Stress and strain

u fi ij, j i

1 2 3

1 2 3

ij i i ii x x x xj

u

Strain

1212

e (u u )pq p,q q,p

u up q( )x xq p

Both the stress and the strain tensor are symmetrice eij ji

11 22 33e e e e u ukk k,k

Stress and strain (again)

cij ijpq pqσ e

Both the stress and the strain tensor are symmetric

e eij ji

Elastic Constants

c cjipq ijpq

c cijqp ijpq

Symmetry:

Thermodynamics:c cpqij ijpq

c ( )ijpq ij pq ip jq iq jp

Lame’s parameters

1 1 2E

( )( )

2 1E( )

Lame’s parameters

Stress and Strain

e e eij pq pq ip jq pq iq jp pq

11 11 22 22 33 33

1 2 312 3

1 2 312 3

e e eij

e e ejq q q qi p , ,

e e eiq q q qj p , ,

e e eij kk ii jj ij ii jj ji

Stress and Strain

2e eij ij kk ij

uu u jk iij ij x x xk j i

And after some algebra…

2 2u u u u u u f

Stress and Strain

VuV

Stress and Strain

2 2u u u u u u f

2

22u u ut

Stress and Strain

2

22u u ut

u

Helmholtz Decomposition Theorem

0" vector"

0" scalar"

22

2

22 2

2

uu

t

uu

t

2

22u u ut

Take the divergence of of both sides

u

2 V

P

2

22u u ut

Take the curl of of both sides

2

2u

ut

u

Shear-wave velocity

2

2

2t

VS

Woelz, 2003

Woelz, 2003