Lecture 22: The mechanism of plastic deformation, part 2

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Lecture 22 The mechanism of plastic deformation,
part 2

• PHYS 430/603 material
• Laszlo Takacs
• UMBC Department of Physics

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Fracture

• Brittle There is very little plastic deformation
  preceding failure. In a uniform tensile sample it
  happens perpendicularly to the loading direction,
  induced by tension rather than shear.
• Ductile Substantial plastic deformation before
  failure. The direct cause is shear. Geometry
  depends on the material and conditions
• Slip at about 45 (or preferred slip direction)
  until failure (typical in single crystals.)
• Necking to a point (very ductile polycrystals.)
• Cup and cone in less ductile polycrystals.
  After some necking, failure starts inside and
  propagates at about 45 outward.
• Failure always happens much before the
  theoretical tensile strength is reached it
  always begins at faults.

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• Technical materials always contain faults, e.g.
  microcracks. Why dont they immediately result in
  failure?
• Create an elliptical microcrack in an infinite
  plate under uniform tension.
• It releases energy due to decreased deformation
• Uelastic -?c2?2/E
• It requires surface energy
• Usurface 4c?
• Uelastic Usurface has a maximum at
• ccrit ?E/??2
• Microcracks smaller than this will heal rather
  than increase. Turned around, if a microcrack of
  length 2c exists inside the plate (or a notch of
  length c on the edge), crack propagation requires
  a minimum stress of
• ?crit (?E/?c)1/2

2c
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The primary mechanism of plastic deformation is
slip due to dislocation motion. The required
shear stress (Peierls stress)
Which slip system is active depends on the
crystal structure Fcc 1 1 1lt1 1 0gt often
split into parallel partials. Hcp 0 0 0 1lt1 1
-2 0gt always available several others, if c/a
lt 1.63. Bcc 1 1 0lt1 1 1gt is the best, but
other slip planes with the same slip direction
are close. More complex structures Larger
Burgers vector makes slip difficult, material
is usually brittle.
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The interaction of dislocations

• Dislocations interact via their elastic stress
  fields. Need to know
• Need to know the force acting on a dislocation
  due to a stress field
• The stress field produced by a dislocation
• Parallel dislocations repel, attract, shift each
  other
• Dislocations on different slip planes must cut
  through each other

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Work of external stress affecting the slip W
(? l1 l3) bWork of force acting on the
dislocation W (F l3) l1Compared F ?
b, where ? is an external stress.In general
geometry F (? b) x s Peach-Koehler
equation.(F is force per unit length.)
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• Except for a core about as wide as a single line
  of atoms, a dislocation can be represented with
  its elastic stress field.
• Edge dislocation Strain is radial.
• Screw dislocation Strain is parallel to the
  dislocation line.
• Strain goes to zero far from the dislocation
  line. With this conditions the stress field can
  be evaluated.

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• For example the stress field of an edge
  dislocation in the z direction is

Here ? is the asimuthal angle in cylindrical
coordinates. Combining this with the P-K equation
for parallel dislocations
The 45 lines are unstable, dislocation move away
from there. The x component shows that
dislocations in the same slip plane (? 0)
repel each other, Fx ?xyb gt0. They form a train
of dislocations. The y component aligns
dislocations into small angle grain boundaries.
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• A general deformation requires that not all
  dislocations are parallel and they move across
  each other on different slip planes. This
  requires extra work a dislocation always moves
  the most freely in a perfectly periodic lattice
• Crossing dislocations create jogs in the
  dislocation lines. (A jog is a step of the
  dislocation line out of the slip plane. Forming
  it requires energy.)
• Some mobile dislocations contained in slip planes
  combine into a locked dislocation that is not
  mobile (Lomer lock).

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The stages of strain hardening

• Stage I Dislocation density is low,
  dislocations move long distances along the
  primary slip plane without meeting an obstacle.
• Stage II Initially few dislocations exist in
  other slip systems, but they start to lead to
  cross-slips and locks, impeding dislocation
  motion. If dislocations are rendered immobile,
  new dislocations must form to continue the
  deformation. The dislocation density and the
  stress increase quickly.
• Stage III Cross slip of screw dislocations
  becomes important. It is a way to avoid obstacles
  and also results in the annihilation of some
  dislocations. The strain hardening rate gets
  smaller.
• The strain hardening rate can be characterized by
  ? d?/d?. The fastest strain hardening (in stage
  II) is about ? G/300 for most metals.

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• A dislocation can overcome an obstacle by
  increased shear stress alone, or thermal
  activation can help. Dislocation motion is easier
  at higher temperature, therefore the elastic
  limit is lower
• Forming metals is easier at high temperature.
  
• Metals become weaker at high temperature
• At low temperature the elastic limit is high, a
  sample might break before plastic deformation
  begins, i.e. it becomes brittle.

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The Frank - Reed source

• A single dislocation can provide a slip of b
  only. For macroscopic deformation many
  dislocations are needed, i.e. it is necessary to
  provide a mechanism for the generation of
  dislocations. Such a mechanism is the F-R source.

Suppose a cross slip generates the dislocation
segment BC. Without stress it is straight. Under
stress it bows out to form an arc of radius R
Gb/2?. As the stress increases, R decreases until
2R BC l is reached at ?0 Gb/l. At this
point the arc becomes unstable, forms a closed
loop and leaves the original line behind. This
cycle can be repeated.
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A Frank - Reed source in Si.Notice that the
loops follow the structure of the lattice rather
than being ideal circles.