Existence of extraordinary transonic states in monoclinic elastic media
This presentation is the property of its rightful owner.
Sponsored Links
1 / 25

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


  • 81 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

Existence of extraordinary transonic states in monoclinic elastic media

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

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


Surface geometry of slowness surface

Surface geometry of slowness surface

Cubic (Cu)

Monoclinic


Existence of extraordinary transonic states in monoclinic elastic media

Surface geometry of slowness surface

Cubic (Cu)

Monoclinic


Zero curvature transonic states

n

m

Zero-curvature transonic states

E1E2E3E4

Barnett, Lothe & Gundersen


Existence of extraordinary transonic states in monoclinic elastic media

Surface geometry of slowness surface

Cubic (Cu)

Monoclinic


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


  • Login