chap_04

Stress and Strain Tensors – Deformation and Strain

MCEN 5023/ASEN 5012

Chapter 4

Fall, 2006

Deformation and StrainDisplacement & Deformation

Deformation: An alteration of shape, as by pressure or stress.

Example:

Displacement: A vector or the magnitude of a vector from the initial position to a subsequent position assumed by a body.

Time 0

Time t

Case 1

Case 2

Case 3

Deformation and StrainDeformation and Strain

Strain characterizes a deformation

Example: 1D strain L0

L

0

0

LLL

I−

=ε

Deformation and StrainKinematics of Continuous Body

2x

1x

3x 1a

2a

3a

Time 0

Undeformed configurationReference (initial) configurationMaterial configuration

Time 0:

Deformed configurationCurrent configurationSpatial configuration

Time t:

ia

Time t

ix

Deformation and Strain

Kinematics of Continuous Body

),,,( 321 taaaxx ii =

OR, due to continuous body

),,,( 321 txxxaa ii =

Lagrangian Description:

Eulerian Description:

The motion is described by the material coordinate and time t.

The motion is described by the spatial coordinate and time t.


),,,( 321 taaaxx ii = ),,,( 321 txxxaa ii =Lagrangian Eulerian

2x

1x1a

2at=0 t=t1 t=t2

(Tacking a material point) (Monitoring a spatial point)

The spatial coordinates of this material point change with time.

Different material points pass this spatial point

Lagrangian vs. Eulerian


Lagrangian vs. Eulerian

Lagrangian Eulerian

Solid Mechanics Fluid MechanicsSolid Mechanics

Tracking a material point. Tracking a spatial point.

Spatial coordinates are fixed butMaterial points keep changing.

Material point is fixed but the spatial coordinates have to be updated.

Good for constitutive model Not good for constitutive model.


Kinematics of Continuous Body

2x

1x

3x 1a

2a

3a

Time 0Time t

ia ixiu

Using undeformed configuration as reference:

iii aaaaxaaau −= ),,(),,( 321321

Using deformed configuration as reference:

),,(),,( 321321 xxxaxxxxu iii −=

Deformation and StrainMeasure the deformation

2x

1x

3x 1a

2a

3a

Time 0Time t

iu

P0

Q0

P

Qiaix

{ }3210 ,, aaaP =

{ }3322110 ,, adaadaadaQ +++=

{ }321 ,, xxxP =

{ }332211 ,, dxxdxxdxxQ +++=



Deformation and StrainStrain Tensor:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

∂∂

= ijj

k

i

kij a

xaxE δ

21

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

−=j

k

i

kijij x

axae δ

21

Green Strain

Almansi Strain

Deformation and StrainStrain Tensor:

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

+∂

∂+

∂∂

=j

k

i

k

i

j

j

iij a

uau

au

auE

21

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

−∂

∂+

∂∂

=j

k

i

k

i

j

j

iij x

uxu

xu

xue

21

Green Strain

Almansi Strain

Applicable to both small and finite (large) deformation.

Deformation and StrainPhysical Explanations of Strain Tensor

2x

1x

3x 1a

2a

3a

Time 0Time t

iu

adxd

P0

Q0

P

Q


2x

1x

3x 1a

2a

3a

Time 0Time t

iu

adxdP0

Q0

P

Q


2x

1x

3x 1a

2a

3a

Time 0Time t

iu

v’v

n n’


If 11 <<∂∂

<<∂∂

j

i

j

i

xu

au small deformation

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=i

j

j

iij a

uauE

21

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=i

j

j

iij x

uxue

21

The quadratic term in Green strain and Almansi strain can be neglected.

Also, in small deformation, the distinction between Lagrangian and Eulerian disappears.

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

==i

j

j

iijij x

uxueE

21

Cauchy’s infinitesimal strain tensor


1

11111 x

uEe∂∂

==

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

==i

j

j

iijij x

uxueE

21

Cauchy’s infinitesimal strain tensor

2

22222 x

uEe∂∂

==

3

33333 x

uEe∂∂

==

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

==2

1

1

21212 2

1xu

xuEe

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

==3

1

1

31313 2

1xu

xuEe

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

==3

2

2

32323 2

1xu

xuEe


If 11 <<∂∂

<<∂∂

j

i

j

i

xu

au small deformation

Note:

small deformation

In most of the cases,

11 <<∂∂

<<∂∂

j

i

j

i

xu

au

But,

Deformation and StrainEngineering Strains

Coordinates: x, y, z Displacements: u, v, w

Normal strains:

11exu

x =∂∂

=ε

22eyv

y =∂∂

=ε

33ezw

z =∂∂

=ε

Deformation and StrainEngineering Strains

Shear Strains:

122exv

yu

xy =∂∂

+∂∂

=γ

232eyw

zv

yz =∂∂

+∂∂

=γ

132exw

zu

xz =∂∂

+∂∂

=γ


Stretches at small deformation


Cauchy’s Shear Strain and Engineering Shear Strains

B

A

C

A’

B’C’

x1

x2

dx2

dx1



B

A

C

A’

B’C’

x1

x2

dx2

dx1

θ1

θ2

yu

xu

∂∂

=∂∂

=2

11θ

xv

xu

∂∂

=∂∂

=1

22θ

21 θθγ +=xy

( )2112 21 θθ +=e



[ ]

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

zyzxz

yzyxy

xzxyx

eeeeeeeee

e

εγγ

γεγ

γγε

21

21

21

21

21

21

332313

232212

131211

Tensor

Not a tensor!!!⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

zyzxz

yzyxy

xzxyx

εγγγεγγγε

Engineering Strain


Transformation of Coordinate System

In general

ijjkikij ee ββ=′


Transformation of Coordinate System – 2D

2X ′

1X ′

1e′

2e′

2X

1X 1e

2e

θ

( ) ( ) 1222

112212

222

12112

22

222

12112

11

sincoscossin

coscossin2sin

sincossin2cos

eeee

eeee

eeee

θθθ

θθθθ

θθθθ

−+−=′

+−=′

++=′


Transformation of Coordinate System – 2D Mohr Circle


Strain Invariants


Strain Deviations

Mean Strain331332211

0θ

=++

=eeee

Strain deviation tensor Iee 0e−=′

ijijij eee δ0−=′

Octahedral Shear Strain

( ) ( ) ( ) ( )231

223

212

21133

23322

222110 6

32 eeeeeeeee +++−+−+−=γ


⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

==i

j

j

iijij x

uxueE

21

Determine Displacement Fields from Strains

Questions: Can the displacements be determined uniquely?


211

1 3xxxu

+=∂∂ 2

12

1 xxu

=∂∂

The strain fields are inconsistent because

312

12

1

1

2

=∂∂

∂=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

xxu

xu

x 121

12

2

1

1

2xxx

uxu

x=

∂∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

21

12

12

12

xxu

xxu

∂∂∂

≠∂∂

∂

Determine Displacement Fields from Strains


Compatibility of Strain Fields

A

B C

D

Undeformed

A

B C

D

Compatible strain fields

A

BC

D

C’

Incompatible strain fields

A

B C

D

C’


Integrability Condition

In general

( )211

1 , xxfxu

=∂∂ ( )21

2

1 , xxgxu

=∂∂

Integrability condition ( Compatibility of strain fields )

12 xg

xf

∂∂

=∂∂

21

12

12

12

xxu

xxu

∂∂∂

=∂∂

∂

Integration of strain fields yields unique displacement components.



1

111 x

ue∂∂

=2

222 x

ue∂∂

= ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=2

1

1

212 2

1xu

xue



0,,,, =−−+ ikjljlikijklklij eeee

St. Venant Equations of Compatibility

Totally 81 equations, but only 6 are essential.

12,1211,2222,11 2eee =+ 12,1313,1211,2323,11 eeee ++−=

23,2322,3333,22 2eee =+

13,1311,3333,11 2eee =+

21,2323,2122,1313,22 eeee ++−=

31,3232,3133,1212,33 eeee ++−=

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