Ideas on Interpreting the Phenomenology of Superalloy Creep. Weertman has argued the natural form of creep has the form:

Transcription

1 Ideas on Interpreting the Phenomenology of Superalloy Creep A key problem in creep (in general) is explaining the stress and temperature dependencies of the creep and linking it to the underlying physical micro-mechanisms. Weertman has argued the natural form of creep has the form: In superalloys, n>3 Q c >Q D $ # ' " = k D eff & ) % E ( How do we rationalize and link to underlying physics?? 3

2 Plan 1. Generalize the world of creep 2. Identify and define obstacle controlled creep 3. Consider phenomenology of 3 somewhat understood cases: 1. Oxide Dispersion Strengthened Metals 2. Pure Metals 3. Metal Matrix Composites 4. Relate this to single xtal superalloy creep

3 Limiting Classes of Plastic Flow Mechanisms 1) Viscous Flow -- Thermal and stress activation of volume defects gives flow with Newtonian viscosity. (e.g., glass above T g ) 2) Diffusional Creep -- Diffusion flux between surface defects (grain boundaries) gives flow with Newtonian character and activation energy of diffusion. 3) Defect Drag Deformation -- Viscous drag of dislocations, or defects captured by dislocations controls strain rate. Can be Non-Newtonian. (e.g., Class I creep, Nabarro climb creep, Mills screw dislocations dragging jogs) 4) Obstacle Controlled Deformation -- Dislocation mobility limited by immobile obstacles. (e.g., Common plasticity, Artz-Rösler DS model, Class II creep, etc.) Note: Some mechanisms may be mixtures (e.g., grain boundary sliding)!

4 Climb Controlled Creep -- A Myth It is often not the thermally activated climb of dislocations over obstacles that controls plasticity (and gives its rate dependence), as commonly assumed. Dislocations tend to stick to obstacles. Other dislocations are common obstacles. The faster of two processes will take place: 1) Dislocations will de-pin from obstacles to accommodate flow. - or - 2) The obstacle field will coarsen, so the pin spacing increases, letting dislocations past.

5 More Generalizations Consider Obstacle Controlled Deformation (from 0 < T < Tm). Creep is governed by 4 fairly independent processes: 1) Dislocation bypass or release from obstacles, 2) Increase in dislocation density with flow, 3) Recovery -- coarsening the obstacle field (some obstacles disappear!) 4) Load-shedding from soft to hard regions. Using only these elements the major phenomenology of 1-D plastic deformation can be recovered, even over a wide range of homologous temperature, and this can be applied to many materials.

6 Case I -- Oxide Dispersion Strengthened Metals -- Very stable, very fine oxides in metal matrix. -- Shows threshold stress and activation energy for creep far greater than that for self-diffusion. -- Strength is strongly related to Orowan Strength. -- For a long, long time a key question was why don t the dislocations simply climb over the dispersoids?? Rösler and Arzt proposed a model of flow that is based on attractive interaction between a particle and dislocation.

7 Obstacle Controlled Deformation Other mechanisms are all better understood than obstacle controlled deformation. Rösler and Arzt have a theoretically appealing and predictive model for dispersion hardened materials. Thermal activation Increasing Obstacle Size In DS materials obstacles can be large (stick dislocations to incoherent dispersoids) and they are stable. Stress activation In simple metals, obstacles may recover away... k= fraction of energy/length compared to as in matrix From Rösler and Arzt (1990)

8 Discrete Obstacles Thermodynamics and kinetics of slip... Good way to link micro and constitutive, but under-utilized, Gives limits of what can be thermally activated. Force athermal breaking force!bl "G* "F* # P slip = exp!"g* & % ( $ kt ' # #!G* =!F * % 1 " % $ $ f obs ˆ k & ( ' p & ( ' q Distance Rules of thumb for!f*--> (see Frost & Ashby).1 µb 3 Solute interaction.5-2 µb 3 Work hardening 10 µb 3 Precipitate interaction Diffusion Activ. Energy ~ 0.25µb 3 Often diffusion is much more rapid than dislocation release!! Dislocations may bow around strong obstacles rather than being released.

9 Case II -- Pure Metals -- Activation energies for creep and self diffusion similar -- n ~5 -- can we rationalize as obstacle controlled?? -- In ODS alloys, obstacle field is fixed. What fixes obstacle density in pure metals??

10 Creep of Simple Pure Metals No accepted model of pure metal like deformation. The Dorn equation is commonly used: #! ss = A Exp "Q c & $ % kt ' ( * ) -, + µ /. We note similar patterns in many metals:! BCC! FCC! HCP! Class II alloys, etc. Q=Q D, n= 4-6, similar strength scaling n! D Overall Trend for Pure Metals (Sherby, 1962) Most models are mechanistic. This is likely inappropriate since these materials are all quite different in many details. 1 5 n=5 We need a universal explanation (which does not depend on details of cross-slip or climb, etc.) "/E

11 Self Selection of Steady-State Length Scale Transient behavior shows effects of structural refinement and coarsening Steady state subgrain size roughly inversely proportional to creep stress Steady state subgrain size / b Applied stress / shear modulus Sherby and Burke (1968) Takeuchi & Argon 1976

12 A Postulate It is often not climb bypass that controls plasticity (and gives its rate dependence), as commonly assumed. Many obstacles are too large to be activated past. Instead, often thermally-activated coarsening that eliminates obstacles and releases dislocations to glide. Coarsening is where the main time and temperature dependence are!

13 What Provides High Temperature Strength? Junctions! Example from McLean (1960) of network formation in iron; <111> type react to <100>type. From Honeycombe. Bulatov, et. al., Nature 2006, Dislocation Multi-Junctions (demonstrated in BCC Nb) Networks by intersection are ubiquitous. These attractive junctions are very hard to break by stress and thermal activation!!f* >>1µb 3 They cannot move by glide, but can be eliminated by recovery-like diffusion-assisted mechanisms.

14 Dislocation Density Accumulation in Simple Annealed Metals 1E+16 1E+15 Good compilation in: J. J. Gilman Micromechanics of Flow in Solids, Dislocation Density (m^-2) ) 1E+14 1E+13 1E+12 1E+11 c=0 c=0.5 Edington Poly V 20 & 230 C Edington Poly V 300 & 400 C Livingston Cu Xtal, multislip Bailey polyxtal Cu Hordon - Cu Xtal Polyslip Hordon Al Xtal Polyslip Hordon Cu xtal polyslip Lawley Mo Xtal 300K "# "$ = M # c 1 [ ( 1$c ) ( ) + " ] o " = M# 1$ c 1$c Ag Polyxtal Orlova; Fe polycreep 1E Plastic Strain assumed density

15 Assumed Recovery Form d (" m ) c = K M( T) dt This form is widely found and studied in modern coarsening literature. m c is coarsening exponent. It is determined by mechanistic path M(T) is the mobility for the rate controlling process. For simple grain growth (Burke and Turnbull, 1950):! m c =2! M(T)=Grain boundary mobility However, Humphreys* notes: There is little evidence for m=2, even in very pure materials. Typically the coarsening exponent is between 3 and 5. We assume there is some pattern that is approximately common to many metals *in Recrystallization and Related Annealing Phenomena (1995).

16 Kinetic Mechanisms & Scaling of Velocity Viscous Boundary Motion Viscous Line Motion LR Diff. / Boundary Motion P 1 P 2 V " d# dt " M$P V " d# dt " M df dl V " d# dt " D$P% # LR Diff. / Line Motion Boundary Diff. / Bdy. Motion Pipe Diff. / Line Motion J vac # V " d# dt " D df dl $ # V " d# dt " D b$p% # 2 V " d# dt " D df dl $ # 3

17 Scaling Analysis of Possible Recovery Paths d (" m ) c = K R( T) dt System morphology Rate Controlling Step Rate Step \ Geom. Viscous Motion m c = 2 Long-Range Lattice Diffusion Diffusion Along Feature Defect E V " #$1 Grains or subgrains M(T) m c = 3 D l (T) m c = 4 * D path (T) E V " #$2 3-D Dislocation Network m c = 3 M d (T) m c = 4 D l (T) m c = 6 D path (T) * Pipe diffusion increases m c by one vs. bulk diffusion because area fraction of dislocation scales with 1/#. If lattice diffusion controls coarsening we expect m c =3 or 4 with a diffusional activation energy.

18 A Proposal... d (" m ) c = K D( T) dt m c =3-4 and diffusion is the rate limiting step (the activation energy for selfdiffusion is expected). K is similar for many structural metals and is an important element in the nearuniversal scaling of metal creep.

19 Replotting of Data 250 C 250 C Small Subgrain Size (µm) 1E C 300 C 325 C 350 C 400 C m=3 Small Subgrain Size (µm) 1E C 300 C 325 C 400 C m=3 K= E+00 1E+01 1E+02 1E+03 1E+04 1E+05 Annealing time (s) 1E-17 1E-16 1E-15 1E-14 1E-13 1E-12 Normalized time (time D(T)) 1E-11 Y. Huang and F. J. Humphreys, Subgrain Growth and Low Angle Boundary Mobility in Aluminum Crystals of Orientation {110}<001>, Acta Mater., 48, (1999).

20 Integration of the Processes for Power-Law Creep

21 Simple Balance of Refinement and Coarsening Refinement d" d# = M" c " = g # Steady-state length scale Coarsening d " m c ( ) = K D( T) dt " ss = % ' & 2 K D(T ) g2#2c M $ ( * ) 1 2+ m c #2c " = sµb # De-pinning (Taylor Equation) " = B D T $ ( )& # µ % ' ) ( n $ = 2 K g 2*2c & % ( ) ( ) ( b s) * 2+ m c *2c M ' ) D T ( ( ) $ # ' & ) % µ ( ( 2+ m c *2c ) Stress exponent, n=2+m c -2c ~ 5, and strain rate scales with self diffusivity.

22 Phenomenon Equation {Range} &/or Assumed value Structure parameter (! or ") Athermal Yield Dislocation Accumulation Recovery (structural coarsening) Other constants " $ D(T) = 2 K g 2#2c & % ( ) d! m c! = g " g={1-20} g=4 $ f = sµb s={ } s=0.5! d" d% = M" c M~2 x10 15 m -2 c=0 ( ) = K M T ( b s) #( 2+m c #2c) ' $ * M ) & ' ) ( % µ ( A model for creep based on microstructural length scale evolution G.S. Daehn, H. Brehm, H. Lee, B.S. Lim, Mat Sci & Engr A, , pp , (2004). Full Model Summary ( ) dt K= 10-6 m ( 2+m c #2c) Subgrain size, # (m) 1E-06 m=3 M(T)=D l (T) b=0.3nm 1E-17 1E C 275 C 300 C 325 C 400 C 1E-15 1E-14 1E-13 Normalized time (time D(T)) 1E-12 1E-11 Dislocation Density (m^-2) (m -2 ) 1E+16 1E+15 1E+14 1E+13 1E+12 1E+11 1E Plastic Strain Edington Poly V 20 & 230 C Edington Poly V 300 & 400 C Livingston Cu Xtal, multislip Bailey polyxtal Cu Hordon - Cu Xtal Polyslip Hordon Al Xtal Polyslip Hordon Cu xtal polyslip Lawley Mo Xtal 300K Ag Polyxtal Orlova; Fe polycreep assumed density

23 The Model is Predictive WITHOUT Input from Creep " $ D(T) = 2 K g 2#2c & % ( ) ( b s) # 2+m c #2c M ( ) ' $ * ' ) & ) ( % µ ( ( 2+m c #2c) m c =3, K=10-6 m, g=4, b=0.3nm, s=0.5, c=0, M= " D(T), cm#2 " D(T) = $ # 2109 & ' ) % µ ( 5 Prediction All inputs are taken from non-creep experiments, and show correspondence to known creep trends " E

24 Case III - Metal Matrix Composites -- SiC particles in Al -- inter-particle spacing too open to provide significant Orowan Strengthening -- These are much stronger than expected Why? What controls flow??

25 Slip versus Coarsening Activation Energies A slightly more comprehensive model of the type analyzed here using: # P slip = exp!"g* & % ( $ kt ' # #!G* =!F * % 1" % $ $ f obs ˆ k & ( ' p & ( ' q shows if!f* for flow is significantly greater than Q D then the activation energy for coarsening is rate controlling; and the activation energy for creep is that for diffusion. Experimental Studies on Al-SiC composites support this view Imagine coarsening is inhibited by the particles... For details see: See H. Brehm and G. S. Daehn, A Framework for Modeling Creep in Pure Metals, Met. Mat. Trans., 33A, (2002).

26 Creep Resistant Material (Al-SiC, Chen&Daehn 1993) 0.03 Little transient on loading 2124Al-15%SiCw Composite 375 C Isothermal Creep Matrix Behavior Composite behavior Strain MPa MPa MPa MPa Al-20%SiCw.400 C 2124Al-20%SiCw.375 C 2124Al-15%SiCw.375 C 2124Al-15%SiCw.425 C 2124Al.400 C (Nieh et al.) 2124Al-F.400 C Al-4%Al2O3.400 C (Milicka et al.) MPa Time (sec) Strain Rate (s -1 ) Composite stress exponent is much greater than five!! Mechanics Solution Applied Stress (MPa) 1

27 Al-SiC Transients Show No Stress-Based Structure Change Al-15%SiCw Composite Multiple Creep Tests at 375 C! 1 True Strain ! 1! !1=25.3 MPa!2=28.6 MPa Al-15%SiCw Time (sec) -1 Strain Rate (s ) 10-6 n=12.7 S S S1 S S1 S S1 S S S S S1=26.3 MPa S2=29.2 MPa Time (sec)

28 Composite Activation Energy > Self Diffusion (Chen) 800 Activation Energy (KJ/mole) ( C) 6061Al-20%SiCp: Nieh et al (57) ( C) 2124Al-20%SiCw: N&S (59) ( C) 2124Al-20%SiCw: this study 2124Al-15%SiCw: this study ( C) 6061Al-30%SiCp: Park et al (62) ( C) ( C) ( C) ( C) ( C) Al self diffusion, Qc=142 KJ/mole Applied Effective Stress (MPa) Y-C Chen, PhD Thesis, Ohio State University, Here, obstacle depinning rather than coarsening controls creep.

29 (PM) Activation Energy (KJ/mole) ( C) ( C) ( C) ( C) ( C) ( C) ( C) ( C) Al self diffusion, Qc=142 KJ/mole Al-20%SiCp: Nieh et al (57) 2124Al-20%SiCw: N&S (59) 2124Al-20%SiCw: this study 2124Al-15%SiCw: this study 6061Al-30%SiCp: Park et al (62) Applied Effective Stress (MPa) Y-C Chen, PhD Thesis, Ohio State University, ( C) Observations: QC>QSD, n>5, strong interactions between dislocations and SiC particles inhibit coarsening... Here, obstacle depinning rather than coarsening controls creep.

30 Case IV - single xtal superalloys -- Many models would lead us to expect: Activation energies for creep like those for self diffusion Low stress exponents. -- Not clear how we model strength -- How do we rationalize big picture?

31 Qc >> QD n>5 Transient behavior not clear Stress dependencies of Q and n not clear.

32 Strong suggestion that dislocation linklength may scale with strength...

33 Superalloy creep is complex! dislocation and phase morphology change in manner that is not self similar. Both dislocation nets and particles are subject to coarsening. Both phases subject to deformation

34 Concluding Remark We can identify a class of behavior of obstacle controlled creep and explain the behavior of several different creep mechanisms. The high stress exponent and activation energies of superalloy creep strongly suggest that dislocations are breaking free from strong pins and this controls creep rate. Strong pin means DG* > 0.25 µb 3. Coarsening also takes place, but at a rate too slow to accommodate creep. If this is true, we expect: constant-structure like transients, Qc will decrease with increasing stress, strength should largely scale with (largest link length) -1. We are a long way from a real quantitative creep model.

35 Can Framework Be Extended?? Phenomenon Harper-Dorn Creep Power-Law Breakdown Varied stress exponent Low temp. plasticity Qc not equal Qsd Anelasticity Bauschinger Effect Primary Creep How to Treat? Recognition that coarsening leads to limiting maximum network dimensions. Enhanced recovery at very small length scales due to dislocation annihilation. Naturally varies somewhat due to changes in hardening and coarsening details. Naturally arises from this framework. Q c > Q d if obstacles not eliminated by coarsening, Qc < Qd if obstacles are relatively weak. Load shedding Load shedding Load shedding in framework