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K.J. Hemker
Materials Come As:
!Amorphous – Glasses, polymers, some metal alloys– Processing can result in amorphous
structures!Crystalline
– Single crystals– Textured crystals– Polycrystalline
K.J. Hemker
Useful crystallography web sites
EMS On Line at http://cimesg1.epfl.ch/CIOL/ems.html
http://www.geo.ucalgary.ca/~tmenard/crystal/crystal.htmlnote: Si is the same structure as diamond, gold is FCC and Fe is BCC
K.J. Hemker
Crystallographic directions :
[100][100]
[001][001]
[010][010]
<100> cube edges
<011> face diagonals
<111> cube diagonals
[110][110]
[101][101]
[011][011][111][111]
K.J. Hemker
Silicon wafers:
Primaryflat
<110>±3°
[010]
[001]
{100} [001]
[00-1]
[010][0-10]
[01-1]
[011]
[0-11]
[0-1-1]
45°
45°
45°
45°
Primary
flat
<110>±3°
[00-1]
[010
{100}
K.J. Hemker
Isotropic Elasticityσ = E ε
τ = G γ
p = -K ∆
ν = εyy / εxx
σyy= λ εxx
All directions the sameAll directions the same
polycrystallinepolycrystalline
σyy
εxx
Lamé Coefficient (λ)
5 constantsonly
2 independent
K.J. Hemker
Anisotropic elasticity (stiffness)
σi = Cij εj
σxx C11 C12 C13 C14 C15 C16 εxx
σyy C21 C22 C23 C24 C25 C26 εyy
σzz = C31 C32 C33 C34 C35 C36 εzz
σyz C41 C42 C43 C44 C45 C46 εyz
σxz C51 C52 C53 C54 C55 C56 εxz
σxy C61 C62 C63 C64 C65 C66 εxy
K.J. Hemker
Complianceεi = Sij σj
εxx S11 S12 S13 S14 S15 S16 σxx
εyy S21 S22 S23 S24 S25 S26 σyy
εzz = S31 S32 S33 S34 S35 S36 σzz
εyz S41 S42 S43 S44 S45 S46 σyz
εxz S51 S52 S53 S54 S55 S56 σxz
εxy S61 S62 S63 S64 S65 S66 σxy
K.J. Hemker
Cij and Sij for Cubic Crystals
C11 C12 C12
C12 C11 C12
Cij = C12 C12 C11
C44
C44
C44
0
0
C11, C12, C44
K.J. Hemker
1-Dimensional LoadingE[100], E[110], and E[111]
E[100] = 1s11
E[110] = 2[s11 + s12 + s44/2]
E[111] = 3[s11 + 2s12 + s44]
K.J. Hemker
Relations :
c11 = s11 + s12(s11 - s12) (s11 + 2s12)
c12 = -s12(s11 - s12) (s11 + 2s12)
c44 = 1s44
s11 = c11 + c12(c11 - c12) (c11 + 2c12)
s12 = -c12(c11 - c12) (c11 + 2c12)
s44 = 1c44
K.J. Hemker
Example of Anisotropic Elasticity:
!To be addressed in homework problem.– Assume that for Si
• C11 = 166 GPa• C12 = 64 GPa• C44 = 80 GPa
– Calculate Young’s modulus (E) along <100>, <110> and <111>
K.J. Hemker
Elasticity of Textured Films:
1. Consider a textured thin film[001]
2. Calculate E from Sij's
3. Estimate Voight and Reuss bounds for E
EReuss = 1
ΣΣΣΣ (((( ViΕΕΕΕi
))))i=1
1000
((((000011111111))))αααα
αααα'
ββββββββ'
EVoight = ΣΣΣΣ ViEii=1
1000
K.J. Hemker
LIGA-Ni texture model
50
100
150
200
250
300
350
400
450
0 50 100 150 200
(011)(111)(001)
You
ng's
mod
ulus
(G
Pa)
Angle (degree)
<011>, 232 GPa
<001>, 136 GPa
<111>, 303 GPa
232 GPa
E(001) = 177 GPa
Emeasured = 180 GPa
• <001> out-of-plane • No in-plane texture
JJOHNS HOPKINSOHNS HOPKINSEENNGGII NNEEEERRII NNGG
Isotropic E
K.J. Hemker
Headlines about elasticity:
! [Cij]’s will not change at micro-scale.
!What out for texture effects.
!Single-crystalline materials require anisotropic elasticity.
K.J. Hemker
Mechanical Strength:
!Ductile Materials (Metals, hi T Si)– Deform plastically or yield
!Brittle Materials (Ceramics, Glass, RT Si)– Fracture
K.J. Hemker
Yield Strength:
!Dislocations lead to plastic deformation!Slow down dislocations = strengthening
– lattice– solute atoms– precipitates– grain boundaries
Note: MEMS is a young field many people still use pure materials.
K.J. Hemker
Stress Magnification:Stress Magnification:
2c
2h
σ
σ
σ
σο
σmax
Inglis 1913
σmax = σ (1+2c / h)for an ellipse: ρ = h2 / cσmax = 2σ (c / ρ)1/2
limρ−>0 (ρ) = ao
σmax = 2σ (10-2 / 10−10)1/2
σmax = 20,000 σ !!!!!
ρ
K.J. Hemker
Basis for Fracture MechanicsBasis for Fracture Mechanics
Geometry and loading
“Stress Intensity Factor”
Material Parameter
“Fracture Toughness”
σ (πc )1/2 = Kc
K.J. Hemker
Key to Fracture MechanicsKey to Fracture Mechanics
1. Determine Kc- measure on specimens of known geometry
2. Calculate K- from current geometry and loading
3. Compare K with Kc- K < Kc is OK- K > Kc will fracture
K.J. Hemker
Comparison of structural members: turbines
Micro Macro
GE 90 Jet Engine :
GE 90 Jet Engine :
Ni superalloys
silicon -> SiC
K.J. Hemker
Design views:
!MEMS view:– “Silicon a wonderful structural material”
!Macro view:– Silicon too brittle and too expensive
Who is right ?
K.J. Hemker
When will Si fracture ?
!Macro world defects ~ 1 mm
σf = KIc/(πc)1/2 = 0.9 MPa m1/2 / (π 10-3 m)1/2 = 16 MPa
!MEMS world defects ~1 µm
σf = KIc/(πc)1/2 = 0.9 MPa m1/2 / (π 10-6 m)1/2
= 508 MPa
K.J. Hemker
How much will Si cost ?Total Cost = Cmaterial + Cmanufacture + Cdisposal
!Macro worldCsteel = $0.20/lb x tons + $
CSi = $4.00/lb x tons + $$$
!MEMS worldCsteel = $0.20/lb x grams + $$$$
CSi = $4.00/lb x tons + $$$