1Mechanical Behavior of Materials
Introduction and Course Syllabus
Ittipon Diewwanit, Sc.D.Department of Metallurgical Engineering,
Chulalongkorn University
Copyright 2013 by Ittipon
Diewwanit
2Textbooks and References
Textbooks
Mechanical Metallurgy, George E. Dieter, SI Metric edition, McGraw-Hill, 1988.
References
Mechanical Metallurgy: Principle and Applications, Marc A. Meyers and Krishan K. Kumar, Prentice Hall, 1984
Metal Forming, W.F. Hosford and R.M. Caddell, 4th ed., Cambridge University Press 2011
Introduction to Dislocations, Derek Hull and David Bacon, Pergamon Press, 1984.
The Mechanics of Crystals and Textured Polycrystals, William F. Hosford, Oxford University Press, 1993.
Fracture and Fatigue Control in Structures: Applications of Fracture Mechanics, Butterworth-Heinemann, 1999
Copyright 2013 by Ittipon
Diewwanit
3Grading Policy
Midsemester Examination 50%
Final Examination 50%
Assignments: weekly handouts of problem sets with solution following.
There will be no grading for assignments.
Copyright 2013 by Ittipon
Diewwanit
4Course Contents
Introduction
Theory of Elasticity
Introduction to Plastic Deformation and Elementary Theory of Plasticity
Theory of Dislocation
Plastic Deformation of Crystalline Materials
Mid-semester Examination
Characterization of Mechanical Behavior
Deformation of Polymeric Materials
Introduction to Fracture Mechanics
Fatigue of Metals
Deformation at Elevated Temperature
Applications to Material Processing and Material Selection
Copyright 2013 by Ittipon
Diewwanit
5Mechanical Behavior
The response of materials to mechanical loads.
Half of the subject deals with the relationship between force and
deformation (displacement) of materials.
The other half deals with internal structure and their influence on material
properties especially mechanical ones.
Copyright 2013 by Ittipon
Diewwanit
6Conceptual Mathematical Space
Force Displacement
Stress Strain
Mechanical Properties
of Materials
measurable
conceptual
1st Rank Tensor: Vector
2nd Rank Tensor
3rd Rank Tensor
Copyright 2013 by Ittipon
Diewwanit
Copyright 2013 by Ittipon
Diewwanit
7
Simple Cantilever Beam
Force in y direction = -9,800 N
No displacement at this end
2 by 2 cm beam made of steel with Youngs modulus of 200 GPa
Copyright 2013 by Ittipon
Diewwanit
8
Normal Stress in the z Direction
Copyright 2013 by Ittipon
Diewwanit
9
Linear Strain in the x Direction
Copyright 2013 by Ittipon
Diewwanit
10
Displacement in the y Direction
11
Definition of Stress
Stress may be described as a mathematical quantity indicating the severity
of mechanical load at a certain location of material.
As defined in continuum mechanics, stress is considered as a second rank
tensor having 9 components.
It may be loosely defined as force divided by area.
The unit of stress is N m-2 or Pascal (abbreviated Pa) in SI system.
Copyright 2013 by Ittipon
Diewwanit
12
Type of Stresses
Stress may be classified to two types based on geometry of applied force with relative to the surface of interest.
If the force is acting normal to the area (surface) of interest, the stress is said to be normal stress.
If the force is acting parallel to the area (surface) of interest, the stress is said to be shear stress.
normal component of stress
shear component of stress
Copyright 2013 by Ittipon
Diewwanit
13Copyright 2013 by Ittipon
DiewwanitFrom M. F. Ashby, Engineering Materials Vol 1
14
From M. F. Ashby, Engineering Materials Vol 1
Copyright 2013 by Ittipon
Diewwanit
15
Type of Stresses (Cont.)
Normal stress may be classified further to two types. Normal stress
creating tension is termed tensile stress and is assigned an algebraic
positive sign.
Normal stress creating compression is termed compressive stress and is
assigned an algebraic negative sign.
Copyright 2013 by Ittipon
Diewwanit
16Copyright 2013 by Ittipon
DiewwanitFrom M. F. Ashby, Engineering Materials Vol 1
Copyright 2013 by Ittipon
Diewwanit
17
Mechanical Behavior of Materials
Definition of Stress
Ittipon Diewwanit
Department of Metallurgical Engineering,
Chulalongkorn University
Elongation: Response of material when subjected to axial tension. Tensile
stress along the direction of tension force is defined as F/A.
Copyright 2013 by Ittipon
Diewwanit
18
Copyright 2013 by Ittipon
Diewwanit
19
Shearing: Response of material when subjected to shear force
Shear stress on the surface is defined as F/A
Copyright 2013 by Ittipon
Diewwanit
20
21
Definition of State of Stress
Stress is a second rank tensor, reference axes must be well defined.
State of stress is a mathematical function dependant on one variable:
position vector.
x, x1, 1
y, x2, 2
z, x3, 3
Static force F
Cantilever beam
with a fixed end
Copyright 2013 by Ittipon
Diewwanit
22
State of Stress in 3-Dimension
x, x1, 1
y, x2, 2
z, x3, 3
zzxz
yz
xxzy
zx
xy
yy
yx
Copyright 2013 by Ittipon
Diewwanit
23
Matrix Representation of Stress
With relative to the reference axes, stress may be written using matrix symbol:
At static equilibrium condition, there are only six independent components according to the relationship:
zzzyzx
yzyyyx
xzxyxx
ij
zyyzzxxzyxxy and ; ;
Copyright 2013 by Ittipon
Diewwanit
24
Static Equilibrium for Rotational
y
x
xy xy
yx
yx
This results in the symmetry of stress tensor matrix: jiij
Copyright 2013 by Ittipon
Diewwanit
25
Components of Stress Tensor
Diagonal components in the matrix represent the three normal stress
according to the orthogonal reference coordinate system.
The rest are shear components.
Due to the symmetry of the matrix under static equilibrium, there are only
six independent components of stress tensor.
Copyright 2013 by Ittipon
Diewwanit
Copyright 2013 by Ittipon
Diewwanit
26
Axes Transformation for 2-D Stress
In 2-D we deal with only 4 components of stress tensor ( )
This condition occurs in many real engineering applications such as thin
wall vessels and other sheet metal components.
Transformation of orthogonal reference coordinate results in the change of
stress components.
Analytically, we can do this by using eq. 2-5 to 2-7 (in Dieters) but we can
also use graphical method called Mohrs circle of stress.
yxxyyyxx ,,,
Copyright 2013 by Ittipon
Diewwanit
27
Axes Transformation for 2-D Stress
2cos2sin2
2sin2cos22
2sin2cos22
xy
xxyy
xyyx
xy
yyxxyyxx
yy
xy
yyxxyyxx
xx
xx
y y
Copyright 2013 by Ittipon
Diewwanit
28
Axes Transformation for 2-D Stress
Principal Stresses occur at a special rotation angle. At this angle of rotation,
the shear components vanishes (eq. 2-8 in Dieters).
yyxx
xy
22tan
2/1
2
2
1max22
xy
yyxxyyxx
2/1
2
2
2min22
xy
yyxxyyxx
Copyright 2013 by Ittipon
Diewwanit
29
Axes Transformation for 2-D Stress
A
B
A
B
2 C
C
D
D
note the direction of rotation
normal
stress
shear
stress
30
Principal Stresses and Maximum Shear Stress
Principal Stresses
Maximum Shear Stress
2/1
2
2
1max22
xy
yyxxyyxx
2/1
2
2
2min22
xy
yyxxyyxx
2/1
2
2
max2
xy
yyxx
Copyright 2013 by Ittipon
Diewwanit
31
Axes Transformation for 2-D Stress
A
B
A
B
2 C
C
D
D
max
12
F
E
E
FCopyright 2013 by Ittipon
Diewwanit
32
Axes Transformation for 3-D Stress
The root of cubic equation [eq.2-14] yields the three values of principal
stresses in 3-D.
The first invariant of stress tensor, I1
0)2(
)()(
222
22223
xyzzxzyyyzxxxzyzxyzzyyxx
xzyzxyzzxxzzyyyyxxzzyyxx
ii
i
iizzyyxx I
3
1
1)(
Copyright 2013 by Ittipon
Diewwanit
33
Transformation (Summary)
The maximum and minimum values of normal stress on three principal
orthogonal planes occur when shear stress on the three planes are zero.
Shear stresses alone occur at angles which are halfway between the three
principal planes.
The value of the maximum shear stress is
2
31max
Copyright 2013 by Ittipon
Diewwanit
Principal Stresses and Maximum Shear Stresses in 3D
From Mechanics of Sheet Metal Forming, Z. Marciniak, J.L. Duncan, S.J. Hu, 2nd ed., Butterworth-Heinemann 2002
34Copyright 2013 by Ittipon
Diewwanit
35
Axes Transformation for 3-D Stress
The second invariant of stress tensor, I2
The third invariant of stress tensor, I3
The value of I3 is equal to the determinant of the stress tensor matrix.
2
222 )( Ixzyzxyzzxxzzyyyyxx
3
222 )2( Ixyzzxzyyyzxxxzyzxyzzyyxx
Copyright 2013 by Ittipon
Diewwanit
36
Mechanical Behavior of Materials
Definition of Strain
Ittipon Diewwanit, Sc.D.Department of Metallurgical Engineering,
Chulalongkorn University
Copyright 2013 by Ittipon
Diewwanit
Copyright 2013 by Ittipon
Diewwanit
37
Deformation of Materials
Deformation behavior of a material may be loosely defined as the response
of the material under applied stresses.
Applied stresses may be external or internal.
Deformation is represented by a measurable vector quantity defined as
displacement.
Strain is a higher rank quantity defined in a differential form based on the
displacement.
Copyright 2013 by Ittipon
Diewwanit
38
Deformation of Materials
Deformation behavior of materials may be divided into two types: elastic and plastic.
Elastic deformation is temporary. Material will resume its original shape and dimensions after removing the applied stresses. The deformation when the material is under applied stresses is termed recoverable elastic deformation.
Plastic deformation is permanent. If the amount of applied stresses exceeds a certain limit (known as elastic limit), material cannot resume its original shape and dimensions after removing the applied stresses. The remaining, permanent deformation is termed plastic deformation.
Copyright 2013 by Ittipon
Diewwanit
39
Engineering Linear Strain
Engineering linear strain (e):
00
0
l
l
l
lle
0l
l
ncontractio 0
extension 0
l
l
y
x
Copyright 2013 by Ittipon
Diewwanit
40
Engineering Shear Strain
Engineering shear strain ():
a
h
tanh
a
y
x
Copyright 2013 by Ittipon
Diewwanit
41
Definition of Strain at a Point
A
BA
B
y
x
x dx
Au
x
u
AB
ABBAe
dxx
udxudxuBA
xx
AB
dxx
uuu AB
Au is the displacement vector of point A
Copyright 2013 by Ittipon
Diewwanit
42
Position and Displacement Vector
A
A
x
z
y
),,(
ofnt vector displaceme
wvu
A
),,(
ofctor positon ve
zyx
A
),,(
ofctor positon ve
zyx
A
Copyright 2013 by Ittipon
Diewwanit
43
Translational Motion
A
A
x
z
y
B
B
Displacement vectors at any point within the body are equal.
Copyright 2013 by Ittipon
Diewwanit
44
Pure Rotational Motion
A
A
x
z
y
B
B
No strain but displacement vector varies as a function of position.
Copyright 2013 by Ittipon
Diewwanit
45
Displacement Vector and Displacement Tensor
Displacement vector is a function of position,
In vector format where
In matrix format
),,(
),,(
),,(
zyxww
zyxvv
zyxuu
jiji
zzzyzx
yzyyyx
xzxyxx
xeu
zeyexew
zeyexev
zeyexeu
or
x
we
x
ve
x
ue zzyyxx
, ,
z
ue
x
ve
y
ue xzyzxy
, ,
zzzyzx
yzyyyx
xzxyxx
ij
eee
eee
eee
e
Copyright 2013 by Ittipon
Diewwanit
46
Displacement, Strain, and Rotation Tensors
Displacement tensor, , can be decomposed into two parts. One is a
symmetric tensor and the other is skew-symmetric tensor.
We define the symmetric tensor as strain tensor
ije
)(2
1)(
2
1jiijjiijij eeeee
)(2
1jiijij ee
Copyright 2013 by Ittipon
Diewwanit
47
Strain Tensor
z
w
y
w
z
v
x
w
z
u
y
w
z
v
y
v
x
v
y
u
x
w
z
u
x
v
y
u
x
u
ij
2
1
2
1
2
1
2
1
2
1
2
1
jiij
Copyright 2013 by Ittipon
Diewwanit
48
Rotation Tensor
02
1
2
1
2
10
2
1
2
1
2
10
y
w
z
v
x
w
z
u
y
w
z
v
x
v
y
u
x
w
z
u
x
v
y
u
ij
jiij
Copyright 2013 by Ittipon
Diewwanit
49
Definition of Strain Tensor
Strain tensor is defined as,
In full matrix form,
ijijij e
z
w
y
w
z
v
x
w
z
u
y
w
z
v
y
v
x
v
y
u
x
w
z
u
x
v
y
u
x
u
ij
2
1
2
1
2
1
2
1
2
1
2
1
total displacement
rigid-body rotation
jiij
Copyright 2013 by Ittipon
Diewwanit
50
Displacement, Strain, and Rotation Tensors
We define the skew-symmetric tensor as rotation tensor
The two tensors play important roles in the analysis of deformation and
motion of bodies.
)(2
1jiijij ee
ijijije
Copyright 2013 by Ittipon Diewwanit 51
General Equation of Motion and Deformation
Combining the decomposition of displacement tensor
with the vector equation for displacement vector
We have a general vector equation describing the motion and deformation
of a body as
ijijije
jiji xeu
jijjiji xxu
Copyright 2013 by Ittipon Diewwanit 52
Stress and Strain Relationship in Shear
Pure rotation without shear
yxxy ee -
x
y
z
ue
x
ve
y
ue xzyzxy
, ,
Copyright 2013 by Ittipon
Diewwanit
53
Stress and Strain Relationship in Shear
x
y
z
ue
x
ve
y
ue xzyzxy
, ,
Simple shear with rotationSimple shear
yxxy ee
0
yx
xy
e
e
Copyright 2013 by Ittipon
Diewwanit
54
Stress and Strain Relationship in Shear
Considering the definition of engineering shear strain which is based on
simple shear, it is obvious that our definition of shear component in the
strain tensor is related to the engineering shear strain by
whereas
ijij 2
z
u
x
w
z
v
y
w
x
v
y
uzxyzxy
and , ,