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
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 - PowerPoint PPT Presentation
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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
Why?
Solid mechanics...
Why?Why?
Why?
A bi-layer of TiNi and SiO2. (From Wang, 2004)
Why is this thing bent?
Thermal actuator produced by Southwest Research Institute
And these?
Why?
A simple piezoelectric actuator design: An applied voltage causes stress in the piezoelectric thin film stress causing the membrane to bend
Membrane is piezoresistive; i.e., the electrical resistance changes with deformation.
Adapted from MEMS: A Practical Guide to Design, Analysis, and Applications, Ed. Jan G. Korvink and Oliver Paul, Springer, 2006
Why?
Hot arm actuator
i
+e-
Joule heating leads to different rates of thermal expansion, in turn causing stress and deflection.
+e-
Zap it with a voltage here… How much does it move here?
ω
Stress and strain
Normal stress
w
w
t
A = w·t
σ = — = —
PP
P
A
P
wt
[F ]
[A ]
[F ]
[L ]2
Dimensions Typical units
N
m 2Pa
δ
ε = — δ
L
L
Dimensions Typical units[L ]
[L ]μ-strain = 10-6
Normal strain
(dimensionless)
Elasticity
How are stress and strain related to each other?
σ
εδ
L
P
PX fracture
X fractureplastic (permanent) deformation
elastic (permanent) deformation
E
E σ = εE
Young’s modulus(Modulus of elasticity,
Elastic modulus)
brittleductile
F = kx
Elasticity
Strain in one direction causes strain in other directions
εy = εx
x
y
-ν
Poisson’s ratio
Stress generalized
Stress is a surface phenomenon.
y
x
z
σy
σx
σz
τxy
τxz
τyx
τyz
τzyτzx
σ : normal stressForce is normal to surfaceσx stress normal to x-surface
τ : shear stressForce is parallel to surfaceτxy stress on x-surface in y-direction
ΣF = 0, ΣMo = 0
τxy = τyx τyz = τzy τzx = τxz
Strain generalized
Essentially, strain is just differential deformation.
Δx
Δy Deforms
Δx + dΔx
Δy + dΔy
dx
dy
u: displacement
ux ux + dx
dux
duy
dxdu
dydu yx
xydxdux
x 21
+=
Break into two pieces:
uniaxial strain shear strain
θ1
θ2 Shear strain is strain with no volume change.
Relation of shear stress to shear strain
Just as normal stress causes uniaxial (normal) strain, shear stress causes shear strain.
dux
duy
θ1
θ2τxy = γxy τxy
τyx
τxy
τyx
G
shear modulus
Magic Algebra Box)1(2
EG
• Si sabes cualquiera dos de E, G, y ν, sabes el tercer.
• Limits on ν:
0 < ν < 0.5ν = 0.5
incompressible
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
εx = [ ] + [ ] + [ ]
x normal strain due to x normal
stress
x normal strain due to y normal
stress
x normal strain due to z normal
stress
Ex
Ey
Ez
-ν -ν τxy = G γxy
Generalized Hooke’s Law
εx =
εy =
εz =
γxy =
γyz =
γzx =
zyxE
1
xzyE
1
yxzE
1
xyG1
yzG1
zxG1
For a general 3-D deformation of an isotropic material, then
Generalized Hooke’s Law
Special cases
• Uniaxial stress/strain
σ = Eε
• No shear stress, todos esfuerzos normales son iguales
σx = σy = σz = σ = K•(ΔV/V)
• Biaxial stressStress in a plane, los dos esfuerzos normales son iguales
σx = σy = σ = [E / (1 - ν)] • ε
bulk modulus
biaxial modulus
volume strain
Elasticity for a crystalline silicon
The previous equations are for isotropic materials. Is crystalline silicon isotropic?
E Cij Compliance coefficients
yz
xz
xy
z
y
x
yz
xz
xy
z
y
x
CC
CCCCCCCCCC
44
44
44
111212
121112
121211
000000000000000000000000
yz
xz
xy
z
y
x
yz
xz
xy
z
y
x
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
yz
xz
xy
z
y
x
yz
xz
xy
z
y
x
For crystalline siliconC11 = 166 GPa, C12 = 64 GPa and C44 = 80 GPa
Te toca a ti
Assuming that elastic theory holds, choose the appropriate modulus and/or stress-strain relationship for each of the following situations. 1. A monkey is hanging on a rope, causing it to stretch. How do you model the
deformation/stress-strain in the rope?
2. A water balloon is being filled with water. How do you model the deformation/stress-strain in the balloon membrane?
3. A nail is hammered into a piece of plywood. How do you model the deformation/stress-strain in the nail?
4. 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?
5. A thin film is deposited on a much thicker silicon wafer. How do you model the deformation/stress-strain in the thin film?
6. A thin film is deposited on a much thicker silicon wafer. How do you model the deformation/stress-strain in the wafer?
7. 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)