Chapter 5:

Imperfections: Interfacial and

Volumetric Defects

Grains in a metal orceramic; the cube depicted in eachgrain indicates the crystallographicorientation of the grain in aschematic fashion.

Grains Orientation

Micrographs showingpolycrystalline (a) tantalum and (b)TiC.

Grains of Tantalum and TiC

Low-anglegrain-boundary observed byhigh-resolution transmissionelectron microscopy. Positions ofindividual dislocations are markedby Burgers circuits. (Courtesy ofR. Gronsky)

Mean Linear Intercept

Low-angle tilt boundary.

Tilt and Twist Boundaries

Low-angle twist boundary

Variation ofgrain-boundary energy γ gb withmisorientation θ. (Adapted withpermission from A. G. Guy,Introduction to Materials Science(New York: McGraw-Hill, 1972), p.212.)

Grain-Boundary Energy with Misorientation

Coincidence latticemade by every seventh atom in thetwo grains, misoriented 22◦ by arotation around the <111> axis.(Adapted from M. L. Kronberg andH. F. Wilson, Trans. AIME, 85(1949), 501.)

Coincidence Boundary

Interface betweenalumina and NiAl2O4 (spinel). (a)High-resolution TEM. (b)Representation of individualatomic positions. (Courtesy ofC. B. Carter.)

Interface between alumina and NiAl2O4

The effect of grain sizeon calculated volume fractions ofintercrystal regions and triplejunctions, assuming a grainboundary thickness of 1 nm.(Adapted from B. Palumbo, S. J.Thorpe, and K. T. Aust, ScriptaMet., 24 (1990) 1347.)

Grain Size vs. Volume Fraction of Intercrystal Regions

Models of ledgeformation in a grain boundary.(Reprinted with permission fromL. E. Murr, Interfacial Phenomena inMetals and Alloys (Reading, MA:Addison Wesley, 1975), p. 255.)

Ledge Formation in Grain Boundary

Grain boundary ledgesobserved by TEM. (Courtesy of L.E. Murr.)

Grain Boundary Ledges

Image and atomicposition model of anapproximately 32◦ [110] tiltboundary in gold; note thearrangement of polygons, whichrepresents the boundary. (FromW. Krakow and D. A. Smith, J.Mater. Res. 22 (1986) 54.)

Tilt Boundary

Schematic of twinningin FCC metals.

Twinning in FCC metals

Deformation twins in(a) iron-silicon (Courtesy of O.V¨ohringer) and (b) stainless steel.

Deformation Twins

Deformation twins in silicon nitride observed by TEM. (a) Bright field.(b) Dark field. (c) Electron diffraction pattern showing spots from two twin variants, Aand B. (Courtesy of K. S. Vecchio.)

Deformation Twins in Silicon Nitride

Serrated stress–straincurve due to twinning in a Cdsingle crystal. (Adapted withpermission from W. Boas and E.Schmid, Z. Phys., 54 (1929) 16.)

Stress-Strain Curve Due to Twinning

Twinning in HCPmetals with c/a ratio more than orDislocation motion at less √3.

Twinning in HCP Metals

Effect of temperature on the stress required for twinning and slip (at low andhigh strain rates). (Courtesy of G. Thomas.)

Twinning Due to Dislocation Motion

(a) Stress–straincurves for copper (which deformsby slip) and 70% Cu–30% Zn brass(which deforms by slip andtwinning). (b) Work-hardeningslope dσ/dε as a function of plasticstrain; a plateau occurs for brass atthe onset of twinning. (After S.Asgari, E. El-Danaf, S. R. Kalidindi,and R. D. Doherty, Met. and Mater.Trans., 28A (1997) 1781.)

Mechanical Effects

Effect of temperatureon twinning stress fore a numberof metals. (From M. A. Meyers, O.Voehringer, and V. A. Lubarda, ActaMater., 49 (2001) 4025.)

Effect of stacking-faultenergy on the twinning stress forseveral copper alloys. (From M. A.Meyers, O. Voehringer, and V. A.Lubarda, Acta Mater., 49 (2001)4025.)

Effect of Temperature and Stacking-Fault Energy on Twinning Stresses

Temperature–strainrate plots with slip and twinningdomains; (a) effect of grain size intitanium; (b) effect of stacking-faultenergy in copper–zinc alloys.(FromM. A. Meyers, O. Voehringer, andV. A. Lubarda, Acta Mater., 49(2001) 4025.)

Temperature-Strain Rate Plots

Hall–Petch plot for anumber of metals and alloys. Y. S.indicates yield strength.

Grain-Size Strengthening

Hall–Petch plot foriron and low-carbon steelextending from monocrystal tonanocrystal; notice the change inslope. (After T. R. Smith, R. W.Armstrong, P. M. Hazzledine, R. A.Masumura, and C. S. Pande, Matls.Res. Soc. Symp. Proc., 362 (1995)31.)

Hall-Petch Plot

σy = σ0 + kD−1/2

σy - is the yield stressσ0 - is a frictional stress required to move dislocations k - is the H--P slopeD is the grain size

Frank–Read sourceoperating in center of grain andproducing two pileups at grain 1boundaries; the Frank–Readsource in grain 2 is activated bystress concentration.

Frank-Read Source

Dislocation activity atgrain boundaries in AISI 304stainless steel (˙ε = 10−3 s−1). (a)Typical dislocation profiles after astrain of 0.15%. (b) Same after astrain of 1.5%. (Courtesy of L. E.Murr.)

Dislocation Activity at Grain Boundaries in Stainless Steel

Sequence of stages in(a) polycrystalline deformation,starting with (b) localized plasticflow in the grain-boundary regions(microyielding), forming (c) awork-hardened grain-boundarylayer that effectively reinforces themicrostructure.

Meyers-Ashworth Theory

Deformation twins inshock-loaded nickel (45 GPa peakpressure; 2 μs pulse duration).Plane of foil (100); twinning planes(111) making 90◦. (Courtesy ofL. E. Murr.)

Strength of wire-drawnand recovered Fe–0.003% C as afunction of transverselinear-intercept cell size. Recoverytemperatures (in ◦C) as indicated.(Adapted with permission from H.J. Rack and M. Cohen, in Frontiersin Materials Science: DistinguishedLectures, L. E. Murr, ed. (New York:M. Dekker, 1976), p. 365.)

Deformation Twins

Gleiter representationof atomic structure of ananocrystalline material; whitecircles indicate grain-boundaryregions. (Courtesy of H. Gleiter.)

Nanocrystalline Materials

Stress–strain curvesfor conventional (D = 50 μm) andnanocrystalline (D = 25 μm)copper. (Adapted from G. W.Nieman, J. R. Weertman, and R.W. Siegel, Nanostructured Materials,1 (1992) 185.)

Hall–Petch relationshipfor nanocrystalline copper. (AfterG. W. Nieman, J. R. Weertman,and R. W. Siegel, NanostructuredMatls., 1 (1992) 185)

Hall-Petch Relationships

Classical Hall–Petchslope compared withMeyers–Ashworth equation andcomputations assuming agrain-boundary region and graininterior with differentwork-hardening curves. As grainsize is decreased, grain-boundaryregion gradually dominates thedeformation process. (From H.-H.Fu, D. J. Benson, and M. A. Meyers,Acta Mater., 49 (2001) 2567.)

Classical Hall-Petch

Voids (dark spotsmarked by arrows) in titaniumcarbide. The intergranular phase(light) is nickel, which was added toincrease the toughness of the TiC.

Voids in Titanium Carbide

(a) Transmissionelectron micrograph illustratingfaceted grain-interior voids withinalumina and (b) voids in titaniumcarbide; dislocations are pinned byvoids.

Voids