15 IntroductiontoViscoelasticity,Creep, andCreepCrackGrowth 15.1INTRODUCTION Sofar,ourdiscussiononthemechanicalbehaviorhasconsideredonlytime- independentdeformation.However,themechanicalbehaviorofmaterials mayalsobetimedependent.Thiscangiverisetotime-dependentstrainsor crackgrowththatcanresult,ultimately,incomponentfailureordamage. Forseveralmaterialsdeformedattemperaturesaboveabout0.3–0.5oftheir meltingtemperatures,T m (inK),time-dependentdeformationcanoccurby creeporstressrelaxation.Thismayresultultimatelyinarangeoffailure mechanismsthatareillustratedschematicallyinFig.15.1.Incrystalline metals and their alloys, creep damage can occur by stress-assisted diffusion and/or dislocation motion. Microvoids may also form and coalesce by the same mechanisms during the final stages of creep deformation. Furthermore, creep damage mechanisms may occur at crack tips, giving rise ultimately to creep crack growth phenomena. Time-dependent creep deformation has also been observed in poly- meric materials by viscous flow processes. These can result in time-depen- dent elastic (viscoelastic) or time-dependent plastic (viscoplastic) processes. Such time-dependent flow can happen at temperatures above the so-called glass transition temperature, T g . Time-dependent deformation may also occur in crystalline materials. Depending on the crystal structure and tem- Copyright © 2003 Marcel Dekker, Inc. perature, these can give rise to stress-assisted movement of interstitials and vacancies, and an elastic deformation. This chapter presents an introduction to time-dependent deformation in crystalline and amorphous materials. Time-dependent deformation/creep of polymers is described along with the temperature dependence of defor- mation in polymers. Phenomenological approaches are then described for the characterization of the different stages of creep deformation. These are followed by an overview of the creep deformation mechanisms. The creep mechanisms are summarized in deformation maps before discussing some FIGURE 15.1 Creep is important in four classes of design: (a) displacement limited; (b) failure-limited; (c) relaxation limited; (d) buckling limited. [From Ashby and Jones (1996) with permission from Butterworth-Heinemann.] Copyright © 2003 Marcel Dekker, Inc. engineeringapproachesforcreepdesignandthepredictionofthecreeplives ofengineeringstructuresandcomponents.Finally,abriefintroductionto superplasticityisthenpresentedbeforeconcludingwithanintroductionto time-dependentfracturemechanicsandthemechanismsofcreepcrack growth. 15.2CREEPANDVISCOELASTICITYINPOLYMERS 15.2.1Introduction Ingeneral,time-dependentdeformationoccursinmaterialsattemperatures between0.3and0.5ofT m ,themeltingpoint(inK).Inthecaseofpolymeric materials,whichhaverelativelylowmeltingpoints,considerabletime- dependentdeformationhasbeenobserved,evenatroomtemperature. Theresultingdeformationinpolymersexhibitsmuchstrongerdependence ontemperatureandtime,whencomparedtothatinmetallicandceramic materials.ThisisduelargelytoVanderWaalsforcesthatexistbetween polymerschains(Fig.1.8).SincetheVanderWaalsforcesarerelatively weak,significanttime-dependentdeformationcanoccurbychain-sliding mechanisms.(Chap.1). 15.2.2MaxwellandVoigtModels Ingeneral,thetime-dependentdeformationofpolymerscanbedescribed intermsofcreepandstressrelaxation,Fig.15.1(a)and(c).Creepisthe time-dependentdeformationthatoccursunderconstantstressconditions, Fig.15.1(a),whilestressrelaxationisameasureofthestressresponse underconstantstrainconditions,Fig.15.1(c).Theunderlyingmechanics ofthetime-dependentresponseofpolymerswillbedescribedinthissec- tion. Time-independentdeformationandrelaxationinpolymerscanbe modeledusingvariouscombinationsofspringsanddashpotsarrangedin seriesand/orparallel.Time-independentelasticdeformationcanbemodeled solelybyspringsthatrespondinstantaneouslytoappliedstress,accordingto Hooke’slaw,Fig.15.2(a).Thisgivestheinitialelasticstress, o , as the product of Young’s modules, E , and the instantaneous elastic strain, " 1 , i.e., o ¼ E" o , where " o is the instantaneous/initial strain. Similarly, purely time dependent strain–time response can be described by the viscous response of a dashpot. This gives the dashpot time-dependent stress, d , as the product of the viscosity, , and the strain rate, d"=dt [Fig. 15.2(b), i.e., d ¼ d"=dt. Copyright © 2003 Marcel Dekker, Inc. The simplest of the spring–dashpot models are the so-called Maxwell and Voigt models, which are illustrated schematically in Fig. 15.3(a) and (b), respectively. 15.2.2.1 Maxwell Model The Maxwell model involves the arrangement of a spring and a dashpot series, Fig. 15.3(a). Under these conditions, the total strain in the system, ", is the sum of the strains in the spring, " 1 , and the strain in the dashpot, " 2 . This gives " ¼ " 1 þ " 2 ð15:1Þ FIGURE 15.2 Schematic illustration of (a) spring model and (b) dashpot model. [From Hertzberg (1996) with permission from John Wiley.] FIGURE 15.3 Schematic illustration of (a) Maxwell model and (b) Voigt model. [From Hertzberg (1996) with permission from John Wiley.] Copyright © 2003 Marcel Dekker, Inc. Since the spring and the dashpot are in series, the stresses are equal in the Maxwell model. Hence, 1 ¼ 2 ¼ . Taking the first derivative of strain with respect to time, we can show from Eq. (15.1) that d" dt ¼ d" 1 dt þ d" 2 dt ¼ 1 E d dt þ ð15:2Þ where " 1 ¼ =E and d" 2 =dt ¼ =. In general, the Maxwell model predicts an initial instantaneous elastic deformation, " 1 , followed by a linear time- dependent plastic deformation stage, " 2 , under constant stress conditions (Fig. 15.4). However, it is important to note that the strain–time response in most materials is not linear under constant stress ¼ o , i.e., creep conditions. Instead, most polymeric materials exhibit a strain rate response that increases with time. Nevertheless, the Maxwell model does provide a good model of stress relaxation, which occurs under conditions of constant strain, " ¼ " o , and strain rate, d"=dt ¼ 0. Applying these conditions to Eq. (15.2) gives 0 ¼ 1 E d dt þ ð15:3Þ Separating variables, rearranging Eq. (15.3), and integrating between the appropriate limits gives ð o d ¼ ð 1 0 À E dt ð15:4aÞ or ¼ 0 exp À Et ð15:4bÞ FIGURE 15.4 (a) Strain–time and (b) stress–time predictions for Maxwell and Voigt models. [From Meyers and Chawla (1998) with permission from Prentice Hall.] Copyright © 2003 Marcel Dekker, Inc. Equation(15.4b)showsthattheinitialstress, o ,decaysexponentially withtime(Fig.15.5).Thetimerequiredforthestresstorelaxtoastressof magnitude=e(eisexp1or2.718)isknownastherelaxationtime,.Thisis givenbytheratio,=E.Equation(15.4)may,therefore,beexpressedas ¼ 0 expÀ t ð15:5Þ Equation(15.5)suggeststhatstressrelaxationoccursindefinitelybyan exponentialdecayprocess.However,stressrelaxationdoesnotgooninde- finitelyinrealmaterials.Inadditiontopolymers,stressrelaxationcanoccur inceramicsandglassesatelevatedtemperature(Soboyejoetal.,2001)and inmetallicmaterials(Bakeretal.,2002).However,themechanismsofstress relaxationinmetalsaredifferentfromthoseinpolymers.Stressrelaxation inmetalsinvolvesthemovementofdefectssuchasvacanciesanddisloca- tions. 15.2.2.2VoigtModel ThesecondmodelillustratedinFig.15.3istheVoigtModel.Thisinvolves the arrangement of a spring and a dashpot in parallel, as shown schemati- cally in Fig. 15.3(b). For the spring and the dashpot in parallel, the strain in the spring and strain in the dashpot are the same, i.e., " ¼ " 1 ¼ " 2 . However, FIGURE 15.5 Effects of increasing molecular weight on the time-dependence of strain in a viscoelastic polymer. [From Meyers and Chawla (1999) with permission from Prentice Hall.] Copyright © 2003 Marcel Dekker, Inc. the total stress is the sum of the stress in the spring and the dashpot. This gives ¼ 1 þ 2 ð15:6Þ where 1 is the stress in the spring and 2 is the stress in the dashpot. Substituting the relationships for 1 and 2 into Eq. (15.6) now gives ¼ E" 1 þ d" 2 dt ð15:7aÞ Furthermore, since " ¼ " 1 þ " 2 , we can write: ¼ E" þ d" dt ð15:7bÞ Under constant stress (creep) conditions, d=dt ¼ 0. Hence, differentiating Eq. (15.7b) now gives d dt ¼ 0 ¼ E d" dt þ d 2 " dt ð15:8Þ Equation (15.8) can be solved by setting v ¼ d"=dt. This gives 0 ¼ Ev þ dv dt ð15:9aÞ or dv v ¼À E dt ð15:9bÞ Integrating Eq. (15.9) between the appropriate limits gives ð v v o dv v ¼ ð t 0 À E dt ð15:10aÞ or ln v v 0 ¼À Et ð15:10bÞ Taking exponentials of both sides of Eq. (15.10) gives v ¼ v 0 exp À Et ð15:11Þ However, substituting v ¼ d"=dt, v o ¼ d" o =dt ¼ =, and ¼ =E into Eq. (15.11) results in _ "" ¼ _ "" 0 exp À t ð15:12aÞ Copyright © 2003 Marcel Dekker, Inc. or d" dt ¼ d" 0 dt exp Àt ¼ exp Àt ð15:12bÞ SeparatingvariablesandintegratingEquation(15.12b)betweenlimitsgives ð " 0 d"¼ ð t 0 exp Àt dtð15:13aÞ or "¼À exp Àt À1 ! ð15:13bÞ Nowsince¼=E,wecansimplifyEq.(15.13b)togive "¼ E 1Àexp Àt ! ð15:13cÞ Thestrain–timedependenceassociatedwithEq.(15.13c)isillustrated schematicallyinFig.15.4.Ingeneral,thepredictionsfromtheVoigtmodel areconsistentwithexperimentalresults.Also,ast¼1the"! o =E. Furthermore,forthestressrelaxationcase,"=" o ¼constant.Hence, d"=dt¼0.Therefore,fromEq.(15.7)wehave ¼E" 0 ð15:14Þ Equation(15.14)givestheconstantstressresponseshownschematicallyin Fig.15.4(b).Itisimportanttonotethatthemolecularweightandstructure ofapolymercanstronglyaffectitstime-dependentresponse.Hence, increasingthemolecularweight(Fig.15.5)orthedegreeofcross-linking in the chain structure tends to increase the creep resistance. This is because increased molecular weight and cross-linking tend to increase the volume density of secondary bonds, and thus improve the creep resistance. Similarly, including side groups that provide structural hindrance (steric hindrance) to the sliding chains will also increase the creep resistance. However, in such polymeric systems, the two-component (Voigt or Maxwell) models do not provide adequate descriptions of the stress–time or strain–time responses. Instead, multicomponent spring and dashpot models are used to characterize the deformation response of such poly- meric systems. The challenge of the polymer scientist/engineer is to de- termine the appropriate combination of springs and dashpots that are needed to characterize the deformation response of complex polymeric structures. Copyright © 2003 Marcel Dekker, Inc. Example15.1 Anexampleofamorecomplexspring–dashpotmodelisthefour-element oneshowninFig.15.6.ThisconsistsofaMaxwellmodelinserieswitha Voigtmodel.Theoverallstrainexperiencedbythismodelisgivenbythe sumoftheMaxwellandVoigtstraincomponents.Thismay,therefore,be expressedas "¼ E 1 þ 1 tþ E 2 1Àexp Àt 2 ! ð15:15Þ Theresultingstrain–timeresponseassociatedwiththecombined modelispresentedinFig.15.7.Uponloading,thisshowstheinitialelastic response, " 1 , associated with the Maxwell spring element at time t ¼ 0. This FIGURE 15.6 Four-element model consisting of a Maxwell model in series with a Voigt model. [From Courtney (1990) with permission from McGraw- Hill.] Copyright © 2003 Marcel Dekker, Inc. is followed by the combined viscoelastic and viscoplastic deformation of the Maxwell and Voigt spring and dashpot elements. It is important to note here that the strain at long times is $ = 1 . This is so because the exponential term in Eq. (15.15) tends towards zero at long times. Hence, the strain–time function exhibits an almost linear relationship at long times. Upon unloading (Fig. 15.7), the elastic strain in the Maxwell spring is recovered instantaneously. This is followed by the recovery of the vis- coelastic Voigt dashpot strain components. However, the strains in the Maxwell dashpot are not recovered. Since the strains in the Maxwell dash- pot are permanent in nature, they are characterized as viscoplastic strains. 15.3 MECHANICAL DAMPING Under cyclic loading conditions, the strain can lag behind the stress. This gives rise to mechanical hysterisis. The resulting time-dependent elastic behavior is associated with mechanical damping of vibrations. Such damp- ing phenomena may cause induced vibrations, due to applied stress pulses, to die out quickly. The hysterisis associated with the cyclic deformation of a FIGURE 15.7 Strain–time response of combined Maxwell and Voigt model (four-element model). [From Courtney (1990) with permission from McGraw-Hill.] Copyright © 2003 Marcel Dekker, Inc. [...]... constant Equation (15.17a) has been shown to apply to a wide range of materials since the pioneering work by Andrade (1911) almost a century ago The extent of primary creep deformation is generally of interest in the design of the hot sections of aeroengines and land-based gas turbines However, the portion of the creep curve that is generally of greatest engineering interest is the secondary creep regime... examined the creep behavior of ODS alloys, Mishra et al (1993) have used the concept of a threshold stress in the analysis of creep data obtained for a number of materials Their analysis also separates out the effects of interparticle spacing from the threshold stress effects Finally in this section, it is important to note that the threshold stress in several aluminum-base particle-hardened systems... initial choice of such a solid, the internal structure of the material can be designed to provide significant resistance to creep deformation In the case of designing against power law creep in metals and ceramics, there are two primary considerations First, the materials of choice are those that resist dislocation motion Hence, materials that contain obstacles to dislocation motion are generally of interest... in materials subjected to low stresses and elevated temperature Material design against creep in such materials may be accomplished by: (1) heat treatments that increase the grain size, (2) the use of grain boundary precipitates to resist grain boundary sliding, and (3) the choice of materials with lower diffusion coefficients Diffusional creep considerations are particularly important in the design of. .. , as a function of the inverse of the absolute temperature 1=T (in K) A typical plot is shown in Fig 15 .16 From Eq (15.18), the negative slope of the line corresponds to ÀQ=R Hence, the activation energy, Q, can be determined by multiplying the slope by the universal gas constant, R Copyright © 2003 Marcel Dekker, Inc FIGURE 15 .16 Determination of the activation energy from a plot of secondary creep... be guaranteed without testing for extended periods of time, which is often impractical Nevertheless, it is common to use measured creep data to estimate the service creep lives of several engineering components and structures One approach involves the use of experimental results of creep lives obtained at different stresses and temperatures An example of such results is presented in Fig 15.26 This shows... tests, then the left-hand side of Eq (15.27) must be equal to a constant for a given creep mechanism This constant is known as the Larsen–Miller parameter (LMP) It is a measure of the creep resistance of a material, and is often expressed as LMP ¼ T ðlog tf þ C Þ ð15:28Þ where log tf is the log to the base 10 of tf in hours, and C is a constant determined by the analysis of experimental data The Larsen–Miller... dependent on molecular weight and the extent of cross-linking of the polymer chains This is illustrated in Fig 15.12, which shows the time dependence of the relaxation modulus This can be exploited in the management of residual stresses in engineering structures and components that are fabricated from polymeric materials 15.5 INTRODUCTION TO CREEP IN METALLIC AND CERAMIC MATERIALS The above discussion has focused... FIGURE 15.28 Melting or softening temperatures for different solids [From Ashby and Jones (1996) with permission from Butterworth-Heinemann.] strengthened materials However, the precipitates or second phases must be stable at elevated temperature to be effective creep strengtheners The second approach to creep strengthening of power law creeping materials involves the selection of materials with high lattice... creep and viscoelasticity in polymeric materials We will now turn our attention to the mechanisms and phenomenology of creep in metallic and ceramic materials with crystalline and noncrystalline structures Creep in such materials is often studied by applying a constant load to a specimen that is heated in a furnace When constant stresses are required, instead of constant load, then the weight can be . Inc. 15.6FUNCTIONALFORMSINTHEDIFFERENTCREEP REGIMES AsdiscussedinSec.15.5,thefunctionalformsofthestrain–timerelation- shipsaredifferentintheprimary,secondary,andtertiaryregimes.Theseare generallycharacterizedbyempiricalormechanism-basedexpressions.This sectionwillfocusonthemechanism-basedmathematicalrelationships. Intheprimarycreepregime,thedependenceofcreepstrain,",ontime, t,hasbeenshownbyAndrade(1911)tobegivenby "¼t 1=3 ð15:17aÞ whereisaconstant.Equation(15.17a)hasbeenshowntoapplytoawide rangeofmaterialssincethepioneeringworkbyAndrade(1911)almosta centuryago.Theextentofprimarycreepdeformationisgenerallyofinterest inthedesignofthehotsectionsofaeroenginesandland-basedgasturbines. However,theportionofthecreepcurvethatisgenerallyofgreatestengi- neeringinterestisthesecondarycreepregime. Inthesecondarycreepregime,theoverallcreepstrain,",maybe expressedmathematicallyas "¼" o þ"½1ÀexpðÀmtÞþ _ "" ss tð15:17bÞ wherethe" o istheinstantaneouselasticstrain,"½1ÀexpðÀmtÞisthepri- marycreepterm, _ "" ss trepresentsthesecondarycreepstraincomponent,mis theexponentialparameterthatcharacterizesthedecayinstrainrateinthe primarycreepregime,"isthepeakstrainintheprimarycreepregime,andt correspondstotime.Similarly,wemayexpressthesecondary/steadystate creepstrainrate, _ "" ss ,as _ "" ss ¼A n exp ÀQ RT ð15:18aÞ whereAisaconstant,istheappliedmeanstress,nisthecreepexponent,Q istheactivationenergy,Ristheuniversalgasconstant(8.317J/molK),and Tistheabsolutetemperatureinkelvins.TakinglogarithmsofEq.(15.18) gives log _ "" ss ¼logAþnlogÀ Q RT ð15:18bÞ Equation(15.18)canbeusedtoextractsomeimportantcreeppara- meters.First,ifweconductconstantstresstestsatthesametemperature, thenallofthetermsinEq.(15.18)remainconstant,exceptfor.Wemay nowplotthemeasuredsecondarystrainrates, _ "" ss ,asafunctionofthe appliedmeanstress,.AtypicalplotispresentedinFig.15.15.This Copyright. Inc. Finally,itisimportanttonotethatthetemperaturedependenceofthe relaxationmodulusishighlydependentonmolecularweightandtheextent ofcross-linkingofthepolymerchains.ThisisillustratedinFig.15.12,which showsthetimedependenceoftherelaxationmodulus.Thiscanbeexploited inthemanagementofresidualstressesinengineeringstructuresandcom- ponentsthatarefabricatedfrompolymericmaterials. 15.5INTRODUCTIONTOCREEPINMETALLICAND CERAMICMATERIALS Theabovediscussionhasfocusedlargelyoncreepandviscoelasticityin polymericmaterials.Wewillnowturnourattentiontothemechanisms andphenomenologyofcreepinmetallicandceramicmaterialswithcrystal- lineandnoncrystallinestructures.Creepinsuchmaterialsisoftenstudied byapplyingaconstantloadtoaspecimenthatisheatedinafurnace.When constantstressesarerequired,insteadofconstantload,thentheweightcan beimmersedinafluidthatdecreasestheeffectiveloadwithincreasing lengthinawaythatmaintainsaconstantappliedtruestress(Andrade, 1911).ThisisachievedbyasimpleuseoftheArchimedesprinciple. Similarly,aconstantstressmaybeappliedtothespeicmenusingavariable leverarmthatappliesaforcethatisdependentonspecimenlength(Fig. 15.13).Inanycase,thestrain–timeresultsobtainedfromconstantstressor constantloadtestsarequalitativelysimilar.However,theprecisionofcon- stantstresstestsmaybedesirableincarefullycontrolledexperiments. Upontheapplicationofaload,theinstantaneousdeformationis elasticinnatureattimet¼0.Thisisfollowedbyathree-stagedeformation process.Thethreestages(I,II,andIII)arecharacterizedastheprimary (stageI),secondary(stageII),andtertiary(stageIII)creepregimes.These areshownschematicallyinFig.15.14(a).Itisalsoimportanttonotethatthe magnitude. 15 IntroductiontoViscoelasticity,Creep, andCreepCrackGrowth 15.1INTRODUCTION Sofar,ourdiscussiononthemechanicalbehaviorhasconsideredonlytime- independentdeformation.However,themechanicalbehaviorofmaterials mayalsobetimedependent.Thiscangiverisetotime-dependentstrainsor crackgrowththatcanresult,ultimately,incomponentfailureordamage. Forseveralmaterialsdeformedattemperaturesaboveabout0.3–0.5oftheir meltingtemperatures,T m (inK),time-dependentdeformationcanoccurby creeporstressrelaxation.Thismayresultultimatelyinarangeoffailure mechanismsthatareillustratedschematicallyinFig.15.1.Incrystalline metals