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Imagine waves in an infinite elastic anisotropic solid. We shall consider waves in pure crystals, where homogeneity and pure anisotropy is assumed. There are many different ways to approach this problem, but for now we consider the following.

Using indicial (or tensor) notation and Newton’s law, set

∂

∂σ^ik =ρ ρ =σ

k

i i ik k

x u or u _, (3.20)

as a governing wave equation, where σ denotes stress. Using Hooke’s law, we have

σik = Ciklmεlm (3.21)

as a constitutive equation. Combining (3.20) and (3.21) yields

ρu C ε ε

From the definition of strain in the strain displacement equations,

εlm l

again, combining (3.22) and (3.21), we find the result:

ρu C u

Note that Ciklm is symmetrical with respect to l and m, so C_iklm = Ciklm = Ckilm. Therefore, we can interchange l and m.

41 3.3 The Christoffel Equation for Anisotropic Media

Let us assume plane harmonic traveling waves to see if a solution of this form is possible. Put

ui = Ai exp{i(kjxj − ωt)}, (3.25) where kj is the wavevector. (Note that Ai = Aαi, where the αi are direction cosines of particle displacement.) Substituting (3.25) into (3.24) gives an eigenvalue problem.

From earlier work in harmonic motion, in considering the amplitude of the second derivative we know that ρüi = ρω²u_i. Note that kjx_j is a single dot product, which is useful in differentiating to obtain

ρüi = −ρω²ui = −Ciklmkkklum.

Tensor analysis can be used to establish that Ciklmu_m.kl is equivalent to Ciklmk_kk_lu_m, with i the free index and with summing over k, l, and m. For one term we can show the equivalence quite easily as follows. If um = A exp i(k₁x₁ + k₂x₂ + k₃x₃ − ωt) = This is the famous Christoffel equation for anisotropic media.

The Christoffel acoustic tensor may be defined as

λim = Γim = Ciklmnknl, (3.27)

For a nontrivial solution, we must set the determinant of the coefficient matrix equal to zero:

where λ11, λ12, λ13, … are obtained from the expression for the acoustic tensor. Recall from (3.27) that λim = Ciklmnknl.

Expand this acoustic tensor carefully for all elements of the matrix. The indices i and m are free; k and l are summers:

We can simplify the results even further by converting Cikjl to Cnm as follows: if i = k then n = i, and if j = l then m = j; if i ≠ k then n = 9 − (i + k), and if j ≠ l then m

Continuing with these computations would allow us to plot a phase profile. Given constants and directions, various c can be calculated in the wavevector k directions.

This is a tedious process that calls for an efficient computer program.

Once the phase velocity values are extracted for specific directions, we would like to see if we have a pure mode – that is, if the particle velocity direction is aligned perfectly with the chosen k direction for the phase velocity computation.

To see if there are pure modes, we must expand and extract all roots of the bi-cubic equation. For a given direction k of propagation, there are three waves, equations to calculate the eigenvectors:

( )

43 3.3 The Christoffel Equation for Anisotropic Media

(λ11 − ρc²)α1 + λ12α2 + λ13α3 = 0,

λ21α1 + (λ22 − ρc²)α2 + λ23α3 = 0, (3.29) λ31α1 + λ32α2 + (λ33 − ρc²)α3 = 0.

Solve three times for each value of c² (the result can be checked, since A is orthogonal).

We can arrange the solution as follows:

c c c

Note that determinant A = +1 for a right-handed Cartesian coordinate system.

3.3.1 Sample Problem

How can we determine phase velocity in a specific direction for a given level of anisotropy? More particularly, we shall discuss two problems with the aid of Cij

matrices as presented in elasticity theory (for waves that are orthotropic, hexagonal, isotropic, etc.; see Appendix B).

(1) Calculate directions for a pure longitudinal wave and two pure transverse waves for a cubic crystal. Note: pure longitudinal, u n× = 0; pure transverse, u n• = 0.

(2) Develop equations for c₁, c₂, and c₃ as a function of the wavevector k directions in the (x1, x2)-plane. Note: n represents the direction cosine of the angle between the wavevector and the x₁, x₂, and x₃ axes. Therefore, we compute Γim

for a cubic crystal. When we solve the determinant for specific n, we must let n vary in the plane for small increments. In general, waves are not pure L or pure S and so are often termed quasi-longitudinal or quasi-shear. However, if k is an eigenvector of λik then the waves are pure.

In order to solve these problems, a specific wavevector must now be considered.

Therefore, imagine plane waves in the [1, 1, 0] direction for a cubic crystal (see Figure 3.1). Next, evaluate the terms of the acoustic tensor,

λim = Ciklmn_kn_l,

where i and m are free indices with double summations over k and l from 1 to 3.

Using tensor-to-matrix notation yields

11 → 1, 22 → 2, 33 → 3, 23, 32 → 4, 31, 13 → 5, 12, 21 → 6.

Figure 3.1. Plane waves in the [1, 1, 0] direction.

The elastic constant matrix for a cubic crystal may be found in Appendix B: By Christoffel’s equations we have and so

λ ρ λ λ

Through factoring, we solve for three c; the eigenvector solution is then required to know which wave is in which direction or alignment with k. Since c_i= C44/ .ρ

45 3.3 The Christoffel Equation for Anisotropic Media

1

To confirm wave type and character (pure or quasi), we must solve for the three eigenvectors:

One wave in closest alignment with n is the pure or quasi-longitudinal wave; the other two are shear or quasi-shear. Although the task is tedious, we must solve the equations three times (for ux, uy, and uz) for each ci. (Note: If ū1 × n = 0 then we have a pure l wave, or use ū × n = |u||n| sin θ. If ū2 • n = 0 or ū3 • n = 0 then we have a pure transverse wave.) Consequently, if αi are direction cosines then we can solve the following.

(1) For c₁:α₁=α₂=1/ 2 and α3 = 0; therefore, c₁ is a pure l wave. (In the [1, 1, 0] direction, the direction cosine vector is in the same direction as the wavevector.)

(2) For c₂: α3 = 1 and α1 = α2 = 0. Therefore, when considering the dot product [0, 0, 1] • [1, 1, 0] = 0, we see that c2 is pure shear.

(3) Finally, for c₃:α₁=α₂=1/ 2,α₂= −1/ 2, and α3 = 0. The dot product is again [1, −1, 0] • [1, 1, 0] = 0; this is also a pure shear wave.

These results may be confirmed by checking the eigenvector direction that leads to three mutually perpendicular directions.

In general, we can solve the eigenvector problem by solving three times for each c (and hence the direction cosines αi for each c):

α1λ11 + α2λ12 + α3λ13 = α1ρc², α1λ21 + α2λ22 + α3λ23 = α2ρc², α1λ31 + α2λ32 + α3λ33 = α3ρc². This general solution was derived by Chistoffel many years ago.

Sample problems can be studied for a variety of different materials. Stiffness coefficients for a few selected materials can be found in Table 3.1.

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Table 3.1. Stiffness coefficients Cij for selected materials

Material Stiffness (10¹⁰ N/m²)

C11 C22 C33 C44 C55 C66 C12 C13 C14 C16 C23 C25

Barium titanate^ab 15.0 14.6 4.4 6.6 6.6

Diamond 102 49.2 25

Lead titanate–zirconate (PZT-2)^ab 13.5 11.3 2.22 6.79 6.81

Quartz^a 8.674 10.72 5.794 0.699 1.191 1.791

Rochelle salt^a 2.8 4.14 3.94 0.666 0.285 0.96 1.74 1.50 1.97

Sapphire 49.4 49.6 14.5 15.8 11.4 −2.3

Cadmium sulfide^a 9.07 9.38 1.504 5.81 5.10

Germanium 12.89 6.71 4.83

Silicon 16.57 7.956 6.39

Aluminum, crystal 10.80 2.85 6.13

Aluminum, polycrystal 11.1 2.5

Gold, crystal 18.6 4.20 15.7

Titanium, crystal 16.2 18.1 4.67 9.2 6.9

Titanium, polycrystal 16.59 4.4

Tungsten, crystal 50.2 15.2 19.9

Tungsten, polycrystal 58.1 13.4

Graphite/epoxy (AS4/3501–6) 16.209 1.63 0.774 0.671 0.795

Sources: Most data from Auld (1990). Data for graphite/epoxy from Mal, Lih, and Bar-Cohen (1994).

a Piezoelectric.

b Poled ceramic. The stiffness matrix has the same form as for the hexagonal crystal system, with Z along the poling axis.

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