slide cơ học vật chất rắn chapter 2 new deformation displacements and strain

.c
om
ng
co
an
th

cu

u

du
o

ng

Chapter 2: Deformation: Displacements
and strain

TDT  University  -­‐  2015
CuuDuongThanCong.com

/>

.c
om

h"p://incos.tdt.edu.vn
 

2.1 General deformations

 

co

ng

2.2 Geometric construction of small deformation theory

an

2.3 Strain transformation

th

2.4 Principal strains

du
o

ng

2.5 Spherical & Deviatoric strains
2.6 Strain compatibility

cu

u

2.7 Curvilinear strain-displacement relations cylindrical coordinates

2
 
CuuDuongThanCong.com

/>

.c
om

h"p://incos.tdt.edu.vn
 

2.1 General deformations
 

co

ng

2.2 Geometric construction of small deformation theory

an

2.3 Strain transformation

th

2.4 Principal strains

du

o

ng

2.5 Spherical & Deviatoric strains
2.6 Strain compatibility

cu

u

2.7 Curvilinear strain-displacement relations cylindrical coordinates

3
 
CuuDuongThanCong.com

/>

h"p://incos.tdt.edu.vn
 

co

ng

This is in contrast to rigid-body motion
where the distance between points
remains the same.

cu

u

du
o

ng

th

an

An elastic solid is said to be deformed or
strained when the relative displacements
between points in the body are changed.

.c
om

Deformations: non-homogeneous

Fig 1. Rigid-body motion

Fig 2. Deformed or strained
4
 

CuuDuongThanCong.com

/>

h"p://incos.tdt.edu.vn
 

.c
om

Small Deformation Theory

ng

- Consider two neighboring material points P0 and P
connected with the relative position vector r as shown
in Fig 3.
- Through a general deformation, these points are
mapped to locations P’0 and P’ in the deformed
configuration.
-  In linear elasticity, only small deformation theory is
necessary.

P

r

an

co

P0

r'

u0

P0′

Fig 3. General deformation
between two neighboring points

ng

th

P′

u

du
o

Taylor series expansion around point P0 to express the
components of u as

cu

u

f ( ui + Δui ) = f ( ui ) + f ,ui ( ui ) Δui

∂u
∂u
∂u
rx + ry + rz
∂x
∂y
∂z
∂v
∂v
∂v
v = v o + rx + ry + rz
∂x
∂y
∂z
∂w
∂w
∂w
w = wo +
rx +
ry +
rz
∂x
∂y
∂z

u = uo +

CuuDuongThanCong.com

The change in the

relative position vector r

prove

Δr = r′ − r

Δri = ui , j rj
5
 
/>

h"p://incos.tdt.edu.vn
 

.c
om

Small Deformation Theory

co

∂u
∂u
∂u
u = u + rx + ry + rz
∂x
∂y
∂z
∂v
∂v

∂v
v = v o + rx + ry + rz
∂x
∂y
∂z
∂w
∂w
∂w
w = wo +
rx +
ry +
rz
∂x
∂y
∂z

ng

f ( ui + Δui ) = f ( ui ) + f ,ui ( ui ) Δui

an

o

P

r

P0

r'

u0

du
o

u

∂u
∂u
∂u
rx + ry + rz
∂x
∂y
∂z
∂v
∂v
∂v
Δry = r ' y − ry = rx + ry + rz
∂x
∂y
∂z
∂w
∂w
∂w
Δrz = r 'z − rz =
rx +
ry +
rz

∂x
∂y
∂z

cu

Δr = r ′ − r
= u − u0

Δrx = r 'x − rx =

CuuDuongThanCong.com

P0′

Fig 3. General deformation
between two neighboring points

ng

th

ui = ui0 + ui, j rj

P′

u

Δri = ui , j rj

/>

h"p://incos.tdt.edu.vn
 

.c
om

Small Deformation Theory
- Tensor ui,j is called the displacement gradient tensor.

th

an

co

ng

∂u ⎤
∂z ⎥⎥
∂v ⎥ 1
1
=
(
u
+
u
)
+

(ui , j − u j ,i ) = eij + ωij
i, j
j ,i
∂z ⎥ 2
2
⎥
∂w ⎥
∂z ⎥⎦

1
⎧
e
=
⎪ ij 2 (ui , j + u j ,i ) , strain tensor
⎨
1
⎪ωij = (ui , j − u j ,i ) , rotation tensor
⎩
2

ng

∂u
∂y
∂v
∂y
∂w
∂y

du

o

⎡ ∂u
⎢ ∂x
⎢
⎢ ∂v
ui , j = ⎢
∂x
⎢
⎢ ∂w
⎢ ∂x
⎣

u

- Choose ri = dxi, we can write the general result in the form

cu

ui = ui0 + ui , j rj = ui0 + eij dx j + ωij dx j

- Using a dual vector ωi = -1/2 εijkωjk, we have
1 ⎛ ∂u ∂u ⎞
1 ⎛ ∂u ∂u ⎞
1 ⎛ ∂u ∂u ⎞
ω1 = ω32 = ⎜ 3 − 2 ⎟ ; ω2 = ω13 = ⎜ 1 − 3 ⎟ ; ω3 = ω21 = ⎜ 2 − 1 ⎟
2 ⎝ ∂x2 ∂x3 ⎠
2 ⎝ ∂x3 ∂x1 ⎠
2 ⎝ ∂x1 ∂x2 ⎠

ω=

1
( ∇ × u)
2
7
 

CuuDuongThanCong.com

/>

.c
om

h"p://incos.tdt.edu.vn
 

2.1 General deformations
 

co

ng

2.2 Geometric construction of small deformation theory

an

2.3 Strain transformation

th

2.4 Principal strains

2.6 Strain compatibility

du
o

ng

2.5 Spherical & Deviatoric strains

cu

u

2.7 Curvilinear strain-displacement relations cylindrical coordinates

8
 
CuuDuongThanCong.com

/>

.c
om

h"p://incos.tdt.edu.vn

 

co
an

(Undeformed Element)

(Rigid Body Rotation)

(Horizontal Extension)

(Vertical Extension)

(Shearing Deformation)

cu

u

du
o

ng

th

Consider the common deformational
behavior of a rectangular element.
Rigid-body motion does not
contribute to the strain field, and

hence does not affect the stresses.
We therefore focus our study on the
extensional and shearing
deformation.

ng

Examples of Continuum Motion & Deformation

Fig 4. Typical deformations of a rectangular element

9
 
CuuDuongThanCong.com

/>

h"p://incos.tdt.edu.vn
 

.c
om

Consider a 2D deformation of a rectangular element with original dimensions dx by dy.
Point A(x,y) with displacement components u(x,y) and v(x,y). Point B has displacement
u(x+dx,y) and v(x+dx,y).

co

ng

In small deformation theory,
(Taylor series expansion)
u(x + dx, y) ≈ u(x, y) + (∂u / ∂x)dx

ng
du
o
u

cu

∂u ⎞ ⎛ ∂v ⎞
⎛
A' B' = ⎜ dx + dx ⎟ + ⎜ dx ⎟
∂x ⎠ ⎝ ∂x ⎠
⎝
2

⎛ ∂u ⎞ ⎛ ∂v ⎞
= dx ⎜1 + ⎟ + ⎜ ⎟
⎝ ∂x ⎠ ⎝ ∂x ⎠
⎛ ∂u ⎞
≈ ⎜ 1 + ⎟ dx
⎝ ∂x ⎠

β

v(x,y+dy)

From the geometry
2

D'
C'

th

The normal strain in x-direction
A ' B '− AB
εx =
AB

∂u
dy
∂y

an


 

2

y

D

C

B'

α

A'

2

dy

∂v
dx
∂x

v(x,y)
A

dx

u(x+dx,y)

B

u(x,y)
x
10
 

Fig 5. Two-dimensional geometric strain deformation
CuuDuongThanCong.com

/>

h"p://incos.tdt.edu.vn
 

.c
om

Fig 5. Two-dimensional geometric strain deformation

v(x,y+dy)

∂v
dx
∂x

D’

C’

!
r

∂u
dy
∂y

/

∂u
dx
∂x

ng

th

an

β

co

ng

∂u
dy
∂y

∂v
dy
∂y

dy

D

B’

A’

α

!
r

u

v(x,y)

cu

y

du
o

C

A

dx

B

u(x+dx,y)

u(x,y)
CuuDuongThanCong.com

∂v
dx
∂x

11
 
/>

.c
om

h"p://incos.tdt.edu.vn
 

Using AB = dx, the normal strain in x - direction reduces to ε x =

D'

an

C'

β

ng

th

v(x,y+dy)

y

du
o

u

cu

∂u
dy
∂y

co


 

A second type of strain is
shearing deformation, which
involves angles changes.
Shear strain is defined as the
change in angle between two
originally orthogonal directions
in the continuum material.
Measured in radians, shear
strain is positive if the right
angle between the positive
directions of two axes

decreases.

∂v
∂y

ng

Similarly, the normal strain in y - direction ε y =

∂u
∂x

D

C

B'

α

A'
dy

∂v
dx
∂x

v(x,y)
A

dx

u(x+dx,y)

B

u(x,y)
x

Fig 5. Two-dimensional geometric strain deformation
12
 

CuuDuongThanCong.com

/>

h"p://incos.tdt.edu.vn
 

γ xy =

.c
om

Shear strains in x- and y-directions can be defined as

π
2

− ∠C ' A ' B ' = α + β

For small deformations, α≈ tanα and β ≈ tanβ, and then

ng

∂u
∂v
dy
dx
∂u
  ∂v
∂y
γ xy = ∂x
+
=
+
∂u
∂v
dx + dx dy + dy ∂y ∂x
∂x
∂y

co

∂u
dy
∂y

an

C'

β

ng

y

du
o

γ xy = γ yx

th

v(x,y+dy)

By interchange of x & y, u & v

cu

u

Similarly, we have
∂u
∂v
∂w
εx = , εy = , εz =
∂x

∂y
∂z
∂u ∂v
∂v ∂w
γ xy = + , γ yz = + ,
∂y ∂x
∂z ∂y
∂w ∂u
γ zx =
+
∂x ∂z

CuuDuongThanCong.com

D'

D

C

B'

α

A'
dy

∂v
dx
∂x

v(x,y)
A

dx

u(x+dx,y)

B

u(x,y)
x

Fig 5. Two-dimensional geometric strain deformation
13
 

/>

h"p://incos.tdt.edu.vn
 

∂u
∂v
∂w
1 ⎛ ∂u ∂v ⎞
1 ⎛ ∂v ∂w ⎞
1 ⎛ ∂w ∂u ⎞
, e y = , ez =
, exy = ⎜ + ⎟ , e yz = ⎜ +

,
e
=
+ ⎟
⎜
⎟ zx
∂x
∂y
∂z
2 ⎝ ∂y ∂x ⎠
2 ⎝ ∂z ∂y ⎠
2 ⎝ ∂x ∂z ⎠

ng

ex =

.c
om

Using the strain tensor eij, the strain-displacement relations can be expressed as

th
ng

y

du
o

u

eyz

cu

exy
ey

β

v(x,y+dy)

The strain is a symmetric
second-order tensor (eij = eji)

⎡ ex
⎢
e = ⎡⎣eij ⎤⎦ = ⎢eyx
⎢ exz
⎣

D'
C'

an

1
eij = (ui , j + u j ,i )
2

1
e = ⎡⎣∇u + (∇u)T ⎤⎦
2

∂u
dy
∂y

co

Using tensor and matrix notation
 

exz ⎤
⎥
eyz ⎥
ez ⎥⎦

D

C

B'

α

A'
dy

∂v

dx
∂x

v(x,y)
A

dx

u(x+dx,y)

B

u(x,y)
x

Fig 5. Two-dimensional geometric strain deformation
14
 

CuuDuongThanCong.com

/>

h"p://incos.tdt.edu.vn
 

.c
om

Example 2-1: Strain and Rotation Examples

cu

u

du
o

ng

th

an

co

ng

Determine the displacement gradient, strain and rotation tensors for the following displacement
field: u = Ax 2 y , v = Byz , w = Cxz 3 , where A, B, and C are arbitrary constants. Also calculate
the dual rotation vector ω = (1/2)(∇×u).

15
 
CuuDuongThanCong.com

/>

h"p://incos.tdt.edu.vn
 

.c
om

Example 2-1: Strain and Rotation Examples

0 ⎤
⎥
By ⎥
3Cxz 2 ⎥⎦
⎡ 2 Axy Ax 2 / 2 Cz 3 / 2 ⎤
1
⎢
⎥
eij = ( ui , j + u j ,i ) = ⎢ Ax 2 / 2
Bz
By / 2 ⎥
2
⎢ Cz 3 / 2 By / 2 3Cxz 2 ⎥
⎣
⎦
2
3
⎡ 0
Ax / 2 −Cz / 2 ⎤
1
⎢
⎥
ωij = ( ui , j − u j ,i ) = ⎢ − Ax 2 / 2
0

By / 2 ⎥
2
⎢ Cz 3 / 2 − By / 2
0 ⎥⎦
⎣
e1
e2
e3
1
1
1
ω = ( ∇ × u ) = ∂ /∂x ∂ /∂y ∂ /∂z = ( − Bye1 − Cz 3e 2 − Ax 2e3 )
2
2
2
Ax 2 y Byz Cxz 3

ng

th

an

Ax 2
Bz
0

cu

u

du
o

⎡ 2 Axy
⎢
ui , j = ⎢ 0
⎢ Cz 3
⎣

co

ng

Determine the displacement gradient, strain and rotation tensors for the following displacement
field: u = Ax 2 y , v = Byz , w = Cxz 3 , where A, B, and C are arbitrary constants. Also calculate
the dual rotation vector ω = (1/2)(∇×u).

CuuDuongThanCong.com

/>
16
 


.c
om

h"p://incos.tdt.edu.vn
 

2.1 General deformations
 

co

ng

2.2 Geometric construction of small deformation theory

an

2.3 Strain transformation

th

2.4 Principal strains

2.6 Strain compatibility

du
o

ng

2.5 Spherical & Deviatoric strains

cu

u

2.7 Curvilinear strain-displacement relations cylindrical coordinates

17
 
CuuDuongThanCong.com

/>

h"p://incos.tdt.edu.vn
 

x3′
exz ⎤
⎥
e yz ⎥
ez ⎥⎦

x1

.c
om

exy
ey
e yz

o
x2

co

x1′

ng

⎡ ex
⎢
e = ⎢exy
⎢ exz
⎣

x3

x2′

⎡ e′x
⎢
e′ = ⎢e′xy
⎢ e′xz
⎣

e′xy
e′y
e′yz

e′xz ⎤
⎥
e′yz ⎥
e′z ⎥⎦

th

where

Qij = cos ( xi′, x j )

y
 
y'
 

du
o

ng

eij′ = QipQ jq e pq

an

Fig 6. 3D rotational transformation

cu

u

ex + e y e x − e y
⎧
′

x'
 
⎪ex = 2 + 2 cos 2θ + exy sin 2θ
θ
 
⎪⎪
e +e e −e
For 2D case, prove ⎨e′y = x y − x y cos 2θ − exy sin 2θ
2
2
⎪
θ
 
e y − ex
⎪e′xy =
x
 
sin 2θ + exy cos 2θ
2
⎪⎩
Fig 7. 2D rotational transformation
18
 
CuuDuongThanCong.com

/>

h"p://incos.tdt.edu.vn
 

0

.c
om

cosθ

0⎤
0⎥
⎥
1 ⎥⎦

ng

sin θ

⎧e′x = ex cos 2 θ + e y sin 2 θ + 2exy sin θ cosθ
⎪
2
2
⎨e′y = ex sin θ + ey cos θ − 2exy sin θ cos θ
⎪e′xy = −ex sin θ cosθ + e y sin θ cosθ + exy (cos 2 θ − sin 2 θ )
⎩

co

⎡ cosθ
Qij = ⎢ − sin θ
⎢
⎢⎣ 0

ng

th

an

eij′ = QipQ jq e pq

cu

u

du
o

ex + e y e x − e y
⎧
′
e
=
+
cos 2θ + exy sin 2θ
⎪ x
2
2
⎪⎪
ex + e y e x − e y
−
cos 2θ − exy sin 2θ

⎨e′y =
2
2
⎪
e −e
⎪e′xy = y x sin 2θ + exy cos 2θ
2
⎪⎩
19
 

CuuDuongThanCong.com

/>

.c
om

h"p://incos.tdt.edu.vn
 

2.1 General deformations
 

co

ng

2.2 Geometric construction of small deformation theory

an

2.3 Strain transformation

th

2.4 Principal strains

2.6 Strain compatibility

du
o

ng

2.5 Spherical & Deviatoric strains

cu

u

2.7 Curvilinear strain-displacement relations cylindrical coordinates

20
 
CuuDuongThanCong.com

/>

h"p://incos.tdt.edu.vn

 

.c
om

eij n j = eni

co

⇒ −e3 + ϑ1e 2 − ϑ2e + ϑ3 = 0

du
o

ng

th

an

ϑ1 = e1 + e2 + e3
ϑ2 = e1e2 + e2e3 + e3e1
ϑ3 = e1e2e3

u

⎡e1 0
eij = ⎢ 0 e2
⎢
⎢⎣ 0 0

cu

=> Principle strains

n1

ng

det ⎡⎣eij − eδ ij ⎤⎦ = 0

n2

0⎤
0⎥
⎥
e3 ⎥⎦

21
 
CuuDuongThanCong.com

/>

.c
om

h"p://incos.tdt.edu.vn
 

2.1 General deformations
 

co

ng

2.2 Geometric construction of small deformation theory

an

2.3 Strain transformation

th

2.4 Principal strains

du
o

ng

2.5 Spherical & Deviatoric strains
2.6 Strain compatibility

cu

u

2.7 Curvilinear strain-displacement relations cylindrical coordinates

22
 
CuuDuongThanCong.com

/>

h"p://incos.tdt.edu.vn
 

co

ng

⎛
⎞ 1
1
eij = ⎜ eij − ekkδ ij ⎟ + ekkδ ij
3
⎝
⎠ 3 $
!#
#"##
$ !"
e%

.c
om

In particular applications it is convenient to decompose the strain

tensor into two parts called spherical and deviatoric strain tensors

eˆij

an
th

represents only volumetric deformation

ng

1
e!ij = ekkδ ij
3

du
o

The spherical strain

ij

cu

u

1
The deviatoric strain eˆij = eij − ekkδ ij accounts for changes in shape of material elements
3
Note: principal directions of the deviatoric strain are the same as those of the strain tensor

23
 
CuuDuongThanCong.com

/>

h"p://incos.tdt.edu.vn
 

.c
om

Example 2-2: Determine the principle, spherical,
deviatoric strains of the following state of strain

cu

u

du
o

ng

th

an

co

ng

⎡ 2 −2 0 ⎤
eij = ⎢ −2 −4 1 ⎥
⎢
⎥
⎢⎣ 0 1 6 ⎥⎦

24
 
CuuDuongThanCong.com

/>

.c
om

h"p://incos.tdt.edu.vn
 

2.1 General deformations
 

co

ng

2.2 Geometric construction of small deformation theory

an

2.3 Strain transformation

th

2.4 Principal strains

du
o

ng

2.5 Spherical & Deviatoric strains

2.6 Strain compatibility

cu

u

2.7 Curvilinear strain-displacement relations cylindrical coordinates

25
 
CuuDuongThanCong.com

/>