To understand this cartoon, you have to be a physicist, but you must also understand a little about baseball! Elastic Properties of Solids Topics Discussed in Kittel, Ch. 3, pages 73-85

Analysis of Elastic Strains Continuum approximation: good for λ > 30A. Description of deformation (Cartesian coordinates): Material point Ref: L.D.Landau, E.M.Lifshitz, “Theory of Elasticity”, Pergamon Press (59/86) Displacement vector field u(r). Nearby point = strain tensor= linear strain tensor

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

Stress Total force acting on a volume element inside solid  f  force density Newton’s 3 rd law → internal forces cancel each other→ only forces on surface contribute This is guaranteed if so that σ  stress tensor σ ik  i th component of force acting on the surface element normal to the x k 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 S α β  elastic compliance constants Elastic energy density: where  elastic stiffness constants i, j, k, l = 1,2,3 α, β = 1,2,…,6  elastic modulus tensor Stress: u ik & u ki treated as independent 21

Elastic Stiffness Constants for Cubic Crystals Invariance under reflections x i → –x i  C with odd numbers of like indices vanishes Invariance under C 3, i.e.,  All C i j k l = 0 except for (summation notation suspended):

where 

Bulk Modulus & Compressibility Uniform dilation: δ = Tr u ik = 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 2 nd law:don’t confuse u i with u α → Similarly 

Dispersion Equation → dispersion equation

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

Waves in the [110] direction → Lonitudinal Transverse