Machine Anisotropy

1. Wave Propagation through
Anisotropic Medium
Project Presentation
ME 691-IX: Wave Motion
Instructed by Prof. K. R. Jayaprakash
Presented by,
Harsh Gupta, Archita Gogoi and Diptangshu Paul

2. 2
What is Anisotropy?
𝒄𝑔
𝒌
𝒌
𝒄𝑔
Wave travels faster in certain directions
The direction of wavevector is not necessarily same
as the direction of the group velocity
James P. Wolfe, ‘Imaging Phonons’, Cambridge University Press, 2005

3. 3
Waves in elastic media
Equation of motion in a material medium:
𝜌
𝜕2𝑢𝑖
𝜕𝑡2 =
𝜕𝜎𝑖𝑗
𝜕𝑥𝑗
𝜌
𝜕2𝑢𝑖
𝜕𝑡2
=
𝜕
𝜕𝑥𝑗
𝐶𝑖𝑗𝑙𝑚𝜀𝑙𝑚 = 𝐶𝑖𝑗𝑙𝑚
𝜕2𝑢𝑙
𝜕𝑥𝑗𝜕𝑥𝑚
Imposing the plane wave solution, 𝒖 = 𝑢0𝒑𝑒𝑖 𝒌∙𝒓−𝜔𝑡 , we obtain, (Note: wavevector 𝒌 = 𝑘𝑖𝑛𝑖,
polarization vector, 𝒑)
𝜌𝜔2𝑝𝑖 = 𝐶𝑖𝑗𝑙𝑚𝑘𝑗𝑘𝑚𝑝𝑙
⇒ 𝜌𝑐2
𝑝𝑙𝛿𝑖𝑙 = 𝐶𝑖𝑗𝑙𝑚𝑛𝑗𝑛𝑚𝑝𝑙
Setting 𝐶𝑖𝑗𝑙𝑚𝑛𝑗𝑛𝑚/𝜌 = Γ𝑖𝑙, we obtain, Christoffel equation,
Γ𝑖𝑙 − 𝑐2𝛿𝑖𝑙 𝑝𝑙 = 0
Which has the eigenvalues i.e., phase velocities, 𝑐𝛼, where 𝛼 = 1,2,3 for a three dimensional medium.

4. 4
Slowness surfaces
A useful way to represent the 𝑐𝛼 values, is the slowness surface.
The representation 𝒌 = 𝑘, 𝜃𝑘, 𝜙𝑘 is used to define
𝑠 𝛼; 𝜃𝑘, 𝜙𝑘 = 1/𝑐𝛼 𝜃𝑘, 𝜙𝑘 , which is called the slowness vector
𝒔 = 𝒌 𝜔 = 𝒏 𝑐. This provides with,
𝐶𝑖𝑗𝑙𝑚𝑠𝑗𝑠𝑚 − 𝜌𝛿𝑖𝑙 𝑝𝑙 = 0
Directivity plot:
A radial plot of 𝑠 𝛼; 𝜃𝑘, 𝜙𝑘 gives a slowness surface for the
mode of propagation, defined by 𝛼, which has the shape of an
iso-frequency surface in 𝒌-space: 𝜔/𝑐𝛼 𝜃𝑘, 𝜙𝑘 .
For example, we saw the modes of propagation 𝛼, in an
isotropic solid, referred to P, SV, and SH waves.
𝜃𝑘
𝑠𝑥
𝑠𝑦
Isotropic medium in 2D medium
𝑐𝑃 = 𝜆 + 2𝜇 𝜌 > 𝜇 𝜌 = 𝑐𝑆

5. 5
Slowness surface and Wave surface
𝒄𝑔
𝒌
Wave surface (from wavefront)
𝒄𝑔
Slowness surface (from iso-frequency contour)
𝜁

6. 6
Waves in a general elastic media
Let us begin at the Christoffel equation, Γ𝑖𝑙 − 𝑐2
𝛿𝑖𝑙 𝑝𝑙 = 0, and the most general expressions are,
Here, Voigt notation has been used as, 𝐶𝐼𝐽 = 𝐶𝑖𝑗𝑙𝑚, where, in three dimensions,
𝐼 or 𝐽 1 2 3 4 5 6
𝑖𝑗 or 𝑙𝑚 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑦𝑧 𝑥𝑧 𝑥𝑦
A. G. Every, ‘General closed-form expressions for acoustic waves in elastically anisotropic solids’, Phys. Rev. B, 22(4) (1980) 1746-1760

7. 7
Invariants of the Christoffel matrix
Trace of Γ, 𝑇 = Γ𝑖𝑖, is the first invariant of Γ. Using, 𝑆 + 𝑇 = 3𝜌𝑐2
in Γ𝑖𝑙 − 𝑐2
𝛿𝑖𝑙 = 0, we obtain,
Λ𝑖𝑙 − 𝑆𝛿𝑖𝑙 = 0
Where, Λ𝑖𝑙 = 3Γ𝑖𝑙 − 𝑇𝛿𝑖𝑙, that has Tr Λ = 0. This results in the following cubic equation,
𝑆3
+ Tr Λ 𝑆2
− 3𝐺𝑆 − 2𝐻 = 0
Having three real roots, 𝑆0, 𝑆1, and 𝑆2. The second invariant is −3𝐺 = Λ𝑖𝑖Λ𝑗𝑗 − Λ𝑖𝑗Λ𝑗𝑖 = 𝑆0𝑆1 + 𝑆1𝑆2 +
𝑆2𝑆0, and the third invariant is 2𝐻 = 𝜖𝑖𝑗𝑘Λ1𝑖Λ2𝑗Λ3𝑘 = 𝑆0𝑆1𝑆2.
Further, the roots 𝑆0, 𝑆1, and 𝑆2, provides three velocities, 𝑐0, 𝑐1, and 𝑐2, related as,
𝑇 = 𝜌 𝑐0
2
+ 𝑐1
2
+ 𝑐2
2
It is interesting to observe that, even if a system lacks a center of inversion, the inversion symmetry is
still valid for the wave propagation. This is because, all elements of Γ are quadratic in 𝑛𝑖.
A. G. Every, ‘General closed-form expressions for acoustic waves in elastically anisotropic solids’, Phys. Rev. B, 22(4) (1980) 1746-1760

8. 8
Isotropic and anisotropic media
Let us use this for isotropic medium, where c0 = 𝜆 + 2𝜇 𝜌, c1,2 = 𝜇 𝜌,
𝑇 = 𝐶11 + 𝐶55 + 𝐶66 𝑛1
2
+ 𝐶22 + 𝐶44 + 𝐶66 𝑛2
2
+ 𝐶33 + 𝐶44 + 𝐶55 𝑛3
2
+2 𝐶56 + 𝐶24 + 𝐶34 𝑛2𝑛3 + 2 𝐶15 + 𝐶46 + 𝐶35 𝑛3𝑛1 + 2 𝐶16 + 𝐶26 + 𝐶45 𝑛1𝑛2
For isotropic medium,
𝐶𝑖𝑗𝑙𝑚 =
𝜆 + 2𝜇 𝜆 𝜆 0 0 0
𝜆 𝜆 + 2𝜇 𝜆 0 0 0
𝜆 𝜆 𝜆 + 2𝜇 0 0 0
0 0 0 𝜇 0 0
0 0 0 0 𝜇 0
0 0 0 0 0 𝜇
Leading to the following verification,
𝑇 = 𝜆 + 4𝜇 𝑛1
2
+ 𝑛2
2
+ 𝑛3
2
= 𝜌
𝜆 + 2𝜇
𝜌
+
𝜇
𝜌
+
𝜇
𝜌
= 𝜌 𝑐0
2
+ 𝑐1
2
+ 𝑐2
2
For materials that lack the isotropic nature, the relations above still hold true, although there are
more number of independent elastic constants depending on the degree of anisotropy.

9. 9
An example of a discrete anisotropic medium
𝑀
𝐾
𝐾
𝑑
Equations of Motion:
𝑀
𝑑2
𝑢 𝑛, 𝑚
𝑑𝑡2 = 𝐾 𝑢 𝑛 − 1, 𝑚 − 2𝑢 𝑛, 𝑚 + 𝑢 𝑛 + 1, 𝑚
+𝐾 𝑢 𝑛, 𝑚 − 1 − 2𝑢 𝑛, 𝑚 + 𝑢 𝑛, 𝑚 + 1
Plane wave solution:
𝑢 𝑛, 𝑚 = 𝑈𝑒𝑖 𝑘𝑥𝑛+𝑘𝑦𝑚
𝑒−𝑖𝜔𝜏
Periodicity Condition:
𝑢 𝑛 ± 1, 𝑚 ± 1 = 𝑢 𝑛, 𝑚 𝑒𝑖 ±𝑘𝑥±𝑘𝑦
Dispersion relation:
𝜔2
=
4𝐾
𝑀
sin2
𝑘𝑥𝑑
2
+
4𝐾
𝑀
sin2
𝑘𝑦𝑑
2

10. 10
Dispersion surfaces and isofrequency contours
MATLAB

11. 11
Isofrequency contours and wave surfaces
MATLAB

12. 12
Low frequency excitation
MATLAB

13. 13
Medium frequency excitation
MATLAB

14. 14
High frequency excitation
MATLAB

15. Thank you!
James P. Wolfe, ‘Imaging Phonons’, Cambridge University Press, 2005
A. G. Every, ‘General closed-form expressions for acoustic waves in elastically anisotropic solids’, Phys. Rev. B, 22(4) (1980) 1746-1760
M. A. Slawinski, ‘On Elastic wave Propagation in Anisotropic Media: Reflection/Refraction Laws, Raytracing and Traveltime Inversion’

16. 16
Anisotropy
𝒄𝑔
Slowness surface (or isofrequency contour)
𝜁
𝒄𝑔
𝒄𝑔
𝜁 = 0
𝜁 ≠ 0
𝒌1
𝒌2
𝒌2
𝒌1

17. 17
Waves in elastic media
A useful way to represent the 𝑐𝛼 values, is the slowness surface.
The representation 𝒌 = 𝑘, 𝜃𝑘, 𝜙𝑘 is used to define 𝑠 𝛼; 𝜃𝑘, 𝜙𝑘 = 1/𝑐𝛼 𝜃𝑘, 𝜙𝑘 , which is called the
slowness vector 𝒔 = 𝒌 𝜔 = 𝒏 𝑐. This provides with,
𝐶𝑖𝑗𝑙𝑚𝑠𝑗𝑠𝑚 − 𝜌𝛿𝑖𝑙 𝑝𝑙 = 0
Directivity plot:
A radial plot of 𝑠 𝛼; 𝜃𝑘, 𝜙𝑘 gives a slowness surface for the mode of propagation, defined by 𝛼, which
has the shape of an iso-frequency surface in 𝒌-space: 𝜔/𝑐𝛼 𝜃𝑘, 𝜙𝑘 .
For example, we saw the modes of propagation 𝛼, in an isotropic solid, referred to P, SV, and SH
waves.