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Stress and strain
u fi ij, j i
1 2 3
1 2 3
ij i i ii x x x xj
u
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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
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Stress and strain (again)
cij ijpq pqσ e
Both the stress and the strain tensor are symmetric
e eij ji
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Elastic Constants
c cjipq ijpq
c cijqp ijpq
Symmetry:
Thermodynamics:c cpqij ijpq
c ( )ijpq ij pq ip jq iq jp
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Lame’s parameters
1 1 2E
( )( )
2 1E( )
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Lame’s parameters
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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
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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
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Stress and Strain
VuV
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Stress and Strain
2 2u u u u u u f
2
22u u ut
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Stress and Strain
2
22u u ut
u
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Helmholtz Decomposition Theorem
0" vector"
0" scalar"
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22
2
22 2
2
uu
t
uu
t
2
22u u ut
Take the divergence of of both sides
u
2 V
P
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2
22u u ut
Take the curl of of both sides
2
2u
ut
u
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Shear-wave velocity
2
2
2t
VS
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Woelz, 2003
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Woelz, 2003