Chapter 4: Dislocations Stress Fields

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Chapter 4 Dislocations Stress Fields

• Dislocations produces deformation via
  incrementally breaking bonds.
• If dislocations don't move, no plastic
  deformation happens!
• 3 types of dislocations, Screw, Edge and Mixed,
  which move differently.

ISSUES TO ADDRESS...
What are the forces on and between
dislocations?

• What is the Peach-Kohler Force?
• How do we calculate the thing?
• Does it agree with estiamtes for simple cases we
  already know?

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BCC Slip Planes and Directions
Principal slip system, but other closed-packed
directions
110 planes in the direction of
?Fe, Mo, W, ? brass
Slip systems 6 x 2 12
211 planes in the direction of
?Fe, Mo, W, Na
Slip systems 12 x 1 12
321 planes in the direction of
?Fe, K
Slip systems 24 x 1 24
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HCP Slip Planes and Directions
Principal slip system can depend on c/a and
relative orientation of load to slip planes
hcp Zinc single crystal
0001 planes in the direction of
Slip systems 1 x 3 3
c/a 1.6333 (ideal)
Cd, Zn, Mg, Ti, Be
Adapted from Fig. 7.9, Callister 6e.
planes in the direction of
Slip systems 3 x 1 3
Ti
planes in the direction of
Adapted from Fig. 7.8, Callister 6e.
Slip systems 6 x 1 6
c/a 1.6333 (ideal)
Mg, Ti
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Energy of Screw Dislocation
Only non-zero displacements for screw.
u3 gives the only non-zero strains
Screw
Roll open cylinder, shear displacement is
Energy/Volume for (elastic) distortion
and
For infinitesimal region,
or
Energy per unit length of screw dislocation
(integrating from r0 to r)
Roughly, due to r dependence
Elasticity theory breaks down for r05b so core
energy is ignored here.
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Energy of Edge Dislocations
For idealized edge component, one entire plane
has been pushed into the other planes above the
glide plane but not below (tensile compressive
stresses). Hence, there is Poisson Effect along
length of line, which yields a (1-v) in
denominator for strain.
Idealized Edge
compression
For many metals, v 1/3, so
Elasticity theory breaks down for r05b so core
energy is ignored here.
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Energy and Forces of Edges
Idealized
Roughly, your expectation should be (as found
from intuition)
b
Energy before 2Gb2 Energy after Gbtot2 0
Should attract
b
b0
b
b
Energy before 2Gb2 Energy after Gbtot2
G(2b)2 4Gb2
Should repel
b2
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Energy of Mixed Dislocation
Energy has components from Screw and Edge.
Energy has component from both types
As in download notes
b
Edge
Screw
Screw
mixed
u
b
Combining (Screw, Mixed, Edge)
b
Edge
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Dislocation Stress Fields Edges
Edges with z-axis line direction best describe
in x-y plane in cartesian coordinates. Use
See book, Hirth and Loath, or on-line notes for
details.
Edge at center with u(001) and b(100)
interacting with edge somewhere nearby
yx
Other stresses are zero!

• General trend
• Above the edge (x0, ygt0), pure compression.
• Below the edge (x0, ylt0), pure tension.
• Along the slip plane (y0), pure shear.
• All other locations, compression tension
  shear.

yx
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Forces between Dislocations
Generally dislocation has screw and edge
character bbe bs. Here the components of b are
b i bx j bz , where bx (bz ) gives edge
(screw) like movement.
In x1-dir. F1 (?21b1e ?23b3s) In x2-dir.
F2 (?11b1e ?13b3s) Total force F i
(?21b1e ?23b3s) j (?11b1e ?13b3s)
GLIDE CLIMB
See more example in on-line notes.
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Forces between Dislocations Peach-Koehler
Equation
See Hirth and Loath, or on-line notes for details.
Generally dislocation has screw and edge
character bbe bs. Total force F i (?21b1e
?23b3s) j (?11b1e ?13b3s) General 3D case
from Peach-Koehler eq force on disl. (2) from
disl. (1).
Example two attracting screws u(1) (001)
u(2) b(1) (001)b b(2)
In cylindrical coordinates, as it is a screw.
Shear stress have cylindrical symmetry
Simple opposing screws have radial force F
?b, as in book
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Forces between Screw Dislocations Cartesian
coords.
Peach-Koehler eq
Example two repelling screws u(1) (001)
u(2) b(1) (001)b b(2).
b1(2)b2(2) 0
b1
Shear stress around screw, back some ways.
xr cos? x/rcos? yr sin? y/rsin? r2x2y2
As before, radial force F ?b for ?0
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Example Peach-Koehler Eq. for Two Edges
u1 (001) u2 and b1 (100)b b2
y x
Shear stress slip
Axial stress climb
y x
y-dir climb force
x-dir glide force
See on-line notes for details.
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Summary

• Dislocation-dislocation interactions, or
  dislocations interacting with other defects, lead
  to higher stresses required to move the
  dislocations further (work-hardening). For
  example, dislocation pile-up, jogs, transfer
  across grain boundaries, etc., all contribute to
  YS increases.
• Dislocation multiplication is responsible for
  the very large increases in YS.
• Dislocations interacting with anything lead to
  other defects (point, planar, volumetric).
• Peach-Kohler Forces Be familiar/conversant with
  how dislocations interact and the consequences,
  including how to get forces between dislocations.