Geometry of Deformation and

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Chapter 6

• Geometry of Deformation and
• Work-Hardening

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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.
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Work-Hardening of a Material
Stressstrain curves (schematic) for an elastic,
ideally plastic a work-hardening and
a work-softening material.
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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.)
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Compression Tests on TiC at Different Temperatures
Stressstrain curves for annealed polycrystalline
TiC deformed in compression at the temperatures
indicated (e 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.)
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Shear stress-Shear Strain Response of Al2O3
Shear stress t vs. shear strain ? for prism plane
slip in Al2O3 at various temperatures ?e 3.5
10-4 s-1 for the solid curves, ?e 1.4 10-4
s-1 for the dashed curves. (Courtesy of T.
E. Mitchell.)
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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.)
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24 Triangles of Stereographic Projections
Standard 001 stereographic projection
divided into 24 triangles.
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Slip Plane and Slip Direction-Schmid Law
Relationship between loading axis and slip plane
and direction.
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Schmids Law and Schmids Factor
Effect of orientation on the inverse of Schmids
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.)
Comparison of Schmid law prediction with
experimental results for zinc. (Adapted
with permission from D. C. Jillson, Trans. AIME,
188 (1950) 1120.)
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Plastic Deformation- Rotation of Slip Direction
Stereographic projection showing the rotation
of slip plane during deformation. Direction P1,
inside stereographic triangle moves towards P2
on boundary 100111. Then, P2 moves toward
211.
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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.)
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Cross-Slip
Generic shear-stressshear-strain curves for FCC
single crystals for two different temperatures.
Model of cross-slip.
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Work-Hardening in Polycrystals
Relationship between flow shear stress and
dislocation density for monocrystalline sapphire
(A13O3) deformed at different temperatures.
(Adapted from B. J. Pletka, A. H. Heuer, and T.
E. Mitchell, Acta Met., 25 (1977) 25.)
Average dislocation density ? as a function of
the resolved shear stress t for copper. (Adapted
with permission from H. Wiedersich, J. Metals, 16
(1964) p. 425, 427.)
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Taylor Model of Work Hardening
? k?b
Taylor model of interaction among dislocations in
a crystal.
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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.
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Kuhlmann-Wilsdorfs 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.
KuhlmannWilsdorf.)
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Load Deformation Curve
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.)
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Softening Mechanism
(a) Compressive true-stresstrue-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-stressshear-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.)
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Shear Bands in Titanium
Shear bands in titanium. (a) Optical
micrograph, showing band. (b) Transmission electro
n micrograph, showing microcrystalline structure,
with grain size approximately equal to 0.2 µm.
The original grain size of the specimen was 50 µm.
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Texture due to Rolling
Perspective view of microstructure of Nickel-200
cold rolled to a reduction in thickness of 60.
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Texture Strengthening
Orientation dependence of yield strength sy and
strain to fracture, ef, of a rolled copper sheet.
Theoretical bounds on the Youngs modulus E of
steel.
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Rolled-brass Sheet
111 pole figure of a rolled-brass sheet.