Existence of extraordinary transonic states in monoclinic elastic media

1 / 25

# Existence of extraordinary transonic states in monoclinic elastic media - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Existence of extraordinary transonic states in monoclinic elastic media' - mostyn

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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
• Existence of space of degeneracy
• Existence of generalized surface waves

n

m

Zero-curvature transonic states

E1 E2 E3 E4

Barnett, Lothe & Gundersen

Problem 1
• Can a slowness curve have zero-curvature locally?
• How flat a slowness curve can be?
Degree of freedom
• Degree of freedom = 6
Wave propagation in monoclinic media
• Elastic stiffness matrix:

k

θ

Christoffel equation

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

θ

k

Curvature in slowness plot

Let

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

θ

k

How to find the eigenvalue ?

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

θ

k

Results - 1

(a) Normal curvature of slowness curve along z-axis

(b) Zero-Curvature along z-axis when d132 = c11∆35 or

(c13+c55)2=c11(c33-c55)

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
• Space of degeneracy in monoclinic media
• Generalized surface waves
Degeneracy of the Stroh eigenvalues

E1 zero-curvature transonic state:

Degeneracy of the Stroh eigenvalues

E4 zero-curvature transonic state:

Result 3

Space of degeneracy vs zero-curvature slowness curve:

Result 4

Space of degeneracy vs generalized surface waves

• Subsonic surface waves
• Supersonic surface waves
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