1. Plastic Deformation Permanent, unrecovered mechanical deformation
s = F/A stress
Deformation by dislocation motion, “glide” or “slip”
Dislocations
Edge, screw, mixed
Defined by Burger’s vector
Form loops, can’t terminate except at crystal surface
Slip system
Glide plane + Burger’s vector
maximum shear stress

2. Crystallography of Slip
Slip system = glide plane + burger’s 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
<1 1 0>
3 per plane (only positive)
12 independent slip systems a1 a2 a3
b = a/2 <1 1 0>
| b | = a/2
[1 1 0]

3. HCP “BCC” b = a/2 <1 1 1> | b | = 3a/2 b = a <1 0 0>
Planes {0 0 1}
1 independent plane
Directions <1 0 0>
3 per plane (only positive)
3 independent slip systems
Planes {1 1 0}
6 independent planes
Directions <1 1 1>
2 per plane (only positive)
12 independent slip systems
Occasionally also {1 1 2} planes in “BCC” are slip planes
Diamond structure type: {1 1 1} and <1 1 0> --- 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 burger’s 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  |b|2
Thermodynamically unfavorable Strong interactions
bonds are under tension
attraction  annihilation
repulsion  pinning
Too many dislocations  become immobile

7. Summary Now on to elastic deformation
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| > |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  R0 R E

9. Elastic Constants F A0 l0 s (stress) e (strain)
Y: Young’s modulus (sometimes E) F
stress =
uniaxial, normal stress A0
material elongates: l0  l
strain =
elongation along force direction l0
observation:
s (stress) Y
e (strain)
material thins/necks: A0  Ai
true stress: use Ai; engineering stress: use A0

10. Elastic Constants Connecting Young’s Modulus to Chemical BondingR0 R0 R E
F = k DR
 k*(length) = Y
stress*area
strain*length
Coulombic attraction
want k in terms of E, R0
observed within some classes of compounds

11. Elastic Constants Bulk Modulus, K apply hydrostatic pressure s = -P
P = F/A
s = -P
measure change in volume
linear response
Can show:
analogous to Young’s modulus
Coulombic:
Useful relationship:

12. Elastic Constants Poisson’s 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
Stress, s: 3  3 symmetric tensor
 6 parameters
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
Some are redundant
 21 independent elastic constants in the most general case
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