Litian Wang

Østfold University College

Existence of extraordinary transonic states in monoclinic elastic media

Litian Wang and Kent Ryne

Østfold University College

1757 Halden Norway

Litian Wang

Østfold University College

Main problems

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

b) Existence of space of degeneracy

c) Existence of generalized surface waves

Litian Wang

Østfold University College

Surface geometry of slowness surface

Cubic (Cu) Monoclinic

Litian Wang

Østfold University College

Surface geometry of slowness surface

Cubic (Cu) Monoclinic

Litian Wang

Østfold University College

Zero-curvature transonic states

E1 E2 E3 E4

Barnett, Lothe & Gundersen

m

n

Litian Wang

Østfold University College

Surface geometry of slowness surface

Cubic (Cu) Monoclinic

Litian Wang

Østfold University College

Problem 1

a) Can a slowness curve have zero-curvature locally?

b) How flat a slowness curve can be?

Litian Wang

Østfold University College

Degree of freedom

• Degree of freedom = 6

Litian Wang

Østfold University College

Wave propagation in monoclinic media

• Elastic stiffness matrix:

11 12 13 16

22 23 26

33 36

44

55

66

0 0

0 0

0 0

0 0

0

IJ

c c c c

c c c

c cC

c

c

c

Litian Wang

Østfold University College

θ k

Litian Wang

Østfold University College

Christoffel equation

))(exp(),( vtxkikAtxu

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

2

22

t

u

xx

uC i

lj

kijkl

AvA

ccd

ccc

dcc2

253333613

362

64442

16

132

162

1555

sinsincossincos

sincossinsin

sincossinsin

Litian Wang

Østfold University College

Curvature in slowness plot

Let

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

2v

0)4(2

1''

0''2

1

)8

1(4

)2

1(

k

k

θk

Litian Wang

Østfold University College

How to find the eigenvalue ?2v

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

AvA

ccd

ccc

dcc2

253333613

362

64442

16

132

162

1555

sinsincossincos

sincossinsin

sincossinsin

θk

Litian Wang

Østfold University College

AvA

ccd

ccc

dcc2

253333613

362

64442

16

132

162

1555

sinsincossincos

sincossinsin

sincossinsin

Perturbation method

AvAV

c

c

c2

33

44

55

00

00

00

0 i i i(H V) A A

θk

Litian Wang

Østfold University College

Whereθk

Litian Wang

Østfold University College

Results - 1

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

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

5535

2133511

1

)(

c

dck

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

(See also Shuvalov et al)

θ k

Litian Wang

Østfold University College

(a) The second derivative of curvature:

Results - 2

)36

)2(34

33()(4

4533511

335

21613

2353616

2134535553311

21345

235

21335

236

41345

145

33555

''1

ccdcc

dcccd

dcdck

(b) Extraordinary zero-curvature along z-axis when (c11c36-d13c16)2=c11

2c55∆45)

)(])([ 554455211

21655133611 ccccccccc θ k

Litian Wang

Østfold University College

Litian Wang

Østfold University College

Litian Wang

Østfold University College

Problem 2

a) Space of degeneracy in monoclinic media

b) Generalized surface waves

Litian Wang

Østfold University College

Degeneracy of the Stroh eigenvalues

E1 zero-curvature transonic state:

Litian Wang

Østfold University College

E4 zero-curvature transonic state:

Degeneracy of the Stroh eigenvalues

Litian Wang

Østfold University College

Result 3

Space of degeneracy vs zero-curvature slowness curve:

Litian Wang

Østfold University College

Result 4

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

Litian Wang

Østfold University College

Conclusions

a) Existence of extraordinary zero-curvature slowness curve

b) Existence of space of degeneracy

c) Existence of supersonic surface wave along the space of degeneracy

d) Existence of generalized subsonic surface wave along the space of degeneracy