1. ANELASTICITY Some Background, Mechanisms and Recent Work Aaron Vodnick MSE 610 4/25/06

2. Why I Care - Thin Cu Film on a Si Substrate - Temperature represents total strain - Stress proportional to elastic strain - From room temp, heat to zero stress and hold. - Stress increases with time So… there’s some anelastic mechanism here I want to understand

3. First – Ideal Elasticity 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

4. 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 2 nd condition for Ideal Elasiticity  t Load Applied Equilibrium strain. Load Removed Complete Recoverability

5. Unique equilibrium Relationship (complete recoverability) InstantaneousLinear Ideal ElasticityYes Nonlinear ElasticityYes No Instantaneous Plasticity NoYesNo AnelasticityYesNoYes Linear VisoelasticityNo Yes Other Behaviors Sometimes people use the term “Anelastic” when it isn’t appropriate

6. Describing Anelasticity: SLS Standard Linear Solid 2 1 Describing stress-strain behavior: E2E2 E1E1 General linear equation describing model

7. SLS Creep Behavior E2E2 E1E1 Apply Constant Stress t   RR Apply Constant Strain 00 RR t Equation Describing Behavior Where  ’s are time constants and E R is the relaxed modulus

8. Dynamic Behavior  is the “loss angle” or “internal friction” –the angle the strain lags the stress.  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

9. Characterization

10. Some Mechanisms

11. 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.

12. 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

13. Grain Boundary Sliding Shear 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: Remember:So: So, the potential relaxation is >50% of the initial strain. (this is big)

14. 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

15. Grain Boundary Sliding Effect 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

16. Dislocations Example 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

17. 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 Internal Friction Determine activation energy from frequency dependence on peak temp. Choi and Nix 2004 and 2006

18. 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