# Fundamentals of Elasticity Theory

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Nov 15, 2013 (4 years and 8 months ago)

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Fundamentals of Elasticity Theory

Reference:
Elasticity Theory Applications and Numerics
,

Theory of Elasticity

Based Upon Principles of Continuum Mechanics, Elasticity Theory Formulates
Stress Analysis Problem As Mathematical Boundary
-
Value Problem for
Solution of Stress, Strain and Displacement Distribution in an Elastic Body.

Governing Field Equations
Model Physics Inside Region

(Same For All Problems)

Boundary Conditions Describe
Physics on Boundary

(Different For Each Problem)

R

S
u

S
t

Value of Elasticity Theory

-

Develops “Exact” Analytical Solutions For Problems of Limited Complexity

-

Provides Framework for Understanding Limitations of Strength of Materials Models

-

Establishes Framework for Developing Linear Finite Element Modeling

-

Generates Solutions for Benchmark Comparisons with FEA Solutions

Deformation and Strain

u
(
x
,
y
)

u
(
x
+
dx
,
y
)

v
(
x
,
y
)

v
(
x
,
y
+
dy
)

dx

dy

A

B

C

D

A'

B'

C'

D'

dy
y
u

dx
x
v

x

y

xy
xy
y
x
x
v
y
u
e
y
v
e
x
u
e

2
1
2
1
Strain Displacement Relations

z
zy
zx
yz
y
yx
xz
xy
x
e
e
e
e
e
e
e
e
e
]
[
e
e
Three
-
Dimensional Theory

Two
-
Dimensional Theory

Deformation and Strain Example

Ax
Cy
z
u
x
w
e
Cx
Cx
y
w
z
v
e
Bx
Bx
x
v
y
u
e
z
w
e
By
y
v
e
Az
x
u
e
C
B
A
Cxy
w
y
x
B
v
Axz
u
zx
yz
xy
z
y
x

2
1
2
1
2
1
0
2
1
2
1
2
0
2
1
2
1
0
2

_
__________
__________
__________
__________
constants

are

,
,

where
,

,
)
(
,
field
nt
displaceme

following

for the
strain

of

components

the
Determine
2
2
Rigid Body Motion

Two
-
Dimensional Example

u
o

v
o

dx

dy

A

B

C

D

y
u
z

x
v
z

x

y

x
v
v
y
u
u
z
o
z
o

*
*
Field
nt
Displaceme

of

Form

General
Zero Strains!

Strain Compatibility

y
x
e
x
e
y
e
xy
y
x

2
2
2
2
2
2
2

3

1

4

Undeformed Configuration

2

3

1

4

Deformed Configuration

Continuous Displacements

2

3

1

4

Deformed Configuration

Discontinuous Displacements

Discretized Elastic Solid

x

y

Compatibility Equation

Strain Compatibility Example

B
A
B
A
B
A
y
x
B
Ax
Ay
y
x
e
x
e
y
e
y
x
Bxy
e
Ax
e
Ay
e
xy
y
x
xy
y
x
3
2
ith
equation w

satisfies
only

3
2
4
6
)
2
2
(
2
6
6
2
__
__________
__________
__________
__________
equation
ity
compatibil

l
dimensiona
-
two
the
satisfies
)
(
,
,

field
strain

following

the
if

see

Check to
2
2
2
2
2
3
3

Body and Surface Forces

Surface Forces:
T
(
x
)

S

Cantilever Beam Under Self
-

Body Forces:
F
(
x
)

Traction and Stress

F

n

A

(Sectioned Body)

P
1

P
2

P
3

p

A
A

F
n
x
T
n
0
lim
)
,
(
Traction Vector

Note that ordinary elasticity theory does not include nor allow
concentrated moments to exist at a continuum point

Stress Components

3
2
1
3
3
2
1
2
3
2
1
1
e
e
e
e
n
x
T
e
e
e
e
n
x
T
e
e
e
e
n
x
T
z
zy
zx
yz
y
yx
xz
xy
x

)
,
(
)
,
(
)
,
(
n
n
n

z
zy
zx
yz
y
yx
xz
xy
x
]
[

3
2
1
e
e
e
T
)
σ
τ
τ
(
)
τ
σ
τ
(
)
τ
τ
σ
(
n
z
z
y
yz
x
xz
z
zy
y
y
x
xy
z
zx
y
yx
x
x
n
n
n
n
n
n
n
n
n

x

z

y

y

x

yx

z

xy

xz

zy

yz

zx

Stress Transformation

3
3
3
2
2
2
1
1
1
)
,
cos(
n
m
l
n
m
l
n
m
l
x
x
j
i
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
2
)
(
2
)
(
2
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
3
3
3
3
3
3
2
3
2
3
2
3
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
2
1
2
1
2
1
n
l
l
n
m
n
n
m
l
m
m
l
n
n
m
m
l
l
n
l
l
n
m
n
n
m
l
m
m
l
n
n
m
m
l
l
n
l
l
n
m
n
n
m
l
m
m
l
n
n
m
m
l
l
l
n
n
m
m
l
n
m
l
l
n
n
m
m
l
n
m
l
l
n
n
m
m
l
n
m
l
zx
yz
xy
z
y
x
zx
zx
yz
xy
z
y
x
yz
zx
yz
xy
z
y
x
xy
zx
yz
xy
z
y
x
z
zx
yz
xy
z
y
x
y
zx
yz
xy
z
y
x
x

e

1

e

3

e

2

e
3

e
2

e
1

x
3

x
1

x
2

x

1

x

2

x

3

)
sin
(cos
cos
sin
cos
sin
cos
sin
2
cos
sin
cos
sin
2
sin
cos
2
2
2
2
2
2

xy
y
x
xy
xy
y
x
y
xy
y
x
x
Two
-
Dimensional Transformation

Three
-
Dimensional Transformation

x

y

x'

y'

0
10
20
30
40
50
60
70
80
90
-0.5
0
0.5
1

(degrees)
Dimensionless Stress
Stress Transformation Example

x

x

s

t

u

cos
sin
cos
2
x
x

2
cos
/
x

cos
sin
/
x
Principal Stresses and Directions

0
0
)
(
)
(
)
(
Solution

Trival
-
Non

Equations,

Algebraic

of

System

s
Homogeneou
0
)
(
)
(
)
(
0
)
(
0
)
(
0
)
(
3
2
2
1
3
3
2
1
3
2
1
3
2
1
2
2
1

I
I
I
n
n
n
n
n
n
n
n
n
n
n
n
z
yz
xz
yz
y
xy
xz
xy
x
z
yz
xz
yz
y
xy
xz
xy
x
z
yz
xz
yz
y
xy
xz
xy
x
Roots of the characteristic equation are the principal stresses

1

2

3

Corresponding to each principal stress is a principal direction
n
1

n
2

n
3

that can be used to construct a principal coordinate system

y

(General Coordinate System)

1

3

2

(Principal Coordinate System)

n
1

n
3

n
2

x

z

y

x

yx

z

xy

xz

zy

yz

zx

1

3

2

I
i

=
Fundamental

Invariants

Equilibrium Equations

yx
xy
y
y
xy
y
x
yx
x
x
M
F
y
x
F
F
y
x
F

0
0
0
0
0
x

y

dx
x
x
x

xy

yx

dy
y
y
y

dx
x
xy
xy

dy
y
yx
yx

x
F
y
F
Body Forces

Equilibrium Equation Example

0
0
0
0
0
2
3
2
3
0
_________
__________
__________
__________
)
1
(
4
3
,
0
,
2
2
3
equations

m
equilibriu

e
satisfy th

stresses

following

that the
show

forces,
body

no

Assuming
3
3
2
2
3

y
x
c
Py
c
Py
y
x
c
y
c
P
c
N
c
Pxy
y
xy
yx
x
xy
y
x
a

a

Hooke’s Law

zx
yz
xy
z
y
x
zx
zx
yz
xy
z
y
x
yz
zx
yz
xy
z
y
x
xy
zx
yz
xy
z
y
x
z
zx
yz
xy
z
y
x
y
zx
yz
xy
z
y
x
x
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
66
65
64
63
62
61
56
55
54
53
52
51
46
45
44
43
42
41
36
35
34
33
32
31
26
25
24
23
22
21
16
15
14
13
12
11
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2

Isotropic Homogeneous Materials

zx
zx
yz
yz
xy
xy
z
z
y
x
z
y
z
y
x
y
x
z
y
x
x
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e

2
2
2
2
)
(
2
)
(
2
)
(

zx
zx
zx
yz
yz
yz
xy
xy
xy
y
x
z
z
x
z
y
y
z
y
x
x
E
e
E
e
E
e
E
e
E
e
E
e

2
1
1
2
1
1
2
1
1
)
(
1
)
(
1
)
(
1

=
Lamé’s constant

=
shear modulus

or
modulus of rigidity

E
=
modulus of elasticity

or
Young’s modulus

v

=
Poisson’s ratio

Orthotropic Materials

(Three Planes of Material Symmetry)

xy
zx
yz
z
y
x
xy
zx
yz
z
y
x
e
e
e
e
e
e
E
E
E
E
E
E
E
E
E
2
2
2
1
0
1
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
12
31
23
3
2
23
1
13
3
32
2
1
12
3
31
2
21
1
Nine Independent Elastic Constants for 3
-
D

Four Independent Elastic Constants for 2
-
D

Physical Meaning of Elastic Constants

(Simple Tension)

(Pure Shear)

p

p

p

(Hydrostatic Compression)

0
0
0
0
0
0
0
0
ij

E
E
E
e
ij
0
0
0
0
0
0

0
0
0
0
0
0
0
ij

0
0
0
0
0
2
/
0
2
/
0
ij
e
xy
xy
e

/
2
/
x
e
E
/

ij
ij
p
p
p
p

0
0
0
0
0
0

p
E
p
E
p
E
e
ij
2
1
0
0
0
2
1
0
0
0
2
1

p
E
e
kk
)
2
1
(
3

k
p
Modulus
Bulk
)
2
1
(
3

p
E
k
Relations Among Elastic Constants

Typical Values of Elastic Constants

Basic Formulation

Fundamental Equations

(15)

-

Strain
-
Displacement (6)

-

Compatibility (3)

-

Equilibrium (3)

-

Hooke’s Law (6)

Fundamental Unknowns

(15)

-

Displacements (3)

-

Strains (6)

-

Stresses (6)

Displacement Conditions

Mixed Conditions

Traction Conditions

R

S

R

S
u

S
t

T
(n)

R

S

u

Typical Boundary Condtions

Basic Problem Formulations

Problem 1 (Traction Problem)

Determine the distribution of displacements, strains and stresses in
the interior of an elastic body in equilibrium when body forces are given and the distribution of the
tractions are prescribed over the surface of the body.

Problem 2 (Displacement Problem)

Determine the distribution of displacements, strains and stresses
in the interior of an elastic body in equilibrium when body forces are given and the distribution of the
displacements are prescribed over the surface of the body.

Problem 3 (Mixed Problem)

Determine the distribution of displacements, strains and stresses in the
interior of an elastic body in equilibrium when body forces are given and the distribution of the
tractions are prescribed over the surface S
t

and the distribution of the displacements are prescribed
over the surface S
u

of the body.

Displacement Conditions

Mixed Conditions

Traction Conditions

R

S

R

S
u

S
t

T
(n)

R

S

u

Basic Boundary Conditions

r

r

r

r

r

x

xy
=T
x

y
=T
y

x

y

x
=T
x

xy
=T
y

y

(Cartesian Coordinate Boundaries)

(Polar Coordinate Boundaries)

Coordinate Boundary Examples

Non
-
Coordinate Boundary Example

x

y

)
,
(
)
(
y
x
F
n
n
T
x
y
xy
x
x
n
x

)
,
(
)
(
y
x
F
n
n
T
y
y
y
x
xy
n
y

n
= unit normal vector

Boundary Condition Examples

Fixed Condition

u

=
v

= 0

Traction Free Condition

x

y

a

b

S

Traction Condition

0
)
(

n
y
T
x

y

l

0
)
(

n
x
T
Fixed Condition

u

=
v

= 0

Traction Condition

(Coordinate Surface Boundaries)

(Non
-
Coordinate Surface Boundary)

Traction Free Condition

S

0
,
)
(
)
(

xy
n
y
x
n
x
T
S
T
S
T
T
y
n
y
xy
n
x

)
(
)
(
,
0
0
,
0
)
(
)
(

y
n
y
xy
n
x
T
T
Symmetry Boundary Conditions

Symmetry Line

0
0
)
(

n
y
T
u
x

y

Rigid
-
Smooth

Boundary Condition

Example Solution

Beam Problem

s
x

-

Contours

x

Saint
-
Venant’s Principle

The Stress, Strain and Displacement Fields Due to Two Different Statically
Equivalent Force Distributions on Parts of the Body Far Away From the Loading
Points Are Approximately the Same.

x

y

P

x

y

P/2

P/2

xy

y

x

xy

y

x

Stresses Approximately Equal

Strain Energy

x
x
x
x
x
e
Ee
E
dxdydz
dU
U
dxdydz
E
dxdydz
E
d
dxdydz
x
u
d
dydz
du
dydz
dx
x
u
u
d
dU
x
x
x
x

2
1
2
2

Volume
Energy
Strain
2
)
(
)
(
2
2
2
0
0
0
0
dx

u

dz

dx
x
u
u

dy

x

y

z

Strain Energy = Energy Stored Inside an

One
-
Dimensional Case

Three
-
Dimensional Case

0
)
2
1
2
1
2
1
(
)
(
2
1
2
1
)
(
2
1
2
2
2
2
2
2
2

zx
yz
xy
z
y
x
z
y
x
ij
ij
zx
zx
yz
yz
xy
xy
z
z
y
y
x
x
e
e
e
e
e
e
e
e
e
e
U
Principle of Virtual Work

0

W
U
dV
u
F
dS
u
T
dV
U
T
i
V
i
S
i
n
i
V
t
Change in Potential Energy (U
T
-
W) During a
Virtual Displacement from Equilibrium is Zero.







V
zx
zx
yz
yz
xy
xy
z
z
y
y
x
x
V
ij
ij
T
dV
e
e
e
dV
e
U
)
(

Energy

Strain

Virtual
The
virtual displacement

u
i

= {

u
,

v
,

w
} of a material point is a fictitious
displacement such that the forces acting on the point remain unchanged. The work
done by these forces during the virtual displacement is called the
virtual work
.

dV
u
F
dS
u
T
W
i
V
i
S
i
n
i
t

s
Body Force

and
Surface
by
Done

Work

Virtual
dV
u
F
dS
u
T
dV
e
i
V
i
S
i
n
i
ij
V
ij
t

Virtual Strain Energy = Virtual Work Done by Surface and Body Forces