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# Class Note for ME 350 with Professor Wei at UA-Static Machine Components (2)

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Date Created: 02/06/15

Timedependent deformation Viscoelasticity and Viscoplasticity Static Machine Component ME35O March 2 2009 Viscoelasticity Elements of polymer structure Linear vs branched Vinyl polymers and substitutions Packing of polymer chains oRandomamorphous Glass transition temperature TG Semicrystalline Crystalline volume fraction melting temperature Amorphous TG Elements of linear Viscoelasticity Creep and relaxation oAnalogue models Idealized Linear Elastic Response Linear elasticity 075 2 E 675 N0 Creep N0 Relaxation INPUT um Gm A 8t A Go 80 t t OUTPUT OUTPUT Em A E 00 A E 00 E E 80 t Note units of E are stress Idealized Linear Viscous Response d6 15 Linear VISCOSItyZ 075 77 d E n t t Creep Relaxation quotm quot39 INPUT ot A 8t A GO E0 gt gt t J t J OUTPUT OUTPUT n 8t ca gt I t Note units of n are stressxtime lg oi g39sna fg g ydlg gfgj njrm eg seCXMPa an infinitesimal time interval and then like the strainrate goes to zero Maxwell Model an Idealized Linear Viscoelastic Response Creep INPUT Gm 575 6elastilt75 viscoust GO 05 Elelastic 39l39 viscoust o39 lttEalttgtn gt J OUTPUT t n oFor times near the finite stress jump all strain occurs in the elastic element E During the hold period all strain occurs in the viscous element Maxwell Model an Idealized Linear Viscoelastic Response t 6eiastict viscoust 639 EeiasticUE viscoust tE 00977 a For times gt o d etdt0 so During the hold period elastic strain is traded for viscous t J strain and stress drops 1 OUTPUT d0 E Gt 0 tgt dt n0 Ego E 0t 0t 0 exp tnE 60 X E exp t77E Relaxation INPUT Characteristic relaxation time InE Real Polymer Relaxation an Idealization Testing of real polymers under relaxation 8quot can be used to extract a timedependent relaxation modulus Et t Shortterm response Erg cm A Longterm response Ere Ext 2 aweo E t Ert gt 0 E Erg glassy Erm Ert gt 00 E E70e equilibrium E0Eg Ere NOTE provided eO is sufficiently small t typically less than 001 the relaxation modulus Ert is approximately independent of so Real Polymer Creep an Idealization Testing of real polymers under suddenly applied constant stress can be used to cit A extract a timedependent creep function JCt INPUT so Shortterm response ch Longterm response J3e a J E 00 OUTPUT a C Jct gt 0 L E ch glassy Jc050choo J00 gt 00 3 J08 equilibrium J4 Note units of Jct 1 stress NOTE provided et remains sufficiently small typically less than 001 the creep t function JCt is approximately independent of 60 Relaxation Modulus Ert and Creep Function JCt Inverse Functions of Time Function Dimensions Trend Jct lstress starts small grows with time ETt stress starts large decays with time QUESTION Are these inverse functions ls JCt Ert 39 1 for all times oANSWERS In general they are not precise inverses However Equilibrium and glassy Jct gt 0 lErt e 0 values are nearly inverse oFor intermediate times t Jct gt 00 1E7t gt 00 but the error in assuming that they are inverse is typically only Jag X Era 1 a few per cent at most Linearity of Response an Idealization If creep response to stress jump 00 is t JCt X 00 then the creep response to stress jump qb x 00 where gs is a proportionality constant is 6t Jct gtlt 00 NOTE similar linear scaling of stress relaxation response applies Superposition of Loading Suppose that the stress history input consists of a sequence of stress jumps Aci applied at successive times t with t00 6t A A62 AG THEN the resulting strain history is given by 6 A0390 J0 l39 A01 J00 tl A0392 J0 t2 2 AUjJCt jo 8tA Special Case LoadUnload INPUT AGO V Correspondence Principle Suppose that a given load P produces displacement vector ux in a linear elastic body having Young s modulus E The displacement vector depends on the position vector xxexyeyzez The magnitudes of the displacement and strain components are proportional to P and inversely proportional to E EXAMPLE Threepoint P Midspan bending PL3 A v 1 L 2 Y Correspondence Principle Now suppose that a given load jump Pt is applied to a geometrically identical linear viscoelastic body having creep function JCt A stress components in the body are timeindependent and spatially vary precisely as they do in an identical linear elastic body subject to the same load The loading produces timedependent displacement vector uxt and corresponding strain components The magnitudes of the displacement and strain components are proportional to both P and JCt EXAMPLE Threepoint p Midspan bending At v1 L2t LI For suddenlyapplied load L I replace 1E with JCt in an elastic solution Correspondence Principle Suppose that a given displacement jump At is applied to a geometrically identical linear viscoelastic body having stress relaxation modulus Ert A displacement and strain components everywhere in the body are timeindependent and precisely equal those in an identical linear elastic body subject to the same applied displacement These boundary conditions produce timedependent loads Pt and stresses The magnitudes of the timedependent load and of the stress components are proportional to both A and to Ert EXAMPLE Threepoint P Midspan bending 48 Pt gtltAgtltErt fhY For suddenlyapplied displacement l L A l replace E with Ert in an elastic solution RATE DEPENDENCE AND RATE INDEPENDENCE INPUT MATERIAL 6 Elastic IE Viscous 6 E ElasticPlastic 6 o Plastic deformation in metals OF RESPONSE STRESSSTRAIN A Rateindependent t Highly ratedependent f quot t Slightly rate A dependent fig l A is thermally activated and inherently rate dependent Creep strain at different time Priertary s d Cr p ecnn afif Stram II Steady State I i Creep I 39 I Tertiary Creep initiai Elastic Strain TI m E During loading under a constant stress the strain often varies as a function of time in the manner shown above Combined stress and temperature dependence of the steadystate creep rate 292 g d fAeX GE 6 608 60 p RT Here and henceforth we drop the subscript 33 on the steady state creep rate 3 pre exponential factor 3 1 creep activation energy Jmol creep exponent reference stress which produces a creep strain rate 6390 0 3lt Dgt Summary of onedimensional creep equations 6 eeI p 66 5 EET E m eyequot 0 Note that the rate of creep or viscoplastic strain increases exponentially with temperature or that the time for a given amount of creep strain decreases exponentially with temperature Example problem stress relaxation 0 Consider a bolt with pre tension a 07 at time t 0 Given that the bolt is maintained at constant temperature determine the the stress in the bolt at some later time t The isothermal constitutive equation for steady state creep is given by p 2 BC with n 7E 1 W at Solution 639 p gtO p sinceezconst in the bolt 39 1d gt0 3Ba gt 0 Ban E Edt 0nd0 EBdt ll 0 t 0nd0 2 EB dt 0139 O U 1 1 001 1 091 1 n 1EBt Schematic of stressrelaxation with time A oE creep stroin elos ric stroin t This shows how the initial elastic strain O39iE is slowly replaced by creep strain and the stress relaxes in the bolt 0 Defining the relaxation time tr such that 00 O39i2 we get t 201 1 1 T n 1EBa 1 High homologous temperatures TgtO3Tm At high temperatures the following flow function is widely used to model creep phenomena 1 Ep 2 A exp m RT 3 Here A pre exponential factor Q m Qlattselfdiff m constant Activation energy R Gas constant m rate sensitivity parameter This form for the flow function is called powerlaw creep At high homologous temperatures m has a reasonably high value m m 02 Note The strain rate sensitivity parameter m 172 where n is the creep exponent Typical deformation map in metals TEMPERATURE I C m mm vmlwm LI39III L Ham um i39 39 ia w ma39 d 39 D l l l a 1 aIa h 39539 FLA STICITY WNAHquot I E nmmawumsnm E g 39 1 E SE 2 5 m7 I D 393 R E I m m D vi 0 3 w IO E 395 U a n E 5 O I I 2 m g m if In 71 w I15I n 02 cu ms 9 a LG HOMOLOGDUS TEMPERATURE 54 Pure nickel of grain size 1 pm workhardened Nickelbased Superalloys an example of a creepresistant material 1Dquot max acceptable 7 5 Cree l f l u g 10 39 as lew mm blade r G elongation in n lCIDD hr reep Strain 1 Rate is WW 1 311 mquot 2 5 a max stress level m m a turbine blade 01 7 1 10 mt Stress MPa As a general rule creep starts to become significant when the homologous temperature is greater than 04 Microstructure of Superalloys in Turbine Blades Incmnlng Roslsuncem Gmp natannauon

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