existence of extraordinary transonic states in monoclinic elastic media
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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