Post on 18-Jan-2016
Plastic Deformation• Permanent, unrecovered mechanical deformation
= 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
• 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
Crystallography of Slip
b = a/2 <1 1 0>| b | = a/2
[1 1 0]
• HCP • “BCC”
– 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
b = a <1 0 0>| b | = a
b = a/2 <1 1 1>| b | = 3a/2
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
Why does the number of independent slip systems matter?
= F/AAre any or all or some of the grains in the proper orientation for slip to occur?
HCP
CCP
• 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
maximum shear stress
Dislocations in Ionic Crystals
like charges touch
like charges do not touch
long burger’s vector compared to metals
1
2
(1) slip brings like charges in contact
(2) does not bring like charges in contact
compare possible slip planes
viewing edge dislocations as the termination defect of “extra half-planes”
Energy Penalty of Dislocationsbonds are compressed
bonds are under tension
R0
tension
R
E
compression
Energy / length |b|2
Thermodynamically unfavorable Strong interactions
attraction annihilation repulsion pinning
Too many dislocations become immobile
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| > |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
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
Elastic ConstantsY: Young’s modulus (sometimes E)
l0
A0
F stress = 0
FA uniaxial, normal stress
material elongates: l0 l
strain = 0
0
l l
l
elongation along force direction
observation:
Y
(s
tre
ss)
(strain)
Y
material thins/necks: A0 Ai elongates: l0 li
true stress: use Ai; nominal (engineering) stress: use A0
true strain: use li; nominal (engineering) stress: use l0
Elastic ConstantsConnecting Young’s Modulus to Chemical Bonding
R0
R
E
1~E
R
Coulombic attraction
F = k R
stress*area strain*length
0
R
dFk
d R
dEF
d R
R0
k / length = Y
want k in terms of E, R0
2
20
( )
R
d E
d R
30 ~ Y R
observed within some classes of compounds
Hook’s Law
Elastic ConstantsBulk Modulus, K
•apply hydrostatic pressure
= -P
•measure change in volume0
V
V
P = F/A
•linear response0
VK
V
Useful relationship:0
x y zV
V
Can show:
0
2
0 2
V V
EK V
V
analogous to Young’s modulus
Coulombic: 1/3
1 1~ ~
E
R V 4/3 4
0 0~ ~ K V R
hydrostatic stress
Elastic ConstantsPoisson’s ratio, •apply uniaxial stress = F/A
•measure ||
||
0,
x
y
l
l
- elongation parallel to force
l0
AF
Rigidity (Shear) Modulus, G
y
x
•measure - thinning normal to force
l0
lF
F
•apply shear stress = F/A
•measure shear strain
= tan
G
||
A
Elastic ConstantsGeneral Considerations
2(1 )
YG
6 parametersStress, : 3 3 symmetric tensor
In principle, each and every strain parameter depends on each and every stress parameter
Strain, : 3 3 symmetric tensor 6 parameters
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 are independent