPEGDMA/tBA

3.1. Constitutive model for the PCO SMP

The constitutive model for the PCO SMP to capture both 1W-and 2W-SM behavior was studied by Westbrook et al (2010).A brief review of this model is provided below.

3.1.1. Mechanics for the PCO SMP. PCO is a crystallizablecrosslinking polymer network with subambient Tg. Therefore,it is assumed that the PCO is a mixture of a rubbery phaseand a crystalline phase. The volume fraction of each phaseis determined by the instantaneous temperature, deformation,corresponding rate and history. A simple large deformationelasticity model is adopted that assumes both the rubberyphase and the crystalline phase follow similar forms ofstress–strain behavior:

σR (λR) = NRkT ln λR, σC (λC) = µ ln λC, (2)

where the subscript ‘R’ and ‘C’ represent the rubbery phaseand the crystalline phase, respectively. NR is the cross-linkdensity, k is Boltzmann’s constant and T is the absolutetemperature. NRkT gives the shear modulus for the rubberyphase of PCO SMP and µ is the shear modulus for thecrystalline phase. λ is the stretch ratio, and ln λ is theHencky strain, which is used as it conveniently convertsthe multiplication of stretches into additive strains and thussimplifies the work of tracking deformation in individualcrystalline phases (Westbrook et al 2010). It should be noted

3

Smart Mater. Struct. 22 (2013) 055009 Q Ge et al

that the stretch in the rubbery phase λR is generally differentfrom the stretch in the crystalline phase λC.

Assuming that the thermomechanical loading starts ata high temperature, at time t = 0, PCO consists of 100%rubbery phase. If a certain load σ 0

total is applied to the PCOSMP, the rubbery phase should deform by λ0

R, therefore:

σ 0total = σR

(λ0

R

). (3)

As the temperature is lowered below the crystallization tem-perature, crystallization begins. Since polymer crystallizationis a relatively slow and continuous process, we assume thatwhen a small fraction of polymer crystals are formed, it isin a stress-free state (Long et al 2010). This stress-free statefor the newly formed crystalline phases was referred to asthe natural configuration by Rajagopal and Srinivasa (1998a,1998b). In the case of PCO, this evolution of the polymercrystalline phase also affects the mechanical deformation andis considered to be the underlying mechanism for the 2W-SMeffect. More details about the effects of crystalline phaseevolution on the mechanics were discussed by Westbrook et al(2010). By considering the volume fractions of the rubberyphase and the individual crystalline phase formed at differenttime increments, the total stress at time t = tc+m1t (tc is thetime when crystallization starts) becomes:

σmtotal = (1− fm) σR

(λm

R

)+

m∑i=1

[1fiσC

(λm

i

)], (4)

where fm =∑m

i=11fi. Note that equation (4) can also beapplied to the case where there is no new crystalline phaseforms simply by setting 1fi = 0.

During heating, the existing crystalline domains start tomelt and recover to the rubbery state. At time t = tm+1t, it isassumed that the most recently formed crystalline phase with1fme melts. Concurrent with melting, a small deformation1λ1

melt is induced. Following the same assumption formechanical deformation during crystallization, at time t =tm + n1t, (tm is the time when melting starts), thecrystalline phase with volume fraction 1fme−n+1 melts andthe deformation is induced by 1λn

melt. The total stressbecomes:

σ ntotal =

(1− fme−n

)σR

[(n∑

k=1

1λkmelt

)λ

meR

]

+

me−n∑i=1

{1fiσC

[(n∑

k=1

1λkmelt

)λ

mei

]}, (5)

where fme−n =∑me−n

i=1 1fi.

3.1.2. Thermal expansion coefficient. Westbrook et al(2010) presented the effective coefficient of thermalexpansion (CTE) at time t = tc + m1t:

αm(T) =

(1−

m∑i=1

1fi

)αR +

m∑i=1

1fiαC + αtran1fi1Tm , (6)

where αR is the CTE of the rubbery phase, αC is theCTE of the crystalline phase, αtran is the volume expansion

ratio during phase transition from the rubbery phase to thecrystalline phase and1Tm is the temperature increment at themth time increment. Thus, the thermal strain of PCO SMP attime t = tc + m1t is:

εT =

m∑i=1

αi(T)1T i. (7)

3.1.3. Evolution rule for the PCO SMP. Westbrook et al(2010) introduced the evolution rule for the PCO SMP, basedon Avrami’s phase transition theory modified by Gent (1954).Crystallization occurs when the temperature is lower than thecrystallization temperature Tc and the crystallization rate is:

fc = kc (f∞ − f ) (Tc − T) (λ− λcrit) ,

if (T < Tc and λ > λcrit), (8)

where f is the volume fraction of the crystalline phase, kc is thecrystallization efficiency factor and f∞ is the saturated volumefraction. Similar to the crystallization process, melting startswhen the temperature is above the melting temperature andthe melting rate is:

fm = kmf (T − Tm) , if (T > Tm), (9)

where km is the melting efficiency factor. The total rate ofcrystalline formation is:

f = fc − fm. (10)

More details about the evolution rule were presented byWestbrook et al (2010).

3.2. Constitutive model for the elastomeric matrix

For the sake of simplicity, the elastomeric matrix follows thesame form of the stress–strain behavior as the PCO SMPrubbery phase:

σMa (λMa) = NMakT ln λMa (11)

where NMa is the cross-link density and NMakT is the shearmodulus for PEGDMA/tBA elastomer.

The thermal strain of the matrix material follows:

εMaT = αMa (T − TH) (12)

where αMa is CTE for the matrix material.

4. Model for the 2W-SMP composite actuator

In this section, the model for the actuator is developed byintegrating the constitutive models for the PCO SMP into alaminate composite theory. For the sake of simplicity, heattransfer is not considered in this case and we assume thetemperature is uniformly distributed inside and outside theactuator. This is a close approximation, as the actuator isgenerally thin and the heating/cooling rates are low.

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Smart Mater. Struct. 22 (2013) 055009 Q Ge et al

Figure 3. The geometry of the actuator: (A) dimensions of the cross section. (B) The schematic of bending of the actuator.

Table 1. Dimensions of the cross section of the actuator.

Parameter Value (mm) Description

Actuator

h1 2.76 Thicknessw1 4.85 WidthL 46.73 Length

Embedded PCO specimen

h2 0.81 Thicknessw2 1.72 WidthhB 0.63 Offset

4.1. Geometry

The actuator is treated as a laminate. Figure 3(A) shows thedimensions of the cross section of the actuator (dimensionslisted in table 1). The xy plane passes through the middleof the PCO SMP strip, where the z-coordinate defines thethickness. The top surface is z1 and the bottom is −z2, soz1 = hT + h2/2, z2 = hB + h2/2 and z1 + z2 = h1. Based onthe Euler–Bernoulli beam theory, taking the strain at z = 0as εo (ε(0) = εo), the strain due to bending along the z-axisfollows:

ε(z) = εo + κ · z (13)

where κ is the curvature of the beam (figure 3(B)). It isemphasized here that z= 0 is not the neutral axis and thereforeit is not necessary for εo to be zero. During actuation, thestrain due to bending is generally small (ε < 1.5%), whichis proved by the simulation shown later. Therefore, duringactuation, we approximate the Hencky strain equal to theengineering strain and in equation (13), the engineering strainε is replaced by the Hencky strain:

ln λ(z) = ln λo + κ · z, (14)

where λo is the stretch in the xy plane and λ(z) is thestretch along z-axis. Equation (14) allows us to unify thestrain description in the PCO SMP and the incremental strainduring actuation. The transverse displacement d is relatedto the curvature of the actuator during bending through therelationship:

d = [1− cos (Lκ)] /κ (15)

where L is the length of the actuator and is listed in table 1.

4.2. Analysis during actuation

In this section, deformations during actuation includingdeformations on the PCO SMP strip and the matrix areanalyzed. We first investigate the material points in the xyplane (z = 0). For portions out of the xy plane, the Henckystrain follows equation (14). Therefore, in the following, allstretches and stresses discussed are those in the xy plane,unless otherwise noted.

Before actuation, since the PCO SMP strip is embeddedunder a load free condition, based on equation (4), the totalstress on the PCO SMP strip is:

σAtotal = (1− f∞) σR

(λ

A0R

)+

me∑i=1

[1fiσC

(λ

A0i

)]= 0, (16)

where f∞ =∑me

i=11fi, λA0R is the strain in rubbery phase and

λA0i is the strain in the crystalline phase formed at t = tc+ i1t.

During actuation, raising the temperature above themelting temperature triggers the shape recovery of the PCOSMP strip, which tends to contract. However, due to theexistence of the matrix, the contraction of the PCO SMPstrip is constrained. Considering thermal expansion, thedeformation on the PCO SMP strip can be decomposedinto:

(I) The mechanical deformation λSM (figure 4), which givesrise to stress acting on the PCO SMP strip.

(II) The thermal expansion λT, where λT = 1+εT and λT = 1at TL = 15 ◦C (figure 4).

Therefore, at time t = tA + n1t during actuation, for theportion of the PCO SMP in the xy plane, the total deformationincluding the part induced at the programming step isλn

TλnSMλ

A0R and the total stress is:

σA→ntotal =

(1− fme−n

)σR

(λn

SMλA0R

)+

me−n∑i=1

[1fiσC

(λn

SMλA0i

)], (17)

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Smart Mater. Struct. 22 (2013) 055009 Q Ge et al

Figure 4. Schematic of deformations of the actuator: the brown barrepresents the PCO SMP strip and the blue bar represents the matrix.Before actuation, the PCO SMP strip and the matrix have the samelength. During actuation, the deformation of the PCO SMP strip canbe decomposed into the mechanical deformation λSM due to the SMeffect, the constraint from the matrix, the thermal deformation λT,and the deformation of the matrix. Furthermore, the deformation ofthe matrix can be decomposed into the mechanical deformation λMastretched by the PCO SMP strip and the thermal expansion λMaT.

where fme−n =∑me−n

i=1 1fi. For the portion of the PCO SMPout of the xy plane, the total stress is:

σA→ntotal (z) =

(1− fme−n

)σR

[λn

SM (z) λA0R

]+

me−n∑i=1

{1fiσC

[λn

SM(z)λA0i

]}, (18)

where λnSM(z) follows equation (14):

ln λnSM(z) = ln λn

SM + κ · z. (19)

Since the matrix constrains the recovery of the PCO SMPstrip, it is pulled back by the PCO SMP strip with λn

Ma,correspondingly, and the stress on the matrix should be:

σA→nMa = σMa

(λn

Ma

). (20)

For portions out of the xy plane, based on equation (14), theHencky strain of the matrix follows:

ln λnMa(z) = ln λn

Ma + κ · z. (21)

Finally, adding the thermal expansion of the matrix λnMaT,

the total strain of the matrix is λnMaTλ

nMa (in figure 4), and

λnMaT = 1+ εn

MaT, λnMaT = 1 at TL = 15 ◦C.

Considering the geometric compatibility in the xy plane,the length of the PCO SMP strip will always be equal to thelength of matrix, both during actuation and at rest (figure 4).

λTλSM = λMaTλMa. (22)

According to equation (22), the Hencky strain of the matrix inthe xy plane can be calculated directly as following:

ln λMa = ln (λTλSM)− ln λMaT. (23)

Based on equations (21) and (23), any Hencky strain withinthe elastomeric matrix along z-axis is:

ln λMa(z) = ln (λTλSM)− ln λMaT + κ · z. (24)

4.3. Solutions

During actuation, there are no external loads or momentsapplied to the actuator. Therefore, at t = tA + n1t, we have:

FA→ntotal = FA→n

S + FA→nMa = 0,

MA→ntotal = MA→n

S +MA→nMa = 0,

(25)

where FA→nS and FA→n

Ma are the forces acting on the crosssection of the PCO SMP strip and the matrix, respectively:

FA→nS = w2

∫ h22

−h22

σA→ntotal

(λn

SM, κ, z)

dz,

FA→nMa = w1

∫ z1

−z2

σA→nMa

(λn

Ma, κ, z)

dz

− w2∫ − h2

2

−h22

σA→nMa

(λn

Ma, κ, z)

dz,

(26)

MA→nS and MA→n

Ma are moments,

MA→nS = w2

∫ h22

−h22

σA→ntotal

(λn

SM, κ, z)

z dz,

MA→nMa = w1

∫ z1

−z2

σA→nMa

(λn

Ma, κ, z)

z dz

− w2∫ − h2

2

−h22

σA→nMa

(λn

Ma, κ, z)

z dz.

(27)

In total, there are three unknown quantities: λnSM, λn

Ma and κ .They can be solved by equations (21), (25) and (27). Detailsabout solving λn

SM, λnMa and κ are listed in the appendix.

5. Results

5.1. Prediction for the actuator characterization experiment

Once κ is solved during the heating–cooling actuation, theanalytical model presents the transverse displacement varyingwith temperature under five thermal cycles. In the model,there are 13 parameters in total (listed in table 2), whichinclude 5 parameters for the thermomechanical behavior ofthe PCO SMP, 6 parameters for the evolution rule of the PCOSMP and 2 parameters for the thermomechanical behaviorof the elastomeric matrix (PEGDMA/tBA). Parameters forPCO SMP are directly from Westbrook et al (2010) andparameters for the elastomeric matrix were identified bysimple tests including uniaxial tension and stress free thermalexpansion tests. Figure 5 presents the model prediction forthe heating–cooling actuation cycles, which shows a goodagreement with the experimental results.

The model successfully predicts three key featurescharacterized in the Introduction. The large hysteresis loopduring actuation is attributed to the crystallization and meltingfor the PCO SMP. The reason for the actuation differencebetween the training cycle and the other cycles is essentiallydue to different loading histories on the PCO SMP stripbetween the programming cycle and later cycles. In figure 6,in the programming step, a constant load (700 kPa) was

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Smart Mater. Struct. 22 (2013) 055009 Q Ge et al

Table 2. Parameters for the analytical model.

Description Parameter Value

Mechanical behavior for the PCO SMP

Polymer crosslinking density for rubbery phase NRk (Pa K−1) 4.7× 103

Crystalline phase modulus µc (MPa) 16

CTEs for the PCO SMP

Rubbery phase CTE αR (◦C−1) 1× 10−4

Crystalline phases CTE αC (◦C−1) 5× 10−4

Phase transition volume expansion ratio αtran (◦C−1) 3× 10−4

Evolution rule for the PCO SMP

Crystallization temperature Tc (◦C) 42Melting temperature Tm (◦C) 47Volume fraction at time t = ∞ f∞ (—) 0.8Critical stretch for crystalline phase λcrit (—) 1.05Crystallization efficiency factor kc (—) 1.0×10−3

Melting efficient factor km (—) 5.5×10−3

Thermomechanical behavior for the elastomeric matrix (PEGDMA/tBA)

Polymer crosslinking density for the matrix NMak (Pa K−1) 35.6×103

Matrix CTE αMa (◦C−1) 1× 10−4

Figure 5. Predictions for the heating–cooling actuation cycles.

applied to achieve the stretched shape during cooling. Afterunloading at TL = 15 ◦C, the strip was immediately placedinto an aluminum mold for the actuator fabrication. Therefore,at the beginning of the training cycle, the stress applied tothe PCO SMP strip was zero. During heating in the trainingcycle, the stress on the PCO SMP strip increased from 0 to1 MPa (higher than the programming stress) with increasingtemperature due to the constrained recovery. During coolingin the training cycle, the crystallization was triggered asthe temperature fell below the crystallization temperature.However, in contrast with the programming cycle, the stressacting on the PCO SMP strip was varied with temperature,as it was caused by the constraint of the matrix and dependenton the actuator bending. Once the temperature arrived at TL =

15 ◦C, the stress on the PCO SMP strip was 0.11 MPa, ratherthan zero, at the beginning of the training cycle. Figure 7

presents the corresponding strain of the actuator during theprogramming step and the actuation. In figure 7(A), theactuator was stretched by 95% under 700 kPa, at TH = 70 ◦Cat the programming step. After unloading at TL = 15 ◦C, thestretched PCO SMP strip returned to 73.95% (figure 7(A)).As no extra loading was applied to the PCO SMP strip duringthe actuator fabrication, at the beginning of the training cycle,the strain on the PCO SMP strip was still 73.95%. Duringheating in the training cycle, the constrained recovery of thePCO SMP strip (from 73.95% to 72.85% in figure 7(B)) led tobending of the actuator. During cooling in the training cycle,as the crystallization was triggered at ∼35 ◦C, the strain ofthe PCO SMP strip decreased with decreasing temperatureand ended up as 73.79% at 15 ◦C (in figure 7(B)), which wasdifferent from the one at the beginning of the training cycle(73.95%). For the opposite motion at the early stage duringheating, where the melting of the PCO SMP has not started,the effective CTE is 4.2 × 10−4 ◦C−1, which is higher thanthe CTE of the matrix material (1 × 10−4 ◦C−1). Thus, thePCO SMP strip expands more than the matrix, causing theactuator to bend in the direction opposite that of the previousactuation (figure 8(B)). Once melting starts, the SM effectleads the PCO SMP strip to contract and the actuator bendsto the positive direction (figure 8(C)).

5.2. Parametric study

The model also provides a good tool to explore the actuatordesign space which is prescribed by the mold geometry,the PCO SMP’s shape programming and the fabrication.In general, the maximum transverse displacement can beimproved by reducing the actuator width, using a softerelastomeric matrix material, or increasing the programmingstress.

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Smart Mater. Struct. 22 (2013) 055009 Q Ge et al

Figure 6. Stress versus temperature on the PCO SMP strip duringactuation including the programming step.

In figure 9, the 3D plots show that the normalizedtransverse displacement at TH = 70 ◦C varies with themodulus of the matrix (EMa from 5 to 15 MPa) and theprogramming stress (σP from 500 to 2000 kPa) under threedifferent width ratios (Rw = 2, 2.82 and 4; where Rw =

w1/w2). It clearly shows that the transverse displacement at70 ◦C is enhanced by decreasing the width ratio, choosinga softer matrix, and/or increasing the programming stress.However, when the modulus of the matrix is close to 5 MPaand the programming stress exceeds∼1.7 MPa, the transversedisplacement begins to decrease with increasing programmingstress (figure 9(A)). Based on equation (15), the maximumtransverse displacement occurs at κcritL = 0.74π , whereκcrit = 0.05 mm−1. When κ is larger than κcrit, the tip of theactuator bends back and the transverse displacement decreaseswhile κ is still increasing. In other words, when the width ratioand the modulus of the elastomeric matrix are low and theprogramming stress is high (such as the case in figure 9(A)),the maximum transverse displacement does not occur at TH,although the maximum κ does. When the width ratio is high

Figure 8. Schematics of actuation (blue strip inside is the PCOSMP strip and the white part around is the matrix): (A) originalposition of the actuator; (B) when the PCO SMP strip expands morethan the matrix, it is defined that the actuator bends in the ‘negative’direction; (C) as the PCO SMP strip contracts due to the SM effect,it is defined that the actuator bends in the ‘positive’ direction.

enough, such as the case in figures 9(B) and (C), both themaximum transverse displacement and the maximum κ occurat TH.

Figure 10 shows the bending angle (κL) of the actuatorunder extreme conditions: low width ratio (Rw = 2), lowmodulus of the elastomeric matrix (EMa from 1 to 5 MPa)and high programming stress (σP from 1500 to 2000 kPa). Theincrease of the programming stress results in an increase of thebending angle under constant width ratio and modulus of thematrix; the decrease of the modulus leads to an increase in thebending angle under constant width ratio and programming

Figure 7. Strain on the PCO SMP strip during programming and actuation: (A) including the programming step; (B) during actuation.

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Smart Mater. Struct. 22 (2013) 055009 Q Ge et al

Figure 9. 3D plots showing the normalized transverse displacement (d/L) at 70 ◦C varying with modulus of the matrix (EMa) and theprogramming stress (σP) under three different width ratios: (A) Rw = 2; (B) Rw = 2.82, the red star represents the current actuator;(C) Rw = 4.

Figure 10. The bending angle (κL) of the actuator under extremeconditions (Rw = 2,EMa = 1–5 MPa and σP = 1.5–2 MPa), wherethe green grid plane represents κL = 2π and the actuator curls intoa completely closed circle.

stress. The green grid plane marks κL = 2π , where theactuator curls into a closed circle.

6. Conclusions

An analytical model for a free-standing shape memorypolymer composite actuator was developed. The actuatorconsists of a programmed PCO SMP strip embedded into anelastomeric matrix. The actuation is triggered by the shapememory effect and the required stress bias is provided by thebending of the matrix. The model successfully captures theobserved actuation behavior and helps to better understandthe underlying phenomena during actuation. As a design tool,the model explores the actuator design space very well. Themodel quantitatively presents the increase of the maximumtransverse displacement by reducing the actuator width, usinga softer elastomeric matrix material and/or increasing theprogramming stress.

Acknowledgments

We gratefully acknowledge the support of an NSF CAREERaward (CMMI-0645219) and an AFOSR grant (FA9550-09-1-0195; Dr B-L ‘Les’ Lee, Program Manager).

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Smart Mater. Struct. 22 (2013) 055009 Q Ge et al

Appendix. Solutions for the actuator

Applying constitutive equations (2)–(17), one would have:

σA→ntotal = A ln λn

SM + B, (A.1a)

where A = (1−fme−n)NRkTn+fme−nµ, B = (1−fme−n)NRkTn

ln(λA0R ) +

∑me−ni=1 [1fiµ ln(λA0

i )] and Tn is the temperature att = tA+ n1t. For portions of the PCO SMP strip out of the xyplane, the total stress becomes:

σA→ntotal (z) = A

(ln λn

SM + κ · z)+ B. (A.1b)

Based on equations (11), (20) and (24) the stress on thematrix at any point along the z-axis is:

σA→nMa (z) = C + D

(ln λn

SM + κ · z), (A.2)

where C = NMakTn[ln(λnTλ

n−1SM )− ln λn

MaT] and D = NMakTn.Through the integral along z-axis, the forces in the PCO

SMP strip and the matrix at t = tA + n1t are:

FA→nS = w2

∫ h22

−h22

σA→ntotal (z) dz = E + F ln λn

SM,

FA→nMa = w1

∫ z1

−z2

σA→nMa (z) dz− w2

∫ h22

−h22

σA→nMa (z) dz

= G+ H ln λnSM + Iκ,

(A.3)

where E = w2h2B,F = w2h2A,G = C (w1h1 − w2h2) ,H =D (w1h1 − w2h2) and I = Dw1h1 (z1 − z2) /2. Since noexternal force was applied to the actuator, the resultant forceis zero:

FA→nS + FA→n

Ma = 0, J + K ln λnSM + Iκ = 0, (A.4)

where J = E + G and K = F + H.Through the integral along z-axis, the moments to y-axis

in the PCO SMP strip and the matrix at t = tA + n1t are:

MA→nS = w2

∫ h22

−h22

σA→ntotal (z)z dz = Lκ,

MA→nMa = w1

∫ z1

−z2

σA→nMa (z)z dz− w2

∫ h22

−h22

σA→nMa (z)z dz

= N + I ln λnSM + Oκ,

(A.5)

where L = Ah32w2,N = Cw1h1 (z1 − z2) /2 and O = D[

w1h1(z2

1 − z1z2 + z22

)/3− w2h3

2/12]. Since no external

moment was applied on the actuator, the resultant moment iszero:

MA→nS +MA→n

Ma = 0, N + I ln λnSM + Pκ = 0, (A.6)

where P = L+ O.Comparing equations (A.4) and (A.6), one can find:[

K I

I P

]{ln λn

SM

κ

}= −

{J

N

}. (A.7)

The incremental stretch λnSM and the curvature of the beam κ

can be solved by equation (A.7) as:{ln λn

SM

κ

}= −

[J H

H P

]−1 {I

L

}. (A.8)

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