6.730 Physics for Solid State Applications
Lecture 4: Vibrations in Solids
February 11, 2004
Outline
• 1-D Elastic Continuum
• 1-D Lattice Waves
• 3-D Elastic Continuum
• 3-D Lattice Waves
Strain E and Displacement u(x)
dx
dx’
u(x)
u(x+dx)
u(L)
δ(dx) = dx’ – dx how much does a differential length change
δ(dx) = u(x+dx) - u(x) difference in displacements
Strain:
1
Uniform Strain & Linear Displacement u(x)
dx
dx’
u(x)
u(x+dx)
u(L)
Linear displacement: u(x) = E0 x
Constant Strain:
More Types of Strain
dx
Non-uniform
E = E(x)
dx’
u(x)
u(x+dx)
u(L)
Zero Strain:
u(x) is constant
Just a Translation
We will ignore this
2
1-D Elastic Continuum
Stress and Strain
uniaxial loading
Lo
L
Stress:
Elongation:
Normal Strain:
If ux is uniform there is no strain, just rigid body motion.
1-D Elastic Continuum
Young’s Modulus
Young’s Modulus For Various Materials (GPa)
from Christina Ortiz
CERAMICS GLASSES AND SEMICONDUCTORS
Diamond (C)
1000
Tungsten Carbide (WC)
450 -650
Silicon Carbide (SiC)
450
390
Aluminum Oxide (Al2O3)
Berylium Oxide (BeO)
380
Magnesium Oxide (MgO)
250
Zirconium Oxide (ZrO)
160 - 241
145
Mullite (Al6Si2O13)
Silicon (Si)
107
94
Silica glass (SiO2)
Soda-lime glass (Na2O - SiO2) 69
METALS :
Tungsten (W)
Chromium (Cr)
Berylium (Be)
Nickel (Ni)
Iron (Fe)
Low Alloy Steels
Stainless Steels
Cast Irons
Copper (Cu)
Titanium (Ti)
Brasses and Bronzes
Aluminum (Al)
406
289
200 - 289
214
196
200 - 207
190 - 200
170 - 190
124
116
103 - 124
69
PINE WOOD (along grain):
POLYMERS :
Polyimides
Polyesters
Nylon
Polystryene
Polyethylene
Rubbers / Biological
Tissues
10
3-5
1-5
2-4
3 - 3.4
0.2 -0.7
0.01-0.1
3
Dynamics of 11-D Continuum
1-D Wave Equation
Net force on incremental volume element:
Dynamics of 11-D Continuum
1-D Wave Equation
Velocity of sound, c, is proportional to stiffness and inverse prop. to inertia
4
Dynamics of 11-D Continuum
1-D Wave Equation Solutions
Clamped Bar: Standing Waves
Dynamics of 11-D Continuum
1-D Wave Equation Solutions
Periodic Boundary Conditions: Traveling Waves
5
3-D Elastic Continuum
Volume Dilatation
Lo
apply load
L
Volume change is sum of all three normal strains
3-D Elastic Continuum
Poisson’s Ratio
ν is Poisson’s Ratio – ratio of lateral strain to axial strain
Poisson’s ratio can not exceed 0.5, typically 0.3
6
3-D Elastic Continuum
Poisson’s Ratio Example
Aluminum: EY=68.9 GPa, ν = 0.35
5kN
5kN
75mm
20mm
3-D Elastic Continuum
Poisson’s Ratio Example
Aluminum: EY=68.9 GPa, ν = 0.35
5kN
5kN
75mm
20mm
7
3-D Elastic Continuum
Poisson’s Ratio Example
Aluminum: EY=68.9 GPa, ν = 0.35
5kN
5kN
75mm
20mm
3-D Elastic Continuum
Shear Strain
Shear loading
Shear plus rotation
2φ
Pure shear
φ
φ
Pure shear strain
Shear stress
G is shear modulus
8
3-D Elastic Continuum
Stress and Strain Tensors
For most general isotropic medium,
Initially we had three elastic constants: EY, G, e
Now reduced to only two: λ, µ
3-D Elastic Continuum
Stress and Strain Tensors
If we look at just the diagonal elements
Inversion of stress/strain relation:
9
3-D Elastic Continuum
Example of Uniaxial Stress
Lo
L
Dynamics of 33-D Continuum
3-D Wave Equation
Net force on incremental volume element:
Total force is the sum of the forces on all the surfaces
10
Dynamics of 33-D Continuum
3-D Wave Equation
Net force in the x-direction:
Dynamics of 33-D Continuum
3-D Wave Equation
Finally, 3-D wave equation….
11
Dynamics of 33-D Continuum
Fourier Transform of 33-D Wave Equation
Anticipating plane wave solutions, we Fourier Transform the equation….
Three coupled equations for Ux, Uy, and Uz….
Dynamics of 33-D Continuum
Dynamical Matrix
Express the system of equations as a matrix….
Turns the problem into an eigenvalue problem for
the polarizations of the modes (eigenvectors) and
wavevectors q (eigenvalues)….
12
Dynamics of 33-D Continuum
Solutions to 33-D Wave Equation
Transverse polarization waves:
Longitudinal polarization waves:
Dynamics of 33-D Continuum
Summary
1. Dynamical Equation can be solved by inspection
2.
There are 2 transverse and 1 longitudinal polarizations for each q
3. The dispersion relations are linear
4. The longitudinal sound velocity is always greater than
the transverse sound velocity
13