《材料力學(xué)性能》大三教學(xué)PPT課件

《材料力學(xué)性能》大三教學(xué)PPT課件,材料力學(xué)性能,材料,力學(xué)性能,大三,教學(xué),PPT,課件 Fatigue FatigueFatigue is a form of failure that occurs in structures subjected to dynamic and fluctuating stresses(e.g.,bridges,aircraft,and machine components).Under these circumstances it is possible for failure to occur at a stress level considerably lower than the tensile strength for a static load.The term“fatigue”is used because this type of failure normally occurs after a long period of repeated stress or strain cycling.SignificanceFatigue is important inasmuch as it is the single largest cause of failure in metals,estimated to comprise approximately 90%of all metallic failure;polymers and ceramics(except for glasses)are also susceptible to this type of failure.Furthermore,it is catastrophic and insidious,occurring very suddenly and without warning.Fatigue failure is brittlelike in nature even in normal ductile metals,in that there is very little,if any,gross plastic deformation associated with failure.The process occurs by the initiation and propagation of cracks,and ordinarily the fracture surface is perpendicular to the direction of an applied tensile stress.Cyclic StressesThe applied stress may be axial(tension-compression),flexural(bending),or torsional(twisting)in nature.In general,three different fluctuating stress-time modes are possible.Reversed stress cycle;Repeated stress cycle;Random stress cycle.Reversed stress cycleRepeated stress cycleRandom stress cycleMean stressRange of stressStress amplitudeStress ratioThe S-N CurveAs with other mechanical characteristics,the fatigue properties of materials can be determined from laboratory simulation tests.A test apparatus should be designed to duplicate as nearly as possible the service stress conditions(stress level,time frequency,stress pattern,etc.).A schematic diagram of a rotating-bending test apparatusSchematic of a pulsator for fatigue tests in tension-compressionTypes of standard test pieces for fatigue testsA series of tests are commenced by subjecting a specimen to the stress cycling at a relatively large maximum stress amplitude(max),usually on the order of two thirds of the static tensile strength;the number of cycles to failure is counted.This procedure is repeated on other specimens at progressively decreasing maximum stress amplitudes.Data are plotted as stress S versus the logarithm of the number N of cycles to failure for each of the specimens.Two distinct types of S-N behavior(1)Two distinct types of S-N behavior(2)Fatigue limit/Endurance limitFatigue limit.For fatigue,the maximum stress amplitude level below which a material can endure an essentially infinite number of stress cycles and not fail.The fatigue limit represents the largest value of fluctuating stress that will not cause failure for essentially an infinite number of cycles.For many steels,fatigue limits range between 35 and 60%of the tensile strength.Fatigue StrengthMost nonferrous alloys(e.g.,aluminum,copper,magnesium)do not have a fatigue limit,in that the SN curve continues its downward trend at increasingly greater N values.Thus,fatigue will ultimately occur regardless of the magnitude of the stress.Fatigue strength.The maximum stress level that a material can sustain,without failing,for some specified number of cycles(e.g.107 cycles).Fatigue LifeFatigue life.The total number of stress cycles that will cause a fatigue failure at some specified stress amplitude.Unfortunately,there always exists considerable scatter in fatigue data,that is,a variation in the measured N value for a number of specimens tested at the same stress level.This may lead to significant design uncertainties when fatigue life and/or fatigue limit(or strength)are being considered.The scatter of results is a consequence of the fatigue sensitivity to a number of test and material parameters that are impossible to control precisely.These parameters include specimen fabrication and surface preparation,metallurgical variables,specimen alignment in the apparatus,mean stress,and test frequency.Fatigue S-N curves similar to those shown before represent“best fit”curves which have been drawn through average-value data points.It should be remembered that S-N curves represented in literature are normally average values,unless noted otherwise.Fatigue S-N probability of failure curves for a aluminum alloyLow-cycle fatigue and high-cycle fatigueThe fatigue behaviors may be classified into two domains.One is associated with relatively high loads that produce not only elastic strain but also some plastic strain during each cycle.Consequently,fatigue lives are relatively short;this domain is termed low-cycle fatigue and occurs at less than about 104 to 105 cycles.For lower stress levels wherein deformations are total elastic,longer lives result.This is called high-cycle fatigue inasmuch as relatively large numbers of cycles are required to produce fatigue failure.High-cycle fatigue is associated with fatigue lives greater than about 104 to 105 cycles.Problems1.Cite five factors that may lead to scatter in fatigue life data.2.Briefly demonstrate that increasing the value of the stress ratio R produces a decrease in stress amplitude a .FractographThree distinct steps of fatigue failure(1)crack initiation,wherein a small crack forms at some point of stress concentration;(2)crack propagation,during which this crack advances incrementally with each stress cycle;and(3)final failure,which occurs very rapidly once the advancing crack has reached a critical size.Crack initiationCrack associated with fatigue failure almost always initiate(or nucleate)on the surface of a component at some point of stress concentration.Crack nucleation sites include surface scratches,keyways,threads,dents,and the like.In addition,cyclic loading can produce microscopic surface discontinuities resulting from dislocation slip steps which may also act as stress raisers,and therefore as crack initiation sites.Crack initiation by slip(P.Neumann)WoodCrack propagationThe region of a fracture surface that formed during crack propagation step may be characterized by two types of markings termed beachmarks and striations.Both of these features indicate the position of the crack tip at some point in time and appear as concentric ridges that expand away from the crack initiation site(s),frequently in a circular or semicircular pattern.Beachmarks(sometimes also called“clamshell marks”)are of macroscopic dimensions,and may be observed with unaided eye.These markings are found for components that experienced interruptions during the crack propagation stage.Each beachmark band represents a period of time over which crack growth occurred.BenchmarksFractographStriationOn the other hand,fatigue striations are microscopic in size and subject to observation with electron microscope(either TEM or SEM).Each striation is thought to represent the advance distance of a crack front during a single load cycle.Striation width depends on,and increases with,increasing stress range.Crack propagation model(Plastic)Crack propagation model(Brittle)It should be emphasized that although both beachmarks and striations are fatigue fracture surface features having similar appearances,they are nevertheless different,both in origin and size.They may be literally thousands of striations within a single beachmark.The presence of beachmark and/or striations on a fracture surface confirms that the cause of failure was fatigue.Nevertheless,the absence of either or both does not exclude fatigue as the cause of failure.Beachmarks and striations will not appear on that region over which the rapid failure occurs.Rather,the rapid failure may be either ductile or brittle;evidence of plastic deformation will be present for ductile,and absent for brittle failure.Crack propagation rateThe total durability of a component is Nf=Ni+Np for Np/Nf90%，studies on crack propagation rate is of very important significance.Crack propagation rate(Paris formula)l is the crack length;N is the numbers of cycle;c,n are the constants of a material depending on the coefficient of cycle asymmetry;K is the amplitude of the stress intensity coefficient.(K=Kmax-Kmin)Factors that affect fatigue lifeThe fatigue behavior of engineering materials is highly sensitive to a number of factors,e.g.mean stress level,geometrical design,surface effects,and metallurgical variables,as well as the environment.Discussion these factors and measures to be taken to improve the fatigue resistance of structural components.MEAN STRESSThe dependence of fatigue life on stress amplitude is represented on the S-N plot.Such data are taken from for a constant mean stress(m),often for the reversed cycle situation(m=0).Mean stress,however,will also affect fatigue life,which influence may be represented by a series of S-N curves,each measured at a different m;this is depicted schematically in following diagram.Increasing the mean stress level leads to a decrease in fatigue life.Influence of mean stress m on S-N fatigue behaviorSURFACE EFFECTFor many common loading situations,the maximum stress within a component or structure occurs at its surface.Consequently,most cracks leading to fatigue failure originated at surface positions,specifically in stress amplification sites.Therefore,it has been observed that fatigue life is especially sensitive to the condition and configuration of the component surface.Design criteria and various surface treatments will lead to an improvement in fatigue life.Design factorsThe design of a component can have a significant influence on its fatigue characteristics.Any notch or geometrical discontinuity can act as stress raiser and fatigue crack initiation site;these design features include grooves,holes,keyways,threads,and so on.The sharper the discontinuity,the more severe the stress concentration.Avoiding structural irregularities or making design modifications.How design can reduce stress stress amplification Surface treatmentsDuring machining operations,small scratches and grooves are invariably introduced into the workpiece surface by cutting tool action.These surface markings can limit the fatigue life.It has been observed that improving the surface finish by polishing will enhance fatigue life significantly.One of the most effective methods of increasing fatigue performance is by imposing residual compressive stresses within a thin outer surface layer.Residual compressive stresses are commonly introduced into ductile metals by shot peening.Schematic S-N fatigue curves for normal and shot-peened steelCase hardeningProblems 1.1 Briefly explain difference between fatigue striations and beachmarks both in terms of(a)size and(b)origin.1.2 List four measures that may be taken to increase the resistance to fatigue of a metal alloy.1.3 The fatigue data for a brass alloy are given as follows:Stress Amplitude(MPa)Cycles to failure31022319116815314313412721051106310611073107110831081109(a)Make an S-N plot(stress amplitude versus logarithm cycles to failure)use these data.(b)Determine the fatigue strength at 5105 cycles.(c)Determine the fatigue life for 200MPa.1.4 Three identical fatigue specimens (denoted A,B,and C)are fabricated from a nonferrous alloy.Each is subjected to one of the maxium-minimum stress cycles listed bellow.The frequency is the same for three tests.Specimen (MPa)(MPa)ABC+450+400+340-350-300-340(a)Rank the fatigue lifetimes of these specimens from the longest to the shortest.(b)Now justify this ranking using a schematic S-N plot1.5 Make a schematic sketch of the fatigue behavior for some metal for which the stress ratio R has a value of+1.華華中中科科技技大大學(xué)學(xué)學(xué)學(xué)生生成成績績記記載載單單（教教師師專專用用）人數(shù)：55人 班班級級名名稱稱 序序號(hào)號(hào)學(xué)學(xué)號(hào)號(hào)姓姓名名平平時(shí)時(shí)成成績績（占占%）考考試試總總成成績績學(xué)學(xué)期期總總成成績績備備注注月平均成績?nèi)帐马?xiàng)功能材料201301班1U201111167鄧澤斌2U201211126趙杰鋒3U201311294李泌軒4U201311295商尚煬5U201311296張浩6U201311297劉全藝7U201311298盛雪軍8U201311299劉元超9U201311300林星10U201311302李京徽11U201311303李令昌12U201311304喬浩13U201311306宋皓升14U201311308彭龍15U201311309陳汝頌16U201311310劉偉平17U201311311歐志威18U201311312盧錦樺19U201311313唐輝20U201311314周輝21U201311315李瑞林22U201311317羅皓23U201311318倪鳳樓24U201311319胥子旺25U201311320趙雪妍26U201311321郭玥27U201311322周君28U201311323王娟功能材料201302班29U201311324張?zhí)爝h(yuǎn)30U201311325沙武鑫31U201311326李警書32U201311327李思航33U201311329宗此暢34U201311331李敏35U201311332吳常剛36U201311334葉鵬37U201311335鄧澤明38U201311336陶鍇39U201311337汪均鑒40U201311339吳西燚41U201311340楊一凡42U201311341莊華鷺43U201311343李國銳44U201311344鄧霽峰45U201311345李緣46U201311346嚴(yán)明坤47U201311347李強(qiáng)48U201311348李軼偉49U201311349毛疏笛50U201311350許可51U201311351徐丹妮52U201311352李水萍53U201311353蔡瑜54U201311354王昕玥55U201311355劉婧Nanomechanics of HallPetch relationshipin nanocrystalline materialsC.S. Pande, K.P. Cooper*Materials Science and Technology Division, Naval Research Laboratory, Washington, DC 20375-5343, USAa r t i c l ei n f oa b s t r a c tClassical HallPetch relation for large grained polycrystals isusually derived using the model of dislocation pile-up first investi-gated mathematically by Nabarro and coworkers. In this paper themechanical properties of nanocrystalline materials are reviewed,with emphasis on the fundamental physical mechanisms involvedin determining yield stress. Special attention is paid to the abnor-mal or inverse HallPetch relationship, which manifests itself asthe softening of nanocrystalline materials of very small (less than12 nm) mean grain sizes. It is emphasized that modeling thestrength of nanocrystalline materials needs consideration of bothdislocation interactions and grain-boundary sliding (presumablydue to Coble creep) acting simultaneously. Such a model appearsto be successful in explaining experimental results provided a real-istic grain size distribution is incorporated into the analysis.Masumura et al. Masumura RA, Hazzledine PM, Pande CS. ActaMater 1998;46:4527 were the first to show that the HallPetchplot for a wide range of materials and mean grain sizes could bedivided into three distinct regimes and also the first to provide adetailed mathematical model of HallPetch relation of plasticdeformation processes for any material including fine-grainednanocrystalline materials. Later developments of this and relatedmodels are briefly reviewed.Prof. Frank Nabarro was a physicist by training, a metallurgist byprofession and a genius by nature, blessed with a unique ability totreat everyone as his equal. During his later years he was verymuch interested in the mechanical properties of nanocrystallinematerials. This review on that topic is our contribution to the spe-cial issue of Progress in Materials Science honoring him.Published by Elsevier Ltd.0079-6425/$ - see front matter Published by Elsevier Ltd.doi:10.1016/j.pmatsci.2009.03.008* Corresponding author.E-mail address: khershed.coopernrl.navy.mil (K.P. Cooper).Progress in Materials Science 54 (2009) 689706Contents lists available at ScienceDirectProgress in Materials Sciencejournal homepage: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6902.Experimental background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6903.Mechanisms of deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6914.Models using lattice dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6935.Role of Coble creep as a competing mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6956.A generalized expression for yield stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6987.Relationship between hardness and yield strength in metals and alloys. . . . . . . . . . . . . . . . . . . . . . . . 7008.Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7031. IntroductionNanocrystalline materials are polycrystalline materials consisting of grains in nanometer range.They have the potential to exhibit outstanding physical, mechanical and chemical properties, whichcould, in principle, lead to new applications and novel technologies (see Refs. 15). There are nowseveral examples of real applications even at present where the outstanding physical mechanicaland chemical properties of nanomaterials are used for commercial products (e.g. soft magnets, coat-ings, structural repair, etc.). These outstanding properties can be due to interface and nano-scale ef-fects due to the high volume fraction of the interfacial phase (up to 50%), and smaller mean grainsizes (not exceeding 100 nm). Of special importance and subject of this review are the unique mechan-ical properties of nanocrystalline materials, especially yield stress, which are essentially different fromthose of conventional coarse-grained polycrystalline materials. For example, bulk nanocrystallinematerials and thin nanocrystalline coatings in some cases show superhardness, extremely highstrength and good fatigue resistance 514, which are desired for many applications. The highstrength however is often accompanied by low ductility at room temperature, which may limit theirpractical utility. (However, more recently, several researchers have claimed substantial strength aswell as high tensile ductility in nanocrystalline materials 1518.) It has been also surmised thatsome nanocrystalline ceramics and metallic alloys may even exhibit superplasticity at lower temper-atures and faster strain rates than their coarse-grained counterparts 1929. There is also experimen-tal evidence 3033 that nanocrystalline materials may show anomalously fast diffusion, which mayin turn explain their deformation behavior. These developments lead one to hope that nanocrystallinematerials with unique combination of high strength and good ductility may provide new structuraland functional applications in the future. The main aim of this paper is to provide a brief overviewof the theoretical models of yield stress in nanocrystalline materials paying special attention to theirmicroscopic mechanisms. Once established, these mechanisms are then expected to provide theoret-ical underpinnings to the mechanical behavior of nanocrystalline materials.2. Experimental backgroundMechanical behavior of nanocrystalline materials has been the theme of over 500 publications andseveral review articles 3,3450. These articles conclude that yield stress and microhardness of nano-crystalline materials can be 210 times higher than the corresponding coarse-grained polycrystallinematerials with the same chemical composition. Similarly, some published values of microhardness ofnanocrystalline composite coatings 13,14 are of the same order as microhardness (HV? 7090 GPa)of diamond. In the range of grain sizes d above about 10 nm, the dependence of the yield stressson ddeviates little from the classical HallPetch relationship given by the formula,ss0 kd?1=21withs0and k being material constants 612. Yield stress may also depend upon on the mode of pro-cessing 51,52. However, any further grain refinement may lead to lower yield stress. Thus, in the690C.S. Pande, K.P. Cooper/Progress in Materials Science 54 (2009) 689706range of smaller grain sizes, heat-treated materials exhibit the so-called inverse HallPetch behavior(softening with further reduction of grain size). Masumura et al. 53 have plotted some of the avail-able data (till 1998) in a HallPetch plot (see Fig. 1). It is seen that the yield stress-grain size exponentfor relatively large grains appears to be very close to ?1/2, as in Eq. (1), and generally this trend con-tinues until the very fine grain regime (?100 nm) is reached. The large scatter of the data for grainsizes below 100 nm could be attributed to problems in preparing these materials or to differencesin thermal treatments. With the advent of better prepared nanocrystalline materials whose grain sizesare of nanometer (nm) dimensions, the applicability and validity of Eq. (1) as well as the underlyingmechanisms became of great interest. In addition, these nanocrystalline materials were found toexhibit, in general, low tensile ductility at room temperature 612. However more recent resultsindicate that nanocrystalline materials with very low porosity 15 or with dendrite-like inclusions17,18 or with bimodal grain size distributions (consisting of both nano- and micron-sized grains)16 show better ductility. Some reports also indicate nanocrystalline materials with high-strain-rate(tensile) superplasticity 2129.As far as microstructures in these materials are concerned, mechanically loaded nanocrystallinematerials are reported to show grain rotations 27,54, formation of shear bands 5559, or emis-sion of (usually) partial lattice dislocations by grain boundaries into grain interiors 27,29,60. Fig. 2shows a HallPetch plot for copper using early data from various researchers. It also defines thethree regions of the HallPetch plot. The limitation of the classical ideas of HallPetch plots is dra-matically demonstrated in Fig. 3 by plotting the yield strength data in terms of grain size instead of(grain size)?1/2.3. Mechanisms of deformationAs early as 1977, Armstrong and coworkers 61 noted the increase in yield stress on grain refine-ment up to the beginning of the nanocrystalline (100 nm) regime. Much effort has been spent totheoretically describe the HallPetch relationship in nanocrystalline materials. Classically, highFig. 1. Scaled yield stress as a function of (grain size)?1/2for several materials 53.C.S. Pande, K.P. Cooper/Progress in Materials Science 54 (2009) 689706691values for yield stress were considered to be related to the effect of increased grain boundaries pro-viding additional obstacles for movement of lattice dislocations.In early theoretical studies, models of nanocrystalline materials were considered as two-phasecomposites consisting of nanograin interiors and grain-boundary regions (see, e.g. Refs. 55,6268). Yield stresssis then accounted for using the so-called rule-of-mixture, yield stresssbeinggiven by some weighted sum of the yield stresses characterizing the grain-interior and grain-bound-ary phases. The ratio of the two phases, of course, strongly depends on the grain size d. In thiscalculation, the yield stress of the grain-boundary phase is assumed to be lower than that of thegrain-interior phase, and with suitable adjustable parameters, the deviations from the conventionalHallPetch relationship can be described roughly in accordance with experimental data. It isobvious that the rule-of-mixture” approach is too approximate and arbitrary and sheds no lighton the actual mechanisms 11.Fig. 2. Compilation of yield stress data for pure copper from various publications 53.Fig. 3. Plots showing limitation of standard HallPetch law at small grain sizes and existence of optimum grain size for yieldstrength 49,53. (a) Schematic of hardness or strength as a function of normalized grain size shows the limitation dramatically.(b) Normalized yield strength plotted against (normalized grain size)?1/2.692C.S. Pande, K.P. Cooper/Progress in Materials Science 54 (2009) 689706Our goal is to briefly review a more precise physical mechanism of plastic flow in nanocrystallinematerials in terms of lattice dislocations, grain-boundary dislocations, vacancies and grain rotationsoccurring in mechanically loaded nanocrystalline materials. At present, there are many theoreticalmodels of the abnormal HallPetch effect based on different deformation mechanisms and claimingagreement with the corresponding experimental data from nanocrystalline materials. The followingplastic deformation mechanisms have been mentioned acting individually or in competition: (1)grain-boundary sliding, (2) grain-boundary diffusional creep, (3) triple junction diffusional creep,(4) rotational deformation (occurring through motion of grain-boundary disclinations) and (5) latticedislocations. The experimental data is usually not precise enough to allow one to select a theoreticalconcept from a variety of theoretical models describing the same experimental data using variousmechanisms. In this context, in next sections of this review article, we will pay special attention totheoretical models of plastic deformation mechanisms in nanocrystalline materials, that can accountfor some additional microstructural results (either experimental or computational) and provide math-ematical results rather than qualitative concepts. Needless to say, the subject is still a matter of somecontroversy.4. Models using lattice dislocationsThe most obvious idea is to use conventional lattice dislocation slip model for nanocrystallinematerials, but taking into account the influence of smaller grain sizes and high-density ensemblesof grain boundaries on the formation of lattice dislocation pile-ups in grain interiors. Thus, this treat-ment extends the classical derivation but assumes that there are very few dislocations available in anyone grain.For this purpose it is instructive to start with a brief discussion of the models describing the clas-sical HallPetch relationship (Eq. (1) in coarse-grained polycrystals. Most of these models use theconcept of dislocation pile-ups (see review by Li and Chou 69). In deriving the HallPetch relation,grain boundaries here are considered as barriers to dislocation motion 70,71, causing stresses to con-centrate and activating dislocation sources in the neighboring grains, thus initiating the slip from grainto grain. In other type of models, though mentioned less often 72,73, the grain boundaries are re-garded as dislocation barriers limiting the mean free path of the dislocations, thereby increasing strainhardening and resulting in a HallPetch type relation. Several variations of these concepts are possible.It is also possible that several dislocation processes could compete or reinforce the deformationprocess.Pande and Masumura 74 were the first to extend mathematically the classical derivation of HallPetch relation to nanocrystalline materials. They assumed that the classical HallPetch dislocationpile-up model is still dominant with the sole exception that the analysis must take into account thefact that in nanocrystalline materials with small grain sizes, the number of dislocations in a pile-upwithin a grain cannot be very large. In the limit at still smaller grain sizes, this mechanism shouldcease when the grains are so small that there are only two dislocations in the pile-up. Mathematically,the model utilizes the fact that the length of the pile-up is no longer proportional to the number ofdislocations in the pile-up if the pile-up is not large.Then, Pande and Masumura 74, by considering the conventional HallPetch model, showed that adislocation theory for the HallPetch effect does not give a linear dependence ofson d?1/2when grainsizes are in the nanometer range. When the number of dislocations in the pile-up falls to one, no fur-ther increase in the yield stress is possible by this mechanism and it saturates. As mentioned before, ifthe number of dislocations n in a pile-up is not too large, the length of the pile-up L is not linear in n.Chou 75 gives the relation between L and n as:L ffiA2s4 n m ? 1 ? 2i12n3?1=3 !?;2where i1= 1.85575 and mb is the Burgers vector of the lead dislocation in the pile-up. (The lead dis-location could be in the grain-boundary itself and, hence, may have a Burgers vector different fromthe rest of the dislocations.)C.S. Pande, K.P. Cooper/Progress in Materials Science 54 (2009) 689706693Pande and Masumura 74 give an improved expression,L ffiA2s2n m ? 11=2?i1e121=3n m ? 11=6i1#;3whereeis a small correction term (e? 1) and can be neglected. Then, following the classical analysis,but using Eq. (3), they find that for small grain sizes there are additional terms to the HallPetchrelation,s k?1=2 c1k?1=25=3 c2k?1=27=3;4where s =s/m/s*, c1= ?0.6881, c2= 0.1339 and l = Lms*/2A. Eq. (4) is expected to be correct for allgrain sizes, as long as dislocation pile-up mechanism is operating. This model thus recovers the clas-sical HallPetch at large grain sizes but for smaller grain sizes theslevels off. This mechanism there-fore cannot explain a drop ins. If, on the other hand, the yield stress is source limited,s? Gb/d, i.e., theyield stress should rise as d?1. Thus, from these arguments, at smaller grain sizes, either the yieldstress should rises faster than d?1/2or it should saturate, but it should not decrease. The HallPetchplot using Eq. (4) is given in Fig. 4 showing the leveling, but not the inverse HallPetch curve.Several researchers 7678 have developed models similar to that of Pande and Masumura 74.On the other hand, Malygin 79 has suggested that dislocations are absorbed in grain boundaries,the effect being larger in nanocrystalline materials. The assumption is that grain boundaries act pre-dominantly as sinks for dislocations (just the opposite to what was proposed by Li 80, who consid-ered grain boundaries as sources for dislocation generation). Malygins model is attractive as adislocation mechanism and should be considered further. However, we point out two problems withthe model. First, it is doubtful if the dislocations play the same role whether the grains are large orsmall. It is more likely that dislocations in ultrafine grains, if present at all, are confined to grainboundaries 81. Second, in Malygins model 79, the stress calculated is a work-hardened flow stressrather than a yield stress and, as with Li 80, it merely assumes a relation between dislocation densityand yield stress, instead of proving it.Lu and Sui 82 assume an enhancement of lattice dislocation penetration through the grain bound-aries and the corresponding softening of nanocrystalline materials. Scattergood and Koch 83 assumethat the yield stress of fine-grained materials is controlled by the intersection of mobile latticedislocations with the dislocation networks at grain boundaries. Zaichenko and Glezer 84 have pro-posed rotational defects (disclinations), formed at triple junctions, as sinks and sources of the latticedislocations moving in grain interiors and causing the plastic flow in nanocrystalline materials (seealso Ref. 85).Fig. 4. Plot of HallPetch dislocation model for nanocrystalline materials 74.694C.S. Pande, K.P. Cooper/Progress in Materials Science 54 (2009) 689706All these models assume the existence of lattice dislocations, which play the same role in nanograininteriors as with conventional coarse grains. This may not be energetically favorable 86,87 in nano-crystalline materials. Transmission electron microscopy experiments 6,88 show reduced density oreven absence of dislocations in nanocrystalline materials. However, one cannot rule out that the mod-els based on lattice dislocation slip may explain to some extent the deformation behavior of nanocrys-talline materials with grain size in the intermediate range (d ? 30100 nm).5. Role of Coble creep as a competing mechanismAs we have seen, at sufficiently small grain sizes, the HallPetch model based upon the lattice dis-locations may not be operative. Clearly, there is a need to consider additional mechanisms for verysmall grain sizes (less than 10 nm). In this section, we consider the deformation mechanisms associ-ated with enhanced diffusion along grain boundaries in nanocrystalline materials. Chokshi et al. 89were the first to suggest grain-boundary diffusional creep (Coble creep) as the dominant deformationmode in nanocrystalline solids. In this region, they have proposed that even at room temperature Co-ble creep may be operative in order to explain their results. Certainly, there is a qualitative order ofmagnitude agreement with this mechanism and the trend is correct, however, the functional depen-dence ofson d given by Choksi et al. 89 is incorrect as pointed out by Neih and Wadsworth 90.Conventional Coble creep demands thats? d3/d?1/26, i.e., thesvs. d?1/2curve falls very steeplyas d?1/2increases. This is not found experimentally 76. Chokshi et al. 89 showed that their datafit better the relation,s b ? K0? d?1=25with b = 937 MPa and K0= 0.027 MPa m1/2instead ofs? d3. Eq. (5) cannot be related simply to anyknown mechanism.Even if the Coble creep argume