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Solid mechanics

Solid mechanics. Define the terms Stress Deformation Strain Thermal strain Thermal expansion coefficient Appropriately relate various types of stress to the correct corresponding strain using elastic theory

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Solid mechanics

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  1. Solid mechanics • Define the terms • Stress • Deformation • Strain • Thermal strain • Thermal expansion coefficient • Appropriately relate various types of stress to the correct corresponding strain using elastic theory • Give qualitative descriptions of how intrinsic stress can form within thin films • Calculate biaxial stress resulting from thermal mismatch in the deposition of thin films • Calculate stresses in deposited thin films using the disk method

  2. Why? • Solid mechanics... Why? Why?

  3. Why? Why is this thing bent? Thermal actuator produced by Southwest Research Institute And these? A bi-layer of TiNiand SiO2. (From Wang, 2004)

  4. Why? Adapted from MEMS: A Practical Guide to Design, Analysis, and Applications, Ed. Jan G. Korvink and Oliver Paul, Springer, 2006 Membrane is piezoresistive; i.e., the electrical resistance changes with deformation. • A simple piezoelectric actuator design: An applied voltage causes stress in the piezoelectric thin film stress causing the membrane to bend

  5. Why? Hot arm actuator + e - Zap it with a voltage here… How much does it move here? + e - ω i Joule heating leads to different rates of thermal expansion, in turn causing stress and deflection.

  6. Stress and strain A= w·t t P P w w δ L δ P P ε =— • Normal stress • Normal strain σ =—=— L A wt • Typical units • Dimensions • Typical units • Dimensions [L ] [F ] [F ] N • μ-strain = 10-6 (dimensionless) Pa [A ] [L ]2 m 2 [L ]

  7. σ Elasticity • How are stress and strain related to each other? P fracture X F = kx plastic (permanent) deformation X fracture E E σ= ε L elastic (permanent) deformation Young’s modulus (Modulus of elasticity, Elastic modulus) brittle ductile E δ ε P

  8. Elasticity Strain in one direction causes strain in other directions y x εy = εx -ν Poisson’s ratio

  9. Stress generalized Stress is a surface phenomenon. z σz ΣF = 0, ΣMo= 0 τxy = τyx τyz= τzy τzx= τxz τzy τzx τyz τxz σy τxy τyx y σx σ : normal stress Force is normal to surface σx stress normal to x-surface τ : shear stress Force is parallel to surface τxy stress on x-surface in y-direction x

  10. Strain generalized Essentially, strain is just differential deformation. Deforms Δy+ dΔy Δy Δx Δx+ dΔx Break into two pieces: dux ux ux + dx = + dy θ2 Shear strain is strain with no volume change. duy dx θ1 shear strain uniaxial strain u: displacement

  11. Relation of shear stress to shear strain • Just as normal stress causes uniaxial (normal) strain, shear stress causes shear strain. dux τyx τxy τxy= γxy θ2 τxy G duy shear modulus τyx θ1 • Si sabescualquiera dos de E, G,y ν, sabes el tercer. • Limits on ν: • 0 < ν < 0.5 Magic Algebra Box ν = 0.5 incompressible

  12. Generalized stress-strain relations • The previous stress/strain relations hold for either pure uniaxial stress or pure shear stress. Most real deformations, however, are complicated combinations of both, and these relations do not hold Deforms τxy= G γxy εx= [ ] + [ ] + [ ] -ν -ν x normal strain due to x normal stress x normal strain due to z normal stress x normal strain due to y normal stress

  13. Generalized Hooke’s Law • For a general 3-D deformation of an isotropic material, then • εx= • εy = • εz = • γxy= • γyz = • γzx = • Generalized Hooke’s Law

  14. Special cases • Uniaxial stress/strain • σ = Eε • No shear stress, todosesfuerzosnormales son iguales • σx= σy = σz= σ = K•(ΔV/V) • Biaxial stress • Stress in a plane, los dos esfuerzosnormales son iguales • σx = σy= σ = [E / (1 - ν)] •ε volume strain bulk modulus biaxial modulus

  15. Elasticity for a crystalline silicon • The previous equations are for isotropic materials. Is crystalline silicon isotropic? • E CijCompliance coefficients For crystalline silicon C11= 166 GPa, C12 = 64 GPaand C44 = 80 GPa

  16. Tetoca a ti • Assuming that elastic theory holds, choose the appropriate modulus and/or stress-strain relationship for each of the following situations. • A monkey is hanging on a rope, causing it to stretch. How do you model the deformation/stress-strain in the rope? • A water balloon is being filled with water. How do you model the deformation/stress-strain in the balloon membrane? • A nail is hammered into a piece of plywood. How do you model the deformation/stress-strain in the nail? • A microparticle is suspended in a liquid for use in a microfluidic application, causing it to compress slightly. How do you model the deformation/stress-strain in the microparticle? • A thin film is deposited on a much thicker silicon wafer. How do you model the deformation/stress-strain in the thin film? • A thin film is deposited on a much thicker silicon wafer. How do you model the deformation/stress-strain in the wafer? • A thin film is deposited on a much thicker glass substrate. How do you model the deformation/stress-strain in the glass substrate? Uniaxial stress/strain Biaxial stress/strain Uniaxial stress/strain Use bulk modulus (no shear, all three normal stresses the same) Biaxial stress/strain Anisotropic stress/strain (Using Cij compliance coefficients) Generalized Hooke’s Law. I.e., ε = (1/E)(σx – ν(σy + σz)) etc.

  17. Thermal Expansion Thermal strain • Most things expand upon heating, and shrink upon cooling. δ(T)= αT (T-T0) • Notes: • αT≈ constant ≠ f(T) • ε(T)≈ ε(T0) + αT (T-T0) • Thermal strain tends to be the same in all directions even when material is otherwise anisotropic. If no initial strain • Thermal expansion coefficient

  18. positive(+) Negative(-) Solid mechanics of thin films • Adhesion • Ways to help ensure adhesion of deposited thin films: • Ensure cleanliness • Increase surface roughness • Include an oxide-forming element in between a metal deposited on oxide • Stress in thin films Tension headache • Tension • Compression

  19. Stress in thin films • Two types of stress • Intrinsic stress • Also known as growth stresses, these develop during as the film is being formed. • Extrinsic stress • These stresses result from externally imposed factors. Thermal stress is a good example. Doping Sputtering Microvoids Gas entrapment Polymer shrinkage

  20. Thermal stress in thin films Consider a thin film deposited on a substrate at a deposition temperature, Td. (Both the film and the substrate are initially at Td.) Initially the film is in a stress free state. The film and substrate are then allowed to cool to room temperature, Tr thin film deposited at Td substrate Since the two materials are hooked together, they both experience the same ____________ as they cool. strain both cooled to Tr εboth = εsubstrate or εfilm ? εmismmatch = αT,s(Tr - Td) - αT,f(Tr- Td) = (αT,f - αT,s)(Td- Tr) εsubstrate = αT,s(Tr - Td) = εfilm= αT,f(Tr - Td) + εmismmatch

  21. Thermal stress in thin films How would you relate σmismatchto εmismatch? Biaxial stress/strain σmismatch = [E / (1-ν)]·εmismatch • = [E / (1-ν)]·(αT,f - αT,s)(Td - Tr) tension • If αT,f > αT,sσmismatch = (+) or (-)  Film is in ___________________. • If αT,f < αT,sσmismatch = (+) or (-)  Film is in ___________________. compression Thin film Initially stress free cantilever σmismatch > 0 σmismatch < 0 Sacrificial layer

  22. Stress in thin films Compression or tension? Compression or tension? (a) (b) (a) Stress in SiO2/Al cantilevers (b) Stress in SiO2/Ti cantilevers [From Fang and Lo, (2000)] αT,Al>, <, = αT,SiO2 ? αT,Ti>, <, = αT,SiO2 ?

  23. Stress in thin films How were these fabricated? (a) (b) (a) Stress in SiO2/Al cantilevers (b) Stress in SiO2/Ti cantilevers [From Fang and Lo, (2000)]

  24. Tetoca a ti • Show that the biaxial modulus is given by • E/(1 – ν) • Pistas: • Remember what the assumptions for “biaxial” are. • In thin films you can always find one set of x-y axes for which there is only σ and no τ.

  25. Measuring thin film stress • Assumptions: • The film thickness is uniform and small compared to the wafer thickness. • The stress in the thin film is biaxial and uniform across it’s thickness. • Ths stress in the wafer is equi-biaxial (biaxial at any location in the thickness). • The wafer is unbowed before the addition of the thin film. • Wafer properties are isotropic in the direction normal to the film. • The wafer isn’t rigidly attached to anything when the deflection measurement is made. The disk method Stressed wafer (after thin film) R Unstressed wafer (before thin film) radius of curvature R = _________________________, T = _________________________ and t = _________________________. wafer thickness Biaxial modulus of the wafer strain at wafer/film interface thin film thickness

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