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10/10/12
1
15. FINITE STRAIN & INFINITESIMAL STRAIN (AT A POINT)
I Main Topics
A The finite strain tensor [E] B DeformaIon paths for finite strain C Infinitesimal strain and the infinitesimal strain tensor ε
10/10/12 GG303 1
15. FINITE STRAIN & INFINITESIMAL STRAIN
10/10/12 GG303 2
The sequence of deformaIon maRers for large strains
10/10/12
2
15. FINITE STRAIN & INFINITESIMAL STRAIN
10/10/12 GG303 3
The sequence of deformaIon does not maRer for small strains
15. FINITE STRAIN & INFINITESIMAL STRAIN
II The finite strain tensor [E] A Used to find the changes
in the squares of distances (ds)2 between points in a deformed body based on differences in their iniIal posiIons
B Displacements by themselves don’t give changes in size or shape
10/10/12 GG303 4
Sides of lower boxes maintain their length, but the diagonals change length
dy dx
ds
10/10/12
3
15. FINITE STRAIN & INFINITESIMAL STRAIN
II The finite strain tensor [E]
C DerivaIon of [E] 1
2
3
4 10/10/12 GG303 5
dxdy
⎡
⎣⎢⎢
⎤
⎦⎥⎥= dX[ ]
ds( )2 = dx( )2 + dy( )2
ds( )2 = dx dy⎡⎣
⎤⎦
dxdy
⎡
⎣⎢⎢
⎤
⎦⎥⎥
dx dy⎡⎣
⎤⎦ = dX[ ]T
15. FINITE STRAIN & INFINITESIMAL STRAIN
II The finite strain tensor [E] C DerivaIon of [E]
1
2
3
4
10/10/12 GG303 6
5
6
7
8
dxdy
⎡
⎣⎢⎢
⎤
⎦⎥⎥= dX[ ]
ds( )2 = dx( )2 + dy( )2
ds( )2 = dX[ ]T dX[ ] = dX[ ]T I[ ] dX[ ],
where I[ ] = 1 00 1
⎡
⎣⎢
⎤
⎦⎥
d ′s( )2 = d ′x d ′y⎡⎣
⎤⎦
d ′xd ′y
⎡
⎣⎢⎢
⎤
⎦⎥⎥
= d ′X[ ]T d ′X[ ]
ds( )2 = dx dy⎡⎣
⎤⎦
dxdy
⎡
⎣⎢⎢
⎤
⎦⎥⎥
dx dy⎡⎣
⎤⎦ = dX[ ]T
d ′X[ ] = F[ ] dX[ ] From lecture 14:
d ′s( )2 = F[ ] dX[ ]⎡⎣ ⎤⎦T F[ ] dX[ ]⎡⎣ ⎤⎦
10/10/12
4
15. FINITE STRAIN & INFINITESIMAL STRAIN
C DerivaIon of [E] (cont.)
9
10
11
12
10/10/12 GG303 7
d ′s( )2 − ds( )2 = dX[ ]T F[ ]T F[ ] dX[ ]− dX[ ]T I[ ]T dX[ ]
d ′s( )2 − ds( )2 = dX[ ]T F[ ]T F[ ]− I[ ]⎡⎣ ⎤⎦ dX[ ]
d ′s( )2 − ds( )2{ }2
=dX[ ]T F[ ]T F[ ]− I[ ]⎡⎣ ⎤⎦ dX[ ]
2
12
d ′s( )2 − ds( )2{ } = dX[ ]T E[ ] dX[ ] E[ ] ≡ 12
F[ ]T F[ ]− I[ ]⎡⎣ ⎤⎦
15. FINITE STRAIN & INFINITESIMAL STRAIN
D Meaning of [E]
1
2
3
But what do the terms of E mean?
10/10/12 GG303 8
12
d ′s( )2 − ds( )2{ } = dX[ ]T E[ ] dX[ ]
E[ ] ≡ 12
F[ ]T F[ ]− I[ ]⎡⎣ ⎤⎦
E[ ] ≡ 12
Ju + I[ ]T Ju + I[ ]− I[ ]⎡⎣
⎤⎦
Given E and dX (the difference in iniIal posiIons) of points), then one can find the difference in the squares of the lengths of lines connecIng the points before and afer deformaIon
10/10/12
5
15. FINITE STRAIN & INFINITESIMAL STRAIN
E Expansion of [E]
1
2
10/10/12 GG303 9
Ju =
∂u∂x
∂u∂y
∂v∂x
∂v∂y
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
E[ ] ≡ 12
Ju + I[ ]T Ju + I[ ]− I[ ]⎡⎣
⎤⎦
E[ ] ≡ 12
∂u∂x
∂u∂y
∂v∂x
∂v∂y
+ 1 00 1
⎡
⎣⎢
⎤
⎦⎥
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
T∂u∂x
∂u∂y
∂v∂x
∂v∂y
+ 1 00 1
⎡
⎣⎢
⎤
⎦⎥
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
− 1 00 1
⎡
⎣⎢
⎤
⎦⎥
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
15. FINITE STRAIN & INFINITESIMAL STRAIN
E Expansion of [E]
3
4
10/10/12 GG303 10
E[ ] ≡ 12
Ju + I[ ]T Ju + I[ ]− I[ ]⎡⎣
⎤⎦
E[ ] ≡ 12
∂u∂x
+1 ∂v∂x
∂u∂y
∂v∂y
+1
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
T∂u∂x
+1 ∂u∂y
∂v∂x
∂v∂y
+1
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
− 1 00 1
⎡
⎣⎢
⎤
⎦⎥
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
E[ ] ≡ 12
∂u∂x
+1⎛⎝⎜
⎞⎠⎟
∂u∂x
+1⎛⎝⎜
⎞⎠⎟+
∂v∂x
⎛⎝⎜
⎞⎠⎟
∂v∂x
⎛⎝⎜
⎞⎠⎟−1 ∂u
∂x+1⎛
⎝⎜⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟+
∂v∂x
⎛⎝⎜
⎞⎠⎟
∂v∂y
+1⎛⎝⎜
⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟
∂u∂x
+1⎛⎝⎜
⎞⎠⎟+
∂v∂y
+1⎛⎝⎜
⎞⎠⎟
∂v∂x
⎛⎝⎜
⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟+
∂v∂y
+1⎛⎝⎜
⎞⎠⎟
∂v∂y
+1⎛⎝⎜
⎞⎠⎟−1
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
The meaning of the terms in E is not intuiIve Imagine what the products of mulIple finite deformaIons yield
10/10/12
6
15. FINITE STRAIN & INFINITESIMAL STRAIN
F Conversion of [E] to [ε] for small strains
1
2
10/10/12 GG303 11
E[ ] ≡ 12
∂u∂x
+1⎛⎝⎜
⎞⎠⎟
∂u∂x
+1⎛⎝⎜
⎞⎠⎟+
∂v∂x
⎛⎝⎜
⎞⎠⎟
∂v∂x
⎛⎝⎜
⎞⎠⎟−1 ∂u
∂x+1⎛
⎝⎜⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟+
∂v∂x
⎛⎝⎜
⎞⎠⎟
∂v∂y
+1⎛⎝⎜
⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟
∂u∂x
+1⎛⎝⎜
⎞⎠⎟+
∂v∂y
+1⎛⎝⎜
⎞⎠⎟
∂v∂x
⎛⎝⎜
⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟+
∂v∂y
+1⎛⎝⎜
⎞⎠⎟
∂v∂y
+1⎛⎝⎜
⎞⎠⎟−1
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
ε[ ] ≡ 12
∂u∂x
⎛⎝⎜
⎞⎠⎟+
∂u∂x
⎛⎝⎜
⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟+
∂v∂x
⎛⎝⎜
⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟+
∂v∂x
⎛⎝⎜
⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟+
∂u∂y
⎛⎝⎜
⎞⎠⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
=12
Ju[ ] + Ju[ ]T⎡⎣
⎤⎦
If the displacement derivaIves are <<1, then their products are Iny and can be neglected
15. FINITE STRAIN & INFINITESIMAL STRAIN
III DeformaIon paths for finite strain Consider two different deformaIons A DeformaIon 1
B DeformaIon 2
10/10/12 GG303 12
F1[ ] = a1 b1c1 d1
⎡
⎣⎢⎢
⎤
⎦⎥⎥
F2[ ] = a2 b2c2 d2
⎡
⎣⎢⎢
⎤
⎦⎥⎥
10/10/12
7
15. FINITE STRAIN & INFINITESIMAL STRAIN
III DeformaIon paths for finite strain Consider two different deformaIons A DeformaIon 1
B DeformaIon 2
C F2 acts on F1
D F1 acts on F2
E The sequence of finite deformaIons maRers – unless off-‐diagonal terms are small
10/10/12 GG303 13
F1[ ] = a1 b1c1 d1
⎡
⎣⎢⎢
⎤
⎦⎥⎥
F2[ ] = a2 b2c2 d2
⎡
⎣⎢⎢
⎤
⎦⎥⎥
F2[ ] F1[ ] = a2a1 + b2c1 a2b1 + b2d1c2a1 + d2c1 c2b1 + d2d1
⎡
⎣⎢⎢
⎤
⎦⎥⎥
F1[ ] F2[ ] = a1a2 + b1c2 a1b2 + b1d2c1a2 + d1c2 c1b1 + d1d2
⎡
⎣⎢⎢
⎤
⎦⎥⎥
15. FINITE STRAIN & INFINITESIMAL STRAIN
10/10/12 GG303 14
[F1]
[F2]
′x′y
⎡
⎣⎢⎢
⎤
⎦⎥⎥= 2 1
0 1⎡
⎣⎢
⎤
⎦⎥
xy
⎡
⎣⎢⎢
⎤
⎦⎥⎥
′x′y
⎡
⎣⎢⎢
⎤
⎦⎥⎥= 1 0
0 2⎡
⎣⎢
⎤
⎦⎥
xy
⎡
⎣⎢⎢
⎤
⎦⎥⎥
1 00 2
⎡
⎣⎢
⎤
⎦⎥ =
1 00 2
⎡
⎣⎢
⎤
⎦⎥
1 00 1
⎡
⎣⎢
⎤
⎦⎥
2 10 1
⎡
⎣⎢
⎤
⎦⎥ =
2 10 1
⎡
⎣⎢
⎤
⎦⎥
1 00 1
⎡
⎣⎢
⎤
⎦⎥
2 10 2
⎡
⎣⎢
⎤
⎦⎥ =
1 00 2
⎡
⎣⎢
⎤
⎦⎥
2 10 2
⎡
⎣⎢
⎤
⎦⎥ = F2[ ] F1[ ]
2 20 2
⎡
⎣⎢
⎤
⎦⎥ =
2 10 2
⎡
⎣⎢
⎤
⎦⎥
1 00 2
⎡
⎣⎢
⎤
⎦⎥ = F1[ ] F2[ ]
F2[ ] F1[ ] ≠ F1[ ] F2[ ]
10/10/12
8
15. FINITE STRAIN & INFINITESIMAL STRAIN
10/10/12 GG303 15
[F1]
[F2]
[F2][F1]
[F1][F2]
′x′y
⎡
⎣⎢⎢
⎤
⎦⎥⎥= 2 1
0 1⎡
⎣⎢
⎤
⎦⎥
xy
⎡
⎣⎢⎢
⎤
⎦⎥⎥
2 10 1
⎡
⎣⎢
⎤
⎦⎥ =
2 10 1
⎡
⎣⎢
⎤
⎦⎥
1 00 1
⎡
⎣⎢
⎤
⎦⎥
′x′y
⎡
⎣⎢⎢
⎤
⎦⎥⎥= 1 0
0 2⎡
⎣⎢
⎤
⎦⎥
xy
⎡
⎣⎢⎢
⎤
⎦⎥⎥
1 00 2
⎡
⎣⎢
⎤
⎦⎥ =
1 00 2
⎡
⎣⎢
⎤
⎦⎥
1 00 1
⎡
⎣⎢
⎤
⎦⎥
′′x′′y
⎡
⎣⎢⎢
⎤
⎦⎥⎥= 2 1
0 2⎡
⎣⎢
⎤
⎦⎥
xy
⎡
⎣⎢⎢
⎤
⎦⎥⎥
′′′x′′′y
⎡
⎣⎢⎢
⎤
⎦⎥⎥= 1 0
0 2⎡
⎣⎢
⎤
⎦⎥
xy
⎡
⎣⎢⎢
⎤
⎦⎥⎥
2 10 2
⎡
⎣⎢
⎤
⎦⎥ =
2 10 2
⎡
⎣⎢
⎤
⎦⎥
1 00 1
⎡
⎣⎢
⎤
⎦⎥
2 20 2
⎡
⎣⎢
⎤
⎦⎥ =
2 20 2
⎡
⎣⎢
⎤
⎦⎥
1 00 1
⎡
⎣⎢
⎤
⎦⎥
[F1]
[F2]
[F2][F1]
[F1][F2] F2[ ] F1[ ] ≠ F1[ ] F2[ ]
15. FINITE STRAIN & INFINITESIMAL STRAIN
IV Infinitesimal strain and the infinitesimal strain tensor [ε] A Infinitesimal strain DeformaIon where the displacement derivaIves in [Ju] are small relaIve to one so that the products of the derivaIves are very small and can be ignored.
B An approximaIon to finite strain
10/10/12 GG303 16
Ju[ ] =∂u∂x
∂u∂y
∂v∂x
∂v∂y
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
Ju[ ]T =
∂u∂x
∂v∂x
∂u∂y
∂v∂y
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
∂v∂x
ε[ ] ≡ 12
∂u∂x
⎛⎝⎜
⎞⎠⎟+
∂u∂x
⎛⎝⎜
⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟+
∂v∂x
⎛⎝⎜
⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟+
∂v∂x
⎛⎝⎜
⎞⎠⎟
∂u∂y
⎛⎝⎜
⎞⎠⎟+
∂u∂y
⎛⎝⎜
⎞⎠⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
ε[ ] = 12
Ju[ ] + Ju[ ]T⎡⎣
⎤⎦
10/10/12
9
15. FINITE STRAIN & INFINITESIMAL STRAIN
IV Infinitesimal strain and the infinitesimal strain tensor [ε] (cont.) C Why consider [ε] if it is
an approximaIon? 1 Relevant to
important geologic deformaIons A Fracture B Earthquake
deformaIon C Volcano
deformaIon
10/10/12 GG303 17
hRp://www.spacegrant.hawaii.edu/class_acts/WebImg/gelaInVolcano.gif
GelaIn Volcano Experiment
15. FINITE STRAIN & INFINITESIMAL STRAIN
C Why consider [ε] if it is an approximaIon? (cont.) 2 Terms of the infinitesimal
strain tensor [ε] have clear geometric meaning
3 Can apply principal of superposiIon (addiIon)
4 Infinitesimal deformaIon is essenIally independent of the deformaIon sequence
5 Amenable to sophisIcated mathemaIcal treatment (e.g., elasIcity theory)
6 QuanItaIve predicIve ability
10/10/12 GG303 18
hRp://www.spacegrant.hawaii.edu/class_acts/WebImg/gelaInVolcano.gif
GelaIn Volcano Experiment
10/10/12
10
15. FINITE STRAIN & INFINITESIMAL STRAIN
C Why consider [ε] if it is an approximaIon? (cont.) 7 Infinitesimal strain example
10/10/12 GG303 19
F3 =1.02 0.010 1.01
⎡
⎣⎢
⎤
⎦⎥
F4 =1.01 00 1.02
⎡
⎣⎢
⎤
⎦⎥
Ju(3) =0.02 0.010 0.01
⎡
⎣⎢
⎤
⎦⎥
Ju(4 ) =0.01 00 0.02
⎡
⎣⎢
⎤
⎦⎥
F4[ ] F3[ ] = 1.01 00 1.02
⎡
⎣⎢
⎤
⎦⎥1.02 0.010 1.01
⎡
⎣⎢
⎤
⎦⎥ =
1.0302 0.01000.0000 1.0302
⎡
⎣⎢
⎤
⎦⎥
F3[ ] F4[ ] = 1.02 0.010 1.01
⎡
⎣⎢
⎤
⎦⎥1.01 00 1.02
⎡
⎣⎢
⎤
⎦⎥ =
1.0302 0.01010.0000 1.0302
⎡
⎣⎢
⎤
⎦⎥
F3[ ] + F4[ ] = 1.02 0.010 1.01
⎡
⎣⎢
⎤
⎦⎥ +
1.01 00 1.02
⎡
⎣⎢
⎤
⎦⎥ =
1.0300 0.01000.0000 1.0300
⎡
⎣⎢
⎤
⎦⎥
Sequence results can be obtained regardless of the order of events, but also by superposiIon
15. FINITE STRAIN & INFINITESIMAL STRAIN
IV Infinitesimal strain and the infinitesimal strain tensor [ε]
D Taylor series expansion We seek U2 given U1 and dX
1
2
3
10/10/12 GG303 20
u2 = u1 + du = u1 +
∂u∂xdx + ∂u
∂ydy
⎛⎝⎜
⎞⎠⎟+…
v2 = v1 + dv = v1 +
∂v∂xdx + ∂v
∂ydy
⎛⎝⎜
⎞⎠⎟+…
u2v2
⎡
⎣⎢⎢
⎤
⎦⎥⎥=
u1v1
⎡
⎣⎢⎢
⎤
⎦⎥⎥+
∂u∂x
∂u∂y
∂v∂x
∂v∂y
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
dxdy
⎡
⎣⎢⎢
⎤
⎦⎥⎥+…⇒ U2[ ] ≈ U1[ ] + dU1[ ] = U1[ ] + Ju[ ] dX[ ]
10/10/12
11
15. FINITE STRAIN & INFINITESIMAL STRAIN
D Taylor series expansion (cont.)
Now split [Ju] into two matrices: the infinitesimal strain matrix [ε] and the anI-‐symmetric rotaIon matrix [ω]
10/10/12 GG303 21
Ju[ ] =∂u∂x
∂u∂y
∂v∂x
∂v∂y
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
Ju[ ]T =
∂u∂x
∂v∂x
∂u∂y
∂v∂y
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
∂v∂x
ε[ ] = Ju[ ] + Ju[ ]T2
=12
∂u∂x
+∂u∂x
∂u∂y
+∂v∂x
∂v∂x
+∂u∂y
∂v∂y
+∂v∂y
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
ω[ ] = Ju[ ]− Ju[ ]T2
=12
0 ∂u∂y
−∂v∂x
∂v∂x
−∂u∂y
0
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
ε[ ] + ω[ ] = Ju[ ]
15. FINITE STRAIN & INFINITESIMAL STRAIN
IV Infinitesimal strain and the infinitesimal strain tensor [ε]
D Taylor series expansion
The deformaIon can be decomposed into a translaIon, a strain, and a rotaIon
10/10/12 GG303 22
U2[ ] ≈ U1[ ] + Ju[ ] dX[ ] = U1[ ] + ε[ ] + ω[ ]⎡⎣ ⎤⎦ dX[ ]
10/10/12
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15. FINITE STRAIN & INFINITESIMAL STRAIN
10/10/12 GG303 23
15. FINITE STRAIN & INFINITESIMAL STRAIN
10/10/12 GG303 24
ε xx =∂u∂x
ε yy =∂v∂y
ε xy =12
Ψ1 − Ψ2( ) = 12
∂u∂y
+∂v∂x
⎛⎝⎜
⎞⎠⎟
ω xy =12
Ψ1 +Ψ2( ) = 12
∂v∂x
−∂u∂y
⎛⎝⎜
⎞⎠⎟
For small angles, Ψ = tanΨ
PosiIve angles are measured about the z-‐axis using a right hand rule. In (c) the angle Ψ2 is clockwise (negaIve), but du is posiIve. In (d) Ψ2 is counter-‐clockwise, and du< 0.
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15. FINITE STRAIN & INFINITESIMAL STRAIN
10/10/12 GG303 25
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