Plastic Deformation

1
Plastic Deformation

• Permanent, unrecovered mechanical deformation

s F/A stress

• Deformation by dislocation motion, glide or
  slip
• Dislocations
• Edge, screw, mixed
• Defined by Burgers vector
• Form loops, cant terminate except at crystal
  surface
• Slip system
• Glide plane Burgers vector

maximum shear stress
2
Crystallography of Slip

• Slip system glide plane burgers vector
• Correspond to close-packed planes directions
• Why?
• Fewest number of broken bonds
• Cubic close-packed
• Closest packed planes
• 1 1 1
• 4 independent planes
• Closest packed directions
• Face diagonals
• lt1 1 0gt
• 3 per plane (only positive)
• 12 independent slip systems

3

• HCP

• BCC

• Planes 1 1 0
• 6 independent planes
• Directions lt1 1 1gt
• 2 per plane (only positive)
• 12 independent slip systems

• Planes 0 0 1
• 1 independent plane
• Directions lt1 0 0gt
• 3 per plane (only positive)
• 3 independent slip systems

Occasionally also 1 1 2 planes in BCC are
slip planes
Diamond structure type 1 1 1 and lt1 1 0gt ---
same as CCP, but slip less uncommon
4
Why does the number of independent slip systems
matter?
Are any or all or some of the grains in the
proper orientation for slip to occur?
s F/A
?
?
HCP
?
?
CCP
maximum shear stress

• Large of independent slip systems in CCP ??
  at least one will be active for any particular
  grain

• True also for BCC

• Polycrystalline HCP materials require more
  stress to induce deformation by dislocation
  motion

5
Dislocations in Ionic Crystals
like charges touch
2
1
like charges do not touch
long burgers vector compared to metals
compare possible slip planes
(1) slip causes like charges to touch
(2) does not cause like charges to touch
6
Energy Penalty of Dislocations
bonds are compressed
E
R0
R
compression
tension
Energy / length ? b2
Thermodynamically unfavorable Strong interactions
bonds are under tension
attraction ? annihilation
repulsion ? pinning
Too many dislocations ? become immobile
7
Summary

• Materials often deform by dislocation glide
• Deforming may be better than breaking
• Metals
• CCP and BCC have 12 indep slip systems
• HCP has only 3, less ductile
• bBCC gt bCCP ? higher energy, lower mobility
• CCP metals are the most ductile
• Ionic materials/Ceramics
• Dislocations have very high electrostatic energy
• Deformation by dislocation glide atypical
• Covalent materials/Semiconductors
• Dislocations extremely rare

Now on to elastic deformation
8
Elastic Deformation

• Connected to chemical bonding
• Stretch bonds and then relax back
• Recall bond-energy curve
• Difficulty of moving from R0
• Curvature at R0
• Elastic constants
• (stress) (elastic constant) (strain)
• stress and strain are tensors ? directional
• the elastic constant being measured depends on
  which component of stress and of strain

9
Elastic Constants
Y Youngs modulus (sometimes E)
F
stress
uniaxial, normal stress
material elongates l0 ? l
strain
elongation along force direction
l0
observation
Y
s (stress)
e (strain)
material thins/necks A0 ? Ai true stress use
Ai engineering stress use A0
10
Elastic Constants
Connecting Youngs Modulus to Chemical Bonding
R0
F k DR
? k(length) Y
stressarea
strainlength
Coulombic attraction
want k in terms of E, R0
observed within some classes of compounds
11
Elastic Constants
Bulk Modulus, K

• apply hydrostatic pressure

P F/A
s -P

• measure change in volume

• linear response

Can show
analogous to Youngs modulus
Coulombic
Useful relationship
12
Elastic Constants
Poissons ratio, n

• apply uniaxial stress

s F/A
F

• measure e

- elongation parallel to force

• measure e?

- thinning normal to force
e
e?
Rigidity (Shear) Modulus, G

• apply shear stress

t F/A
A
Dl

• measure shear strain

F
?f
tanf
?? f
l0
y
F
x
13
Elastic Constants
General Considerations
? 6 parameters
Stress, s 3 ? 3 symmetric tensor
Strain, e 3 ? 3 symmetric tensor
? 6 parameters
In principle, each and every strain parameter
depends on each and every stress parameter
? 36 elastic constants
? 21 independent elastic constants in the most
general case
Some are redundant
Material symmetry
? some are zero, some are inter-related
Isotropic material
? only 2 independent elastic constants
normal stress
? only normal deformation
shear stress
? only shear deformation
Cubic material
? G, Y and n are independent