Existence of extraordinary transonic states in monoclinic elastic media
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Existence of extraordinary transonic states in monoclinic elastic media. Litian Wang and Kent Ryne Østfold University College 1757 Halden Norway. Main problems. Existence of extraordinary transonic states associated with extraordinary zero-curvature slowness curve

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Existence of extraordinary transonic states in monoclinic elastic media

Existence of extraordinary transonic states in monoclinic elastic media

Litian Wang and Kent Ryne

Østfold University College

1757 Halden Norway


Main problems
Main problems

  • Existence of extraordinary transonic states associated with extraordinary zero-curvature slowness curve

  • Existence of space of degeneracy

  • Existence of generalized surface waves




Zero curvature transonic states

n

m

Zero-curvature transonic states

E1 E2 E3 E4

Barnett, Lothe & Gundersen



Problem 1
Problem 1

  • Can a slowness curve have zero-curvature locally?

  • How flat a slowness curve can be?


Degree of freedom
Degree of freedom

  • Degree of freedom = 6


Wave propagation in monoclinic media
Wave propagation in monoclinic media

  • Elastic stiffness matrix:


Existence of extraordinary transonic states in monoclinic elastic media

k

θ


Christoffel equation
Christoffel equation

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


Curvature in slowness plot

θ

k

Curvature in slowness plot

Let

Curvature k and its second derivative k’’ in the neighborhood of z-axis are given by


Existence of extraordinary transonic states in monoclinic elastic media

θ

k

How to find the eigenvalue ?

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


Perturbation method

θ

k

Perturbation method


Existence of extraordinary transonic states in monoclinic elastic media

Where

θ

k


Results 1

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)


Results 2

θ

k

Results - 2

(a) The second derivative of curvature:

(b) Extraordinary zero-curvature along z-axis when (c11c36-d13c16)2=c112c55∆45)


Problem 2
Problem 2

  • Space of degeneracy in monoclinic media

  • Generalized surface waves


Degeneracy of the stroh eigenvalues
Degeneracy of the Stroh eigenvalues

E1 zero-curvature transonic state:


Degeneracy of the stroh eigenvalues1
Degeneracy of the Stroh eigenvalues

E4 zero-curvature transonic state:


Result 3
Result 3

Space of degeneracy vs zero-curvature slowness curve:


Result 4
Result 4

Space of degeneracy vs generalized surface waves

  • Subsonic surface waves

  • Supersonic surface waves


Conclusions
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