1. Elastic Properties of Solids, Part II Topics Discussed in Kittel, Ch
Elastic Properties of Solids, Part II Topics Discussed in Kittel, Ch. 3, pages A lecture found on the internet!
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2. Analysis of Elastic Strains
Ref: L.D.Landau, E.M.Lifshitz, “Theory of Elasticity”, Pergamon Press (59/86)
Continuum approximation: good for λ > 30A.
Description of deformation (Cartesian coordinates):
Displacement vector field u(r).
Material point
Nearby point
= strain tensor
= linear strain tensor

3. Dilation
uik is symmetric → diagonalizable →  principal axes such that
(no summation over i ) → 
Fractional volume change 
 Trace of uik

4. Stress Total force acting on a volume element inside solid 
f  force density
Newton’s 3rd law → internal forces cancel each other
→ only forces on surface contribute
This is guaranteed if
σ  stress tensor
so that
σik  ith component of force acting on the surface element normal to the xk axis.
Moment on volume element 
Only forces on surface contribute →
(σ is symmetric)

5. Elastic Compliance & Stiffness Constants
σ and u are symmetric → they have at most 6 independent components
Compact index notations (i , j) → α :
(1,1) → 1, (2,2) → 2, (3,3) → 3, (1,2) = (2,1) → 4, (2,3) = (3,2) → 5, (3,1) = (1,3) → 6
Elastic energy density:
i , j , k, l = 1,2,3
α , β = 1,2,…,6 21
where
 elastic stiffness constants
 elastic modulus tensor
uik & uki treated as independent
Stress:
S α β  elastic compliance constants

6. Elastic Stiffness Constants for Cubic Crystals
Invariance under reflections xi → –xi  C with odd numbers of like indices vanishes
Invariance under C3 , i.e.,
 All C i j k l = 0 except for (summation notation suspended):

7. where 

8. Bulk Modulus & Compressibility
Uniform dilation:
δ = Tr uik = fractional volume change
B = Bulk modulus
= 1/κ
κ = compressibility
See table 3 for values of B & κ .

9. Elastic Waves in Cubic Crystals
Newton’s 2nd law:
don’t confuse ui with uα → 
Similarly

10. Dispersion Equation →
dispersion equation

11. Waves in the [100] direction→
Longitudinal
Transverse, degenerate

12. Waves in the [110] direction→
Lonitudinal
Transverse
Transverse

13. Prob 3.10