CMU MSE 27301 - The Effect of Grain Size on Strength and Toughness

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1ObjectiveGrain SizeVaristorsHall-PetchCreepMicrostructure-Properties: ILecture 5AThe Effect of Grain Size onStrength and Toughness27-301Fall, 2007A. D. Rollett2ObjectiveGrain SizeVaristorsHall-PetchCreepBibliography•Mechanical Behavior of Materials (1966), F. McClintock andA. S. Argon, Addison Wesley.•Mechanical Behavior of Materials, T.H. Courtney, McGraw-Hill, ISBN 0-07-013265-8, 620.11292,C86M•Microstructure and Properties of Materials, J.C.M. Li, editor,World Scientific, ISBN 981-02-2403-6• Leslie, WC, The Physical Metallurgy of Steels, HemispherePress, McGraw-Hill• Llewellyn, DT & Hudd, RC, Steels, Metallurgy andApplications, Butterworths-Heinemann• http://www.steeluniversity.org/content/html3ObjectiveGrain SizeVaristorsHall-PetchCreepObjective• This lecture is concerned with the effects of grainsize on properties.• Two examples will be given:• (4A) The effect of grain size on mechanicalproperties (Hall-Petch effect, Nabarro-Herringcreep).• (4B) The effect of grain size on resistance inceramics used for varistors (e.g. in surgeprotectors).• If time permits, the discussion will be extended tomagnetic hardness also.4ObjectiveGrain SizeVaristorsHall-PetchCreepKey Concepts• Grain boundaries (effectively) have properties that differ fromthe matrix.• Properties of polycrystal depend on the content of planardefects, i.e. grain boundaries, i.e. grain size.• Grain boundaries in semiconductors used to make varistorshave a one-way voltage barrier.• The Hall-Petch effect quantifies the trend of increasingstrength and toughness with decreasing grain size. That is tosay, fine grain size strengthens the material.• Creep rates (Coble creep) increase with increasing grainboundary area (per unit volume), hence decreasing grain size.Therefore grain size has the opposite effect at hightemperatures where fine grain size weakens the material.• Low temperature service optimized by fine grain size, but hightemperature service optimized by use of single crystals.5ObjectiveGrain SizeVaristorsHall-PetchCreepNotation•σy : yield strength•σ0 : friction stress•K : Hall-Petch coefficient•d : grain size•n : number of dislocations ina pile-up• ν : Poisson’s ratio• G : shear modulus• τ : shear stress• J : vacancy flux• D : diffusion coefficient• T : temperature• Tm : melt T• Ω : atomic volume• σ : stress• N : concentration ofvacancies• Qm : Activation energy formigration• QVacancy : Activation energyfor vacancy formation• kT : Boltzmann’s constantmultiplied by temperature6ObjectiveGrain SizeVaristorsHall-PetchCreepHall-Petch Effect• The Hall-Petch effect isremarkably simple toexpress but still difficult toexplain in fundamentalterms.• At ambient conditions (nocreep), yield strengthrises as the grain sizedecreases.• The variation in strengthcan be described by apower-law relationship:σy = σ0 + kd-1/2The Hall-Petch effect is named for E.O. Hall and N.J. Petchfrom their papers of the early 1950’s, e.g. “The CleavageStrength of Crystals” N.J. Petch, J. Iron & Steel Inst., 174,25-28.http://www.steeluniversity.org/content/htmłeng/default.asp?catid=171&pageid=2081271789 - after:“The yield stress-grain size relation in iron substitutional alloys (Grain sizeeffects on iron substitutional alloys yield stress, investigating Hall-Petchrelation)”, MORRISON, W B; LESLIE, W CMetallurgical Transactions. Vol. 4, pp. 379-381. Jan. 19737ObjectiveGrain SizeVaristorsHall-PetchCreepDislocation Pile-ups• The central idea is that dislocations are forced topile up at grain boundaries, either because there isa barrier to crossing over into the next grain, orbecause a source must be activated in the nextgrain.[Courtney]8ObjectiveGrain SizeVaristorsHall-PetchCreepDislocation Pile-up at a Boundary• The classical explanation for theHall-Petch effect is that somestress concentration in a givengrain is required to initiate slip inits neighboring grain. That stressconcentration is most plausiblyobtained through a dislocationpile-up, see figure 5.5. Theessence of the argument is thatstress is higher as the number ofdislocations increases. Thus thelarger the grain size, the morequickly (in terms of macroscopicstrain) is the critical stress reachedat which slip is initiated in theneighboring grain. The form of theequation describing the pile-upstress contains a term in √(d/r)where d is the grain diameter andr is the (average) distance to thesource in the neighboring grainfrom the boundary.! Number of (edge) dislocations in pile - up :n =(1"#)$%d4GbStress at the head of the pile - up :%c= n%=(1"#)$%2d4GbNow invert the relationship to find the shear stress to overcome the criticalbreaking stress of the grain boundaryand insert a friction stress (e.g. solutes) :%applied=%i+4Gb%c$d=%i+ kd"1/ 29ObjectiveGrain SizeVaristorsHall-PetchCreepStress concentration• The square root term is analogous to the stress concentration at thetip of a penny-shaped crack (in fracture mechanics). Thus,(τapplied - τi) √(d/4r) = τ*,where τ* is the critical stress for dislocation source activation, τi isthe resistance to dislocation motion in each grain, r is the tip radius,and τapplied is the applied shear stress. Again, the larger the diameter(of the crack, or, in this case, grain), d, the more dislocations in thepile-up for a given applied stress (minus the resistance).Rearranging, we get {Courtney - Eq. 5.8}. τapplied = τi + 2 τ* √ r d-1/2 = τi + kd-1/2.10ObjectiveGrain SizeVaristorsHall-PetchCreepMaterial Dependence• The Hall-Petch constant, k in the equation, variesconsiderably amongst materials. This in itselfraises some questions about the mechanism(s)underlying the effect. The explanation given ispurely geometrical and although the materialdependence could be explained through the ratiod/r, it is not clear why this should be so!• Solutes tend toenhance themagnitude of the Hall-Petch effect.[Courtney]11ObjectiveGrain SizeVaristorsHall-PetchCreepGrain Size and Fracture• Grain size also has a marked effect on fracture,which was, in fact, part of Petch’s originalcontribution.McClintock & Argon12ObjectiveGrain SizeVaristorsHall-PetchCreepNanocrystalline materials• All this suggests that remarkably strong materials can be generated ifvery small grain sizes can be achieved. This, of course, is one aim ofnanocrystalline materials in which grain sizes are obtained that arewell less than one micron. The processing (in metals)


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CMU MSE 27301 - The Effect of Grain Size on Strength and Toughness

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