15.%FINITE%STRAIN%&%INFINITESIMAL%STRAIN… ·...

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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

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15. FINITE STRAIN & INFINITESIMAL STRAIN

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The sequence of deformaIon maRers for large strains

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2


10/10/12 GG303 3

The sequence of deformaIon does not maRer for small strains


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

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Sides of lower boxes maintain their length, but the diagonals change length

dy dx

ds

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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


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[ ]⎡⎣ ⎤⎦

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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[ ]⎡⎣ ⎤⎦


D Meaning of [E]

1

2

3

But what do the terms of E mean?

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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

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E Expansion of [E]

1

2

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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

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥


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

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F Conversion of [E] to [ε] for small strains

1

2

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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


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

⎡

⎣⎢⎢

⎤

⎦⎥⎥

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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

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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

⎡

⎣⎢⎢

⎤

⎦⎥⎥


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[ ]

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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[ ]


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⎡⎣

⎤⎦

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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


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

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hRp://www.spacegrant.hawaii.edu/class_acts/WebImg/gelaInVolcano.gif

GelaIn Volcano Experiment

10/10/12

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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


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[ ]

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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[ ]


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

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U2[ ] ≈ U1[ ] + Ju[ ] dX[ ] = U1[ ] + ε[ ] + ω[ ]⎡⎣ ⎤⎦ dX[ ]

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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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