Elastic Properties of Solids, Part II Topics Discussed in Kittel, Ch Elastic Properties of Solids, Part II Topics Discussed in Kittel, Ch. 3, pages 73-85 A lecture found on the internet! To understand this cartoon, you have to be a physicist, but also understand a little about baseball!

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

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

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)

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

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):

where 

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 .

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

Dispersion Equation → dispersion equation

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

Waves in the [110] direction → Lonitudinal Transverse Transverse

Prob 3.10