12 Mechanical Properties 12.1 INTRODUCTION It is usually material properties, in addition to cost and availability, that determine which class of materials–polymers, metals, or ceramics—and which particular member within that class are used for a given application. Many commodity thermoplastics, for example, begin to soften around 100 C, and this essentially limits their use to temperatures that are a few tens of degrees Celsius below this value. A major factor in favor of polymers, though, is their low density (by a factor of 4 or 5) relative to metals; the possibility of a large weight savings, coupled with high strength, makes plastics very attractive for automotive, marine, and aerospace applications. In terms of choosing a specific polymer, however, it is necessary to consider whether the application of interest is structural or nonstructural. In the former case, mechanical properties such as tensile strength, stiffness, impact strength, and chemical resistance might be relevant, whereas important considerations in the latter case might include surface finish, ease of painting, and the influence of humidity and ultraviolet radiation on the tendency of the material to crack. In this chapter, we will consider mechanical properties of polymers at small strains as well as large strains. In general, the mode of deformation could be tension, compression, shear, flexure, torsion, or a combina- tion of these. To keep the discussion manageable, we will restrict ourselves to tension and shear. Note, however, that we can use viscoelasticity theory [1], 487 Copyright © 2003 Marcel Dekker, Inc. especiallyatsmallstrains,topredictthebehaviorinonemodeofdeformation frommeasurementsmadeinanothermodeofdeformation.Aswithmetals,we expectthatthemeasuredpropertiesdependonthechemicalnatureofthepolymer andthetemperatureofmeasurement.However,whatmakesdataanalysisand interpretationbothfascinatingandchallengingarethefactsthatresultsalso dependontimeofloadingortherateofdeformation,polymermolecularweight, molecular-weightdistribution,chainbranching,degreeofcross-linking,chain orientation,extentofcrystallization,crystalstructure,sizeandshapeofcrystals, andwhetherthepolymerwassolutioncastormeltprocessed.Thesevariablesare notallindependent;molecularweight,forexample,candeterminechain orientationandcrystallinityinaparticularprocessingsituation.Toexplainthe separateinfluenceofsomeofthesevariables,wepresentdataonpolystyrene,a polymerthatcanbesynthesizedinnarrowmolecular-weightfractionsusing anionicpolymerization.Methodsofimprovingpolymermechanicalproperties areagainillustratedusingpolystyrene.Thischapterthereforefocusesonthe (glassy)behaviorofpolymersbelowtheirglasstransitiontemperature. 12.2STRESS^STRAINBEHAVIOR WhendiscussingthetheoryofrubberelasticityinChapter10,wewereconcerned withfairlylargeextensionsorstrains.Thesearosebecausepolymermolecules coulduncoilattemperaturesaboveT g .Formaterialsusedasstructuralelements (suchasglassypolymers),weusuallycannottoleratestrainsofmorethana fractionof1%.Therefore,itiscustomarytoemploymeasuresofinfinitesimal strain.Inatensiletest,weusuallytakeaspecimenwithtabsattheendsand stretchit,asshowninFigure12.1.Oneendofthesampleistypicallyfixed, whereastheotherismovedoutwardataconstantvelocity.TheforceFnecessary tocarryoutthestretchingdeformationismonitoredasafunctionoftimealong withtheinstantaneoussamplelength,L.Fromthemeasuredloadversus extensionbehavior,wecancalculatethestressandstrainasfollows: StressðsÞ¼ ForceðFÞ Cross-sectionalarea ð12:2:1Þ Ifthecross-sectionalareaistheundeformed,originalcross-sectionalarea,the stressiscalledengineeringstress,andiftheactual,instantaneousareaisused,the truestressismeasured. StrainðEÞ¼ LL 0 L 0 ð12:2:2Þ 488Chapter12 Copyright © 2003 Marcel Dekker, Inc. whereL 0 istheinitialsamplelength,andthestrain,sodefined,isknownasthe engineeringstrain.NotethatthisstrainisrelatedtotheHenckystrain,alsocalled thetruestrain,asfollows: E true ¼lnð1þE eng Þð12:2:3Þ andthetwostrainmeasuresareidenticalforsmallstrains. Asthesampleisstretchedinthezdirection,itscross-sectionalarea decreases,andthisimpliesthatthematerialsuffersanegativestraininthex direction,whichisperpendiculartothestretchingdirection.Thisisquantified usingthePoissonration,definedas E x ¼nE z ð12:2:4Þ Forincompressiblematerialssuchasrubber,itiseasytoshowthatPoisson’sratio equals0.5.Forglassypolymersthesamplevolumeincreasessomewhaton stretching,andPoisson’sratiorangesfrom0.3to0.4. Typicalstress–straindataforglassypolystyreneareshowninFigure12.2in bothtensionandcompression[2].Theslopeofthestress–straincurveevaluated attheoriginistermedtheelasticmodulus,E,andistakentobeameasureofthe stiffnessofthematerial.Itisseeninthisparticularcasethatthemodulusin tensiondiffersfromthatincompression.Thetwocurvesendwhenthesample fractures.Thestressatfractureiscalledthestrengthofthematerial.Because materialsfractureduetothepropagationofcracks,thestrengthintensionis usuallylessthanthatincompressionbecauseacompressivedeformationtendsto healanycracksthatform(providedthesampledoesnotbuckle).Thestrainat FIGURE12.1Typicalspecimenforatensiletest. Mechanical Properties 489 Copyright © 2003 Marcel Dekker, Inc. fracture is known as the elongation-to-break; the larger the value of this quantity, the more ductile is the material being tested. Glassy polystyrene is not ductile in tension; indeed, it is quite brittle. Finally, the area under the stress–strain curve is called the toughness and has units of energy per unit volume. For design purposes, the materials generally sought are stiff, strong, ductile, and tough. For materials that are liquidlike, such as polymers above their softening point, it is easier to conduct shear testing than tensile testing. This conceptually involves deforming a block of material, as shown in Figure 12.3. The force F is FIGURE 12.2 Stress–strain behavior of a normally brittle polymer such as polystyrene under tension and compression. (Reprinted from Nielsen, L. E., and R. F. Landel: Mechanical Properties of Polymers and Composites, 2nd ed., Marcel Dekker, Inc., New York, 1994, p. 250, by courtesy of Marcel Dekker, Inc.) FIGURE 12.3 Shear defor mation. 490 Chapter 12 Copyright © 2003 Marcel Dekker, Inc. againmonitored,butnowasafunctionofthedisplacementDu x .Stressandstrain arenowdefinedasfollows: ShearstressðtÞ¼ ForceðFÞ Surfacearea ð12:2:5Þ ShearstrainðgÞ¼ Du x Dy ð12:2:6Þ Attemperaturesabovethepolymerglasstransitiontemperature,sheartestingis doneusingavarietyofviscometers(seeChap.14).Wemight,forexample,keep thesampleintheannularregionbetweentwoconcentriccylindersandmeasure thetorquewhilerotatingonecylinderrelativetotheother.Stress–straindatain shearlookqualitativelysimilartothetensiledatashownearlierinFigure12.2. The initial slope is called the shear modulus, G. For elastic materials the moduli in shear and tension are related by the following expression: E ¼ 2Gð1 þnÞð12:2:7Þ so that E equals 3 G for incompressible, elastic polymers. Note that when material properties are time dependent (i.e., viscoelastic), the modulus and strength increase with increasing rate of deformation [3], whereas the elongation-to- break generally reduces. Viscoelastic data are often represented with the help of mechanical analogs. Example 12.1: A polymer sample is subjected to a constant tensile stress s 0 . How does the strain change with time? Assume that the mechanical behavior of the polymer can be represented by a spring and dashpot in series, as shown in Figure 12.4. Solution: The stress-versus-strain behavior of a Hookean spring is given by s ¼ EE FIGURE 12.4 A Maxwell element. Mechanical Properties 491 Copyright © 2003 Marcel Dekker, Inc. ForaNewtoniandashpot,therelationis s¼Z dE dt ThetermsEandZarethespringmodulusanddashpotviscosity,respectively. Forthespringanddashpotcombination,oftencalledaMaxwellelement, thetotalelongationorstrainisthesumoftheindividualstrains.Thestressforthe springandforthedashpotisthesame, Totalstrain¼ s 0 E þ s 0 Z t anditisseenthatthestrainincreaseslinearlywithtime.Thisbehaviorisknownas creep.AlthoughasimplemechanicalanalogsuchasaMaxwellelementcannotbe expectedtoportraytruepolymerbehavior,itdoesillustratetheusuallyundesir- ablephenomenonofcreep.Abettermodelforthequantitativerepresentationof creepisafour-elementmodelwhichisalinearcombinationofaMaxwellelement andaVoigtelement;thelatteriscomposedofaspringandadashpotinparallel. Apolymersamplecreepsbecausepolymermoleculesareheldinplaceby secondarybondsonly,andtheycanrearrangethemselvesundertheinfluenceof anappliedload.Thisisespeciallyeasyabovethepolymerglasstransition temperature,butitalsohappensbelowT g andstraingaugeshavetobeemployed foraccuratemeasurements.Toillustratethelatterpoint,weshowlong-termcreep data,intheformofcirclesinFigure12.5,onsamplesofpolyvinylchloride (PVC) at constant values of tensile stress, temperature, and relative humidity [4]. Note that data for the first 1000 h are shown separately, followed by all of the data using a compressed time scale. It is seen that the total creep can be several percent, and a steady state is not reached even after 26 years! These and similar data can be represented by the following simple equation shown by solid lines in Figure 12.5: EðtÞ¼E 0 þ E þ t n ð12:2:8Þ in which E 0 , E þ , and n are constants. Although n is often independent of temperature and imposed stress, the other two constants are stress and tempera- ture dependent. If creep is not arrested, it can lead to failure, which may occur either by the process of crazing or by the formation of shear bands; these failure mechanisms are discussed later in the chapter. Equation (12.2.8) is an empirical equation that is known as the Findley model. It may sometimes contain a second time-dependent term if failure can occur by two different mechanisms. Creep can generally be reduced by lowering the test temperature, raising the polymer T g , cross-linking the sample, or adding either particulates or short fibers. Conversely, anything that lowers the T g , such as exposure to atmospheric moisture, promotes creep. Physical aging (described later) also affects the extent of creep. 492 Chapter 12 Copyright © 2003 Marcel Dekker, Inc. FIGURE 12.5 Creep curves for polyvinyl chloride at 75 F, 50% relative humidity. (From Ref. 4.) Mechanical Properties 493 Copyright © 2003 Marcel Dekker, Inc. 12.2.1 In£uence of Variables such as Molecular Weight and Temperature The strength and stiffness of one glassy polymer can be expected to differ from that of another glassy polymer due to differences in intermolecular forces as a result of differences in chemical structure and the presence or absence of secondary bonds (e.g., hydrogen-bonding). Given these differences, the two variables that influence the mechanical properties of amorphous polymers the most are molecular weight and temperature. However, the elastic moduli and other small-strain properties of strain-free glassy polymers such as polystyrene (PS) are found not to depend on the molecular weight or molecular-weight distribution, except at very low molecular weights [5–7]. The tensile strength, s f , of polymers having a narrow molecular-weight distribution, however, is negli- gible at low molecular weight, increases with increasing molecular weight, and, ultimately reaches an asymptotic value [8]. This behavior can often be repre- sented by the following equation [6, 9]: s f ¼ A B=M n ð12:2:9Þ where A and B are constants. Data for polystyrene, shown in Figure 12.6, support these conclusions [10]. From an examination of this figure, it is obvious that the addition of a low-molecular-weight fraction is bound to affect the tensile strength of any polymer. However, for polydisperse samples, data do not follow Eq. (12.2.9) exactly; results vary with the polydispersity index, even when the number-average molecular weight is held fixed. The data just discussed are related to amorphous polymer samples for which the polymer chains were randomly oriented. One method of increasing both strength and stiffness is to use samples wherein polymer chains are oriented FIGURE 12.6 Tensile strength of monodisperse polystyrene as a function of molecular weight. From Hahnfeld, J. L., and B. D. Dalke: General purpose polystyrene, in Encyclopedia of Polymer Science and Engineering, 2nd ed., vol. 16, H. F. Mark, N. M. Bikales, C. G. Overberger, and G. Menges (eds.) Copyright # 1989 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc. 494 Chapter 12 Copyright © 2003 Marcel Dekker, Inc. alongthestretchingdirection.Byusingthistechnique,wecanverysignificantly increasethemodulusofpolystyreneandhopetogetstrengththatapproachesthe strengthofprimarychemicalbonds[11].Indeed,asdiscussedinChapter11, mechanicalpropertyenhancementusingchainalignmentisthereasonforthe popularityofpolymersthatpossessliquid-crystallineorder.Propertiesina directionperpendiculartothechainaxis,however,arelikelytobeinferiorto thosealongthechainaxis. WhentheYoung’smodulusofanypolymerisplottedasafunctionof temperature,wefindthatthisquantityisoftheorderof10 5 –10 6 psiatlow temperaturesanddecreasesslowlywithincreasingtemperature.Thisregionis knownastheglassyregion.AttheglasstransitiontemperatureT g (seealsoChap. 2),whichvariesfordifferentpolymers,themodulusdropssuddenlybyatleast threeordersofmagnitudeandcanreachextremelylowvaluesforlow-molecular- weightpolymers.Figure12.7showstheYoung’smodulusofpolystyreneina temperaturerangeof200 Cto25 C[12].Figure12.8showsshearstressversus shearstraindataforanentangledpolystyreneinatemperaturerangeof160 C– 210 C[13].IfwedisregardthenumericaldifferencebetweentheYoung’s modulusandtheshearmodulusandnotethat1MPaequals145psi,wefind thatthemoduluscalculatedfromdatainFigure12.8isseveralordersof magnitudesmallerthanthenumberexpectedonthebasisofextrapolatingthe curveinFigure12.7.ThishappensbecausetheT g ofpolystyreneis100 C.The behavioroftheYoung’smodulus,inqualitativeterms,issketchedinFigure12.9 overatemperaturerangethatincludesT g .Ifthepolymermolecularweightis abovethatneededforentanglementformation(forpolystyrene,thisisapproxi- mately35,000),thepresenceoftheseentanglementstemporarilyarreststhefallin modulusoncrossingT g .Thisregionofalmostconstantmodulusiscalledthe rubberyplateau,andtheresultisarubberypolymer.Becausecrystalsactina mannersimilartoentanglements,themodulusofasemicrystallinepolymerdoes FIGURE12.7EffectoftemperatureonYoung’smodulusofpolystyrene.(FromRef. 12.) Reprinted with permission from J. Appl. Phys., vol. 28, Rudd, J. F., and E. F. Gurnee: Photoelastic properties of polystyrene in the glassy state: II. Effect of temperature, 1096– 1100, 1957. Copyright 1957 American Institute of Physics Mechanical Properties 495 Copyright © 2003 Marcel Dekker, Inc. FIGURE 12.8 Effect of temperature on the stress–strain cur ves of polystyrene melts. (From Ref. 13.) FIGURE 12.9 Qualitative effect of temperature on the elastic modulus of polymers. 496 Chapter 12 Copyright © 2003 Marcel Dekker, Inc. [...]... 497 not fall as precipitously as that of amorphous polymers for temperatures between the Tg and the melting point of the crystals Of course, if chemical cross-links are present, the polymer cannot flow and the temperature variation of the modulus above Tg is given by the theory of rubber elasticity Understanding and relating mechanical properties of a semicrystalline polymer to the different variables... regions of dramatically different polymer properties In particular, a polymer behaves like a hard, brittle, elastic solid below Tg In this glassy region, the motion of polymer chains is frozen and strain occurs by the stretching of bonds The elastic modulus decreases with increasing temperature On heating above Tg , an entangled, amorphous polymer displays a rubbery region in which it is soft and pliable... to be of the order of 105 – 106 ergs=cm2 to obtain agreement with experimental data [2] 12.7 CRAZING AND SHEAR YIELDING Nonlinearities in the stress–strain curve of a glassy polymer usually indicate the presence of irrecoverable deformation Although the extent of yielding depends on the test temperature and the rate of strain, it is generally true that most glassy polymers do show some amount of plastic... Properties of Glassy Polymers, Treatise Mater Sci Technol., 10B, 541–598, 1977 Sternstein, S S., and F A Myers, Yielding of Glassy Polymers in the Second Quadrant of Principal Stress Space, J Macromol Sci Phys., B8, 539–571, 1973 Argon, A S., J G Hanoosh, and M M Salama, Initiation and Growth of Crazes in Glassy Polymers, Fracture, 1, 445–470, 1977 Kramer, E J., Microscopic and Molecular Fundamentals of Crazing,... the temperature of the glass transition, it can be used to study the curing behavior of thermosetting polymers and to measure secondary transitions and damping peaks These peaks can be related to phenomena such as the motion of side groups, effects related to crystal size, and different facets of multiphase systems such as miscibility of polymer blends and adhesion between components of a composite... obtain any small-strain property of a polymer Example 12.3: Use Figures 12.17 and 12.18 to determine the storage modulus at 200 C and 1 rad=sec of the polystyrene fraction labeled L27 Compare the result with that obtained with the use of Figure 12.16 Solution: From Figure 12.18, the value of aT at 200 C is 0. 013 Thus, oaT is 0. 013 rad=sec and logðoaT Þ equals À1:89 The use of Figure 12.17 then reveals... it from a rigid to a more flexible material The glass transition temperatures of common polymers are listed in the Polymer Handbook [20], and selected values are given in Table 12.1 One of the most convenient methods of measuring Tg is through the use of a differential scanning calorimeter (DSC) [21] The principle of operation of this instrument is shown schematically in Figure 12.12 A DSC contains two... these objectives For homopolymers, improving mechanical properties has largely been a process of relating the internal structure of the polymer to its properties For amorphous polymers, we seek to align all of the polymer chains in the same direction; this anisotropy results in a higher glass transition temperature and an increase in both stiffness and strength in the direction of molecular chain orientation... effect of these different variables using polymers such as nylons, polyethylene terephthalate, and polypropylene; a considerable body of knowledge now exists [47,48] This has allowed for the production of not only high-quality textile yarns but also of highstrength, high-modulus fibers from conventional polymers such as high-density polyethylene of high molecular weight (> 106 ) By careful control of molecular... quintessential method of improving the strength and stiffness of polymers is to form reinforced composites by adding filler particles, whiskers, short fibers, or long fibers to polymer matrices such as epoxies, unsaturated polyesters, and vinyl esters [49,55] Composite materials containing 50–70% by weight of fibers of glass, carbon, or polyaramid in thermoplastic or thermosetting polymer matrices can be . Inc. especiallyatsmallstrains,topredictthebehaviorinonemodeofdeformation frommeasurementsmadeinanothermodeofdeformation.Aswithmetals,we expectthatthemeasuredpropertiesdependonthechemicalnatureofthepolymer andthetemperatureofmeasurement.However,whatmakesdataanalysisand interpretationbothfascinatingandchallengingarethefactsthatresultsalso dependontimeofloadingortherateofdeformation,polymermolecularweight, molecular-weightdistribution,chainbranching,degreeofcross-linking,chain orientation,extentofcrystallization,crystalstructure,sizeandshapeofcrystals, andwhetherthepolymerwassolutioncastormeltprocessed.Thesevariablesare notallindependent;molecularweight,forexample,candeterminechain orientationandcrystallinityinaparticularprocessingsituation.Toexplainthe separateinfluenceofsomeofthesevariables,wepresentdataonpolystyrene,a polymerthatcanbesynthesizedinnarrowmolecular-weightfractionsusing anionicpolymerization.Methodsofimprovingpolymermechanicalproperties areagainillustratedusingpolystyrene.Thischapterthereforefocusesonthe (glassy)behaviorofpolymersbelowtheirglasstransitiontemperature. 12.2STRESS^STRAINBEHAVIOR WhendiscussingthetheoryofrubberelasticityinChapter10,wewereconcerned withfairlylargeextensionsorstrains.Thesearosebecausepolymermolecules coulduncoilattemperaturesaboveT g .Formaterialsusedasstructuralelements (suchasglassypolymers),weusuallycannottoleratestrainsofmorethana fractionof1%.Therefore,itiscustomarytoemploymeasuresofinfinitesimal strain.Inatensiletest,weusuallytakeaspecimenwithtabsattheendsand stretchit,asshowninFigure12.1.Oneendofthesampleistypicallyfixed, whereastheotherismovedoutwardataconstantvelocity.TheforceFnecessary tocarryoutthestretchingdeformationismonitoredasafunctionoftimealong withtheinstantaneoussamplelength,L.Fromthemeasuredloadversus extensionbehavior,wecancalculatethestressandstrainasfollows: StressðsÞ¼ ForceðFÞ Cross-sectionalarea ð12:2:1Þ Ifthecross-sectionalareaistheundeformed,originalcross-sectionalarea,the stressiscalledengineeringstress,andiftheactual,instantaneousareaisused,the truestressismeasured. StrainðEÞ¼ LL 0 L 0 ð12:2:2Þ 488Chapter12 Copyright. Inc. notfallasprecipitouslyasthatofamorphouspolymersfortemperaturesbetween theT g andthemeltingpointofthecrystals.Ofcourse,ifchemicalcross-linksare present,thepolymercannotflowandthetemperaturevariationofthemodulus aboveT g isgivenbythetheoryofrubberelasticity.Understandingandrelating mechanicalpropertiesofasemicrystallinepolymertothedifferentvariablesthat characterizeitsstructurehasbeendiscussedinChapter11andistreatedindetail bySamuels[14]. 12.3THEGLASSTRANSITIONTEMPERATURE AsdiscussedinChapter2,theglasstransitiontemperatureseparatesregionsof dramaticallydifferentpolymerproperties.Inparticular,apolymerbehaveslikea hard,brittle,elasticsolidbelowT g .Inthisglassyregion,themotionofpolymer chainsisfrozenandstrainoccursbythestretchingofbonds.Theelasticmodulus decreaseswithincreasingtemperature.OnheatingaboveT g ,anentangled, amorphouspolymerdisplaysarubberyregioninwhichitissoftandpliable duetotheabilityofpolymerchainsegmentsandentirepolymerchainstomove pasteachotherinareversiblemanner.Inthisregion,theelasticmoduluscan increasewithanincreaseintemperature;thispropertyhasbeenexplained theoreticallyinChapter10.StructuralapplicationsclearlyrequireapolymerT g aboveroomtemperature,whereasapplicationswherematerialflexibilityis important,suchasinfilmsusedforpackaging,requirethattheT g bebelow roomtemperature. Althoughwecanuseobservationsofthechangeinmechanicalproperties asameansofmeasuringT g ,wealsofindthatthermodynamicpropertieschange slopeongoingthroughtheglasstransition.Thus,ifweplotthevolumeofa sampleoritsenthalpyasafunctionoftemperature,behaviordepictedqualita- tivelyinFigure12.10isobserved:Theslopeintheliquidphaseislargerthanthe slope. Inc. mentionedinChapter2.Torecapitulate,thepolymerfreevolumeisthedifference inthesamplevolumeandtheactualvolumeoccupiedbytheatomsand molecules.Thefreevolumeiszeroatabsolutezerotemperatureanditincreases asthetemperatureincreases.Slowcoolingallowsforacloserapproachto equilibriumandalowerfreevolumerelativetomaterialsubjectedtorapid cooling.Thus,theslowlycooledsamplehastobeheatedtoahighertemperature inorderthattherebeenoughfreevolumeforthemoleculestomovearound,and thisimpliesahigherT g .InadditiontochangesinT g withcoolingrate,wealso observevolumerelaxationwhenapolymersamplethatwasrapidlycooledis subsequentlyheatedtoatemperatureclosetoT g andheldthereforsometime. Materialshrinkagealsooccurs,accompaniedbychangesinthemechanical propertiesofthesolidpolymer.Thephenomenonisknownasphysicalaging [15]andisthesubjectofconsiderableresearchbecauseofitsinfluenceon propertiessuchascreep[16]. Theglasstransitiontemperatureofapolymerdependsonanumberof factors,includingthepolymermolecularweight.Themolecular-weightdepen- dencecanbeseeninFigure12.11,wheretheT g ofpolystyreneisplottedasa functionofthenumber-averagemolecularweight[3,17].Thesedatacanbe representedmathematicallybythefollowingequation[18]: T g ¼T g1 K MM n ð12:3:1Þ ThisvariationofT g withmolecularweightcanagainberelatedtothefreevolume [19].Asthemolecularweightdecreases,thenumberdensityofchainends increases.Becauseeachchainendisassumedtocontributeafixedamountoffree FIGURE12.10Variationofvolumeorenthalpyofpolymerswithtemperature. 498Chapter