mechanical behaviour of materials chapter 4

Chapter 4Imperfections: Point and Line

Defects

Dimensional Range for Different Classes of Defects

Types of Imperfections:

1. Point Defects – 0-Dimensional ImperfectionsLocalized (foreign atoms, vacancies, extra or missing e)

2. Line Defects – 1-Dimensional ImperfectionsExtend through crystal on a line (dislocations)

3. Interfacial Defects – 2-Dimensional or Planar ImperfectionsBoundaries between regions of order (order can be atomic,

magnetic, electronic, or chemical)

4. Bulk Defects – 3-Dimensional ImperfectionsMacroscopic or large scale defects (voids, cracks and inclusions)

Stress Required to Shear a Crystal

Theoretical Shear Strength of Some Materials

Theoretical Shear Strength of Some Materials (adapted from Hosford, W. F., Mechanical Behavior of Materials, Cambridge University Press

(2008).)

Atomic point defects.

Two most common point defects in compounds:

Schottky and Frenkel defects.

Point Defects

Atomic point defects.

Two most common point

defects in compounds:Schottky and Frenkeldefects.

Point Defects(adapted from Barrett, Nix and Tetelman, The Principles of Engineering Materials, Prentice Hall, Inc. (1973).)

Interstices in FCC structure. (a)

Octahedral void. (b) Tetrahedral void.

Interstices in the BCC structure. (a)

Octahedral void. (b) Tetrahedral void.

Interstices in the HCP structure. (a)

Octahedral void. (b) Tetrahedral void.

Point Defects

Formation of point defects by the annihilation of

dislocations. (a) Row of vacancies. (b) Row of interstitials.

Formation of Point Defects

Shear stress-versus-strain curves for aluminum

single crystals. The crystallographic orientation isshown in the stereographic triangle. (Adapted with permission from A. H. Cottrell, Phil. Mag., 46 (1955) p. 737.)

Shear stress-Shear Strain Curves for Aluminum Single Crystal

Seeger model of damage produced by

irradiation. P indicates the position where the first “knock-on” terminates.(Reprinted with permission fromA. Seeger, in Proc. Symp. Radiat.Damage Solids React., Vol. 1,

(Vienna, IAEA, 1962) pp. 101, 105.)

Voids formed in nickel irradiated using 400

keV 14N2+ ions to a dose of 40 dpa at 500 ◦C; notice the voids with polyhedral shape; dpa = displacements per atom. (Courtesy of L. J. Chen andA. J. Ardell.)

Radiation Damage

Stress–strain curves for irradiated and

unirradiated Zircaloy. (Adapted with permission from J. T. A. Roberts, IEEE Trans. Nucl. Sci., NS-22, (1975) 2219.)

Radiation Damage

Stress-free dilation in AISI 316 steel (20% cold

worked). (Adapted with permission from J.T. A. Roberts, IEEE Trans. Nucl. Sci., NS-22, (1975) 2219.)

Dependence of fast neutron-induced dilation

in stainless steel (Fe–Cr–Ni) as a function of Ni and Cr amounts. (Adapted with permission

from W. B. Hillig, Science, 191 (1976) 733.)

Radiation Damage

(a) Rug with a fold.

Caterpillar with a hump.

Line Defects

Arrangement of atoms in an edge dislocation and the Burgers vector b

that produces closure of circuit ABCDE.

Edge and Screw Dislocations

Arrangement of atoms in a screw dislocation with “parking garage”

setup. Notice car entering garage.

Edge and Screw Dislocations

.

(a) Perfect crystal. (b) Edge dislocation. (c) Screw dislocation.

Plastic deformation of a crystal by the

movement of a dislocation along a slip plane.

Plastic Deformation

Shear Produced by Dislocation Movement

(adapted from Barrett, Nix and Tetelman, The Principles of Engineering Materials, Prentice Hall, Inc. (1973).)

Mixed dislocation obtained from cut-

and-shear operation; notice the anglebetween b and dislocation line.

Mixed Dislocation

(a) Titanium. (Courtesy of B. K. Kad.) (b) Silicon.

Dislocations in Metals

Dislocations in (a) Al2O3 and (b) TiC. (Courtesy of J. C. LaSalvia.)

Dislocations in Al2O3 and TiC

Atomic resolution transmission electron micrograph of dislocation inmolybdenum with a Burgers circuit around it. (Courtesy of R. Gronsky.)

Dislocation in Molybdenum

Square Dislocation Loop

Elliptic dislocation loop. (a) Intermediate position. (b) Final (sheared) position. (c) TEM of shear

loop in copper. (Courtesy of F. Gregori and M. S. Schneider.)

Elliptic Dislocation Loop

Prismatic loop produced by the introduction

of a disk into metal. (a) Perspective view. (b) Section AAAA. (c) Section BBBB.

Prismatic Loop

Slip produced by the movement of dislocation.

(a) Positive and negative edge dislocations. (b) Positive and negative screw dislocations.

Movement of Dislocation

Expansion of a Dislocation Loop

Stresses due to Dislocations

Screw Dislocation Edge Dislocation

Stress fields around an edge

dislocation. (The dislocation line is Ox3), (a) σ11; (b) σ22; (c) σ33; (d) σ12. (Adapted with permission from J. C. M. Li, in Electron Microscopy and Strength of Crystals, eds. G. Thomas and J. Washburn (New York: Interscience Publishers, 1963).)

Stress Fields Around a Edge Dislocation

Energy of a Dislocation

Schematic representation of an idealized dislocation array (a) in two dimensions (b) in three dimensions; note that dislocations on three perpendicular atomic planes define a volume V.

Dislocation Array

Bending of a Dislocation

Dislocations in an FCC Crystal

Peach-Koehler Equation

Decomposition of a dislocation b1 into two partial dislocations b2

and b3, separated by a distance d0.

Decomposition of Dislocation

Stacking Fault Energies of Some Metals

Short segment of stacking fault in AISI

304 stainless steel overlapping with coherent twin boundary. Differences in the nature of these defects are illustrated by fringe contrast differences.

Stacking Fault and Partial Dislocations

Dislocations in AISI 304 stainless steel splitting

into partials bounded by short stacking-fault region. Partials spacing marked as d. (Courtesy of L. E. Murr.)

Effect of stacking-fault energy on dislocation

substructure. (a) High-stacking-fault-energy material (pure copper); (b) Low-stacking-fault-energy material (copper–2 wt% aluminum).

Both materials were laser-shock compressed with an initial pressure of 40 GPa and pulse duration of 3 ns. (Courtesy of M. S. Schneider.)

Effects of Stacking-Fault Energy on Dislocation Substructure

Frank or Sessile dislocations.

(a) Intrinsic. (b) Extrinsic.

Frank or Sessile Dislocations

Cottrell–Lomer lock.

Stairway dislocation.

Cottrell –Lomer and Stairway Dislocations

Basal, pyramidal, and prism plane in HCP structure.

Important Planes in HCP Structure

Temperature for Macroscopic Plasticity in Some Ceramics

Slip Systems and Burgers Vectors in Some Ceramics

Screw Dislocation

Edge Dislocation

General Form

Expressions for Energy of Dislocation

Basal Plane in Al2O3

Elastic Energy for Dislocations in Ceramics

(a) Dislocations, dipoles, and loops in sapphire.

(b) Interaction between dislocations insapphire. (From K. P. D. Lagerdorf, B. J. Pletka, T. E. Mitchell, and A. H. Heuer, Radiation Effects, 74 (1983) 87).

Dislocations in Sapphire

Hexagonal array of dislocations in titanium diboride. (Courtesy of D. A. Hoke and G. T. Gray.)

Stacking faults in GaP.

(Courtesy of P. Pirouz.)

Dislocations in Titanium Diboride

Homogeneous Nucleation of Dislocations

Emission of dislocations from ledges in grain boundary, as observed in transmission electron microscopy during heating by electron beam.

(Courtesy of L. E. Murr.)

Grain Boundary as a Source of Dislocations

Effect of oxide layer on the tensile properties of niobium.(Reprinted with permission fromV. K. Sethi and R. Gibala, ScriptaMet. 9 (1975) 527.)

Effect of Oxide Layer on the Tensile Properties of Niobium

Formation of dislocation loop by the Frank–Read mechanism.

Frank-Read Mechanism

Frank–Read source formed by cross-

slip.

Dislocation Source: Cross Slip

Epitaxial growth of thin film. (a) Substrate.

(b) Start of epitaxial growth. (c) Formation ofdislocations.

Epitaxial Growth

Pileup of dislocations against a

barrier.

Pileup of dislocations against grain

boundaries (or dislocations being emitted from grain boundary sources?) in copper observed by etch pitting.

Dislocation Pileups

)a) Edge dislocation traversing “forest”

dislocation. (b) Screw dislocation traversing “forest” dislocations.

Dislocation Interactions

(a) Kink and jog in edge dislocation. (b)

Kink and jog in screw dislocation.

Loop being pinched out when jog is left behind

by dislocation motion.

Kinks and Jogs in Dislocations

Orowan’s Equation

k b

(a) Movement of dislocation away from its equilibrium position. (b) Variation of Peierls–Nabarro stress with distance. (Reprinted with permission from H. Conrad, J. Metals, 16 (1964), 583.)

Peierls-Nabarro Stress

Overcoming of Peierls barrier by Seeger kink pair mechanism.

(a) Original straight dislocation. (b) Dislocation with two kinks. (c) Kinks moving apart.

Overcoming of Peierls Barrier

Effect of temperature on Young’s modulus. (Adapted from J. B. Wachtman Jr.,W. E. Tefft, D. G. Lam, Jr., and C. S. Apstein, J. Res. Natl. Bur. Stand., 64A (1960) 213; and J. Lemartre and J. L. Chaboche, Mechanics of Solid Materials, Cambridge: Cambridge

University Press, 1990, p. 143.)

Temperature Effect on Young’s Modulus

Flow stress as a function of temperature for (a) an idealized material, (b) BCC metals, and (c) FCC metals. Notice the greater temperature dependence for Ta and Fe (BCC).

Flow Stress as a Function of Temperature

Stresses and dislocations generated at film-substrate interface; (a) Film and substrate with different lattice parameters; (b) elastic (coherent) accommodation of strains by film;(c) elastic + dislocation (semi-coherent) accommodation of strains at a film thickness greater than hc.(Adapted from W. D. Nix, Met. Trans., 20A (1989)2217.)

Dislocations on Film-Substrate Interface

Critical film thickness as a function of misfit strain;

the greater fraction Ge, the greater the misfit stain and the smaller hc. Predictions from van der Merwe Matthews theory; measurements from J. C. Bean, L. C. Feldman, A. T. Fiory, S. Nakahara, and I. K. Robinson, J. Vac. Sci. Technol. A, 2 (1984) 436.(Adapted from W. D. Nix., Met. Trans., 20A (1989) 2216.)

Critical Film Thickness vs. Atomic Fraction of Ge

Mechanisms of misfit dislocation generation; (a)

Freund mechanism in which a “threading”dislocation preexisting in substrate lays over interface creating misfit dislocation; (b) Nix mechanism, in which a surface source creates

half-loops that move toward interface.

Misfit Dislocation Generation