Chapter 6
Geometry of Deformation and
Work-Hardening
Common Metal Working Methods
Common metalworking methods. (a)
Rolling. (b) Forging (open and closed die).
(c) Extrusion (direct and indirect).
(d) Wire drawing. (e) Stamping.
Work-Hardening of a Material
Stress–strain curves (schematic) for an
elastic, ideally plastic; a work-hardening;
and a work-softening material.
Engineering Stress-Strain Curves for Nickel
Engineering-stress– engineering-strain curves for nickel. (a)
Nickel subjected to 0, 20, 40, 60, 80, and 90% cold-rolling
reduction. (b) Nickel cold rolled to 80%, followed by
annealing at different temperatures. (From D. Jaramillo, V.
S. Kuriyama, and M. A. Meyers, Acta Met. 34 (1986) 313.)
Compression Tests on TiC at Different Temperatures
Stress–strain curves for annealed
polycrystalline TiC deformed in
compression at the temperatures indicated
(ε = 1.7 × 10−4 s−1). (Adapted from G. Das,
K. S. Mazdiyasni, and H. A. Lipsitt,
J. Am. Cer. Soc., 65 (Feb. 1982) 104.)
Shear Stress-Shear Strain Response of Al2O3
Shear stress τ vs. shear strain γ for prism
plane slip in Al2O3 at various temperatures; έ
= 3.5 × 10−4 s−1 for the solid curves, έ = 1.4 ×
10−4 s−1 for the dashed curves. (Courtesy of
T. E. Mitchell.)
Stereographic Projections
(a) Representation of crystallographic directions and poles
(normals to planes) for cubic structure. (b) Standard [100]
stereographic projection. (Reprinted with permission from C.
S. Barrett and T. B. Massalski, The Structure of Metals, 3d
ed. (New York: McGraw-Hill, 1966), p. 39.)
Standard Stereographic Projection
Standard [001] stereographic projection
divided into 24 triangles.
Slip Plane and Slip Direction-Schmid Law
Relationship between loading axis and slip
plane and direction.
Schmid’s Law and Schmid Factor
Comparison of Schmid’s law
prediction with experimental
results for zinc. (Adapted with
permission from D. C. Jillson,
Trans. AIME, 188 (1950) 1120.)
Effect of orientation on the inverse of
Schmid factor (1/M) for FCC metals.
(Adapted with permission from G. Y.
Chin, “Inhomogeneities of Plastic
Deformation,” in The Role of Preferred
Orientation in Plastic Deformation
(Metals Park, OH: ASM, 1973), pp. 83,
85.)
Plastic Deformation- Rotation of Slip Plane
Stereographic projection showing the rotation of
slip plane during deformation. Direction P1,
inside stereographic triangle, moves toward P2
on boundary [100]–[111]. Then, P2 moves
toward [211].
Shear-Stress vs. Shear-Strain Curve for Nb (BCC)
Shear-stress vs. shear-strain curves for Nb (BCC) monocrystals
at different crystallographic orientations; arrows indicate
calculated strain at which conjugate slip is initiated. (From T. E.
Mitchell, Prog. App. Matls. Res. 6 (1964) 117.)
Cross-Slip
Generic shear-stress–shear-strain curves
for FCC single crystals for two
different temperatures.
Model of cross-slip.
Shear Stress-Shear Strain Curves
for FCC Single Crystals
Generic shear-stress–shear-strain curves for FCC
single crystals for two different temperatures.
Cross-Slip
Work-Hardening in Polycrystalline Cu
Average dislocation density ρ as a function of the resolved
shear stress τ for copper. (Adapted with permission from H.
Wiedersich, J. Metals, 16 (1964) p. 425, 427.)
Work-Hardening in Polycrystalline Alumina
Relationship between flow shear stress and
dislocation density for monocrystalline sapphire
(A12O3) deformed at different temperatures.
(Adapted from B. J. Pletka, A. H. Heuer, and T. E.
Mitchell, Acta Met., 25 (1977) 25.)
Taylor Model of Work Hardening
Taylor model of interaction among dislocations in a crystal.
Dislocation Cells
Development of substructure of Nickel-200 as a
function of plastic deformation by cold rolling. (a)
20% reduction. (b) 40% reduction. (c) 80% reduction.
Kuhlmann-Wilsdorf’s Work Hardening Theory
Schematic representation of dislocation cells of size L, with
activation of dislocation sources from the cell walls and bowing out
of loops into the cell interior. (Courtesy of D. Kuhlmann–Wilsdorf.)
Load-Deformation Curve fro Concrete
Typical load deformation curve for
concrete under uniaxial compression;
the specimen was unloaded and
reloaded at different stages of
deformation. (From G. A. Hegemier
and H. E. Reed, Mech. Mater., 4
(1985) 215; data originally from A.
Anvar.)
Work Softening
(a) Compressive true-stress–true-strain curves for
titanium at different strain rates; notice the onset of
softening at the arrows. (Adapted from M. A.
Meyers, G. Subhash, B. K. Kad, and L. Prasad,
Mech. Mater., 17 (1994) 175.)
(b) Schematic linear shear-stress–shear-strain
curves for titanium at different temperatures, with
superimposed adiabatic curve constructed from
isothermal curves by incrementally converting
deformation work into heat (and a consequent rise in
temperature.) (Adapted from M. A. Meyers and H. R. Pak, Acta Met., 34 (1986) 2493.)
Shear Bands in Titanium
Shear bands in titanium. (a) Optical micrograph,
showing band. (b) Transmission electron
micrograph, showing microcrystalline structure,
with grain size approximately equal to 0.2 μm. The
original grain size of the specimen was 50 μm.
Rolling Texture
Perspective view of microstructure of Nickel-200 cold
rolled to a reduction in thickness of 60%.
Texture Strengthening
Theoretical bounds on the Young’s
modulus E of steel.
Orientation dependence of yield strength and
strain to fracture of a rolled copper sheet.
Common Wire and Sheet Textures
Rolled-Brass Sheet
[111] pole figure of a rolled-brass sheet.