modeling of shape memory induction and recovery in heat-shrinkable polymers

Modeling of Shape Memory Induction and

Recovery in Heat-Shrinkable Polymers

Jalil Morshedian, Hossein A. Khonakdar,* Sorour Rasouli

Department of Polymer Processing, Iran Polymer and Petrochemical Institute (IPPI), P. O. Box 14965-115, Tehran, IranFax: þ98 21 4580021; þ98 21 4580022; E-mail: [email protected], [email protected]

Received: December 27, 2004; Revised: May 9, 2005; Accepted: May 31, 2005; DOI: 10.1002/mats.200400108

Keywords: heat shrinkable; mechanical model; shape memory; strain recovery; stress relaxation

Introduction

Heat shrinkable polymers are important materials which

have wide applications in the packaging industry, the elec-

tronic and cable industries and in the preparation of heat

shrinkable tubes and connections. The application of heat

will shrink these materials to more or less their original

dimensions, or they will mold themselves into a permanent

skin tight protective covering around the object. The built-

in recovery of the initial shape potential is referred to as an

elastic memory.[1–4] Semi-crystalline polymers, such as

different grades of polyethylene and its copolymers and

blends, are among a number of common polymers widely

used in the industries mentioned above.[5–7]

The stretching ratio applied to the production is usually

between 2/1 and 4/1. However, the heat shrinkability is not

the same and full recovery does not happen. The reasons for

this have not yet been explored, except to point out the

avoidance of post curing during the process of stretching-

holding-cooling.[8,9]

Indeed, the heat shrinkable property is gained in two

steps, as follows.

Step 1: preforming, which is carried out by first making

the required crosslinkable compound and then extruding

or molding the compound into the required shape or form

and crosslinking it by chemical means or high energy

irradiation.

Step 2: expansion or shape (elastic) memory induction,

which is done by heating and stretching or expanding the

compound and then quenching under strain to ‘‘freeze-in’’

the oriented molecules and the resulted elastic restoring force

or build up of elastic memory. During application, heating

results in the melting of the plastic crystalline domains and

the stored elastic stress is therefore released.[10–11]

The mechanism of the elastic memory build up and heat

shrinkability, the experimental preparation of the com-

pound and its characterization, shrinkage behavior and

other properties have already been reported.[12–15]

Kumar et al. have studied thermally recoverable cross-

linked polyethylene and investigated the effect of the

Summary: A mechanical model was developed to describequalitatively and quantitatively the stress-strain-time behav-ior of a prepared shape memory crosslinked polyethyleneduring hot stretching, stress relaxation under 200% strain athigh temperature and strain recovery of the heat shrinkablepolymer. The stress-strain, the stress relaxation and the ir-recoverable strain behavior of the model were established bydriving the constitutive equation, which could qualitativelyrepresent the behavior of the real material. By choosing sig-nificant values for the parameters of the proposed model, anexcellent fit was obtained between the experimental behaviorof the polymer and that predicted by the model. It was alsorevealed that the main source responsible for the imperfectrecovery of the induced strain observed was the stressrelaxation occurring during the stretch holding-cooling timestep.

Stress relaxation of crosslinked polyethylene under 200%strain at 160 8C.

Macromol. Theory Simul. 2005, 14, 428–434 DOI: 10.1002/mats.200400108 � 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

428 Full Paper

irradiation dose on the gel fraction and recovery percentage

of the stored stress.[16]

In another work, Wayne used a model to quantify the

thermal shrinkage behavior of polyester shrink film under

isothermal and non-isothermal conditions. This model

employed straightforward first order reaction kinetics for an

otherwise microscopically complex relaxation process. It

offered three unique parameters for quality control and

material specification of the shrink film. Under isothermal

conditions, induction time, shrink constant and ultimate

shrinkage were the three parameters of the study. Under

non-isothermal conditions, shrink temperature, shrink cons-

tant and ultimate shrinkage were the investigated param-

eters.[17] Pakula and Trznadel used a four state model that

described temperature dependence, time dependence and

the induction time of shrinkage forces in amorphous poly-

mers.[18–21] Bhattacharyya and Tobushi derived the iso-

thermal mechanical response of a 4 element rheological

model for shape memory polymers under conditions of

constant stress, constant strain, constant stress rate, constant

strain rate and periodic strain. The effect of shape memory

strain was modeled by the friction element.[22]

However, there is little systematic information available

concerning the modeling of shape memory induction and

recovering of heat shrinkable polymers.

In this work, shape memory is induced into crosslinked

LDPE by a cyclic heating-stretching-cooling process. The

modeling of stress-strain behavior during the tensile

process and the stress relaxation of the stretched polymer

at high temperatures above the crystalline melting point was

carried out, as well as the prediction of the irrecoverable

strain observed in the heat shrinkage of real products and

determining its origin.

The Mechanical Model

This work was based on the behavior of heat shrinkable

products which regain their original shape upon heating. We

aimed to establish a mechanical model with a similar res-

ponse. The proposed simple model combined a Kelvin unit

(the spring,E, and the dashpot, Z1, in parallel) and a dashpot

unit, Z2, in series Z2 >> Z1. Then, the behavior of the model

for the simulation of elastic restoring force build up (the

final step in making shape memory) and its release to give

heat dimensional recovery were determined. The model and

the steps in making a heat shrinkable system and its heat

recovery are illustrated in Figure 1(a)–(d).

It should be noted that none of the basic mechanical

models, such as Maxwell, Kelvin, Standard Linear Solid

(SLS) and four element models, can reproduce heat shrink-

able material internal mechanisms. Only this type of

arrangement of spring and dashpot units, using a minimum

number of them as shown in Figure 1, is capable of

representing the main features of behavior of this type of

material. In Figure 1(b), the model is stretched at a high

temperature where both dashpots are in the liquid state.

Then, after gaining a predetermined strain, it is cooled

under stretched conditions. Before complete solidification

of dashpots and the model coming to a standstill frozen

state, some stress relaxation occurs due to expansion of

the lower dashpot and contraction of the Kelvin element,

maintaining the overall strain constant (Figure 1(c)). When

the frozen model (Figure 1(c)) is reheated, it shrinks at the

melting temperature of dashpot Z1 to a somewhat longer

length (Figure 1(d)) than its initial length (Figure 1(a)).

In this regard, the constitutive equation of the proposed

model is derived and then is solved for tension at constant

rate to obtain the stress-strain behavior. It is also solved for

stress relaxation to find out the viscous flow build up during

the stress relaxation time and the heat recovery behavior.

The constitutive equation is derived as follows:

Strain : gkelvin ¼ g1; gdashpot ¼ g2 ð1Þ

Total strain : g ¼ g1 þ g2 ð2Þ

Stress : s ¼ s1 ¼ s2 ð3Þ

In dashpot : s2 ¼ Z2

dg2

dt! g2 ¼ s2t

Z2

ð4Þ

Figure 1. Proposed mechanical model for shape memory induction and heat recovery.

Modeling of Shape Memory Induction and Recovery in Heat-Shrinkable Polymers 429

Macromol. Theory Simul. 2005, 14, 428–434 www.mts-journal.de � 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In Kelvin : s ¼ Eðg� g2Þ þ Z1

dðg� g2Þdt

ðwhereE is the spring modulusÞ ð5Þ

s ¼ Eg� Eg2 þ Z1

dgdt

� Z1

dg2

dtð6Þ

s ¼Egþ Z1

dgdt

1 þ Z1Z2

þ t

t

ð7Þ

t ¼ Z2

Eðwhere t is relaxation time of the modelÞ ð8Þ

Stress-Strain Behavior in Tension withConstant Strain Rate

The model is stretched with a constant strain rate, K, until

it reaches a specified strain when it is held constant

(Figure 1(b)). The relation between stress and strain can be

expressed as follows:

t ¼ 0 :d gd t

¼ 0 ð9Þ

t > 0 :dgdt

¼d L�L0

L0

� �

dt¼ dL

L0dt¼ K 0

L0

! gK 0t

L0

¼ Kt ð10Þ

where L0 is the specimen initial gauge length and K0 is the

tensile speed of the testing machine.

Therefore, Equation (7) can be rearranged to give:

s ¼ Egþ Z1K

1 þ Z1

Z2

þ gKt

ð11Þ

Figure 2 depicts the stress-strain behavior of the model,

which is a typical elastic response.

Substituting the following boundary condition in Equa-

tion (11) yields Equation (13) for initial stress, s0, at the

beginning of the stress relaxation test at a constant

predetermined strain, g0.

g ¼ g0 ! t ¼ g0K

s ¼ s0

ð12Þ

s0 ¼ Eg0 þ Z1K

1 þ Z1

Z2

þ g0

Kt

ð13Þ

In other words, s0 is the stress generated in the sample or

model as soon as it is elongated by a specified strain (g0).

Stress Relaxation Behavior

The stress relaxation equation is obtained from the cons-

titutive equation (Equation (7)) with the following bound-

ary condition and Equation (13) after the model has been

stretched and kept in a stationary state by a specified strain,

g0 (Figure 1(c)).

g ¼ g0 ! s ¼ Eg0

1 þ Z1

Z2

þ t

t

ð14Þ

t ¼ 0 ! s ¼ s0 ! s0 ¼ Eg0

1 þ Z1

Z2

¼ Eg0 þ Z1K

1 þ Z1

Z2

þ g0

Kt

! Eg0

¼1 þ Z1

Z2

� �ðEg0 þ Z1KÞ

1 þ Z1

Z2

þ g0

Kt

! s ¼ s0

1 þ Z1

Z2

1 þ Z1

Z2

þ t

t

ð15Þ

where s0 is substituted from Equation (13). Figure 3 depicts

the stress relaxation behavior of the model.

It can be seen that stress diminishes from the initial value,

s0, to zero after sufficient time. Like a real viscoelastic

polymer, which softens during the stress relaxation process

(i.e., the stress relaxation modulus decreases with time), the

rigidity of the model which decreases with time is

proportional to the ratio of decaying stress to constant

strain.

Figure 2. Stress-strain curve predicted by the proposed model.Figure 3. Stress relaxation curve predicted by the proposedmodel.

430 J. Morshedian, H. A. Khonakdar, S. Rasouli


Viscous Strain Build Up During the StressRelaxation Time

The strain in the model is constant during the stress

relaxation process. Therefore, contraction of the Kelvin unit

with respect to time must be equal to the expansion of the

single dashpot, which are both equal to the build up of an

irrecoverable strain (Figure 1(c)). The stress in Equation

(15) has been replaced by a stress term in the Kelvin

constitutive equation (Equation (16)) to yield Equation (17)

from which the viscous or irrecoverable strain can be

calculated. Since the analytical solution of Equation (17) is

too complex, it is solved by numerical solution (Runge-

Kutta) with the MATLAB program. The result for viscous/

irrecoverable strain is depicted in Figure 4, which shows an

S-shape increase of viscous strain by increasing the stress

relaxation time.

Zdgdt

þ Eg ¼ s ð16Þ

Z1

dgdt

þ Eg ¼ s0

1 þ Z1

Z2

1 þ Z1

Z2

þ t

t

ð17Þ

Experimental Part

Materials

All the materials used in this study were commercial productswhich were used as received without further treatment.

Low density polyethylene (LDPE) grade MG20, with anMFI of 20 g � (10 min)�1 and a density of 923 kg �m�3, wassupplied by Qapco, Qatar. Carbon black masterbatch (con-taining 40% carbon black) with a grade of 5003 was preparedby New Particle Color Corporation of Taiwan. 2,5-Di(tert-butylperoxy)hexane peroxide was purchased from AkzoNobel, The Netherlands, and was used in this work as acrosslinking agent with a half life of 1 h at 134 8C. Theantioxidant Irganox 1010 was purchased from Ciba Geigy Co.

Sample Preparation and Testing

LDPE, peroxide and the other additives were mixed in aninternal mixer (Haake) at 60 rpm and 120 8C. A hot press was

used to prepare cured sheets from which dumbbelled speci-mens were hollow-die punched. The conditions for achieving acomplete crosslinking reaction during hot press molding werefound using Differential Scanning Calorimetry (DSC) andtorsional rheometry. DSC revealed an exothermic cure reactionpeak at about 180 8C and the rheometer monitored a maximumsteady torque in less than ten minutes at 180 8C. An optimumamount of nearly 70% gel was measured by hot p-xyleneextraction according to ASTM D2765.

The cured specimens were then stretched by a tensilemachine at a rate of 5 mm �min�1 at 160 8C. For making heatshrinkable specimens and studying the stress relaxation beh-avior, the tensile machine, equipped with a heating chamber,was stopped when 200% strain was gained at 160 8C. Inpractice, as soon as the expansion is completed, air coolingstarts which, because of the poor thermal conductivity of thepolymer, it takes a few minutes for crystal formation andsolidification. The heat shrinkable product is then ready.

In this work, the cooling process was carried out instantlyusing salt-ice water and also at predetermined times, vis. 0, 8,15, 180 and 480 min, after reaching 200% strain in specimens,at 160 8C. This allowed some stress relaxation to occur, givingrise to some viscous flow before rapid cooling. Thus, at theabove mentioned times, samples were suddenly quenched tofreeze the build up of molecular orientation and consequentlythe stresses on samples monitored by the tensile machinealmost disappeared and the samples could be unclamped andremoved without contraction. When the heat shrinkable sam-ples prepared as above were reheated up to a temperature abovetheir crystalline melting point, they were suddenly shrunk.After cooling down and allowing for complete dimensionalstability the irrecovered strain was measured.

Results

The observed stress-strain curve in tension of crosslinked

specimens at 160 8C is illustrated in Figure 5. Since the

polymer is in a semi-rubbery state at this temperature due to

its crosslinked networks, it shows no yield point, with

similar trends to rubber behavior in Figure 2.

Figure 6 shows the stress relaxation of the same specimen

as soon as it has elongated three times at 160 8C and the

200% strain was kept stationary. It can be seen that, after an

initial invariance, stress decayed to zero after nearly 30 h. At

this time, as expected, when the specimen was unclamped

no recovery was observed. On the other hand, full retarded

Figure 4. Viscous/irrecoverable strain during stress relaxationtime predicted by the proposed model.

Figure 5. Stress-strain curve of crosslinked polyethylene at160 8C.



recovery at 160 8C was experienced for the specimen un-

clamped immediately after it had been elongated three

times. Quenching was carried out at different times inter-

vals (Figure 6) to make heat-shrinkable specimens of

different kinds. The heat shrinkage temperatures obtained

for these specimens by Thermal Mechanical Analysis

(TMA) were almost the same, being about 100 8C which is

near the crystalline softening temperature of LDPE (see

Figure 7). Moreover, shrinkage occurred over a narrow tem-

perature range. On the other hand, Figure 8 shows the ob-

served irrecovered strain of heat shrunk specimens of

primarily different allocated stress relaxation times before

quenching. It can be seen that for a specimen which was

quenched rapidly as soon as it had reached 200% strain, full

recovery occurred.

This indicates that no stress relaxation occurred during

the stretching process which could in turn result in viscous

flow. Secondly, some strain irrecovery was observed which

increased for specimens with higher allocated times for

stress relaxation. Finally, for the specimen which had pri-

marily experienced long enough stress relaxation before

quenching, no shrinkage occurred by heating. This means

that irrecoverability is closely related to the stress relax-

ation which occurs during the strain hold/cooling time.

Interpretation and Comparison of the Model’sPredictions with Experimental Results

It has been shown that the proposed mechanical model

(Figure 1(a)) can qualitatively represent all the features of

the real material in the elastic memory induction step, heat

shrinkability and its shape recovery, i.e., the stress-strain

behavior in tension at a constant strain rate, the stress relax-

ation behavior and the irrecoverable strain in heat shrunk

specimens.

Since it was concluded that no stress relaxation occurred

during stretching, the viscosity of the single dashpot (Z2),

whose displacement resulted in viscous/irrecoverable

strain, must be much higher than that in the Kelvin unit

(Z1) in Figure 1(a). Thus during hot stretching, the Kelvin

unit which had no stress relaxation was expanded while the

single dashpot was frozen and hardly moved (Figure 1(b)).

After some stress relaxation time, when the model was

held under constant strain, the single dashpot had the op-

portunity to move, resulting in viscous flow and changed

part of the strain into an irrecoverable one. At the same time,

the Kelvin unit started retracting (Figure 1(c)). If the

temperature dropped below the freezing point of the Kelvin

dashpot, the stress upon the model would be diminished to

zero. As with the shape memory polymer, a stable stretched

model which was heat shrinkable upon liquidification of the

Kelvin dashpot, or indeed above the polymer crystalline

melting point, would be obtained because of elastic restor-

ing forces (Figure 1(c)). It should be noted that upon rehe-

ating, while the Kelvin unit recovers totally to its original

length, the single expanded dashpot cannot retract and gives

rise to some irrecoverable strain (Figure 1(d)). During the

stress relaxation period, if the time is long enough, the

Kelvin unit completely recovers and the whole strain is

transferred to the single dashpot. In this situation, if the

model is quenched and reheated, no shrinkage occurs at all.

In this part, we have tried to examine the capability of the

model to describe the observed experimental results quanti-

tatively as well as qualitatively by attributing significant

Figure 6. Stress relaxation of crosslinked polyethylene under200% strain at 160 8C.

Figure 7. Recovery behavior of heat shrinkable specimenshowing the shrinkage temperature.

Figure 8. Irrecovered strain in heat shrunk specimens withprimarily different allocated times for stress relaxation



values to its parameters. The initial tangent modulus of the

stress-strain response of the model (Figure 2) is equal to

the spring modulus, E. Hence, E is approximated by the

Young’s modulus of the material obtained from the initial

tangent modulus of the experimental stress-strain curve at

160 8C (Figure 5), i.e., E� 30 000 Pa.

The viscosity of the Kelvin element dashpot during hot

stretching and stress relaxation was estimated using the

initial elongational viscosity of the molten PE compound at

160 8C, i.e., Z1� 105 to 106 Poise.[23] It has been assumed

that Z1 changes with temperature according to the Arrhenius

equation, i.e., Z1¼Aee/RT. It increases as the temperature is

lowered to a leathery-solidification temperature which is

about 100 8C, near to the onset of the LDPE crystallization

temperature. Under further cooling, the solid dashpot flow

viscosity (Z1) takes a very high value. Hence, the value of

the Arrhenius constants becomes E¼ 17 kcal �mol�1 and

A¼ 0.001 Pa � s.

The single dashpot viscosity (Z2) is taken to be consi-

derably higher than Z1. Therefore, it is in the solid state at the

liquidification or flow temperature of dashpot Z1, which is at

about 100 8C, and even at the stretching temperature of

160 8C. However, it follows the stress-time-temperature

superposition, i.e., the dashpot (Z2) softens and flows at

160 8C, sometimes after being under stress. Its flow gives

rise to the build up of an irrecoverable strain. The Z2 value at

160 8C is taken as the flow viscosity of a solid near to its

softening point, i.e., Z2� 6� 107 Poise.

Figure 9 demonstrates the variation of the viscosity of the

Kelvin dashpot against temperature by showing the solid

and liquid states. The model simulates elastic memory build

up and its recovery as follows.

When the model is stretched up to a fixed strain at 160 8C,

the single dashpot hardly moves. This observation corre-

sponds to full recovery upon heating of the heat shrinkable

specimen prepared by rapid cooling as soon as it was

stretched three times (no stress relaxation occurred during

employed method of rapid cooling). It must be noted that

when the Kelvin element is stretched, first of all the dashpot

retards extension of the parallel spring, and secondly no

stress relaxation occurs in the element while being stretched

or when it is held under constant strain.

Upon freezing, the applied stress on the model dimini-

shes to zero, which corresponds to stress vanishing from the

tensile machine as soon as the specimen was quenched well

below its crystalline melting point. Upon reheating, the

model fully recovers its original length.

When the model is held in the stretched condition at

160 8C, after some period of time which is closely matched

to the relaxation time of the model, the single dashpot solid

softens under a combination of stress, time and temperature.

At this time, the single dashpot slowly expands while the

Kelvin element slowly contracts, giving rise to the occur-

rence of an irrecoverable strain equal to displacement or

indeed the viscous flow of the dashpot, Z2.

When cooling is applied, Z2 first freezes and then

Z1 increases logarithmically from its initial value at

160 8C to about 107–108 Poise at its solidification point

near 100 8C. Further cooling freezes the extended model,

followed by fading of the stress and a rise in the viscosity as

high as 1012 Poise and thus building-up of heat shrinkable

model.

Upon heating of this model, as soon as the Kelvin dashpot

viscosity, Z1, is dropped to around its flow temperature

(about 100 8C), due to release of the elastic restoring force

of the spring, the Kelvin element shrinks to its original state

but the single dashpot, Z2, never recovers its original dimen-

sions, even when melted with further heating. Prepared heat

shrinkable specimens shrink on heating at about 100 8Cwhen their crystallites are melted and the entropic restoring

forces of extended chain segments between the crosslinks

become free to act. Some of the observed length irrecovery

is a function of the allocated time under strain before

quenching or stress-relaxation, which happens when the

sample or the product has not yet cooled below the polymer

crystallization temperature.

Figure 10, 11 and 12, which show respectively the

comparison of stress-strain, stress-relaxation, and irrecov-

ered strain behavior of real materials with quantitative

curves of the model, clearly indicate that the proposed

Figure 9. Variation of the viscosity of the Kelvin dashpot againsttemperature showing both solid and liquid states.

Figure 10. Comparison of experimental and model results forstress-strain behavior in stretching.



model with the estimated parameters predicts with high

precession the observed behavior.

Conclusion

No stress relaxation occurs during hot stretching of cross-

linked polyethylene, while it does occur during the strain

holding/cooling time step before the formation of crystal-

lites at the solidification point. If the elapsed time before

crystallization has been 30 h, the residing elastic restoring

force completely releases, resulting in zero shrinkage.

Accordingly it was revealed that the loss of the full

dimensional recovery of heat shrinkable materials origi-

nated from the occurrence of some stress relaxation during

molecular orientation freeze-up in the stretch holding-

cooling time step.

A combination of the Kelvin unit and a dashpot in series

was the proposed mechanical model used. Initial significant

values were selected and then varied. The mechanical

model was capable of describing the experimental results

observed in shape memory induction and dissipation

excellently.

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Figure 11. Comparison of experimental and model results forstress relaxation behavior.

Figure 12. Comparison of experimental and model results forirrecoverable strain build up during strain hold-on time.



modeling of shape memory induction and recovery in heat-shrinkable polymers

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