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(1)

Stress and Strain Tensors – Deformation and Strain

MCEN 5023/ASEN 5012 Chapter 4

(2)

Deformation and Strain

Displacement & 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

(3)

Deformation and Strain

Deformation and Strain

Strain characterizes a deformation

Example: 1D strain L0 L 0 0

L

L

L

I

=

ε

(4)

Deformation and Strain

Kinematics of Continuous Body

2

x

1

x

3

x

a

1 2

a

3

a

Time 0 Undeformed configuration

Reference (initial) configuration Material configuration Time 0: Deformed configuration Current configuration Spatial configuration Time t: i

a

Time t i

x

(5)

Deformation and Strain

Kinematics of Continuous Body

)

,

,

,

(

a

1

a

2

a

3

t

x

x

i

=

i

OR, due to continuous body

)

,

,

,

(

x

1

x

2

x

3

t

a

a

i

=

i

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.

(6)

Deformation and Strain

)

,

,

,

(

a

1

a

2

a

3

t

x

x

i

=

i

a

i

=

a

i

(

x

1

,

x

2

,

x

3

,

t

)

Lagrangian

Eulerian

2

x

1

x

1

a

2

a

t=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

(7)

Deformation and Strain

Lagrangian vs. Eulerian

Lagrangian

Eulerian

Solid Mechanics Fluid Mechanics

Solid Mechanics Tracking a material point. Tracking a spatial point.

Spatial coordinates are fixed but Material points keep changing. Material point is fixed but the spatial

coordinates have to be updated.

(8)

Deformation and Strain

Kinematics of Continuous Body

2

x

1

x

3

x

a

1 2

a

3

a

Time 0 Time t i

a

x

i i

u

Using undeformed configuration as reference:

i i

i

a

a

a

x

a

a

a

a

u

(

1

,

2

,

3

)

=

(

1

,

2

,

3

)

Using deformed configuration as reference:

)

,

,

(

)

,

,

(

x

1

x

2

x

3

x

a

x

1

x

2

x

3

u

i

=

i

i

(9)

Deformation and Strain

Measure the deformation

2

x

1

x

3

x

a

1 2

a

3

a

Time 0 Time t i

u

P0 Q0 P Q i

a

i

x

{

1 2 3

}

0

a

,

a

,

a

P

=

{

1 1 2 2 3 3

}

0

a

d

a

,

a

d

a

,

a

d

a

Q

=

+

+

+

{

x

1

,

x

2

,

x

3

}

P

=

{

x

1

dx

1

,

x

2

dx

2

,

x

3

dx

3

}

Q

=

+

+

+

(10)

Deformation and Strain

(11)

Deformation and Strain

(12)

Deformation and Strain

Strain Tensor:

=

ij j k i k ij

a

x

a

x

E

δ

2

1

=

j k i k ij ij

x

a

x

a

e

δ

2

1

Green Strain Almansi Strain

(13)

Deformation and Strain

Strain Tensor:

+

+

=

j k i k i j j i ij

a

u

a

u

a

u

a

u

E

2

1

+

=

j k i k i j j i ij

x

u

x

u

x

u

x

u

e

2

1

Green Strain

Almansi Strain

(14)

Deformation and Strain

Physical Explanations of Strain Tensor

2 x 1 x 3 x a1 2 a 3 a Time 0 Time t i u

a

d

x

d

P0 Q0 P Q

(15)

Deformation and Strain

Physical Explanations of Strain Tensor

2 x 1 x 3 x a1 2 a 3 a Time 0 Time t i u a d x d P0 Q0 P Q

(16)

Deformation and Strain

Physical Explanations of Strain Tensor

2 x 1 x 3 x a1 2 a 3 a Time 0 Time t i u v’ v n n’

(17)

Deformation and Strain

If

1

<<

1

<<

j i j i

x

u

a

u

small deformation

+

=

i j j i ij

a

u

a

u

E

2

1

+

=

i j j i ij

x

u

x

u

e

2

1

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 i ij ij

x

u

x

u

e

E

2

1

(18)

Deformation and Strain

1 1 11 11

x

u

E

e

=

=

+

=

=

i j j i ij ij

x

u

x

u

e

E

2

1

Cauchy’s infinitesimal strain tensor

2 2 22 22

x

u

E

e

=

=

3 3 33 33

x

u

E

e

=

=

⎟⎟

⎜⎜

+

=

=

2 1 1 2 12 12

2

1

x

u

x

u

E

e

⎟⎟

⎜⎜

+

=

=

3 1 1 3 13 13

2

1

x

u

x

u

E

e

⎟⎟

⎜⎜

+

=

=

3 2 2 3 23 23

2

1

x

u

x

u

E

e

(19)

Deformation and Strain

If

1

<<

1

<<

j i j i

x

u

a

u

small deformation

Note:

small deformation

In most of the cases,

1

1

<<

<<

j i j i

x

u

a

u

But,

(20)

Deformation and Strain

Engineering Strains

Coordinates:

x, y, z

Displacements:

u, v, w

Normal strains: 11

e

x

u

x

=

=

ε

22

e

y

v

y

=

=

ε

33

e

z

w

z

=

=

ε

(21)

Deformation and Strain

Engineering Strains

Shear Strains:

12

2

e

x

v

y

u

xy

=

+

=

γ

23

2

e

y

w

z

v

yz

=

+

=

γ

13

2

e

x

w

z

u

xz

=

+

=

γ

(22)

Deformation and Strain

(23)

Deformation and Strain

Cauchy’s Shear Strain and Engineering Shear Strains

B

A

C

A

B

C

x

1

x

2

dx

2

dx

1

(24)

Deformation and Strain

Cauchy’s Shear Strain and Engineering Shear Strains

B

A

C

A

B

C

x

1

x

2

dx

2

dx

1

θ

1

θ

2

y

u

x

u

=

=

2 1 1

θ

x

v

x

u

=

=

1 2 2

θ

2 1

θ

θ

γ

xy

=

+

(

1 2

)

12

2

1

θ

+

θ

=

e

(25)

Deformation and Strain

Cauchy’s Shear Strain and Engineering Shear Strains

[ ]

=

=

z yz xz yz y xy xz xy x

e

e

e

e

e

e

e

e

e

e

ε

γ

γ

γ

ε

γ

γ

γ

ε

2

1

2

1

2

1

2

1

2

1

2

1

33 23 13 23 22 12 13 12 11

Tensor

Not a tensor!!!

z yz xz yz y xy xz xy x

ε

γ

γ

γ

ε

γ

γ

γ

ε

Engineering Strain

(26)

Deformation and Strain

Transformation of Coordinate System

In general

ij

jk

ik

ij

e

e

=

β

β

(27)

Deformation and Strain

Transformation of Coordinate System – 2D

2

X

1

X

1

e

2

e

2 X 1 X

e

1 2

e

θ

(

)

(

2 2

)

12 11 22 12 22 2 12 11 2 22 22 2 12 11 2 11

sin

cos

cos

sin

cos

cos

sin

2

sin

sin

cos

sin

2

cos

e

e

e

e

e

e

e

e

e

e

e

e

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

θ

+

=

+

=

+

+

=

(28)

Deformation and Strain

(29)

Deformation and Strain

(30)

Deformation and Strain

Strain Deviations Mean Strain

3

3

1 33 22 11 0

θ

=

+

+

=

e

e

e

e

Strain deviation tensor

e

e

I

0

e

=

ij ij ij

e

e

e

=

0

δ

Octahedral Shear Strain

(

) (

) (

)

(

2

)

31 2 23 2 12 2 11 33 2 33 22 2 22 11 0

6

3

2

e

e

e

e

e

e

e

e

e

+

+

+

+

+

=

γ

(31)
(32)

Deformation and Strain

+

=

=

i j j i ij ij

x

u

x

u

e

E

2

1

Determine Displacement Fields from Strains

(33)

Deformation and Strain

2 1 1 1

x

3

x

x

u

+

=

2 1 2 1

x

x

u

=

The strain fields are inconsistent because

3 1 2 1 2 1 1 2 = ∂ ∂ ∂ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ x x u x u x 1 2 1 1 2 2 1 1 2x x x u x u x ∂ ∂ = ∂ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ 2 1 1 2 1 2 1 2

x

x

u

x

x

u

(34)

Deformation and Strain

Compatibility of Strain Fields

A B C D Undeformed A B C D

Compatible strain fields

A B

C D C’

Incompatible strain fields

A

B C

D C’

(35)

Deformation and Strain

Integrability Condition In general

(

1 2

)

1 1

f

x

,

x

x

u

=

(

)

2 1 2 1

g

x

,

x

x

u

=

Integrability condition ( Compatibility of strain fields )

1 2

x

g

x

f

=

2 1 1 2 1 2 1 2

x

x

u

x

x

u

=

(36)

Deformation and Strain

Compatibility of Strain Fields

1 1 11

x

u

e

=

2 2 22

x

u

e

=

⎟⎟

⎜⎜

+

=

2 1 1 2 12

2

1

x

u

x

u

e

(37)

Deformation and Strain

Compatibility of Strain Fields

0

, , , ,kl

+

kl ij

ik jl

jl ik

=

ij

e

e

e

e

St. Venant Equations of Compatibility

Totally 81 equations, but only 6 are essential.

12 , 12 11 , 22 22 , 11

e

2e

e

+

=

e

11,23

=

e

23,11

+

e

12,13

+

e

13,12 23 , 23 22 , 33 33 , 22

e

2e

e

+

=

13 , 13 11 , 33 33 , 11

e

2e

e

+

=

21 , 23 23 , 21 22 , 13 13 , 22

e

e

e

e

=

+

+

31 , 32 32 , 31 33 , 12 12 , 33

e

e

e

e

=

+

+

References

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