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

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Main problems

1. Existence of extraordinary transonic states
   associated with extraordinary zero-curvature
   slowness curve
2. Existence of space of degeneracy
3. Existence of generalized surface waves

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Surface geometry of slowness surface
Cubic (Cu)
Monoclinic
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Surface geometry of slowness surface
Cubic (Cu)
Monoclinic
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Zero-curvature transonic states
E1 E2 E3 E4
Barnett, Lothe Gundersen
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Surface geometry of slowness surface
Cubic (Cu)
Monoclinic
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Problem 1

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

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Degree of freedom

• Degree of freedom 6

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Wave propagation in monoclinic media

• Elastic stiffness matrix

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Christoffel equation
Where d13c13c55, ?15c11-c55, ?64c66-c44,
?53c55-c33,
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Curvature in slowness plot

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

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• How to find the eigenvalue ?

Where d13c13c55, ?15c11-c55, ?64c66-c44,
?53c55-c33,
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Perturbation method
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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
(c13c55)2c11(c33-c55)
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Results - 2
(a) The second derivative of curvature
(b) Extraordinary zero-curvature along z-axis
when (c11c36-d13c16)2c112c55?45)
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Problem 2

1. Space of degeneracy in monoclinic media
2. Generalized surface waves

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Degeneracy of the Stroh eigenvalues
E1 zero-curvature transonic state
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Degeneracy of the Stroh eigenvalues
E4 zero-curvature transonic state
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Result 3

• Space of degeneracy vs zero-curvature slowness
  curve

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Result 4

• Space of degeneracy vs generalized surface waves
• Subsonic surface waves
• Supersonic surface waves

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Conclusions

1. Existence of extraordinary zero-curvature
   slowness curve
2. Existence of space of degeneracy
3. Existence of supersonic surface wave along the
   space of degeneracy
4. Existence of generalized subsonic surface wave
   along the space of degeneracy