Existence of extraordinary transonic states in monoclinic elastic media

1. Existence of extraordinary transonic states in monoclinic elastic media Litian Wang and Kent Ryne Østfold University College 1757 Halden Norway

2. Main problems • Existence of extraordinary transonic states associated with extraordinary zero-curvature slowness curve • Existence of space of degeneracy • Existence of generalized surface waves

3. Surface geometry of slowness surface Cubic (Cu) Monoclinic

4. Surface geometry of slowness surface Cubic (Cu) Monoclinic

5. n m Zero-curvature transonic states E1 E2 E3 E4 Barnett, Lothe & Gundersen

6. Surface geometry of slowness surface Cubic (Cu) Monoclinic

7. Problem 1 • Can a slowness curve have zero-curvature locally? • How flat a slowness curve can be?

8. Degree of freedom • Degree of freedom = 6

9. Wave propagation in monoclinic media • Elastic stiffness matrix:

10. k θ

11. Christoffel equation Where d13=c13+c55, ∆15=c11-c55, ∆64=c66-c44, ∆53=c55-c33,

12. θ k Curvature in slowness plot Let Curvature k and its second derivative k’’ in the neighborhood of z-axis are given by

13. θ k How to find the eigenvalue ? Where d13=c13+c55, ∆15=c11-c55, ∆64=c66-c44, ∆53=c55-c33,

14. θ k Perturbation method

15. Where θ k

16. θ k Results - 1 (a) Normal curvature of slowness curve along z-axis (See also Shuvalov et al) (b) Zero-Curvature along z-axis when d132 = c11∆35 or (c13+c55)2=c11(c33-c55)

17. θ k Results - 2 (a) The second derivative of curvature: (b) Extraordinary zero-curvature along z-axis when (c11c36-d13c16)2=c112c55∆45)

20. Problem 2 • Space of degeneracy in monoclinic media • Generalized surface waves

21. Degeneracy of the Stroh eigenvalues E1 zero-curvature transonic state:

22. Degeneracy of the Stroh eigenvalues E4 zero-curvature transonic state:

23. Result 3 Space of degeneracy vs zero-curvature slowness curve:

24. Result 4 Space of degeneracy vs generalized surface waves • Subsonic surface waves • Supersonic surface waves

25. Conclusions • Existence of extraordinary zero-curvature slowness curve • Existence of space of degeneracy • Existence of supersonic surface wave along the space of degeneracy • Existence of generalized subsonic surface wave along the space of degeneracy