of 12
erte
MD 21
Article history:Received 5 January 2010Received in revised form 18 September2010Available online 25 January 2011
Keywords:
of Liu et al. (2007). The permanent shape is preserved inthe equilibrium conguration of the crosslinked network,while shape storage and recovery stem from mechanismsof the glass transition. An example of a class I SMP is a familyof photopolymerized networks produced by the copolymer-
to meet specic stress and strain requirements duringrecovery by varying the weight fraction and molecularweight of the crosslinking agents (Yakacki et al., 2008;Ortega et al., 2008).
While the relationship between the polymer structureand thermomechanical properties have been demon-strated for a number of class I SMPs (see also Liu et al.(2007) and Safranski and Gall (2008)), few studies have
0167-6636/$ - see front matter 2011 Elsevier Ltd. All rights reserved.
Corresponding author. Tel.: +1 410 516 4538E-mail address: [email protected] (Thao. D. Nguyen).
Mechanics of Materials 43 (2011) 127138
Contents lists available at ScienceDirect
Mechanics of
elsedoi:10.1016/j.mechmat.2011.01.0011. Introduction
Thermally activated shapememory polymers (SMPs) area diverse group of materials that can be manufactured tomemorize a permanent shape and programmed thermome-chanically to hold a temporary shape. Thematerial recoversits permanent shape when exposed to a specic tempera-ture stimulus. The simplest type of SMPs are covalentlycrosslinked amorphous networks, class I in the taxonomy
ization of tert-butyl acrylate (tBA) monomers with di(ethelyne glycol) dimethacrylate (DEGDMA), and/or poly(ethelyne glycol) dimethacrylate (PEGDMA) crosslinkingagents developed by Gall et al. (2005) and Yakacki et al.(2007) for cardiovascular applications. The tBA, DEGDMA,and PEGDMAhomopolymers havewidely different thermo-mechanical properties, including the glass transitiontemperature and rubberymodulus. Consequently, the ther-momechanical properties of the copolymer can be tailoredShape memory polymersUnconstrained recoveryConstrained recoveryConstitutive modelThermoviscoelasticityViscoelasticityThis work investigated the inuence of material properties and loading conditions on therecovery performance of amorphous shape memory polymers using a recently developedthermoviscoelastic model. The model incorporated the time-dependent mechanisms ofstress and structural relaxation and viscoplastic ow to describe the glass transition ofthe material from a soft viscoelastic rubber to a hard viscoplastic glass. The model capturedmany important features of the unconstrained strain recovery response and of the stresshysteresis observed in the constrained recovery response. A parameter study was devel-oped that varied the model and loading parameters one-by-one to compare their effectson the start and end temperatures and recovery rate of the unconstrained recoveryresponse and on the stress hysteresis of the constrained recovery response. The loadingparameters included the cooling rate, the annealing time, and the high and low tempera-tures of the programming stage and the heating rate of the recovery stage. The results con-rmed experimental observations that viscoelasticity is the underlying mechanism of theunconstrained recovery response. In contrast, the constrained recovery response was inu-enced by the interaction of many different mechanisms, including thermal expansion andstructural and stress relaxation. For the loading parameters, the cooling rate of the pro-gramming stage and the heating rate of the recovery stage had the largest inuence onboth the constrained and unconstrained recovery response.
2011 Elsevier Ltd. All rights reserved.a r t i c l e i n f o a b s t r a c tInuence of thermoviscoelastic propthe recovery performance of shape m
Xiang Chen, Thao. D. Nguyen Department of Mechanical Engineering, Johns Hopkins University, Baltimore,
journal homepage: www.ies and loading conditions onmory polymers
218, USA
Materials
vier .com/locate /mechmat
deformations by Diani et al. (2006) and Nguyen et al.(2008) using an internal state variable framework. Nguyenet al. (2008) incorporated the time-dependent effects ofviscoelasticity, structural relaxation, and viscoplastic owbelow the glass transition temperature. The nonlinearAdamsGibbs model was used to model the transition inthe temperature dependence of the viscosity from WLFabove Tg to Arrhenius below Tg. This allowed the modelto predict many important features of the recovery re-sponse, including the start temperature of strain recovery,and the peak stress and its associated temperature of theconstrained recovery stress response.
128 X. Chen, Thao. D. Nguyen /Mechanics of Materials 43 (2011) 127138examined the relationship between the thermomechanicalproperties and the shape memory performance. Yakackiet al. (2008) undertook this investigation for the free andxed strain recovery response of methyl methacrylate(MMA) copolymerized with PEGDMA. The MMA-co-PEG-DMA materials exhibited glass transition temperaturesranging from Tg = 5692 C and rubbery moduli varyingfrom 9.323 MPa. The study found that the strain recoveryresponse was insensitive to the rubbery modulus. In con-trast, a similar study by Yakacki et al. (2007) for tBA-co-PEGDMAmaterials with a lower range of rubbery modulus,1.511.5 MPa, achieved faster strain recovery for higherrubbery moduli. The strain recovery initiated near the on-set temperature of the glass transition, while the con-strained recovery response ended near the mid-pointtemperature of the glass transition region. The dependenceof the recovery behavior on the glass transition tempera-ture suggests that viscoelasticity strongly inuences therecovery response (Yakacki et al., 2008). Buckley et al.(2007) studied the effect of crosslink density on the uncon-strained recovery response of triol-crosslinked polyure-thanes. Their results showed that an increase in thecrosslink density increased the retardation times and thetemperature of the peak recovery rate for free strain recov-ery. Crosslinking agents with higher molecular weight alsodecreased the temperature span of free strain recovery.
The present work investigates the effects of thermome-chanical properties and loading conditions on the uncon-strained and constrained recovery behavior of class ISMPs by applying the recently developed thermoviscoelas-tic model of Nguyen et al. (2008). Efforts to model the per-formance of SMPs have accelerated in recent years,motivated by the need for an efcient design tool forincreasingly sophisticated materials and devices (Matheret al., 2009). The majority of SMP models conform to eithera phase transition or thermoviscoelastic approach. In thephase transition approach, the SMP is conceptualized as amixture of a rubbery phase, which fully occupies the mate-rial at temperatures T Tg, and a glassy phase, whichdominates at T Tg (e.g., Liu et al. (2006), Qi et al.(2008), Chen and Lagoudas (2008)). The glass transitionis modeled as the temperature driven change in the vol-ume fraction of each phase. Shape storage occurs whendeformation incurred by the compliant rubbery phase athigh temperatures becomes locked in the stiff glass phaseduring cooling. The permanent shape is recovered whenthe stored deformation is released back into the rubberyphase during heating. In contrast, the thermoviscoelasticapproach uses rheological concepts to model shape mem-ory behavior. A temperature dependence is prescribed forthe viscosity and moduli parameters to describe the transi-tion between a compliant, mobile rubbery behavior at hightemperature and a stiff, immobile glassy behavior at lowtemperatures. The earliest thermoviscoelastic models wereone-dimensional small-strain models. Tobushi et al.(1997), Tobushi et al. (2001), Morshedian et al. (2005),and Khonakdar et al. (2007) used the Arrhenius equationto describe the decrease the mobility during coolingthrough the glass transition, while Buckley et al. (2007)used the an empirically determined equation. Generalizedthermoviscoelastic models have been developed for niteFor the present work, the thermoviscoelastic model ofNguyen et al. (2008) was applied to study the effects ofmaterial properties describing structural relaxation, hightemperature viscoelasticity, low temperature viscoplastic-ity, and thermal expansion on the recovery performance.The study also examined the effects of the thermomechan-ical loading conditions during the programming and recov-ery stage, such as strain rate, cooling and heating rate, andthe annealing time. The results conrmed experimentalobservations that stress and structural relaxation werethe underlying mechanisms of free strain recovery. In con-trast, constrained recovery was inuenced by a complexcombination of stress relaxation, structural relaxation,thermal expansion. Neither recovery response was inu-enced by the viscoplastic ow mechanism of the glassymaterial. For the loading parameters, the applied strainand cooling rate during the programming stage and theheating rate during the recovery stage had the largest im-pact on the recovery performance.
2. Methods
2.1. Model formulation
A detailed development of the generalized, nite defor-mation, thermoviscoelastic model was presented by Ngu-yen et al. (2008). Here, we summarize the importantfeatures of the model for uniaxial stress simulations of therecovery response. The model formulation is an extrapola-tion of the rheological model in Fig. 1 to nite deformation.The rheological model consists of a thermal element inseries with a mechanical element. The thermal elementexperiences structural relaxation upon a temperaturechange DT, and the time-dependent thermal strainresponse is described by a spring in series with a Voigtelement. In nite deformation, this arrangement is
Fig. 1. A rheological representation of the constitutive model for class ISMPs. The model has thermal and mechanical elements. The thermalelement experiences structural relaxation upon a temperature changewhile the mechanical element undergoes stress relaxation at hightemperature and viscoplastic ow at low temperatures.
nM 13k2M1 k2M2 k2M3 1=2
;
neM 13
ke2
M1 ke2M2 ke
2
M3
1=2; 3
we can express the stress response as,
ri 1J leqk2Mi n2M|{z}
seqi
1Jlneq ke2Mi ne
2
M
|{z}
sneqi
1JjHM 1|{z}
p
;
leq lNkLnML1
nMkL
; 4
where J =HMHJ = k1k2k3 is the total volumetric deforma-tion. The parameters lN is the shear modulus of the
Table 1Parameters of the thermoviscoelastic model
Modelparameter
Baselinevalue
Physical signicance
Trefg C 25 Glass transition temperature forqcool = 1 C/min
srefR (s) 1100 Structural relaxation time at T TrefgsrefS (s) 34.9 Deviatoric stress relaxation time at
T TrefgC1 17.44 First WLF constantC2(C) 90 Second WLF constantar(104/C) 7.67 Rubbery coefcient of volumetric
thermal expansionleq (MPa) 0.88 Equilibrium shear modulusj (MPa) 611 Bulk moduluskL 4.0 Limiting chain stretch of equilibrium
networkag(104/C) 3.85 Glassy coefcient of volumetric
thermal expansionlneq (MPa) 406 Non-equilibrium shear modulus of
glassy materialsy0 MPa 40 Steady-state yield strengthQS=sy0 K=MPa 101 Activation parameter for viscous
owsyss =sy0 0.43 Ratio of initial to steady-state yield
strengthh (MPa) 250 Flow softening modulus
103T
onset
X. Chen, Thao. D. Nguyen /Mechanics of Materials 43 (2011) 127138 129describedby themultiplicativedecompositionof the stretchinto thermal andmechanical parts, ki kTikMi , where i = 1, 2,3 for the loading and two lateral directions. The thermaldeformation is assumed to be isotropic and kTi H1=3T ,whereHT is the volumetric thermal deformation. The ther-mal element experiences structural relaxation upon a tem-perature change DT. Thus, the thermal dilatation isdecomposed additively as, HT = 1 + agDT + d, where d(t,T)is an internal variable related to the nonequilibrium volu-metric deformation. The internal thermal deformationevolves to equilibrium according to the following rst orderrate equation,
_d 1sRd ar agT T0; dt 0 0; 1
where ar and ag are the coefcients of thermal expansion(CTE) of the rubbery and glassymaterials and T0 is the initialtemperature. The nonlinear AdamGibbs model for thestructural relaxation time sR was used to describe the tran-sition in the temperature dependence of the chain mobilityfrom the rubbery WLF behavior to the glassy Arrheniusbehavior (Adam and Gibbs, 1965; Scherer, 1984; Hodge,1987). The model for sR can be expressed in terms of theWLF constants, C1 and C2, and the reference structural relax-ation time, srefR , at the glass transition temperature T
refg as,
sRT; T f srefR exp C1log e
C2T T f T T f Trefg
T C2 T f Trefg
0@
1A
24
35;
Tf 1ar ag d T0; 2
where Tf is the ctive temperature rst introduced by Tool(1946) to describe the effects of heat treatments on glass.Here, the ctive temperature scales linearly with d and actsas an equivalent representation of the nonequilibriumvolu-metric deformation. The parameter Trefg in Eq. (2) is the glasstransition temperaturemeasuredat a reference cooling rate.In general, the glass transition temperaturewill changewiththe cooling/heating rate because of structural relaxation.
The mechanical element, which consists of a spring inparallel with a Maxwell element, describes the time andtemperature dependent viscoelastic behavior of the rub-bery material and the viscoplastic behavior of the glassymaterial. In nite deformation, the mechanical stretch forthe deformation of the Maxwell element is decomposedinto elastic and viscous parts, kMi keMik
vMi. We further
decompose the mechanical stretch into volumetric parts,HM kM1kM2kM3 and HeM keM1k
eM2keM3 , and distortional
parts, kMi H1=3M kMi and keMi He1=3M k
eMi. Correspondingly,
the stress response is decomposed additively into an equi-librium distortional part, seqi , a nonequilibrium distortionalpart, sneqi , and an elastic volumetric part p. The Arruda andBoyce (1993) network model with Langevin chain statisticsis applied for the equilibrium part to describe the compli-ant entropic elastic behavior of the rubbery material, whilea NeoHookean model is applied for the nonequilibriumpart to describe the stiff response of the glassy material.Introducing the effective stretch of a representative vol-ume element of the polymer network and a correspondingelastic effective stretch as,0 20 40 60 80 100100
101
102
Temperature (oC)
Stor
age
Mod
ulus
(MPa
)
Tterm
S
Fig. 2. Dening the onset and termination temperatures, Tonset and Tterm,and slope S of the glass transition region for the temperature dependenceof the storage modulus.
rubbery network, kL is the limiting stretch correspondingthe contour length of the network chains, j is the bulkmodulus, and leq + lneq is the shear modulus of the glassymaterial. For the uniaxial stress problem where i = 1 is theloading direction, the traction free boundary conditions,r2 = 0 and r3 = 0, are used to solve for the lateral strainsk2 and k3.
A modied Eyring ow rule is used to model time evo-lution of the viscous strains to capture both the viscoelasticrelaxation at high temperatures and viscoplastic ow atlow temperatures,
and the compliant entropic elastic behavior of the rubberymaterial. As the material cools, the molecular mobilitydecreases and the stress response begins to exhibit notice-able viscoelasticity. The decrease in the molecular mobilityis modeled by the increase in the relaxation times sR and sSwith decreasing temperature. From a rheological point ofview, this temperature dependence causes the deforma-tion imposed at high temperatures to become locked inthe dashpot of the Maxwell element at low temperature.Structural relaxation changes the temperature dependenceof the relaxation times from a WLF to Arrhenius depen-
Appl
ied
Com
pres
sive
Stra
in
Constrained Recovery
s cona unc
Table 2Thermomechanical loading parameters
Loadingparameter
Baselinevalues
Signicance
qcool (C/min) 1 Cooling rate during programmingqheat (C/min) 1 Heating rate during recovery_e s1 102 Compressive true strain-rate during
programmingemax 30% Applied compressive strain during
programmingThigh (C) 80 Initial and nal temperatureTlow (C) 10 Annealing temperatureteq (min) 1 Equilibration time at Thigh before
coolingtan (min) 30 Annealing time at Tlow before
recovery
_kvikvi 1
2grefSexp
C1log e
C2T T f T T f Trefg
T C2 T f Trefg
0@
1A
0@
1A syT
QSsinh
QST
ssy
2435
|{z}_cv
sneqis
; kv0 1
s 12
sneq2
1 sneq2
2 sneq2
3
1=2:
5
130 X. Chen, Thao. D. Nguyen /Mechanics of Materials 43 (2011) 127138qcool
Fig. 3. Applied strain and temperature history. The compressive strain waconstrained recovery simulation, while no constraints were placed in theThe variable s is the ow stress, sy is the yield strength, andQS scales with the activation energy. The resulting stressrelaxation time, sS = gS/lneq, where gS is the shear viscos-ity, has the same temperature and structure dependenceas the structural relaxation time sR. To model post-yieldsoftening, the yield strength is assumed to evolve withthe viscous strain rate as follows,
_sy h 1 sysyss
_cv ; syt 0 syss ; 6
where sy0 is the initial yield strength, Syss is the steady stateyield strength, and h is the softening modulus.
In summary, the material model exhibits thermoelasticbehavior at high temperatures, such that for T Tg, thethermal strain and stress response is given by the CTEqheat
strained to be greater than or equal to emax during the heating stage in aonstrained recovery simulation.
dence during the glass transition. The nonlinear depen-dence of sS on the ow stress allows the material to yieldthen ow plastically when deformed in the glassy state.
2.2. Parameter study
The model parameters were determined from a set ofthermomechanical experiments for a tBA-co-PEGDMA
SMP using a procedure described in detail in Nguyenet al. (2008). The experiments included dynamic mechani-cal analysis (DMA) at 1 Hz for the temperature dependenceof the storage modulus, thermal expansion experiments ata cooling rate of qcool = 1 C/min, and isothermal compres-sion experiments at strain rates of _e 0:1=s and_e 0:01=s to characterize the rate-dependent and temper-ature-dependent viscoplastic behavior of the glass. The
0
10
20
30
40
50
60
Sref R
ref
C1 C2
1100
2200550
34.9
17.4
5
69.8
17.4
48.
72
34.8
8 9045 180
T onse
t(o C
)
0
20
40
60
80
100
Sref
Rref
C1 C2
1100
2200550
34.9
17.4
5
69.8
17.4
48.
72
34.8
8 9045 180
T ter
m (o C
)
0.6
0.8
1
C1
C2
2200
S (lo
g(MPa
)/oC)
nset, (b
0 20 40 60 80 0.0
0.17
0.33
0.5
0.66
0.83
1
Temperature ( oC)
Rec
over
ed s
train
ratio
(1-
/m
ax)
Tstart
Tend
k
0 20 40 60 80
1.2
1
0.8
0.6
0.4
0.2
0
0.2
Temperature ( oC)
(M
Pa)
CoolingHeating
peak
Tpeak
T0.5peak
Fig. 4. Recovery behavior calculated for the baseline parameters: (a) strain-temperature curve for unconstrained recovery, and (b) stresstemperaturecurve for constrained recovery.
X. Chen, Thao. D. Nguyen /Mechanics of Materials 43 (2011) 127138 1310
0.2
0.4 S
ref Rref
1100550
34.9
17.4
5
69.8
Fig. 5. Effects of stress and structural relaxation parameters on (a) To17.4
48.
72
34.8
8 9045 180
) Tterm, and (c) slope S of the glass transition of the storage modulus.
glass transition temperature Trefg was determined from thethermal expansion experiments for a cooling rate ofqcool = 1 C/min. The WLF parameters C1 and C2 should bedetermined from timetemperature superposition testsfor the master curve of the frequency dependence of thestorage modulus. However, the model assumed that theviscoelastic behavior was described by a single characteris-tic relaxation time. In reality, the SMP exhibited a broadspectrum of relaxation times. Because of this simplica-tion, the WLF parameters were treated as phenomenologi-cal parameters determined along with the characteristicrelaxation time to t the temperature range of the glasstransition of the storage modulus. We are currently work-ing to incorporate a broad relaxation spectrum in the mod-el to more accurately describe the viscoelastic behavior ofSMPs (Nguyen et al., 2010).
Table 1 summarizes the parameters of the thermovisco-elastic models, their physical signicance, and valuesdetermined for the tBA-co-PEGDMA material. These wereused as the baseline values for the parameter study, whichvaried the parameters one-by-one from 0.5 to 2 times thebaseline value. The effects of the CTE, moduli parameters,and limiting stretch on the thermal dilatation and stressresponse of the SMP are clear. The viscoplasticity parame-ters (sy0 ; syss ;Qs, h0) determine the stress response for largedeformations and temperatures below the glass transition
temperature. The parameters C1, C2, srefS , and srefR describesthe time-dependence of the stress response for tempera-tures near and greater than the glass transition tempera-ture. To understand the effects of the WLF parametersand characteristic stress and structural relaxation timeson the viscoelastic behavior of the SMP, the temperaturedependence of the storage modulus was calculated alsofor parameter variations of a factor of 0.5 and 2 from theirbaseline values. Fig. 2 plots the storage modulus calculatedfor uniaxial compression at 1 Hz using the baseline param-eters. For the purpose of comparison, we dened three fea-tures of the glass transition region: the onset temperatureTonset, termination temperature Tterm, and the slope S asshown in Fig. 2.
2.3. Numerical simulations of recovery
The thermoviscoelastic model was implementednumerically into Matlab and used to simulate the uncon-strained strain recovery and the constrained recoverystress response under uniaxial compression. Fig. 3 illus-trates the strain and temperature loading history for bothrecovery simulations. Both began with the material equili-brated at the temperature Thigh = 80 C. At time t = 0, thestrain was applied at a constant rate to a compressive truevalue of emax = 0.3. The compressed material was held at
30
40S
ref Rref
C1 C2 r
g neq eq sy0
t(o C)
30
40
50
60
Sref R
ref
C1 C2
r
gneq eq sy0
(o C)
C2
90 180
(c) th
132 X. Chen, Thao. D. Nguyen /Mechanics of Materials 43 (2011) 1271380
10
20
T sta
r
1100
2200550
34.9
17.4
5
69.8
17.4
48.
72
34.8
8 9045 180
406
812
203 804020
0.88
0.44
1.75
7.67
3.85
7.70
1.92
15.3
4
3.84
Sref R
ref
C1k (o C
-1 )
1100
220055
0
34.9
17.4
5
69.8
17.4
48.
72
34.8
8 45
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Fig. 6. Effects of model parameters on (a) Tstart, (b) Tend, and0
10
20
T end
1100
2200550
34.9
17.4
5
69.8
17.4
48.
72
34.8
8 9045 180
406
812
203 804020
0.88
0.44
1.75
7.67
3.85
7.70
1.92
15.3
4
3.84
r
gneq eq sy0
406
812
203 804020
0.88
0.44
1.75
7.67
3.85
7.70
1.92
15.3
4
3.84
e maximum slope k of the unconstrained recovery response.
Thigh for teq = 1 min, cooled at a rate qcool = 1 C/min toTlow = 10 C, then annealed at Tlow for tan = 30 min. Ther-mal contraction during cooling caused the compressivestrain to exceed emax. In an experiment, this would corre-spond to the SMP specimen contracting away from thecompression platens. For the constrained recovery simula-tions, the material was reheated at a rate of qheat = 1 C/minto Thigh while constraining the compressive strain to beeP emax. This corresponded to allowing the specimen toexpand thermally back into contact with the compressionplaten. In the strain recovery simulations, the samplewas reheated without any constraints placed on the strain.Table 2 summarizes the loading parameters and their base-line values. These were varied one-by-one by a factor of 2and 0.5 in the parameter study, except for the high and lowtemperatures. The ranges of those were set to Thigh =(52.5,80,135) C and Tlow = (45,10,7.5) C.
Fig. 4(a) shows the recovered true-strain ratio duringthe heating stage of the unconstrained recovery simula-tion, and Fig. 4(b) plots the true-stress response from con-strained recovery for the entire temperature cycle. Thebaseline model and loading parameters were used for thesimulations. Both results agreed well with experimentsas discussed in Nguyen et al. (2008). In particular, the mod-el was able to predict the onset temperature of strain
recovery, and the peak stress and its temperature of theconstrained recovery case. To compare the impact of thedifferent parameters, three characteristic features were de-ned for each recovery response. For the unconstrained re-sponse, the maximum tangent k described the strainrecovery rate. The start and end temperatures were de-ned from the intersection of a line denoting the thermalexpansion of glassy material for Tstart and of the thermalexpansion of the rubbery with material for Tend with a lineof slope k for the recovered strain. For the constrainedrecovery simulations, Drpeak referred to the maximumstress overshoot encountered during the heating stage atthe temperature Tpeak. The DT0.5peak described the temper-ature span of the stresstemperature hysteresis curve cal-culated where Dr = 0.5Drpeak. The parameter studycompared the sensitivity of these features of the recoveryresponse to the parameter variations.
3. Results and discussion
3.1. Impact of thermoviscoelastic model parameters
Fig. 5 compares the effects of the WLF parameters, C1and C2, and the characteristic relaxation times, srefR andsrefS , on the storage modulus. The parameter srefS
0.4
0.6
0.8
1
d st
rain
ratio
(1
/ m
ax) R
ref = 550 s
Rref
= 1100 s
Rref
= 2200 s
0.4
0.6
0.8
1
d st
rain
ratio
(1
/ m
ax) S
ref = 17.45 s
Sref
= 34.9 s
Sref
= 69.8 s
s, (b)
X. Chen, Thao. D. Nguyen /Mechanics of Materials 43 (2011) 127138 13320 25 30 35 40 45 500
0.2
Temperature (oC)
Rec
over
e
20 30 40 50 600
0.2
0.4
0.6
0.8
1
Temperature (oC)
Rec
over
ed s
train
ratio
(1
/ m
ax)
C1=8.72
C1 = 17.44
C1=34.88
Fig. 7. Comparing the effects of different (a) structural relaxation timeunconstrained recovery response.20 25 30 35 40 45 500
0.2
Temperature (oC)
Rec
over
e
20 30 40 50 600
0.2
0.4
0.6
0.8
1
Temperature (oC)
Rec
over
ed s
train
ratio
(1
/ m
ax)
C2=45oC
C2=90oC
C2=180oC
stress relaxation times, and WLF parameters (c) C1 and (d) C2 on the
corresponds to the molecular resistance to viscous ow. Alarger srefS produced a higher glass transition temperatureand broader glass transition temperature range (smallerS). The structural relaxation time srefR describes howquickly the nonequilibrium structure, represented by d,and thus the structure and temperature dependent stressrelaxation time responds to a temperature change. A largersrefR produced a more sluggish response, which increasedthe onset temperature Tonset and the slope S of the glasstransition region. The storage modulus was most sensitiveto the WLF parameters, though C1 and C2 had opposite ef-fects on the glass transition. A lower C2 and higher C1shifted the glass transition to lower temperatures and nar-rowed its temperature range. Decreasing C2 by half in-creased S by more than a factor of ve.
Fig. 6 shows the effects of some key thermoviscoelasticparameters on the unconstrained recovery response. As ex-pected, the rubbery and glassy CTE did not contribute tothe unconstrained recovery response. The viscoplasticparameters also had negligible impact as shown in Fig. 6for the initial yield strength, sy0. The same result was ob-tained for the steady state yield strength, syss, the activa-tion parameter, QS, and the softening modulus, h (notshown). The programmed deformation was applied atThigh Tg where the material could ow viscoelastically
in response to an applied stress that was much smallerthan the yield strength. This viscous deformation wasstored in the dashpot of the Maxwell element during cool-ing then released during heating (Fig. 1). The stress stateremained signicantly smaller than the yield stressthroughout the programming and recovery process. As aresult, both were insensitive to the parameters for yielding.Both the rubbery and nonequilibrium shear modulus, leq
and lneq, had a noticeable effect on the unconstrainedrecovery response. A higher rubbery modulus increasedthe recovery rate, which agreed with the experimentalobservations of Yakacki et al. (2007), and caused strainrecovery to start at a lower temperature. The bulk modulushad no effect on the response (not shown) because the rub-bery material was modeled as quasi-incompressible atThigh.
Fig. 7 compares the unconstrained recovery responsefor different values of the WLF parameters and relaxationtimes, srefS and srefS . Increasing the structural relaxationtime srefR caused the stress relaxation time to respond moresluggishly to a temperature rise. This shifted strain recov-ery to higher temperatures and resulted in a higher recov-ery rate k. Similar features were observed in Fig. 5 for thetemperature dependence of the storage modulus. Thestress relaxation time, srefS , and WLF parameters, C1 and
34
36
38
40
T sta
rt (o C
)
40
45
50
55
T end
(o C)
g(MPa(c)
of the
134 X. Chen, Thao. D. Nguyen /Mechanics of Materials 43 (2011) 12713830 35 40 45 50 5528
30
32
Tonset
(oC)
(a)
0 0.2 0.4
0.2
0.4
0.6
0.8
1.0
1.2
S (lo
k (1/o
C)
Fig. 8. Comparing the temperature dependence20 40 60 80 100 12030
35
Tterm
(oC)
(b)
0.6 0.8 1)/oC)
storage modulus and strain recovery response.
C2, also had similar impact on the unconstrained recoveryresponse as observed for the storage modulus. The recov-ery rate k was most sensitive to the WLF parameters C1and C2, which controlled the temperature range of the glasstransition region S. These results suggested that the fea-tures of the unconstrained recovery response correlatedwith those of the temperature dependence of the storagemodulus.
Fig. 8 compares the features of the temperature depen-dence of the storage modulus and strain recovery responsefor cases involving the WLF parameters, srefS , and srefR . TheTend of the unconstrained recovery response increasednearly linearly with the termination temperature Tterm ofthe glass transition of the storage modulus. Similarly, therecovery slope k correlated well with the slope S of theglass transition region of the storage modulus, and Tstartof the recovery response generally trended with Tonset.The correlation between Tonset and Tstart were not as goodas for the other features because of the signicant inu-ences of structural relaxation, which was most active atthe start of recovery. These results strongly suggests thatstructural and stress relaxation are the underlying mecha-nisms of unconstrained shape recovery.
The results of the parameter study for constrainedrecovery in Fig. 9 showed that the stress response was
inuenced by thermal expansion as well as stress andstructural relaxation mechanisms. Fig. 10 compares theconstrained stress response during the heating stage fordifferent values of the glassy CTE and nonequilibriummodulus, two parameters which produced a large effect.The peak stress overshoot, Drpeak, was strongly correlatedwith the glassy CTE and was dependent on the nonequilib-rium shear modulus. These results showed that the over-shoot in the stress early in the heating stage was causedby the constrained thermal expansion of the stiff glassymaterial. At higher temperatures near the glass transition,the material became more compliant and the stress re-laxed from the peak stress to the programmed stress atThigh. The peak stress increased with the rubbery modulussince the programmed stress scaled with leq. However, theincrease was not proportional, and a larger leq produced asmaller ratio of the peak stress rpeak to the programmedstress. The peak stress overshoot also increased withdecreasing C1 and C2 and increasing srefS . Larger srefS andsmaller C1 either shifted the glass transition temperatureto higher temperatures or produced a broader glass transi-tion temperature range. Both resulted in the constrainedthermal expansion of a stiffer material early during heat-ing. The temperature Tpeak of the peak stress also stronglydepended on the structural and stress relaxation times
Sref R
ref
C1
C2 r
g
sy0
pea
k (MPa
)
0.6
0.8
1
eq
neq
10
15
20
Sref R
ref C1 C2
rg neq eq sy0
T pea
k(oC)
C2
9045 180
peak, a
X. Chen, Thao. D. Nguyen /Mechanics of Materials 43 (2011) 127138 135
1100
220055
0
34.9
17.4
5
69.8
17.4
48.
72
34.8
8 9045 180
406
812
203 804020
0.88
0.44
1.75
0
0.2
0.4
7.67
3.85
7.70
1.92
15.3
4
3.84
0
5
10
15
20
Sref R
ref
C1
T0.
5pea
k(oC)
1100
220055
0
34.9
17.4
5
69.8
17.4
48.
72
34.8
8
Fig. 9. Effects of model parameters on (a) Drpeak, (b) T5
1100
220055
0
34.9
17.4
5
69.8
17.4
48.
72
34.8
8 9045 180 804020
0.88
0.44
1.7540
681
2
203
7.67
3.85
7.70
1.92
15.3
4
3.84
r
g neq eq sy0
406
812
203 804020
0.88
0.44
1.75
7.67
3.85
7.70
1.92
15.3
4
3.84
nd (c) DT0.5peak of the constrained recovery response.
and the WLF parameters. A larger srefS increased the glasstransition temperature, which caused the material to re-main glassy longer during heating. This led to a higherpeak stress at a higher temperature, Tpeak. The parameterC1 had the largest effect on the temperature span DT0.5peak.As with the unconstrained recovery response, the
viscoplasticity parameters, sy0, syss, QS, and h, did not affectthe constrained recovery response, because the stress stateof the deformed specimen remained signicantly belowthe yield strength during the programming and heatingstages for the selected model parameters and loadingcondition.
0 10 20 30 40
2
1.5
1
0.5
0
Temperature (oC)
(M
Pa)
g=1.92e4/oC
g=3.85e4/oC
g=7.7e4/oC
0 10 20 30 40
1.8
1.4
1
0.6
0.2
Temperature (oC)
(M
Pa)
neq=203MPaneq=406MPaneq=812MPa
Fig. 10. Comparing the effects of different (a) glassy CTEs and (b) nonequilibrium moduli on the constrained recovery response.
33
34
35
36
37
qcool
qheat
max Thigh Tlow tan
T sta
rt(o C
)
39
40
41
42
43
qcool
qheat
max
Thigh Tlow tan
T en
d(oC)
136 X. Chen, Thao. D. Nguyen /Mechanics of Materials 43 (2011) 12713829
30
31
32
1.0
0.5
2.0
1.0
0.5
2.0
0.30
0.15
0.60 8052.5
135
-10
-45 7.5 3015 600.0
0.5
1.0
1.5
2.0
2.5
qcool
qheat max
1.0
0.5
2.0
1.0
0.5
2.0
0.30
0.15
0.60
k (o C
-1 )
Fig. 11. Effects of the thermomechanical loading parameters on (a) Tstart, (b) Te36
37
38
1.0
0.5
2.0
1.0
0.5
2.0
0.30
0.15
0.60 8052.5
135
-10
-45 7.5 3015 60Thigh Tlow tan
8052.5
135
-10
-45 7.5 3015 60
nd, and (c) the maximum slope k of the unconstrained recovery response.
6ma
X. Chen, Thao. D. Nguyen /Mechanics of Materials 43 (2011) 127138 1370
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8q
coolqheat
max
Thigh1.
00.
5
2.0
pe
ak (M
Pa)
Tlow tan
1.0
0.5
2.0
0.30
0.15
0.60 8052.5
135
-10
-45 7.5 3015 60
4
6
8
10
12
14
16q
coolqheat
T0.
5pea
k (oC)3.2. Impact of thermomechanical loading parameters
Both the unconstrained and constrained recovery re-sponses were unaffected by the strain rate _e and equilibra-tion time teq of the programming stage. The relaxation timewas negligibly small at Thigh = 80 C such that stress re-sponse attained equilibrium nearly instantaneously.Fig. 11 shows the effects of the remainder of the thermo-mechanical loading parameters on the unconstrainedrecovery response. The start and end temperatures andthe recovery rate were all signicantly affected by theheating and cooling rate, which demonstrated the impor-tance of structural relaxation to the strain recovery re-sponse. A faster heating rate shifted the glass transitionand thus the onset of strain recovery to higher tempera-tures. It also increased the recovery rate because relaxationoccurs faster at higher temperatures. A lower Tstart andslower recovery rate were also observed for faster coolingrate. A slower cooling rate allowed the material structureto evolve more towards equilibrium during programming.This caused the molecular mobility to be lower (i.e., therelaxation times to be higher) at the start of the heatingstage. The parameters Thigh and Tlow also signicantly af-fected the start and end temperatures of the unconstrainedrecovery response, while the applied strain emax had littleeffect.
0
2
1.0
0.5
2.0
1.0
0.5
2.0
0.30
0.15
Fig. 12. Effects of thermomechanical loading parameters on (a) Drpea0
2
4
1.0
0.5
2.0
1.0
0.5
2.0
0.30
0.15
0.60 8052.5
135
-10
-45 7.5 3015 60
x Thigh Tlow tan8
10
12
14
16
18
20q
cool
qheat max Thigh Tlow t
an
T pea
k (oC)The cooling and heating rates also had a large effect onthe constrained recovery response as shown in Fig. 12. Afaster heating rate increased the glass transition tempera-ture, while a slower cooling rate allowed the materialstructure to evolve closer to equilibrium during cooling.Both effects delayed the softening of the material duringheating, which led to a greater stress overshoot. These re-sults were consistent with the experimental measure-ments of Castro et al. (2010), which showed a slightincrease in the stress overshoot with decreasing coolingrate and a larger increase in the stress overshoot withincreasing heating rate. As expected, the peak stress over-shoot increased with the programmed strain emax, thoughnot proportionally. In general, emax and leq produced sim-ilar effects on the constrained recovery response as shownin Figs. 9 and 12.
4. Conclusions
A parameter study was developed to investigate therelationship between the thermomechanical materialproperties and loading conditions during programmingand recovery on the unconstrained strain recovery re-sponse and the constrained recovery stress response ofamorphous shape memory polymers. The study used a
0.60 8052.5
135
-10
-45 7.5 3015 60
k, (b) Tpeak, and (c) D T0.5peak of the constrain recovery response.
thermoviscoelastic model recently developed by Nguyenet al. (2008) that incorporated the time-dependent effectsof stress and structural relaxation and viscoplastic ow of
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Influence of thermoviscoelastic properties and loading conditions on the recovery performance of shape memory polymersIntroductionMethodsModel formulationParameter studyNumerical simulations of recovery
Results and discussionImpact of thermoviscoelastic model parametersImpact of thermomechanical loading parameters
ConclusionsReferences
