Chapter 6: Dislocation Slip and Strengthening Mechanisms

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Chapter 6 Dislocation Slip and Strengthening
Mechanisms
ISSUES TO ADDRESS...
Why are dislocations observed primarily in
metals and alloys? What are slip systems? How to
use Schmids Law to get initial active slip
systems?
How are strength and dislocation motion
related?
How do we increase strength? e.g.,
Strain-hardening, grain-boundary and solute
hardening, and solid-solution strengthening.
How can heating change strength and other
properties? e.g., recrystallization and grain
growth.
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Plastic Deformation to Fracture Engineering is
in between
Stress-strain and microscopic factors can be used
to engineer materials.
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Slip plane and Directions (Burgers vectors)

• Each crystal structure (e.g., fcc, bcc, and hcp)
  has different allowed slip planes, occurring at
  specific angles to applied stress, and different
  slip directions, occurring at other angles.
• Active slip planes and directions depend upon
  max. shear stresses.
• Shear stress (not normal stress) is what causes
  planar slip to occur.
• Active slip plane is typically the most
  CLOSE-PACKED Planes.
• Active slip direction is the most CLOSE-PACKED
  Directions.

FCC Slip plane and Directions
(111) planes in the direction of
Slip systems 4 x 3 12
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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 1
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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Slip and Shear Stress in a Tensile Sample

• Consider Plane at an angle f to the applied
  load F.
• NORMAL Force FN F cosf
• Consider Slip along direction at angle ?.
• SHEAR Force FS F cosl.
• Area of f-planes Af A/cosf (Af gt A)
• NORMAL Stress in f-planes
• sN FN / Af (Fcosf)/ (A/cosf) s cos2f
• SHEAR Stress in f-planes
• ts Fs / Af (Fcosl)/ (A/cosf) s cosf cosl
• Shear stress is what causes the slip to occur.
• Slip is not in same direction as tilt of plane!

e.g. fcc
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Slip of atomic plane via applied tensile stress
HCP Zinc
Resolved Shear Stress
Maximum Resolved Shear Stress
Occurs at ?? 450.

• one slip system is general favored initially.

Critical Resolved Shear Stress
Typically 10-4 to 10-2 GPa
Slip is on planes 45o from the applied stress.

• ?CRSS is the min. shear stress to initiate slip.
• Dislocation move at sys when ?R gt ?CRSS.
• ?R will vary from one crystal to another.

• Plastic flow is initiated when tRSS reaches a
  critical value, characteristic of the material,
  called critical RSS, when m tRSS ?ys (Schmids
  law).

• Schmids Law is only approximate, i.e., works
  for FCC, basil slip in HCP, not BCC.

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Critical Resolved Shear Stress
Condition for dislocation motion
Crystal orientation can make it easy or
hard to move disl.
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CRSS versus temperature
BCC has larger ?
bcc
covalent
fcc
Solute effect
Ionic w/ inherent less str.

• For T lt 0.7 Tmelt, ?CRSS ?a ?,
• (which depend upon microstructure) with athermal
  and thermal dependent parts.
• Athermal ?a from dislocation-dislocation
  interaction (long-range stress fields).
• Thermal ? arises from Peierls stress due to
  impurities, kinks and jogs (short-range barriers
  ).
• Small impurities are more mobile at intermediate
  T and can catch up to dislocations, repinning,
  etc. Impurity atoms increase CRSS. (PLC
  effect).

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Simple Shear direction of Slip is in plane of
tilt

• Simple shear occurs when ? ? 90o (a minimum
  value, which happens only when direction are
  coplanar),
• i.e., Normal direction is along Y' and Shear
  direction is along X'.
• Then, ?s ? cosf sin?, (Where is
  maximum? Plot ?s ?/m .)
• which is the same as you may have seen in
  continuum mechanics using Mohr's Circle for
  transforming stress to new coordinate system.
• MAXIMUM Resolved Shear Stress occurs when ? ?
  45o
• called tRSS,max. Slip is on the planes 45o from
  the applied stress.
• Then ?RSS, max s cos2? s /2 at ? l 45o.

Isotropic case ?max (?max - ?min)/2 ?max/2
Uniaxial test
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Slip in Metal Polycrystals
Slip lines on surface of polycrystalline Cu (173
x)
Requires motion of dislocations
unstressed
stressed
load
Grains rotate and elongate
Slip planes directions (l, f) change from one
crystal to another. tR will vary from one
crystal to another. The crystal with the
largest tR yields first. Less favorably
oriented crystals yield later.
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Recall Slip Systems in FCC are 111lt110gt

• 12 slip systems in FCC 4 111 planes and 3
  lt110gt Directions

Planes

• For a given direction of APPLIED Stress, there
  are different
• angles to the SLIP PLANE , f, and SLIP
  DIRECTIONS, l.

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For slip, need angles between load and plane
normal and load and slip direction

• For a given direction of APPLIED Stress, there
  are different
• angles to the SLIP PLANE , f, and SLIP
  DIRECTIONS, l.

a
q

• Angles from geometry or

b

• Consider (101) plane with load along 100.
• n 101 and l100

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Example FCC Cu with Loading axis 112

• What is most likely initial slip system?
• If CRSS is 50 MPa, what is the tensile stress
  at which Cu will start to deform plastically?

smallest stress to cause slip (yielding)
Initial Slip Systems (plane, direction) are then
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Self-Assessment Example
Crystal with simple cubic structure slip planes
100 and slip directions lt010gt Load is applied
along 010. Determine Schmid factor, M, and
what slip occurs.
M
Resolved Shear stress
Is there any slip? Why? If no slip, what
must happen finally to material as load is
increased?
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Self-Assessment on Slip

• What are slip planes and slip directions of BCC
  crystals?
• What are the most close-packed planes?
• What are the most close-packed directions?
• Are there other planes possible? Why?

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Bicrystals and Compatibility
Each bicrystal must have six strain components 3
tensile and 3 shear ?ii, (tensile) and ?ij
(shear) with i1,3 and j1,3. To have
compatibility of strains at interface (grain
boundary) must have ?11A ?11B , ?33A ?33B
, and ?13A ?13B . Because one grain has a
larger value of cos ? cos ? smaller Schmid
factor (1/m), the above constraints restrict the
deformation of this more favorably oriented grain
and result in a higher YS (greater work-hardening
response) of the bicrystal.
For poly-xtals, more restrictive constraint are
required than those above. While each grain has 3
tensile and 3 shear components, plastic
deformation has volume constraint, or ?11 ?22
?330. Thus, 5 independent slip systems are
needed in vicinity of g.b. to have compatibility!
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Bicrystals and Slip

• If A B comprise 1 single crystal of the same
  materials, e.g. bcc Nb, then it is compatible by
  definition.
• If A B comprise 2 bi-crystals of the same
  materials, e.g. bcc Nb, then it can be highly
  compatible
• boundary is less significant
• primary slip is active
• If A B material are not compatible, then
• boundary is significant
• secondary slip is activated near g.b.

Additional strengthening effect is associated
with reduced grain size, as (1/d)n, where
n1/2-1/3
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Example Bicrystal Slip Systems - must consider
both halves!

• Note orientation of crystal halves.
• Slip systems must be from slip system is
  100lt100gt.
• Not all slip directions are possible from
  diagram, you see that
• (a) slip direction 100 does not lie in slip
  plane (100)
• (b) slip direction 010 does not lie in slip
  plane (010)
• (c) slip direction 001 does not lie in slip
  plane (001)
• So, these must be ignored (also true for the
  negative of these).

• A bicrystal (marked as half 1 and 2) with a
  simple cubic crystal structure is oriented as
  shown.
• (a) What are the load axis directions for 1 and
  2?
• (b) From cos? cos?, determine which crystal will
  slip first if the slip system is 100lt100gt ?
• (Make a table of load axes, slip plane and
  direction, cosines, and 1/m for both crystals)
• (c) What are the initial active slip systems?

• In half 1, the load axis is
• What is it for half 2?
• With this load axis for 1, there are 3 slip
  planes
• Some slip direction can be ignored, such as
• Show primary slip system with (max.) cos? cos?
  1/2 is
• (c) Show that 2 has no primary slip systems!
  Hence only 1 will exhibit initial slip!

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Summary

• Dislocations are observed primarily in metals
  and alloys.
• Initial active slip can be predicted.
• During tensile test, sample must rotate by
  constraint from equipment.
• Geometric dislocations can also be created.
• Grain boundaries not only affect dislocation
  motion bit may also generate dislocation when
  stress is high enough.

Strength is increased by making dislocation
motion difficult. Next topic.