Transcript Zero curvature transonic states and supersonic surface waves

Extraordinary degeneracy and space of degeneracy
in transversely isotropic elastic media
Litian Wang
Østfold University College
1757 Halden Norway
Litian Wang
Østfold University College
Main goals
Relationship between the extraordinary degeneracies
and existence of the space of degeneracy
The Stroh formalism is applied to the transversely
isotropic media.
We will show that
(a) the space of degeneracy can be regarded as an
extension of the static degeneracy.
(b) the static degeneracy can span a continuous
space of degeneracy in the static limit.
Litian Wang
Østfold University College
Transversely isotropic elastic media
Litian Wang
Østfold University College
TiB2
The Stroh formalism
 uk
 ui
2
C ijkl
x j xl
2
 
t
2
u ( x , t )  a exp{ik [( m  p n )  x  vt ]}
{( m m )  p [( m n )  ( nm )]  p ( nn )} a   v a
2
2
w here
( m m )  m i c ijkl m l , ( m n )  m i c ijkl n l , ( nn )  n i c ijkl n l ,
Litian Wang
Østfold University College
The Stroh formalism
u ( x , t )  a exp{ik [( m  p n )  x  vt ]}
 
Reference plane: ( m , n )
n
m
Traction:
b   [( m n )  p ( nn )] a
Litian Wang
Østfold University College
The Stroh formalism
a 
  
b 
1

( nn ) ( nm )

 ( mn )( nn ) 1 ( nm )   v 2 I

Litian Wang
Østfold University College
1

   p   
1 
( mn )( nn ) 
( nn )
Transversely isotropic elastic media
C IJ
 c11








c12
c13
0
0
c11
c13
0
0
c 33
0
0
c 44
0
c 66  ( c11  c12 ) / 2
Litian Wang
Østfold University College
c 44
0 

0

0 

0 
0 

c 66 
Transversely isotropic elastic media
Litian Wang
Østfold University College
TiB2
Three symmetrical configurations (m,n)
γ-configuration
β-configuration
α-configuration
φ
φ
φ
φ
Litian Wang
Østfold University College
(Chadwick)
γ-configuration
φ
m  ex
n  cos  e z  sin  e y
{( m m )   v  p [( m n )  ( nm )]  p ( nn )}a  0
2
Litian Wang
Østfold University College
2
β-configuration
m  cos  e x  sin  e z
φ
n  ey
{( m m )   v  p [( m n )  ( nm )]  p ( nn )}a  0
2
2
Litian Wang
Østfold University College
(Alshits)
α-configuration
φ
φ
m  cos  e z  sin  e x
n  sin  e z  cos  e x
{( m m )   v  p [( m n )  ( nm )]  p ( nn )}a  0
2
Litian Wang
Østfold University College
2
φ
Space of degneracy
in the γ-configuration
Characteristic equation:
Litian Wang
Østfold University College
Space of degneracy
in the β-configuration
φ
Characteristic equation:
Litian et
Wang
(Shuvalov
al)
Østfold University College
Space of degneracy
in the α-configuration
φ
φ
Characteristic equation:
Litian Wang
Østfold University College
Properties of the space of degeneracy
Result 1: Evolution of the space of degeneracy
γ
β
p1=p2=p3=i
Extraordinary degeneracy
Litian Wang
Østfold University College
α
Properties of the space of degeneracy
Result 2: Characteristic of the space of degeneracy
γ
β
p1=p2=p3=i
α
Extraordinary degeneracy D2
Semisimple degeneracy
Litian Wang
Østfold University College
Non semisimple degeneracy
Properties of the space of degeneracy
Result 3: Existence of the space of degeneracy
Litian Wang
Østfold University College
Properties of the space of degeneracy
Result 4: Existence of the space of extraordinary degeneracy
Im p
p1=p2=p3≠i
Litian Wang
p1=p2=p3=i
Østfold University College
p1=p2=p3=i
Properties of the space of degeneracy
Result 5: Space of degeneracy at the static limit (v=0)
Litian Wang
Østfold University College
Conclusions
(a) A space of degeneracy (semisimple) can exist in both
supersonic and subsonic regime.
(b) A space of degeneracy (nonsemisimple) will end up at a
type E1 zero-curvature transonic state.
(c) A space of degeneracy (extraordinary) can bifurcate into
a number of ordinary spaces of degeneracy.
(d) A space of degeneracy can anchor or trespass acoustic
axes with same type degeneracy.
Litian Wang
Østfold University College
Litian Wang
Østfold University College
Litian Wang
Østfold University College
Litian Wang
Østfold University College
Litian Wang
Østfold University College