ANELASTICITYSome Background, Mechanisms and Recent Work
Aaron Vodnick
MSE 610
4/25/06
Why I Care
0 100 200 300 400 500
-200
0
200
400
600
Film
Str
ess
(MP
a)
Temperature (°C)
(111) grains (100) grains Substrate Curvature
500 nm Passivated Cu FilmWith Oxygen
0 30 60 900
20
40
60
80
Ch
an
ge
in S
tre
ss (
MP
a)
Time (Minutes)
(111) grains (100) grains
500 nm Passivated Cu FilmWith Oxygen
- Thin Cu Film on a Si Substrate
- Temperature represents total strain
plasticelastictot
- Stress proportional to elastic strain
- From room temp, heat to zero stress and hold.
- Stress increases with time
elastic plastic
So… there’s some anelastic mechanism here I want to understand
First – Ideal Elasticity
E
klijklij C
Hooke’s Law
Isotropic:
Anisotropic:
Conditions for Ideal Elasticity
1) Each level of applied stress has a unique equilibrium value of strain
2) The equilibrium response is achieved instantaneously (phonon velocity)
3) The response is linear (doubling the stress doubles the strain)
- Easing conditions allows us to generalize elastic behavior
E
t
t
Anelasticity
Conditions for Ideal Elasticity
1) Each level of applied stress has a unique equilibrium value of strain
2) The equilibrium response is achieved instantaneously (phonon velocity)
3) The response is linear (doubling the stress doubles the strain)
Conditions for Anelasticity
1) Each level of applied stress has a unique equilibrium value of strain
2) The equilibrium response is achieved only after the passage of sufficient time
3) The response is linear (doubling the stress doubles the strain)
Relax the 2nd condition for Ideal Elasiticity
t
Load Applied
Equilibrium strain.
Load Removed
Complete Recoverability
Unique equilibrium Relationship
(complete recoverability)Instantaneous Linear
Ideal Elasticity Yes Yes Yes
Nonlinear Elasticity Yes Yes No
Instantaneous Plasticity
No Yes No
Anelasticity Yes No Yes
Linear Visoelasticity No No Yes
Other Behaviors
Sometimes people use the term “Anelastic” when it isn’t appropriate
Describing Anelasticity: SLSStandard Linear Solid
2
1
21
111 E
Describing stress-strain behavior:
21
E2
E1
2E
2121 bbaa General linear equation describing model
2222 E
SLS Creep Behavior
E2
E1
2E
Apply Constant Stress
t
R
Apply Constant Strain
0
R
t
RE
RE
Equation Describing Behavior
RE
Where ’s are time constants and ER is the relaxed modulus
Dynamic Behavior
tie 0 tie0
is the “loss angle” or “internal friction” –the angle the strain lags the stress.
0 2 4 6 8 10
0
Ste
ss/S
trai
n
Time
Stress Strain
Common Measurement methods:
• Resonant Vibrations
• Wave propagation
It is a measure of energy absorbed in each cycle
Dynamic tests give behavior over short times – but can relate to relaxations
Can calculate activation energies by measuring internal friction as a function of temperature
Characterization
kT
Hpeak exp
Some
Mechanisms
Snoek Relaxation
• Interstitial Relaxation
•Defect Symmetry:
- For point defect relaxations, defects must have a symmetry less than lattice
- BCC Octahedral interstitial have tetragonal symmetry (not cubic)
- Creates an “Elastic Dipoles” (three types)
- Dipole can “feel” external stresses
• These types of point defects don’t exist in FCC crystals. Can get relaxations with point defect pairs.
• Consider a tensile stress along the Z axis of a [001] crystal
• Tetragonal axis of z-sites elongates
• Tetragonal axis of x,y-sites shortens
• Driving force to diffuse to low energy sites
• Kinetic process
Snoek Relaxation
Equal distribution
Diffusion to z-sites
Saturation
time
Grain Boundary SlidingShear stresses act across grain boundaries
(Grain)
-Viscous slip occurs at boundary (x)
- Grain corners sustain more of shearing force
- Stress at corners provide driving force for reverse slip
The potential relaxation strength is given by:
6.0575721 2 UR EE
RE
0UE 0UR EERemember: So:
So, the potential relaxation is >50% of the initial strain. (this is big)
Grain Boundary Sliding
Relationship with Stacking Fault Energy
• Grain boundaries composed of dislocations
• Sliding may be associated with dislocation motion
• Stacking fault energy represents dislocation “width” when it spreads
-- These models are not very realistic because it ignores strong interactions of dislocations with boundaries
Grain Boundary SlidingEffect of Solutes
• Second peak appears and grows with impurity addition
• Boundaries contain steps/ledges
• Migration smoothes boundaries
• Occurs by solute drag at high concentration
• Rate controlling step is slower of two
Cu – 0.1% Ni
Cu – 0.5% Ni
Pure Metal
Solid Solution
Self Diffusion
migration
Sliding
DislocationsExample of dislocation in thin metal film
Final Configuration
Pinning points could also be things such as dragged solute atoms
• Dislocation is anchored at film surfaces
• Segments bow and exert force f on Jogs
• Diffusion occurs to drag jogs to final configuration
• Line tension restores initial configuration upon removal of stress
Choi and Nix, 2006
kT
Hpeak exp
Thin Film Measurement• Si cantilevers microfabricated and
coated with films to be tested
• Electrostatic force from AC voltage vibrates cantilever
• AC voltage turned off, decay of velocity is measured
• Internal friction from rate of amplitude decay
Inte
rnal
Fri
ctio
n
• Determine activation energy from frequency dependence on peak temp.
Choi and Nix 2004 and 2006
Final Statements
• Anelasticity is in fact mind numbing
• Few people have cared about it since before the seventies
• There is some new interest in determining mechanisms governing material behaviors on small scales
• Any time-dependent, reversible, processes can cause anelasticity