A Review of the Recent Development in Machining Parameter Optimization
1. 1
Solid Mechanics Review
Deformation and the concept of
stress and strain
MCE 508 – Advanced Mechanics of Material
1
Deformation
Deformation: Change of microstructure resulting from loading.
Loading types: Tension, compression, and torsion or combination of
two or all.
Mechanical property: Measured material response to applied loads.
2
2. 2
Deformation
Materials are usually assumed to be
Continuous (no cracks or voids).
Homogeneous (identical properties at all points).
Isotropic (particular property does not vary with
direction). Anisotropy in property exists if the
property changes with the direction.
Most polycrystalline materials especially metals that
have equiaxed grains show isotropy in mechanical
properties.
In deformation, two measured property are very
important: stress and strain.
3
Stress
A
dA
A
P
dA
P
σ is the average stress and A is the cross-sectional area.
Internal resisting force
4
3. 3
Stress
If the applied load is normal to the acted area then the
stress is called normal stress and represented by S or σ
(engineering and true stresses).
If the load is parallel to the area then the stress is called
shear stress and represented by t.
For both stresses, normal and shear, the stress is simply
found dividing the load by the area.
It is assumed that stress is constant, therefore the
calculated stress is an average stress, which does not
represent stress in micro scale such as stress around voids
and etc.
5
Stress
Fracture is normally driven by normal stresses and
plastic deformation (viscous flow-change in shape) is
driven by shear stresses (see Fig. given below, points
reorient atoms).
6
6. 6
Tensile test
1. ELASTIC DEFORMATION
2. PLASTIC DEFORMATION 11
Elastic deformation
•Stress proportional to strainknown as Hooke’s Law:
Material obeying Hooke’s law called linear elastic or
Hokean solids.
Hooke’s Law is not valid for every material, such as Rubber
E is the Elastic Modulus or Young’s Modulus.
12
7. 7
Elastic deformation
Reversible Process; upon releasing the load strain goes back to
zero. Immediate response Linear elastic.
time dependent elastic recovery viscoelastic materials.
13
Elastic deformation
During elastic deformation volume of the material changes: atomic
distance increases/decreases with increasing loading
0
V
E is structure insensitive property; metallurgical processes have
little effect on it. It is mainly determined by the binding forces
between atoms. (E decreases with increasing temperature)
For shear stress and strain (t) Hooke’s law still valid but in
this case the relation becomes
t
t G
.
G
E
14
8. 8
Onset of plastic deformation
Elastic limit: requires
precise instruments
such as strain gages
and strain measuring
systems.
Proportional limit:
stress at which S-e
curve deviates from
linearity.
Yield strength: (offset yield strength or proof strength) stress
that produces a small amount of permanent deformation,
generally at 0.002 (0.2%) offset strain. 15
Plastic deformation
Permanent deformation
(irreversible)
Density changes little with
plastic deformation and
therefore it is assumed that
volume change is zero
(compare with elastic
deformation). Plastic
deformation changes only the
shape of the material.
p
e
t e
e
e e
t
p e
e
e
0
V
16
9. 9
Plastic deformation
Plastic deformation
proceeds by breaking
and reestablishing the
atomic bonds.
In metals, plastic
deformation occurs via
dislocation motion.
17
Necking
Necking starts
when the maximum
load is reached
(Ultimate Tensile
Strength (UTS)).
Deformation then
proceeds in the
necked region until
the material fails
(fracture).
0
dA
.
d
.
A
dP
A
P
A
dA
d
18
10. 10
True stress-strain
L
dL
d
o
L
o
L L
L
ln
L
dL
(true or logarithmic strain)
)
e
1
ln(
L
)
L
(
ln
o
o
L
L
A
A
A
A
L
L
AL
L
A
0
V o
o
o
o
o
o
)
1
( e
S
L
L
Ao
P
A
P
o
For small strains
=S and =e
19
Ductility
Percent elongation
Percent reduction in area
It shows material ability to deform plastically
o
o
f
L
L
L
E
%
o
f
o
A
A
A
A
%
20
11. 11
Stress at a point
3-Normal stresses:
x,y,z
6 shear stresses: txy, txz
,tyx ,tyz ,tzx ,tzy
Notation: x: Normal
stress acting in x-
direction. txy: x is the
plane on which the stress
acts and y is the
direction. Plane Direction Value
- - +
+ - -
- + -
+ + +
Value of shear stress
21
Stress at a point
Summation of the moments of forces about z axis;
y
z
x
yx
x
z
y
xy )
(
)
(
t
t yx
xy t
t zy
yz
zx
xz , t
t
t
t
State of stress at a point is described completely by six
components of stress; 3 normal and 3 shear stresses.
22
12. 12
Plane stress
plane stress condition: z=0, txz=tyz=0, L and w >>t
0
Fx
t
sin
m
,
cos
l
sin
.
A
.
cos
.
A
.
A
S yx
x
x
m
.
A
.
l
.
A
.
A
.
S yx
x
x t
l
.
A
.
m
.
A
.
A
.
S xy
y
y t
l
.
m
.
S xy
y
y t
m
.
S
l
.
S
sin
.
S
cos
.
S
' y
x
y
x
x
m
.
l
.
S yx
x
x t
(A)
23
t
sin
.
cos
.
.
2
sin
.
cos
.
)
(
' xy
2
y
2
x
x
Plane stress
m
).
m
.
l
.
(
l
).
l
.
m
.
(
m
.
S
l
.
S yx
x
xy
y
x
y
'
y
'
x t
t
t
t
t cos
.
sin
).
(
)
sin
(cos
)
( x
y
2
2
xy
'
y
'
x
)
2
sin(
).
2
cos(
.
.
2
)
2
(
sin
.
)
2
(
cos
.
)
2
(
' xy
2
y
2
x
x
t
t
sin
.
cos
.
.
2
cos
.
sin
.
)
(
' xy
2
y
2
x
y
2
2
cos
1
sin
2
1
2
cos
cos 2
2
2
cos
sin
cos
2
sin
cos
.
sin
.
2 2
2
t
2
sin
.
2
cos
.
2
2
)
(
' xy
y
x
y
x
x
t
2
sin
.
2
cos
.
2
2
)
(
' xy
y
x
y
x
y
t
t 2
cos
.
2
sin
.
2
)
( xy
x
y
'
y
'
x 24
13. 13
-40
-20
0
20
40
60
80
100
0 40 80 120 160
stress(MPa)
Teta
max
min
max
min
t
t
45
o
45
o
Max. and min normal stresses occur when t=0
Max. and min. values and t occur 90o apart
tmax at an angle halfway between max and min
and t change in the form of sine wave, with a period of 180o
25
Example
-15
-10
-5
0
5
10
15
0 40 80 120 160
stress(MPa)
Teta
max
min
max
min
t
t
45
o
26
14. 14
Example
-40
-20
0
20
40
60
0 60 120 180
stress(MPa)
Teta
max
min
max
min
t
t
45
o
25 MPa
x=50 MPa
27
Principal planes and stresses
Principle Plane: plane on which no shear stress
acts (shear stress=0), plane of maximum and
minimum normal stresses.
Normal stresses acting on principle planes are
called principle stresses.
For 2D case: max=1 and min=2
For 3D case: max=1 and min=3
The direction of principal stresses are the
principal directions and shown as 1, 2, and 3.
28
15. 15
Principal planes and stresses
Since there is no shear stresses on the principal
plane;
2
2
2
tan
0
2
cos
.
2
sin
2
)
(
1
2
1
'
'
t
t
t
n
and
root
two
with
y
x
xy
xy
x
y
y
x
29
Principal planes and stresses
o
o
o
o
y
x
xy
and 05
.
172
90
5
.
82
5
.
82
90
95
.
7
95
.
7
15
2
70
20
10
80
)
10
.
2
(
2
2
tan
2
,
1
2
,
1
t
x=80 MPa
y=10 MPa
txy=-10 MPa
o
o
max 180
135
between
o
2
o
1 5
.
82
and
05
.
172
30
16. 16
Principal planes and stresses
y
x
xy
t
2
2
tan
2
y
x
2
xy
xy
)
2
(
2
sin
t
t
2
y
x
2
xy
y
x
)
2
(
2
)
(
2
cos
t
2
y
x
2
xy
xy
xy
2
y
x
2
xy
y
x
y
x
y
x
x
)
2
(
.
)
2
(
2
)
(
.
2
2
)
(
'
t
t
t
t
2
y
x
2
xy
y
x
2
,
1
min
max, )
2
(
2
t
2
31
Maximum and minimum shear
stresses
To find tmax,min
0
2
sin
.
.
2
)
(
0
d
d
xy
y
x
'
y
'
x
t
t
xy
y
x
s
t
2
2
tan
2
y
x
2
xy
min
max,
s
'
y
'
x )
2
(
)
(
t
t
t
max
min
max
2
t
32
17. 17
Example
x=80 MPa
y=10 MPa
txy=-10 MPa
2
y
x
2
xy
y
x
2
,
1
min
max, )
2
(
2
t
2
)
2
10
80
(
)
10
(
2
10
80 2
2
,
1
4
.
36
45
2
,
1
MPa
4
.
81
4
.
36
45
1
MPa
6
.
8
4
.
36
45
2
MPa
4
.
36
min
max,
t
33
Strain at a point
x
u
dx
dx
dx
x
u
dx
exx
y
v
eyy
z
w
ezz
z
e
y
e
x
e
w
z
e
y
e
x
e
v
z
e
y
e
x
e
u
zz
zy
zx
yz
yy
yx
xz
xy
xx
j
ij
i x
e
u where i free suffix and jdummy suffix (x,y,z)
34
18. 18
Strain at a point
z
w
y
w
x
w
z
v
y
v
x
v
z
u
y
u
x
u
e
e
e
e
e
e
e
e
e
e
zz
zy
zx
yz
yy
yx
xz
xy
xx
ij
exy=eyxPure shear
exy=-eyxpure rotation
ij
ij
ji
ij
ji
ij
ij )
e
e
(
2
1
)
e
e
(
2
1
e
xy
i
j
j
i
ij )
x
v
y
u
(
2
1
)
x
u
x
u
(
2
1
xy
i
j
j
i
ij )
x
v
y
u
(
2
1
)
x
u
x
u
(
2
1
Displacement
tensor
35
Strain at a point
x
v
y
u
xy
x
w
z
u
xz
z
v
y
w
yz
ij
ij 2
z
w
)
y
w
z
v
(
2
1
)
x
w
z
u
(
2
1
)
y
w
z
v
(
2
1
y
v
)
x
v
y
u
(
2
1
)
x
w
z
u
(
2
1
)
x
v
y
u
(
2
1
x
u
zz
zy
zx
yz
yy
yx
xz
xy
xx
ij
0
)
y
w
z
v
(
2
1
)
x
w
z
u
(
2
1
)
y
w
z
v
(
2
1
0
)
x
v
y
u
(
2
1
)
x
w
z
u
(
2
1
)
x
v
y
u
(
2
1
0
zz
zy
zx
yz
yy
yx
xz
xy
xx
ij
36
19. 19
Poissons’ ratio
x
x E
E
v
v
E x
x
z
y
x
x
Where v Poisson’s Ratio
Perfectly isotropic material v=0.25
For most metals v~0.33
x
y
v
37
Uniaxial tension or compression.
Constitutive Equations
)
(
v
E
1
z
y
x
x
)
(
v
E
1
x
z
y
y
)
(
v
E
1
y
x
z
z
xy
xy G
t
yz
yz G
t
xz
xz G
t
)
m
z
y
x
z
y
x
E
v
E
v
3
2
1
2
1
38
20. 20
Bulk and Shear Modulus
Stress-strain relation for isotropic material
involves three constants, E, G, and v.
Bulk Modulus (K)
1
P
strain
volume
pressure
c
hydrostati
K m
)
v
2
1
(
3
E
K
Shear
Modulus (G): )
v
1
(
2
E
G
39
Constitutive Equations
)
(
E
v
E
)
v
1
(
E
v
E
v
E
v
E
v
E
z
y
x
x
x
x
z
y
x
x
ij
kk
ij
ij
E
v
)
E
v
1
(
)
1
)(
(
E
v
E
)
v
1
(
zz
yy
xx
xx
xx
i=j=x
)
(
v
E
1
zz
yy
xx
xx
)
0
)(
(
E
v
E
)
v
1
(
2
kk
xy
xy
xy
t
i=x and j=y
)
v
1
(
2
E
G
)
v
1
(
2
E
E
)
v
1
(
2
xx
xy
xy
xy
t
t
40
21. 21
Constitutive Equations
)
(
)
v
2
1
(
E
)
( z
y
x
z
y
x
)
(
E
v
E
)
v
1
(
z
y
x
x
x
)
(
)
v
2
1
)(
v
1
(
vE
)
v
1
(
E
z
y
x
x
x
ij
kk
ij
ij
)
v
2
1
)(
v
1
(
vE
)
v
1
(
E
t
tan
cons
s
'
Lame
)
v
2
1
)(
v
1
(
vE
ij
kk
ij
ij
)
v
1
(
E
x
z
y
x
x
x G
2
)
1
)(
(
)
v
1
(
E
'
G
2
'
v
1
E
' ij
ij
ij
kk
kk
ii K
3
v
2
1
E
41
Constitutive Equations
kl
ijkl
ij S
kl
ijkl
ij C
where Sijkl is the compliance and Cijkl stiffness tensors.
S and C are fourth rank tensors with 81 (34) constants.
12
31
23
33
22
11
12
13
23
33
22
11
G
0
0
0
0
0
0
G
0
0
0
0
0
0
G
0
0
0
0
0
0
)
v
1
)(
v
2
1
(
)
v
1
(
E
)
v
1
)(
v
2
1
(
)
v
1
(
E
)
v
1
)(
v
2
1
(
)
v
1
(
E
0
0
0
)
v
1
)(
v
2
1
(
)
v
1
(
E
)
v
1
)(
v
2
1
(
)
v
1
(
E
)
v
1
)(
v
2
1
(
)
v
1
(
E
0
0
0
)
v
1
)(
v
2
1
(
Ev
)
v
1
)(
v
2
1
(
Ev
)
v
1
)(
v
2
1
(
)
v
1
(
E
t
t
t
42
