stress and strain
DESCRIPTION
Stress and strain. Strain. Both the stress and the strain tensor are symmetric. Stress and strain (again). Both the stress and the strain tensor are symmetric. Elastic Constants. Symmetry:. Thermodynamics:. Lame’s parameters. Lame’s parameters. Stress and Strain. Stress and Strain. - PowerPoint PPT PresentationTRANSCRIPT
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