components of microstrain in polyethylene

Components of Microstrain in Polyethylene

NORMAN BROWN, STEPHEN RABINOWITZ

Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104

Received 29 January 2002; revised 20 August 2002; accepted 26 August 2002Published online 00 Month 2002 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/polb.10322

ABSTRACT: Tensile microstrain was measured in high-density polyethylene with aprecision of 2 � 10�7 for strains up to 10�4 in the temperature range of 17–28 °C overa range of strain rates. The total strain was partitioned into three of the followingcomponents: (1) elastic, (2) amorphous, and (3) dislocation. Also, their respective de-pendence on stress was determined. Measurements of the area of the hysteresis loopsgave the energy loss per cycle from which the frictional stress on the amorphous flowand the dislocations was determined. Although the amorphous strain originated at zerostress and was dependent on the temperature and strain rate, the dislocation motionwas activated above acritical stress and independent of temperature and strain ratewithin the scope of these experiments. The viscosity of the amorphous flow was deter-mined. The effect of 5 Mrd � irradiation on the micro-deformation behavior was notappreciable. About 80 Mrd reduced the amorphous flow by about 30% and increased thestress to activate the motions of the dislocations. © 2002 Wiley Periodicals, Inc. J Polym SciPart B: Polym Phys 40: 2693–2701, 2002Keywords: microstain; polyethylene; room temperature

INTRODUCTION

Polyethylene (PE) has a complex structure con-sisting of amorphous and crystalline regions. Thetotal strain in the material consists of three of thefollowing components: (1) elastic deformation pro-duced by the stretching of intermolecular bonds,(2) viscous flow in the amorphous region, and (3)crystal shear by the motion of dislocations. Thepurpose of this article is to quantitatively parti-tion the total strain into these components. Themethod measures the strain in the microstrainregion with a precision of 2 � 10�7 and limits thetotal strain to 10�4 so that the structure of thematerial is unaltered after successive cycles ofloading and unloading. The method was devel-oped by Brown1 to measure the reversibility andinitiation of dislocation motion in metals. Young’s

modulus of PE in the vicinity of zero strain wasinvestigated by Rabinowitz and Brown.2 For atotal strain of less than 20 � 10�6, the stress-strain curve in simple tension and constant strainrate agreed with the model for a standard linearsolid that consists of two Hookean springs and aviscous element governed by Newtonian flowwhose constitutive equation is given by

��̇ � � � Mr� � �Mu�̇ (1)

The solution to eq 1 for the case of a constantstrain rate, which was first presented by Rabi-nowitz and Brown,2 is

� � Mr� � ��̇ �Mu � Mr� �1 � e��/��̇� (2)

where � is stress, � is strain, �̇ is strain rate, � isrelaxation time under constant strain, and Muand Mr are the unrelaxed and relaxed Young’smodulus.

Correspondence to: N. Brown (E-mail: [email protected])Journal of Polymer Science: Part B: Polymer Physics, Vol. 40, 2693–2701 (2002)© 2002 Wiley Periodicals, Inc.

2693

Consequently, the stress-strain curve was non-linear at the lowest possible strain that could bemeasured. At higher strains, where eq 1 did notapply an additional strain component arose thatmay be attributed to strain within the crystallineregion probably from dislocations. One purpose ofthis article is to investigate and measure thisdislocation strain and to isolate the dislocationstrain from the viscous component. It was neces-sary to quench the viscous strain by increasingthe strain rate until it was not measurable withinthe time frame of the experiment. A critical stresswas required to initiate the dislocation strain.The energy loss during tensile loading and theunloading cycle was measured. The energy losswas partitioned between the amorphous flow andthe dislocation strain. The frictional stress thatwas determined from the energy loss was greaterfor the dislocation motion than for viscous flow.

EXPERIMENTAL

Details of the mechanical testing procedure havebeen described by Rabinowitz and Brown2 for PEand originally by Brown1 for metals. These exper-iments are more difficult with polymers than withmetals because the temperature must be moreaccurately controlled because the metals have amuch lower coefficient of thermal expansion.Specimens with a nominal 50-mm-gauge lengthand 5.7-mm diameter were deformed in tension at

a constant strain rate ranging from 10�6 to 10�4

s�1. A strain sensitivity as low as 2 � 10�7 wasobtained with a capacitance extensometer. Thelinearity of the system was calibrated by insert-ing a steel specimen. The system was linear forstrains up to 10�3. Generally the strain ampli-tude was less than 10�4. Because the capacitanceextensometer was attached to the shoulders of thespecimen, there was an uncertainty in the abso-lute value of the strain on the basis of the uncer-tainty of the effective gauge length. Previous ex-perience with this problem indicated that the ab-solute value of the strain may be off by as much as10%. The repeatability of the measurements issuch that the relative values of the strain for thesame setting of the extensometer vary within� 3% as shown in Figure 1. The loss in energyduring one cycle of loading and unloading wasobtained by measuring the area of the hysteresisloop with a planimeter. Five measurements weremade on each loop.

To avoid erratic behavior in the vicinity of zeroload, it is important to properly seat the grips sothat the specimen is aligned with the tensileforce. The alignment procedure depends on theweight of the grips. Thus, the lowest stress abovewhich reproducible stress-strain curves could beobtained was 0.14 MPa, at which point zero strainwas set. After the specimen was unloaded fromthe maximum load, the specimen was not re-loaded until the strain returned to zero. During

Figure 1. Repeatability of the modulus measurements.

2694 BROWN AND RABINOWITZ

any load–unload cycle the temperature was con-stant within � 0.01 °C.

The high-density PE (HDPE) was injection-molded in the form of 12.5-mm rods at 190 °C andsubsequently annealed in a vacuum at 130 °C for24 h. The resultant density was 973 kg/m3 (num-ber-average molecular weight: 11,500 and weight-average molecular weight 144,000). The HDPE inthe annealed condition was irradiated in vacuumwith Co60 at room temperature to doses of 5 and80 Mrad at respective rates of 5 � 104 and 7.75� 106 rad/h. These specimens are designated asHDPE-5 and HDPE-80.

RESULTS

Micro-stress-strain curves for various strain ratesare illustrated in Figure 2. The initial slope des-ignated by Mu is the unrelaxed modulus as deter-mined in by Rabinowitz and Brown.2 Figure 3shows the effect of the stress amplitude on thestress-strain curve.

The nonlinear component of the strain wasmeasured, that is, the deviation of the stress-strain curve from the unrelaxed modulus curve.This nonlinear strain is called �n. At a stress of0.24 MPa �n is plotted against strain rate at 17

Figure 2. Stress versus strain at 28 °C.

Figure 3. Stress versus strain at various stress amplitudes at 28 °C.

COMPONENTS OF MICROSTRAIN 2695

and 28 °C in Figure 4. By extrapolating thesecurves to an infinite strain rate, the asymptoticvalue, the unrelaxed strain (called �nu) was de-termined. �nu � 1.32 � 10�6 and 1.42 � 10�6 at17 and 28 °C, respectively, at 0.24 MPa. In Figure5 the stress (�) was plotted against �nu at 17, 23,

and 28 °C. All three curves coincide within theexperimental uncertainty. Thus, �nu is indepen-dent of strain rate and temperature. The strain-rate-dependent part of the nonlinear strain iscalled �a � (�n � �nu) and is plotted against stressin Figure 6 at 17 and 28 °C. Figure 6 suggests �adepends on temperature and is larger than �nu,which is also plotted in Figure 6. Thus, �a consists

Figure 4. Nonlinear strain versus strain rate at 17 and 28 °C.

Figure 5. Stress versus the strain-rate independentpart of the nonlinear strain.

Figure 6. Stress versus the amorphous part of thestrain.


of the temperature and strain-rate part of thetotal strain. It is important to note another dif-ference between �a and �nu. �a is initiated at zerostress in accordance with the observations byRabinowitz and Brown2 and as described by eq 1for the standard linear solid. Figure 5 shows that�nu is initiated at a stress of 0.19 MPa as if it isassociated with a yield point that is essentiallyindependent of strain rate and temperature overthe ranges in these experiments.

It is of interest to quantitatively determine theamounts of the three components of the totalstrain and their dependence on the stress. Thetotal strain, �, consists of the elastic part, �e, theamorphous part, �a, so-called because it dependson strain rate and temperature and the temper-ature and strain-rate-independent part, �nu,where

� � �e � �a � �nu (3)

Figure 7. Nonlinear part of the strain versus strain rate after 80 Mrd irradiation.

Figure 8. Energy loss versus strain rate at 17 and 28 °C.


From Figure 5 and 6 and the values of the mod-ulus as determined by Rabinowitz and Brown,2

the dependence of each of these strain compo-nents on stress is now presented in units of 10�6.The stress is in units of megapascals,

�e � �/Mu, �e � 257� at 17 °C,

�e � 363� at 28 °C (4a)

�nu � 143 �� 0.19�3/2 (4b)

�a � 350 �2 at 17 °C; �̇ � 2 � 10�6s�1

and �a � 875�2 at 28 °C; �̇ � 2 � 10�6s�1 (4c)

It was found that 5 Mrd had no significanteffect on the microstrain behavior. Figure 7 de-scribes how �n varies with strain rate forHDPE-80 as compared with the un-irradiatedmaterial. For HDPE-80, �nu was essentially zeroat stresses up to 0.26 MPa, whereas �nu was ac-tivated above 0.19 MPa for HDPE. The stress toinitiate �nu in HDPE-80 was not measured. At astrain rate of 2 � 10�6 s�1, �a for HDPE-80 wasabout 70% of the value for the un-irradiated ma-terial.

The energy loss per cycle, W, was measured asa function of strain rate as shown in Figure 8 at17 and 28 °C. The asymptotic value of W at infi-nite strain rate, Wu, depends on stress as shownin Figure 9. Wu does not depend on temperatureand becomes evident at a stress above 0.19 MPa,which is like the dependence of �nu on stress inFigure 5. Figure 10 portrays the dependence ofWa � (W � Wu) on stress; this behavior is similarto that of �a in Figure 6. Thus, it is expected thatthere is a unique relationship between Wu and �nuthat is independent of strain rate and tempera-ture between Wa and �a.

DISCUSSION

The physical nature of the three components ofthe strain is now discussed. The linear elasticcomponent is associated with stretching of thevan der Waals bonds. The magnitude of the elas-tic strain is determined by the modulus of elastic-ity. The standard linear model represented by eq1 consists of two linear springs, one of which isuncoupled and corresponds to the unrelaxed mod-ulus, Mu, and the other is coupled to the viscous

element, which we call Mc. In terms of Mu and Mr,the unrelaxed and relaxed moduli are

1/Mc � 1/Mr � 1/Mu (5)

Using the values of Mr and Mu measured by Rabi-nowitz and Brown,2 Mc was determined as shownin Table 1. Thus, the part of the modulus, Mc, thatis coupled to the viscous region is much largerthan the part, Mu, not being coupled and is free torespond to an instantaneous change in stress. Itis suggested that Mc produces the stress thatrestores the strain, �a, when the specimen is un-loaded and associated with the stretching of vander Waals bonds within the amorphous region.Mu is associated with stretching of van der Waalsbonds in the crystalline region.

The amorphous region has a viscosity, V, thatcan be calculated as follows from the dependenceof the amorphous strain on stress as measured ineq 4c

Figure 9. Stress versus the energy loss that was in-dependent of strain rate.


�a � �0

t

�̇adt (6)

�̇a � �/V (7)

t � �/�̇ and �̇ � �̇M (8)

where �̇ is the total strain rate, and M is modulus.Combining eqs 6–8, let V be a constant that isequal to its average value

�a � �2/2V�̇M (9)

The experimental results in eq 4c exhibit thesame dependence on stress as the theoretical eq 9.By equating eqs 4c and 9 and with the moduli of3.5 and 3.0 GPa at 17 and 28 °C, respectively, atthe strain rate of 2 � 10�6s�1, V � 200 and 120GPa s at 17 and 28 °C, respectively.

The strain, �nu, that is independent of strainrate and temperature is probably caused by themotion of dislocation motion in the crystalline

region. There is a variety of possible dislocationsin PE that have been investigated by Keith andPassaglia.3 Frank et al.4 also investigated varioustypes of dislocation in PE involving slip, twinning,and a stress-induced phase transformation. Thisinvestigation cannot distinguish which of the pos-sible dislocations produce �nu. However, all typesof dislocation motion are associated with a fric-tional stress and with an internal stress field thatcan reverse their motion. Also, the strain associ-ated with dislocations is expected to be small ascompared with �e and �a and much less sensitiveto temperature and strain rate as compared withviscous flow in the amorphous region. The revers-ible motion of dislocations in the microstrain re-gion and the associated energy loss has been an-alyzed by Roberts and Brown5 for metals. Here anadditional term is added for the amorphous flow.The work done is given by

W � ��ed�e � ��id�nu � ��dfd�nu � ��afd�a (10)

where �e is the stress that produces the elasticdeformation, �i is the internal stress produced bythe dislocations, and �df and �af are the frictionalstresses on the dislocations and the amorphousflow. The first two integrals in eq 10 are zerobecause these stresses do not change their signduring tensile loading and unloading, but the cor-responding strain increments do change. Thethird and fourth integrals represent the energyloss because the frictional stresses resist the mo-tion of the dislocations and amorphous flow

Wu � ��dfd�nu (11)

Wa � ��dad�a (12)

The plot of Wu against �nu in Figure 11 at 17 and28 °C produced a linear relationship with a slopeof 0.18 MPa that is independent of temperature.The frictional stress �df � 0.09 MPa, which isone-half this slope because half the energy is lostduring loading of the specimen and half duringunloading. The stress to activate the dislocation

Table 1. Elastic Moduli for High-Density PE

T (°C) Mu (GPa) Mr (GPa) Mc (GPa)

17 3.9 3.1 15.328 2.75 2.2 11.0

Figure 10. Stress versus the energy loss associatedwith amorphous flow.


strain (Fig. 5) is twice the frictional stress. Theexplanation is as follows. Once the dislocationdensity has been generated during the very firstloading of the specimen at a certain stress level,then this dislocation density and its associatedinternal stress field remain constant during sub-sequent loading and unloading cycles at stresslevels that are less than the original stress levelthat generated the dislocation array. Thus, at

zero applied stress, the dislocations are in equi-librium between the internal stress field and thefrictional stress. When the specimen is reloadedin tension, the stress to activate the dislocationsmust overcome both the frictional stress and theinternal stress. If the specimen had been reloadedin compression, the stress to activate the disloca-tions would have been zero. This is the fundamen-tal explanation for the Bauschinger effect.

Figure 11. Energy loss that is independent of strain rate versus component of strainthat is independent of strain rate.

Figure 12. Energy loss from amorphous flow versus strain produced by amorphousflow.


Wa is a linear function of �a as shown in Figure12 where the slope is 0.12 MPa at 17 and 28 °C.This result is surprisingly independent of temper-ature because the frictional stress on the amor-phous flow should vary inversely with the viscos-ity, and the viscosity should decrease when thetemperature increases. Probably the temperaturerange of the experiment was too narrow for ob-serving a difference. The frictional stress againstthe amorphous flow is 0.06 MPa. From this fric-tional stress, the viscosity can be calculated as

V � �af/�̇a and �̇a � ��̇, where � � 0.2 (13)

For �af � 0.06 MPa and �̇ � 2 � 10�6s�1, V � 150GPa s, which is consistent with the previous cal-culation of V on the basis of eq 9. There is afundamental difference between the frictionalstress on a dislocation and the one that resists theamorphous flow. The former acts when the dislo-cation is not moving or is moving at a very lowvelocity, and the latter only acts during the amor-phous flow and is very sensitive to the strain rate.

The effect of irradiation at the 5 Mrd level wastoo small to be observed. At 80 Mrd there was asignificant decrease in the amorphous strain.This result is expected because � irradiationcrosslinks the amorphous region and increasesthe viscosity. Although the dislocation strain, �nu,was activated at 0.18 MPa in the un-irradiatedmaterial, it could not be activated at 0.26 MPa inHDPE-80. Presumably the irradiation pinned thedislocations. The pinning of dislocations in non-polymeric crystals is caused by the formation ofvacancies and interstitials that are attracted tothe dislocations. In polymers the details of thepinning of dislocations by � rays have not beendetermined.

CONCLUSIONS

The following conclusions are listed:

1. The microstrain in HDPE was measuredwith a precision of 2 � 10�7.

2. The total strain was partitioned into threecomponents—elastic strain associated withstretching of van der Waals bonds, amor-phous flow that varies as �2 and is activatedat zero stress, and dislocation motion thatvaries as �3/2 and is activated at 0.18 MPa.

3. From the energy loss per cycle, the frictionalstress that resisted the motion of disloca-tions was 0.09 MPa, and for the amorphousflow it was 0.060 MPa.

4. The viscosity of the amorphous region isabout 150 GPa s.

5. The effect of 5 Mrd of � irradiation had noappreciable effect on the micro-deformation.However, 80 Mrd reduced the amorphousflow by 30% and pinned the dislocations sothat they could not be activated at 0.26MPa.

This work was completed at the Laboratory for Re-search on the Structure of Matter and supported by theNational Science Foundation under award numberDMR 0079909.

REFERENCES AND NOTES

1. Brown, N. In Microplasticity; McMahon, C. J., Ed.;Wiley-Interscience: New York, 1968; pp 45–73.

2. Rabinowitz, S.; Brown, N. J Polym Sci Part B: PolymPhys 2001, 39, 2420–2429.

3. Keith, H. D.; Passaglia, E. J Res Natl Bur St and1964, 68, 513.

4. Frank, F. C.; Kcllcr, A.; O’Connor, A. Philos Mag1958, 3, 64.

5. Roberts, J. M.; Brown, N. Trans Metall Soc AIME1960, 218, 454.


components of microstrain in polyethylene

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