MSE 450 MSE 450
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This 20 page Class Notes was uploaded by Mrs. Jazmyne Kulas on Thursday October 15, 2015. The Class Notes belongs to MSE 450 at North Carolina State University taught by Staff in Fall. Since its upload, it has received 46 views. For similar materials see /class/223749/mse-450-north-carolina-state-university in Materials Science Engineering at North Carolina State University.
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Date Created: 10/15/15
Creep and Stress Rupture Ch 13 3 4 5 6 7 8 91112 13 15optional De nition of Creep and Creep Curve 133 def Creep is the timedependent plastic strain at constant stress and temperature Creep curve Fig 134 steadystate creeprate e S or simply e Temperature and Stress Dependencies Fig 136 Fig 138 total creep curve e 80 8p as 80 instantaneous strain at loading elastic anelastic and plastic as steadystate creep strain constantrate viscous creep 839 St 8p primary or transient creep AndradeB ow or 13 rd law Btl3 primary or transient creep AndradeB ow or 13 rd law 8p Btl3 ltgt problem as taO Garofalo Dorn Equation 8p at 1 e39rt r is related to 120 e S Dorn gt Both primary and steadystate follow similar kinetics temperature compensated time 6 t e39 QCRT single universal curve with t replaced by 6 or est Or creep strain 8 80 at 1 e39g St 5 St ltgt see SherbyDorn Al Murty Zr SherbyDorn 6parameter l 1 N 0 531 K 1 I U A m 1 7721 quot 55 o 5 ame K 4 a 7 s aw H It 0 i D 42 4 1 3rd STAGE I er hCry creepl il39 39w9m quot o 4 r W W W W 39 39 39 W W e I g g I 39 E 1quot I ff Z 0 3 p 1 quotquotquotquot E 1quot 2 5 134 71 02 I 7 t 155545 w o39 f39fl O f L a T l M E v h 0 U 1 5 smite93 Creep curves for A1 at Sherby amp Dorn 1956 A single curve demonstrating the 3000 psi and at three different temperatures validity of Gparameter KL Murty MSE 450 page 1 397 5 g amm f sxf F H39nf fdrrf f i 1 I 337594 Eh r 59quot J gquot quot r 7 r 3quot 39 ast Creep data in Zircaloy at varied temperatures F and stresses ksi fall into a single curve demonstrating the validity of Dom equation Murty et al 1976 Figs 1317 1318 0 SherbyDorn Parameter I PLM Z I PSD t CQRT l K L Murty MS Thesis 1967 0 ZenerHolloman Z eQRT Stress Rupture Test 134 6 vs tr 0 Representation of engineering creep rupture data 1312 1313 TlogtC Fig 13 19 21 LarsonMiller Parameter PMH Z log t log ta TTa MansonHaferd Parameter these parameters are for a given stress and are functions of 6 Fig 1320 MonkmanGrant str K F Eq 1324 KL Murty Demonstration of MonkmanGrant Relationship in Cu Feltham and Meakin 1959 page 2 MSE 450 Creep Under Multiaxial Loading text 1 4 1 4 Use LevyMises Equations in plasticity Geff L 61622 62632 63602 x dee Geff since creep is plastic deformation 12 appears as in plasticity Similarly dez and d83 Dividing by dt get the corresponding creeprates eff Geff One first determines the uniaxial creeprate equation gs A on eQRT 1 and del 01 5 0263 1 31 01 5 6263 etc n and assume the same for effective strainrate 8 eff A Geff e39QRT l 1 so that 11 A GEE e39QRT 51 5 6263 etc Stress Relaxation As noted in section 811 the stress relaxation occurs when the deformation is held constant such as in bolt in ange where the constraint is that the total length of the system is xed 9 at 8E acreep const Here 8E E d8 ldo do Thus EOEa s Or a E SEAon fixedT Integration from o to t gives 0f do 5 Data 39um HW 878 t EAfdt EAt 0 stress MPa 0i G m 0r 0t 00 1 AEn 1o g 1t quot 1 KL Murty MSE 450 page 3 0 Deformation Creep Mechanisms Introduction structural changes 135 Slip difficult to observe slip lines folds etc are usually noted Subgrains GBS excess deformation induced vacancies Two important relationships 39 62 Orowan equatlon pbv and Taylor equatlon M Thermally Activated Dislocation Glide at low T andor high strainrates 9 A eBG eQimT where Q is the activation energy for the underlying mechanisms Peierls mechanism bcc metals Intersection mechanism fee and hcp metals Dislocation creep lattice diffusion controlled glide and climb Diffusion creep viscous creep mechanisms mainly due to point defects at low stresses and high temperatures GrainBoundary Sliding GBS intermediate stresses in small grained materials and ceramics where matrix deformation is difficult Many different mechanisms may contribute and the total strainrate parallel mechanism series mechanisms fastest controls dominates slower controls dominates 71 l i l E 7 Z 3 i E 1 El Uncrept specimen Crept at 5500 psi to 215 strain KL Murty MS thesis Cornell University 1967 KL Murry MSE 450 page 4 Dislocation Creep Pure Metals ClassM alloys Experiments 5 A SH e39QCRT n z 5 Q z QL QD edge J glide climb model WeertmanClimb model Weertman PillBox Model sequential processes L average distance a dislocation glides tg time for glide motion h average distance a dislocation climbs to time for climb J L J ih FR4 L gt LomerCottrell Barrier A7 strain during glideclimb event AYg AVG z AYg p b L N L t t1me of ghdechmb event tg to to VC vc chmb ve1001ty Al L t hVC PbhVC where vc oc ACV e39EmkT Em activation energy for vacancy migration Here ACV C C39V C3 eGVkT C3 eGVkT C3 2 Sinh L L 839 ocpb Hvcoc pb H C3 e39EmkTZ Sinh At low stresses Sinh 2 so that Garofalo Eqn L o V 8A1pb CV eEmkTE T L v L eAlpbgDLE TzA2pog DL Or 8 A03D ltgt natural creeplaw L 15 45 Weertman H OCG EZAG D as experimentally observed in Al In general 839 AT 6n Powerlaw n is the stress exponent fxal structure F 8 A D SinhBGn I Powerlaw breakdown Power law I creep l l a l HarperDorn 5 creep 5 l 1 4 I I 102 D I 5 N r r r r I I 39 4 m also known as Norton s Equation 11 is Norton index At high stresses 6 Z 10393 E Sinhx ex 8 AH e136 D Powerlaw breakdown KL Murty MSE 450 page 5 huh If 3 at Jimquot ably j ey 39f quot Mina Fig 313 Dieter Sherby What happens if we keep decreasing the stress say to a level at and below the TFR As O is decreased gt reach a point when G S GFR dislocation density would become constant independent of G 3 oc G Viscous creep known as HarperDom creep HarperDom creep occurs at G E ginpg 2105 90106ch In D 2 HD creep is observed in large grained 1 materials metals ceramics etc 9 HD 2 AHD DLG lnG Characteristics of Climb Creep ClassA4 0 large primary creep regions 1 subgrain formation 5 oc E dislocation density oc 62 0 independent of grain size KL Murty MSE 450 page 6 0 Effects of Alloying classA Solidsolution decreases rate of glide r glide controlled creep although annihilation due to climb still occurs microcreep viscous glide creep Viscous glide controlled creep decreased creeprates e g Ag DS 63 Ds is solute diffusion A1 5 Al3Mg 3 classA little or no primary creep no subgrain formation a p 0c 62 grainsize independent logIISUBJiILI bej 10gstress At low stresses for large grain sizes HarperDorn creep dominates what happens as grain size becomes small lt As grainsize decreases and at low stresses diffusion creep due to point defects becomes important due to migration of vacancies from tensile boundaries to compressive boundaries G NabarroHerrmg Creep d1ffus10n through the latt1ce NH ANH DL g G Coble Creep d1ffus10n through gramboundaries cO AcO Db g NabarroHerring Creep vs Coble Creep Coble creep for small grain sizes and at E C ble low temperature g 1 2 NH creep for larger gram sizes and at 39a 2 NH high temperatures a HarperDom no 1 at very large gram s1zes HarperDorn 9 creep dominates log grainsize At small grainsizes GBS dominates at intermediate stresses and temperatures 2 l G 39 3 GBS AGES Db E ltgt superplastzczty KL Murty MSE 450 page 7 Effect of dispersoids Dispersion Strengthening Precipitate Hardening recall Orowan Bowing at high temperatures climb of dislocation loops around the precipitates controls creep ppt Appt D 68 39 20 IRules for Increasing Creep Resistancel Formability Improvement Large Grain Size directionally solidi ed superalloys I Small stable Equiaxed Grain Size superplasticity Low Stacking Fault Energy Cu vs CuAl alloys 39 Strengthen Matrix ie increase GBS ceramics Solid Solution Alloying Al vs AlMg alloys 39 Stoichiometry especially Ceramics Dispersion Strengthening Ni vs TDNi KL Murty MSE 450 page 8 71 Summary of Creep Mechanisms at NH 2 Coble HD 3 GBS as e g n Dorn Equation A g DE b E Mechanism D n A Climb of edge dislocations DL 5 6X107 Pure Metals and classM alloys n function of Xal structure amp F Lowtemperature climb D J 7 2XlO8 Viscous glide ClassI alloys microcreep DS 3 6 b NabarroHerrlng DL 1 14 a 2 b Coble Db l 100 a 3 HarperDom DL 1 3XlO3910 b GBS superplastlclty Db 2 200 a 2 DL lattice diffusivity DS solute diffusivity D L core diffusivity Db GrainBoundary Diffusivity b Burgers vector d grain size Gb 52 5 subgrain size 10 T and p W where G is the shear modulus gtkn increases with decreasing F stackingfault energy KL Murty MSE 450 page 9 Deformation Mechanism Maps Visual picture of the domains 6 T Where various mechanisms dominate Theoretical strength TEMPERATURE C Dislocation glide 10 1 910quot A PLASTiCiTV E Dislocation 339 gt 4 creep 0G 10 E E 17gt 10 3 x O a o lt M 1 Coble creep U 4 10 en 10 5 a m DRAIN PIPE NabarroHerring 3 i E creep ltI HOT WATER m g PiPE II 0 10 lt 2 Lu 5 10 l l I O 02 04 06 08 10 6 lb 9 Homologous temperature T TM 100 HOMOLOGOUS TEMPERATUR i AshbyMap E Aquot DE 39 L STRENGTH 9 0511 NORMALISED SHEAR STRESS SHEAR STRESS AT 300K MNm2 Lead pipes on a 75yearold building in southern England The creepinduced curvature of these pipes is typical of Victorian lead water piping Frost and Ashby 02 01 quot 05 06 10 HOMOLOGOUS TEMPERATURE VTquot Other examples W filament light bulbs turbind blade Nibased alloy DS by Ni3TiAl KL Murty MSE 450 page 10 WEER TMAN PILL BOX MODEL Pure Metals Glide faster Climbcontrolled creep n25 39 l d C 1 is l Alloys Glide slower Glidecontrolled ereep n23 13 an my 22 3 5 39cz gm stress inlermlma guess ag ngll 11835 lfmgm i fxsmusglllle i 9r l 9mm g 4kaan A l lt l l 5 l saw lm l limexpl d M E l l l l g sawmill names 3 M 1 aquot 525 w w aw Mom Mm M lrmsxlaen J T mlanl I l l mm g WW all m 3 a3 a l Wm 214545333 u l a l w l 2 llgl y lasemALA liiastrala m sf l l6 leg 3quot millilme mile i s a versus quot sfquot v 1 Tm smallish Solid Solution Alloys will was and Dem 285 m WWqu may em mm A2 Mg laml at al 5332 l ms 3 Ml W4 0 363 mi Creep Transitions for Alloy Class 10395 10 T lnE 3 Murty and Turlik 1992 10 KL Murty MSE 450 page 11 KL Murty MSE 450 Fracture 7 1814 1 4 Brittle vs Ductile cgt Relative terms ductile acture implies appreciable plastic deformation prior to fracture in general bcc and hcp metals exhibit brittle fracture while fcc are ductile some fcc metals can be so ductile point fracture akin to superplastic materials Fracture in tensile testing Fig7l and fracture characterization strain to fracture ductile vs brittle mode crystallographic shear vs cleavage appearance of fracture surface fibrous vs granular Figs 7789 dimpled vs faceted Fracture stress 6f Cohesive strength om similar to theoretical yield strength except here in tension h 39 N E see text for derivation 6108 Slve E Eq 75 In a perfect brittle material all the elastic strain energy is expended in creating the cohesive N Y two new surfaces omax S a0 is interatomic spacing Eq 78 a0 Stresses in a Cracked Body presence of cracks reduces 6f due to oconcentration at the crack tip recall MAT 201 see section 215 de ne c half crack length interior crack crack length of surface crack Fig39 7394 p radius 0f curvature so that C C cmw2126 A P here 6 is the applied nominal stress X cohesive Fracture occurs when om Gmax c 2 C c Eys Or 6 at fracture 6f 2 5f P a0 With p a0sharpest crack possible EYS OF I page 1 KL Murty MSE 450 Griffith Theory of Brittle Fracture 74 A crack will propagate when the decrease in elastic strain energy is at least equal to the energy required to create new crack surface Increase in surface energy decrease in elastic strain energy expended in fracture A dAU ZEYs OrUEUS U j dc OrGf We stress as c increases of decreases I ZEYS for thicker plate or plane strain case 0f 1V2m note these are valid for completely brittle material with no plastic deformation Eq 715 for a thin plate plane amp Vs may be altered by environment corrosion etc If there is some plastic deformation Griffith Model needs to be modified Orowan suggested 71 75 7p p for plastic work required to extend the crack 2E 7 7 2E7 so that Of N since lpgtgtls Read 76 and 77 Metallographic Aspects of Fracture SEM Fractography In general cracks are formed during plastic flow dislocation pileups Fig 710 lead to cracks fracture involves 3 stages a plastic deformation to produce cracks b crack initiation nucleation c crack propagation 3 Types of Fracture Cleavage g Quasicleavage DimpledRupture brittle ductile flat facets dimples tear sides dimples around inclusions around facets microvoid coalescence Fig 77 Fig 78 Fig 79 page 2 KL Murty MSE 450 Effects of Notch amp Crystallography 0n Fracture Notch effects 710 p 268 increased tendency for brittle fracture produce high local stresses introduce triaXial tensile stress state produce high local strain hardening and cracking Temperature Effects depend on crystal structure all exhibit ductile fracture at HT while bcc metals exhibit DBTT effect of T on of vs 6y bcc metals show temperature sensitive of 6y at low Ts If LEE 739 ittic quot LL J How to determine fracture energy amp DBTT Charpy and Izod impact tests recall MAT 201 CV vs T Fig 143 CV vs T for bcc vs fcc amp hcp metals 139 3939quot39 39 quot quot 39 9 far j 7Z Mammy 7 i lnstrumented Charpy Testing fracture initiation propagation page 3 KL Murty MSE 450 Fracture Mechanics Ch 11 Recall Griffith Criterion for brittle fracture 2 ZEY 6f S a is half cracklength 15a from competition between the decrease in elastic strain energy and increase in surface 1c 262 62 energy U61 a per un1t plate thickness E X volume 2Ttaz and Us 2 2a Vs dU 2 U U61 Us amp 5 hi 0 conditlon for stable crack growth This implies that at a given 6 if a af cracks start propagating as long as the load or stress is applied ie what stable means 2EYSYE N 2EYE N EYR Cf Orowan modification of Tca Tca a plane stress VS plane strain a factor of lV2 in the denominator Unfortunately 7p is not a measurable quantity 2 Irwin s Energy Balance Approach Fracture occurs at a 6 corresponding to a Critical Crack Extension Force g also known as Strain Energy Release Rate BUB nacZ where g a 2E 39 2 appears s1nce crack extends both Sldes for a central crack a When g gc crack extension occurs E QC I c P a1 a2 gt a1 39 5 Thus of 2 compare With 1 Griffith criterion gc 2757p N 2lp A Can determine G as in Fig lll 5 singleedgenotch specimen Fig 111 P M 8 or 8 C P where M is elastic stiffness and C is elastic compliance P ax 9C Measure Pmax at which unstable crack starts 2 Ge 2 a a Eq 118 page 4 KL Murty MSE 450 Crack Tip Stresses 113 Crack Deformation Modes Fig 113 Mode I Opening mode Mode II Sliding mode Mode III Tearing mode Stresses around the crack tip Mode I Eq 119 Fig 112 using elasticity a a a Plane stress ox o E f19 6y o E f29 Txy o E f39 these stresses are valid near the crack tip for a gt r gt p note straight ahead of the crack tip 90 ox 6y o g and Txy 0 note as r gt 0 6x and CY gt 00 implies there exists a zone where these elastic stress eld equations are not valid indeed at r rc where ox 6y 50 yield stress yielding occurs and elasticity does not hold plastic or process zone close to the crack tip Irwin defines stress intensity factor P unitsofK I B P 0 net section stress WaB here model gt K1 6 Tca Now can rewrite the stress fields in Eq 119 in terms of K K is a convenient way of describing the stress distribution around a flaw if two flaws of different geometry have the same values of K then the stress fields around them are identical In general K Y o Tca where Y is a geometry factor or in the text W 15a Y 1 for a crack in an infinite size body amp Y E ta W Eq 1113 can find equations for Y in ASTM standards for various geometries page 5 KL Murty MSE 450 Now we can relate K to g energy release rate crack extension force n 62 recall g E and since K Tca 2 K2 GB for plane stress Eq 1114 2 2 LE amp K 1V2 plane strain Eq 1115 Fracture Criterion basic concept K zone vs Process or plastic zone near the cracktip plastic flow microvoids since cannot get Sup gt oo due to plastic Process one deformation at o 50 gt process zone rp occurs when K G OIGZ I ZG X y 0 2 zone orr p 271173 so Kzone is the zone around the craktip where 6 s are scaled by K If Kzone gtgt processzone ie rp is small failure or bond rupture in process zone will be determined by stresses in Kzone ie by K or LEFM 2 fracture criterion is K K1c 2 critical fracture toughness KR is a material parameter depends on T and strainrate but not on geometry for large thickness Plane Strain Critical Fracture Toughness Fig 117 That means in terms of fracture stress 6f K1c Y Gf39 T5a Fracture Design Basis for a given load or stress 6 in a material with a crack or flaw of size a K Y o Tca and ifK lt K1c no failure Or can find ac for given K1c and 6 so that for a lt ac no failure would occur if ac gt B thickness then have the condition of Leak Before Crack implying that the crack will pass through the wall without catastrophic brittle fracture if ac lt B get brittle fracture lt avoid bad design page 6 KL Murty MSE 450 Plastic Zone Size The above approach using Kfields is not valid for ductile materials where processzone is large since rp corresponds to where 6y so i see g K atr 6 60 p y V2an 1 K Or rp 6 0 2 Modified stress distribution elastic plastic Note as oo rp U approach LEFM 7 39 Otherwise we need to use EPFM 9 or may use plasticity corrections Fig IHO Before we discuss plasticity corrections consider how to determine KIC 115 using CT 3point bend or notched round specimen Plane Strain vs Plane Stress ee Thick Plate Thin Sheet LK2 LK2 rquot 6it 62 lap 271363 see View Graph Relation between 60 and KC M 0433 M W m Plasticity Corrections 1 Irwin replace a by ae a rp so that Keg o xix aeff o it arp rp fKoO K appears both sides gt solve by iterative process itK2 2 Dugdale plastic zone in the form of narrow strips Fig 1111 of s1ze R 860 so that aeff a R gt calculate Keff There is a limit to the extent to which K can be adjusted page 7 KL Murty MSE 450 Need EPFM Elastic Elastic Eracture Mechanics l CTOD CrackTipOpeningDisplacement considers material ahead of the crak as a no of miniature tensile specimens and crack propagates as each of these tensile specimens fractures Fig 1112 8 8 since if T0 g Gauge length 2p fracture criterion is 8c 2p8f for planestress thinplate 8c 8ft t thickness can show that g 50 8 Eq 1132 or Q 7 50 8c Eq 1133 here 7 depends on where CTOD is measured unfortunately CTOD or COD method is very sensitively dependent on the specimen geometry cannot have a single number better approach is to use JIntegral Method 1 18 2 Jlntegral Mehod Rice has shown that Jintegral defined u as in Eq 1137 is path independent J I W dy 39 T X d5 Eq 1 13937 around the crack F J MNm or lVlPa m y J is the potential energy difference Fig39 3913 between two identically loaded r specimens with different crack lengths Fig lll4 gt X aU K2 J g g E Eq 1138 dS ASTME81381 J Test Procedure 2A for a 3point bend specimen J g A area under Load vs displacement curve b W a remaining ligament B specimen thickness page 8 KL Murty MAT 450 Spring 96 Multiple specimens with different crack lengths measure crack growth Aa due to loading to a given displacement level and noting the load P calculate J vs Aa see fig llle Aa is meaured after the test by heat tinting and fracturing different initial crack sizes a would give different b values and give the JR curve R resistance Fig 111 5a determine the critical crack initiation elasticplastic fracture toughness ch Single Specimen and determine a as a function of P a is measured from compliance or using Potential Drop technique or unloading compliance method In either case obtain data as in Fig 1115 amp find ch still dimensional criterion is checked for the test to be valid but size limitation is not as stringent as in KC J 1C Eq pg 367 for 3pomt bend spec1men b 2 25 6 0 for A533B steel Valid JIC b 05 in vs Valid KIC B 2 ft page 9
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