Litian Wang

Østfold University College

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

TiB2

Litian Wang

Østfold University College

The Stroh formalism

( , ) exp{ [( ) ]}u x t a ik m p n x vt ����������������������������

2

22

t

u

xx

uC i

lj

kijkl

2 2{( ) [( ) ( )] ( )}

where

( ) , ( ) , ( ) ,i ijkl l i ijkl l i ijkl l

mm p mn nm p nn a v a

mm m c m mn m c n nn n c n

Litian Wang

Østfold University College

The Stroh formalism

( , ) exp{ [( ) ]}u x t a ik m p n x vt ����������������������������

),( nm

Reference plane:

Traction:

m

n

[( ) ( )]b mn p nn a

Litian Wang

Østfold University College

The Stroh formalism

a

b

pnnmnIvnmnnmn

nnnmnn

121

11

))(()())((

)()()(

Litian Wang

Østfold University College

Transversely isotropic elastic media

11 12 13

11 13

33

44

44

66

0 0 0

0 0 0

0 0 0

0 0

0

IJ

c c c

c c

cC

c

c

c

2/)( 121166 ccc

Litian Wang

Østfold University College

Transversely isotropic elastic media

TiB2

Litian Wang

Østfold University College

Three symmetrical configurations (m,n)

γ-configuration β-configuration α-configuration

(Chadwick)

φ

φ

φ

φ

Litian Wang

Østfold University College

φ

γ-configuration

cos sin x

z y

m e

n e e

2 2{( ) [( ) ( )] ( )} 0mm v p mn nm p nn a

Litian Wang

Østfold University College

β-configuration

cos sin x z

y

m e e

n e

2 2{( ) [( ) ( )] ( )} 0mm v p mn nm p nn a

(Alshits)

φ

Litian Wang

Østfold University College

α-configuration

cos sin

sin cos z x

z x

m e e

n e e

2 2{( ) [( ) ( )] ( )} 0mm v p mn nm p nn a

φ

φ

Litian Wang

Østfold University College

Space of degneracy in the γ-configuration

Characteristic equation:

φ

Litian Wang

Østfold University College

Space of degneracy in the β-configuration

Characteristic equation:

(Shuvalov et al)

φ

Litian Wang

Østfold University College

Space of degneracy in the α-configuration

Characteristic equation:

φ

φ

Litian Wang

Østfold University College

Properties of the space of degeneracy

p1=p2=p3=i

Extraordinary degeneracy

Result 1: Evolution of the space of degeneracy

αβγ

Litian Wang

Østfold University College

p1=p2=p3=i

Semisimple degeneracy

Non semisimple degeneracy

Extraordinary degeneracy D2

Result 2: Characteristic of the space of degeneracy

Properties of the space of degeneracy

αβγ

Litian Wang

Østfold University College

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

p1=p2=p3=i p1=p2=p3=i

p1=p2=p3≠i

Im p

Litian Wang

Østfold University College

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