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and Creep Stress Rupture : Ch. 13 : 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15(optional) Definition of Creep and Creep Curve : (13-3) def. Creep is the time-dependent plastic strain at constant stress and temperature Creep curve : Fig. 13-4 steady-state creep-rate ( D s or simply D ) : Temperature and Stress Dependencies - Fig. 13-6 Fig. 13-8 - total creep curve : = o + p + s o = instantaneous strain at loading (elastic, anelastic and plastic) s = steady-state creep strain (constant-rate viscous creep ) = D st p = primary or transient creep : Andrade- flow (or 1/3 rd law) : t1/3 primary or transient creep : Andrade- flow (or 1/3 rd law) : p = t1/3 problem as t 0 Garofalo / Dorn Equation : p = t (1 - e-rt ) , r is related to Dorn Both primary and steady-state follow similar kinetics - temperature compensated time ( = t e- Qc/RT) - single universal curve with t replaced by or st > i (~1-20) > s D Or, creep strain - o = t (1 - e- st ) + D st see Sherby-Dorn (Al), Murty (Zr) Sherby-Dorn -parameter Creep curves for Al at Sherby & Dorn (1956) (3,000 psi) and at three different temperatures KL Murty MSE 450 A single curve demonstrating the validity of -parameter page 1 st Creep data in Zircaloy at varied temperatures (F) and stresses (ksi) fall into a single curve demonstrating the validity of Dorn equation (Murty et al 1976) (K. L. Murty, M.S. Thesis, 1967) Zener-Holloman : Z = De Q / RT Stress Rupture Test : (13-4) vs tr Representation of engineering creep / rupture data (13-12, 13-13) - Figs. 13-17, 13-18 Sherby-Dorn Parameter : Larson-Miller Parameter : PS-D = t e-Q/RT PL-M = T (log t + C) a Fig. 13-19-21 T - Ta Manson-Haferd Parameter : PM-H = log t - log t --- these parameters are for a given stress and are functions of (Fig. 13-20) --- Monkman-Grant : Cs t r = Eq. 13-24 Demonstration of Monkman-Grant Relationship in Cu (Feltham and Meakin 1959) KL Murty MSE 450 page 2 Creep Under Multiaxial Loading (text 14-14) Use Levy-Mises Equations in plasticity 1 (1-2)2 + (2-3)2 + (3-1)2 eff = 2 deff 1 [1 - 2 (2+3) ] , and d1 = eff since creep is plastic deformation 1/2 appears as in plasticity. Similarly, d2 and d3. Dividing by dt, get the corresponding creep-rates, eff 1 D 1= [1 - 2 (2+3) ], etc. eff One first determines the uniaxial creep-rate equation, D s = A n e-Q/RT n and assume the same for effective strain-rate : D eff = A eff e-Q/RT n-1 1 1 = A eff e-Q/RT [1 - 2 (2+3)] so that etc. Stress Relaxation As noted in section 8-11, the stress relaxation occurs when the deformation is held constant such as in bolt in flange where the constraint is that the total length of the system is fixed. t = E + creep = const. Here, E = E . dt 1 d d Or, dt = - E D s = - E A n @ fixed T Thus dt = 0 = E dt + D s Integration from o to t gives, f t d n = - E A dt = - E A t o i 100 Data from "HW #8-8" 80 60 final or (t ) = o n [1 + AE (n 1) o 1t ]1 /( n 1) 40 20 0 1000 2000 3000 4000 5000 time, hr KL Murty MSE 450 page 3 Deformation / Creep Mechanisms : Introduction - structural changes (13-5) - Slip (difficult to observe slip lines / folds etc are usually noted) Subgrains GBS - excess (deformation induced) vacancies Two important relationships : Orowan equation : D =bv and Taylor equation : = 2 2G2b2 Thermally Activated Dislocation Glide (at low T and/or high strain-rates) D = A eB e-Qi/RT where Qi is the activation energy for the underlying mechanisms Peierls mechanism (bcc metals) Intersection mechanism (fcc 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 Grain-Boundary Sliding - (GBS) - intermediate stresses in small grained materials and ceramics (where matrix deformation is difficult) Many different mechanisms may contribute and the total strain-rate : parallel mechanism (fastest controls / dominates) D = Di i series mechanisms (slower controls / dominates) D = 1 i 1 Slip following creep deformation in -iron Uncrept specimen Crept at 5500 psi to 21.5% strain (K.L. Murty, MS thesis, Cornell University, 1967) KL Murty MSE 450 page 4 Dislocation Creep : Pure Metals / Class-M alloys: Experiments : D = A n e-Qc/RT ,n 5, Qc QL (QD) (edge ) glide - climb model Weertman-Climb model (Weertman Pill-Box Model) sequential processes L = average distance a dislocation glides h tg = time for glide motion h = average distance a dislocation climbs FR L Lomer-Cottrell tc = time for climb Barrier = strain during glide-climb event = g + c g = b L h t = time of glide-climb event = tg + tc tc = v , vc = climb velocity c L b L D = t = h/v = b ( h ) vc c where Cv vc e-Em/kT , Em = activation energy for vacancy migration Here, Cv = Cv - Cv = Cv eV/kT - Cv e-V/kT = Cv 2 Sinh( kT ) L L V D = b ( h ) vc = b ( h ) Co e-Em/kT 2 Sinh( kT ) v At low stresses, Sinh() so that Garofalo Eqn. L D = A D (sinhB)n V D = A1 b ( ) Co e-Em/kT v kT h L L V D = A1 b ( ) DL A2 ( h ) DL kT h Or D = A 3 D natural creep-law L Weertman: h 1.5, D = A 4.5 D as experimentally observed in Al In general D = A(T) n Power-law - n is the stress exponent {f(xal structure, )} + o o o V also known as Nortons Equation (n is Norton index) At high stresses ( 10-3 E), Sinh(x) ex, D = AH eB D (Power-law breakdown) KL Murty MSE 450 page 5 Experimental Observations - Dislocation Creep Fig. 13-13 (Dieter) (Sherby) What happens if we keep decreasing the stress, say to a level at and below the FR? As is decreased reach a point when FR , dislocation density would become constant (independent of ): D - viscous creep known as Harper-Dorn creep Harper-Dorn creep occurs at 10-5 , o 106cm-2 E b H-D creep is observed in large grained materials (metals, ceramics, etc.) D HD = AHD DL ln 1 2 ln Characteristics of Climb Creep (Class-M) : large primary creep regions 1 subgrain formation ( ) 2 dislocation density independent of grain size KL Murty MSE 450 page 6 Effects of Alloying : (class-A) Solid-solution - decreases rate of glide A glide controlled creep although annihilation due to climb still occurs (micro-creep / viscous glide creep) viscous glide controlled creep : (decreased creep-rates) D g = Ag Ds 3 , Ds is solute diffusion (Al) class-M 1 1 log(stress) 5 3 (Al-3Mg) class-A little or no primary creep no subgrain formation 2 grain-size independent At low stresses (for large grain sizes), Harper-Dorn creep dominates what happens as grain size becomes small As grain-size decreases (and at low stresses) diffusion creep due to point defects becomes important : (due to migration of vacancies from tensile boundaries to compressive boundaries) Nabarro-Herring Creep (diffusion through the lattice) : D NH = ANH DL 2 d Coble Creep (diffusion through grain-boundaries) : D Co = ACo Db 3 d Nabarro-Herring Creep vs Coble Creep : Coble creep for small grain sizes and at low temperature NH creep for larger grain sizes and at high temperatures at very large grain sizes, Harper-Dorn creep dominates 3 1 2 Coble N-H 1 Harper-Dorn log (grain-size) At small grain-sizes, GBS dominates at intermediate stresses and temperatures : D GBS = AGBS Db 2 d 2 superplasticity 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 D ppt = Appt D 8 - 20 Rules for Increasing Creep Resistance Large Grain Size (directiona...