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