Deformation: Displacement & Strain

mi@seu.edu.cn

Outline

• Generalized Displacement（位移概念）• Small Deformation Theory（小变形理论）• Continuum Motion & Deformation（运动与变形）• Strain & Rotation（应变与旋转）• Principal Strains（主应变）• Spherical and Deviatoric Strain（平均应变与偏应变）• Strain Compatibility（应变相容性）• Domain Connectivity（区域连通性）• Cylindrical Strain and Rotation（柱座标应变与旋转）• Spherical Strain and Rotation（球座标应变与旋转）

2

Generalized Displacement

• Generalized concept of displacement:

coordinate difference of the same

material point in two reference states.

• Generalized displacement

= Rigid-body translation

+ Rigid-body rotation

+ Strain deformation

• Rigid-body motion: the distance between points remains the same.

• Strain deformation: an elastic solid is said to be deformed or

strained when the relative displacements between points in the

body are changed.

• We are not concerned with rigid-body motions in Elasticity Theory.3

Small Deformation Theory

,

,

o

i i i j j

o

i i i i j j

d

u u u dx

dx u u u dx

ou u u x

(Deformed) (Undeformed)

o

o

o

u u uu u dx dy dz

x y z

v v vv v dx dy dz

x y z

w w ww w dx dy dz

x y z

• Taylor expansion of u w.r.t. uo:

dx d x

od d d x x x u u

4

, , , , ,

1 1( ) ( )

2 2i j i j j i i j j i ij ij

u u u

x y z

v v vu u u u u

x y z

w w w

x y z

, ,

, ,

1 1( ); , strain tensor (symmetric)

2 2

1 1( ); , rotation tensor (anti-symmetric)

2 2

ij i j j i

ij i j j i

u u

u u

ε u u

ω u u

Small Deformation Theory

• Displacement gradient (Jacobean)

• Generalized displacement

o

i i ij ij ju u dx 5

o

i i ij j ij ju u dx dx

• Components of general displacement

General

displacement

Rigid-body

displacement

Strain

displacement

Rigid-body

rotationExamples of Continuum Motion & Deformation

(Undeformed Element) (Rigid Body Rotation)

(Horizontal Extension) (Shearing Deformation) (Vertical Extension)

Continuum Motion & Deformation

6

Two-dimensional Geometric Deformation

, :

, , ,

, , ;

, :

, , ,

, , .

B x dx y

uu x dx y u x y dx

x

vv x dx y v x y dx

x

C x y dy

uu x y dy u x y dy

y

vv x y dy v x y dy

y

2 2 2

1 2u v u u

A B dx dx dx dxx x x x

2v

x

1 ;

tan , tan .

udx

x

v uv u u vdx dx dx dy dy dy

x x y yx y

1

,

1

.

x xx

y yy

udx dx

A B AB ux

AB dx x

vdy dy

yA C AC v

AC dy y

• Normal strain

;2

xy

v uC A B

x y

• Engineering shear strain

• Shear strain

1 1.

2 2xy xy

v u

x y

Two-dimensional Geometric Deformation

, ,

1

2

1

2

1

2

x y z

xy

yz

zx

u v w

x y z

u v

y x

v w

z y

w u

x z

• 3-D Strain-displacement

relationship

8

Two-dimensional Rigid-body Rotation

• Rigid-body rotation around z-axis

1

2z

v u v u

x y x y

• 3-D rigid-body rotation

*

*

*

o z y

o x z

o y x

u u y z

v v z x

w w x y

,1 2 1 2i ijk jk ijk k ju

9

*

*

o z

o z

u u y

v v x

• Integrate for constant rotation

Determine the displacement gradient, strain and rotation tensors for the following displacement

field: 32 ,, CxzwByzvyAxu , where A, B, and C are arbitrary constants. Also calculate

the dual rotation vector = (1/2)(u).

2

,

3 2

2 3

2

, ,

3 2

2 3

2

, ,

3

2 3

2

2 0

0

0 3

2 / 2 / 21

/ 2 / 22

/ 2 / 2 3

0 / 2 / 21

/ 2 0 / 22

/ 2 / 2 0

1 1/ / /

2 2

i j

ij i j j i

ij i j j i

Axy Ax

u Bz By

Cz Cxz

Axy Ax Cz

u u Ax Bz By

Cz By Cxz

Ax Cz

u u Ax By

Cz By

x y z

Ax y Byz Cxz

1e e e

ω u 3 2

2 3

3

1

2By Cz Ax

1e e e

Sample Problem

10

3-D Principal Strains by Eigen-equation

12

3

3 2

1 2 3

1

2

3

det 0

0

1

2

det

x xy xz

yx y yz

zx zy z

ij j n i

ij n ij

n n n

kk

ii jj ij ji

ij

n n

I I I

I

I

I

11

cos sin

sin cosijQ

11 1 1

2 2

2 2

2 2

cos sin 2 sin cos

sin cos 2 sin cos

sin cos sin cos (cos sin )

x x y xy

y x y xy

xy x y xy

Q Q

Q Q

cos2 sin 22 2

cos2 sin 22 2

sin 2 cos22

x y x y

x xy

x y x y

y xy

x y

xy xy

2-D Principal Strains by Transformation

2 2 2

2

2

ave ;2 2

x ave xy

x y x y

xy

R

R

max,min

2; ; tan 2

xy

x y x y ave p

x y

R

12

Spherical and Deviatoric Strain

ˆij ij ij

• Decomposition of the strain tensor

• Spherical (mean) strain tensor: volumetric deformation + isotropic

• Deviatoric (octahedral) strain tensor: shape change

1 1

3 3ij m ij kk ij x y z ij

ij ij ij

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆij j n i ij m ij j n i ij j n m in n n n n n

• Relationships among principal strains and directions

ˆ

ˆ

i i

ij j n i

n n m

n nn n

13

• Normally we want continuous and single-valued displacements;

i.e. a mesh that fits perfectly together after deformation.

Undeformed State Deformed State

Strain Compatibility - Concept

14

1 1 1, , , , ,

2 2 2x y z xy yz zx

u v w u v v w w u

x y z y x z y x z

• Given the three displacements: We have six equations to

easily determine the six strain components.

• Given the six strains: We have six equations to

determine three displacement components. This is an

over-determined system and in general will not yield

continuous single-valued displacements unless the strain

components satisfy some additional relations.

Strain Compatibility – Mathematical Context

• Strain-displacement relationship

15

Strain Compatibility – Physical Interpretation

16

Compatibility Equations

,, ,

, ,

,

,

,

, ,, ,

,

,

1 1; ;

1 2 2

1 12; .

2 2

l ki jkl

i kjl

j ikl

j li

ij kl kl ij

ij i j j i

ik jl jl ik

ij

l ji

k lij

kk jl ki

u

u

u

u

uu

u u

u u

• Differentiating twice the strain-displacement relationship

, , , , 0ij kl kl ij ik jl jl ik

• The continuity of displacements implies the

interchangeability of partial derivatives

• This strain compatibility condition forms the necessary and

sufficient condition for continuous and single-valuedness

displacements (up to a rigid-body motion) in simply connected

regions.

• For multiply connected regions, strain compatibility is necessary

but no longer sufficient. Additional conditions must be imposed.17

Domain Connectivity

• Simply connected: all simple closed curves drawn in the region can

be continuously shrunk to a point without going outside the region.

18

• In 2-D, only 1 out of the 16 equations is meaningful and

independent.

Compatibility Equations

, , , , 1111 1112

2222 2212

1122

1212

1122 11,22 22,

1111 2222

111211 12,12 12, 11,1 ,2 12 12

0

symm., , , 1,2

0;

;

ijkl ij kl kl ij ik jl jl ik

ijkl jikl ijlk klij

i j k l

1212 12,12 12,12 11,22 22,

11 11,12 12,11

2212 22,12 12,22 21,22 22,21

2 22

11

2 2

1122

0;

; 0.

2y xyx

y x x y

19

Compatibility Equations

1111 1112 1113

2222 2212

1133

2233 2213

12

2

1122

121

223, , , ,

3333 3313 33233

1123

1 23

13

1

1

3

2 2

2

13

0

, , , 1,2,3symm.

ijkl ij kl kl ij ik jl jl ik

ijkl jikl ijlk klij

i j k l

3

1133 11,33 33,

2323

111122 11,22 22,11 12,12 12,12

2233 22,33 33,22 23,23 23,

13,13 13,

1111 2222 3333

1112 11,12 12,11 11,12 12,1123

1323

13

11

0;

; ;

; 0;

13 11,13 13,11 11,13 13,11

2212 22,12 12,22 21,22 22,2 2213 22,13 13,22 21,23

1123 11,23 23,11 12,13 13,1

3312 33,12 12,33

23,211

2223 22,23 2 33,22 22,23 23,22

20; ;

0; ;

0;

2323 23,23 23,23 22,33

1,32

1212 12,12 12,

32,31

1323 13,23

33,

131

22

1223 1

12

2,23 23,12 12

3 13,13 13

1213 12,13 13,12 11,23 23,

,23 23,12

,13 11,33 33,11

11

1,22 2 , 12 11

;

; ;

; ;

;

23,13 12,33 33,12;

• In 3-D:

20

2 22

2 2

2 2 2

2 2

2 22

2 2

2

2

2

1 : 2 ;

2 : 2 ;

3 : 2 ;

4 : ;

5 : ;

6 :

y yzz

z x zx

y xyx

xy yzz zx

yz xyx zx

y xyzx

z y y z

x z x z

y x x y

x y z z x y

y z x x y z

z x y y z

.

yz

x

Compatibility Equations

• Only 6 out of the 81

are meaningful

21

• These 6 equations may be

obtained by letting k=l

11,1

, , , ,

1

0

1, 1 1 2 3 :

ij kk kk ij ik jk jk ikk l

i j

11,22 11,33 11,11 22,11 33,

1 ,11

11

12

12,12 13

12,1

,13

1

0;

2, 2 2 3 1;

3, 3 3 1 2 ;

1, 2 4 4 :

i j

i j

i j

12,22 12,33 11,12 22,12 33,12

11,21

12,22 13,23 21,11 22,12 23,13 0;

1, 3 5 5 ;

2, 3 6 6 .

i j

i j

42 2 2 2 3

2 2 2 2 2

42 2 2 2 3

2 2 2 2 2

42 2 2 2 3

2 2 2 2 2

1 2 3 4 : ;

2 3 1 5 : ;

3 1 2 6 :

xy yz zxz

yz xyx zx

y

x y z x y x y x y z z x y

y z x y z y z x y z x y z

z x y z x z x x y z

.

xy yzzx

y z x

• Further reductions are possible. Only 3 out of the 81 are

independent.

Compatibility Equations

22

Cylindrical Strain and Rotation

c

1 1; ; ;

2 2

1 1

1

1 10,

2

r z

r r rr r r r z r r

z z zz z r z z z

rr z r

u u u

u u u u uu u

r r z r r

u u u u

z r r z

u u u

r r

r θ zε u u ω u u u e e e

e e e e e e e e e e

u

e e e e e e e e

,

1 1 1, ;

2 2

1 1 1, , , ,

2

1 1 1, .

2 2

z z rz zr

r z rr r z r

z z rz zr

r

u u u u

z r r z

u u u u u uu

r r z r r r

u u u u

z r r z

23

Spherical Strain and Rotation

s

1 1; ; ;

2 2

1 1

sin

cot1 1

sin

1 1

sin

R R

R R RR R R R

RR

RR

u u u

u uu u u

R R R R R

u u uuu

R R R R R

u u uu

R R R R

ε u u ω u u u e e e

e e e e e e

u e e e e e e

e e e e

cot

1 10, ,

2 sin

cot1 1 1 1 1, ;

2 sin 2

co1 1, ,

sin

RR R

RR

R R RR

u

R

u uu

R R R

u u uu u u

R R R R R R

u uu u u

R R R R R

e e

t 1 1, ,

2

cot1 1 1 1 1, .

2 sin 2 sin

RR

RR

u u uu

R R R R

uu u u uu

R R R R R R

Outline

• Generalized Displacement（位移概念）• Small Deformation Theory（小变形理论）• Continuum Motion & Deformation（运动与变形）• Strain & Rotation（应变与旋转）• Principal Strains（主应变）• Spherical and Deviatoric Strain（平均应变与偏应变）• Strain Compatibility（应变相容性）• Domain Connectivity（区域连通性）• Cylindrical Strain and Rotation（柱座标应变与旋转）• Spherical Strain and Rotation（球座标应变与旋转）

25