Mechanics of Materials 1
Chapter 2
Basics of Materials
and Mechanics
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Mechanics of Materials 2
2.1 Stress
2.1.1 Definition of stress
- P
S
ΔSΔS
φ
P- P
- P
P
P
P2
PP1
内力
V
V1
V2
hypothetical internal
surface S Body is in
equilibrium
Sum of the internal
forces on S
=Applied force P
Stress Vector
Quantity to express the
intensity of internal
force
SS
Pp
0lim (1)
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Mechanics of Materials 3
Unit of Stress ・・・・Pa(=N/m2), kgf/m2, psi (=lb/inch2),etc
Stress depends not only the internal force (vector) but
also the direction of the surface considered.
dS
dPp
dS
dPp
2
1
sin
cos
Normal component:Normal Stress
Inplane component:Shear Stress
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Mechanics of Materials 4
Positive stresses are defined as shown above. In the other sides
of the cube where the outward normals are opposite to the
coordinate axes, the directions of positive stresses are opposite.
Definition of Stresses in the Cartician
Coordinate System
x
y
z
P
B
A
O
C dx
dy
dz x xyxz y
yx
yz
z
zx
zy
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Mechanics of Materials 5
Stress in an arbitrary surface
Equilibrium of the force for infinitesimal tetrahedron
Area of ABC=dS
Outward normal of ABC : =(l,m,n)
l, m, n are directional cosines
OA=dx, OB=dy, OC=dz
OBC=dydz/2=dS・l,
OCA=dzdx/2=dS・m,
OAB=dxdy/2=dS・n
n
x
y
z
x
y
z
O A
B
C pdS
-pxldS
-pymdS
-pzndS
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Mechanics of Materials 6
zzyzxz
xzyyxy
xzxyxx
ZYX
,,
,,
,,
,,
p
p
p
p
Stress vectors
Equilibrium of the force when no body force exists.
0 ndSmdSldSdS zyx pppp
ndSmdSldSdS
Z
Y
X
z
zy
zx
yz
y
yx
xz
xy
x
x
y
z
OA
B
C pdS
-pxldS
-pymdS
-pzndS
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Mechanics of Materials 7
n
m
l
Z
Y
X
zyzxz
zyyxy
zxyxx
Then
Stress vector of an arbitrary surface
→Nine components are necessary to express the stress state.
Note) From the equilibrium of an infinitesimal cube in moment,
zy=yz , xz=zx, yx=xy
Six components are practically independent.
yxxyxzzxzyyzzyx ,,,,,,,,pp
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Mechanics of Materials 8
2.1.2 Transformation of Coordinate system
Let (x, y, z) and (x', y', z') be two Cartesian coordinate systems
with a common origin. The cosines of the angles between the six
coordinate axes are represented in the following tabular form:
x y z
x’ a11 a12 a13
y’ a21 a22 a23
z’ a31 a32 a33
x
y
z
O
p
i’
j
i
x’ y’
z’ k
k’
j’
Normal to the
x’ axis
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Mechanics of Materials 9
333231
232221
131211
332313
322212
312111
'''''
'''''
'''''
aaa
aaa
aaa
aaa
aaa
aaa
zyzxz
zyyxy
zxyxx
zzyzx
yzyyx
xzxyx
The stresses referring to (x', y', z') coordinate
system can be derived from the stresses
referring to (x, y, z) coordinate system .
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Mechanics of Materials 10
2.1.3 Principal stress
The values of stresses at a point depend on the plane where the
stresses are considered. The direction of the plane where the normal
stress is extremum is called principal axis and the normal stress is
called principal stress. The principal directions and principal stresses
are obtained through the following eigenvalue analysis.
0
zyzxz
zyyxy
zxyxx
222
3
2222
1
xyzzxyyzxzyx
xyzxyzyxxzzy
zyx
J
J
J
The coefficients of the equation are called stress invarient.
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Mechanics of Materials 11
2.1.4 Maximum Shear Stress
Shear Stress becomes maximum at the surface whose unit
normal is 0,2/1,2/1,, nml
212
1
212
1
The normal stress at the surface of maximum shear stress is
where the coordinate system accords to the principal directions. 1 is
maximum principal stress and 2 is minimum principal stress
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Mechanics of Materials 12
2.1.5 Equilibrium Equation
0
0
0
Zzyx
Yzyx
Xzyx
zyzxz
zyyxy
zxyxx
x y
z
P
B
A
O
C dx
dy
dz
dxx
xx
dxx
xyxy
dxxxz
xz
dyy
yy
dyy
yxyx
dyy
yzyz
dzz
zz
dz
zzx
zx
dz
z
zyzy
From the equilibrium of the force acting on the
small rectangular parallelepiped
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Mechanics of Materials 13
A’
B’ C
D’
C’
A B
dz
D
dy
dx
x
y
z
2.2 Strain 2.2.1 Definition of strain
Intensity of Deformation
Normal strain
Shear strain
x
xxx
i
ir
AB
ABBA
''
xy
2
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Mechanics of Materials 14
Normal strain represents change of length.
When the length is increased, it is called
tensile strain. When the length is decreased,
it is called compressive strain. Normal strain
is usually positive if tensile. Strain being
defined as length/length is non-dimensional
value.
Shear strain represents change of shape. It is
defined as reduction of angle between two
coordinate axes which were normal each
other.
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Mechanics of Materials 15
Relations between the strain components and
displacements (u,v,w)
x
w
z
w
x
v
z
v
x
u
z
u
x
w
z
u
z
w
y
w
z
v
y
v
z
u
y
u
z
v
y
w
y
w
x
w
y
v
x
v
y
u
x
u
y
u
x
v
z
w
z
v
z
u
z
w
y
w
y
v
y
u
y
v
x
w
x
v
x
u
x
u
zx
yz
xy
z
y
x
222
222
222
2
1
2
1
2
1
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Mechanics of Materials 16
When u and v is infinitesimal,
x
w
z
u
z
v
y
w
y
u
x
v
z
w
y
v
x
u
zxyzxy
zyx
,,
,,
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Mechanics of Materials 17
2.2.2 Motion of elastic body
Displacement of P
dz
dy
dx
dz
dy
dx
w
v
u
zzyzzx
yzyxy
zxxyx
zxy
xz
yz
drp
2/2/
2/2/
2/2/
0
0
0
0
0
0
0 uuuu
y
u
x
v
x
w
z
u
z
v
y
wzyx
2
1,
2
1,
2
1
where
, : deformation
: rotation
O
/2
P
O'
P'/2
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Mechanics of Materials 18
2.2.3 Transformation of strain component
under change of coordinate system
Two rectangular Cartesian
coordinate systems x, y, z
and x’, y’, z’
333231
232221
131211
21
21
21
21
21
21
332313
322212
312111
'''21
''21
''21
'''21
''21
''21
'
aaa
aaa
aaa
aaa
aaa
aaa
zyzxz
zyyxy
zxyxx
zzyzx
yzyyx
xzxyx
Table of directional cosines
x y z
x’ a11 a12 a13
y’ a21 a22 a23
z’ a31 a32 a33
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Mechanics of Materials 19
zyxzyx
zyxyxz
zyxxzy
zxzx
yzyz
xyxy
xyzxyzz
xyzxyzy
xyzxyzx
zxxz
yzzy
xyyx
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2.2.4 Compatibility Equation
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Mechanics of Materials 20
2.3 Generalized Hooke’s Law and elastic constants
Hooke’s Law refers that normal strain is directly
proportional to the relating normal stress. However, in
general stress state, the whole strain components are
dependent on whole stress state as
xy
zx
yz
z
y
x
xy
zx
yz
z
y
x
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
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Mechanics of Materials 21
For three dimensional isotropic elastic body
Generalized
Hooke’s law
yxzz
xzyy
zyxx
E
E
E
1
1
1
G
G
G
xyxy
zxzx
yzyz
zyxzz
zyxyy
zyxxx
E
E
E
211
211
211
xyxy
zxzx
yzyz
G
G
G
Stresses are expressed by Strains
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Mechanics of Materials 22
One dimensional tension
Young’s modulus
strain Tensile
stress TensileE
L
T
strainTensile
direction rsein transvestraineCompressiv
Poisson’s ratio
Shear modulus
zyx
zyxK
strainVolumetric
3pressureAverage
Bulk modulus
strainShear
stressShearG
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Mechanics of Materials 23
An isotropic elastic material has only two independent
elastic constants.
The four elastic constants mentioned in the preceding
slide are not independent each other. Each elastic constant
can be expressed by the other two constants as follows.
213
12
EK
EG
When we measure the load and longitudinal and transverse
strains in a tensile test of a round bar, we can obtain
Young’s modulus and Poisson ratio simultaneously.
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Mechanics of Materials 24
2.4 Strain Energy
Let’s consider a bar of Young’s modulus E, length l and uniform
cross-sectional area A. The energy done by a force P0 under which the
bar is elongated by l .
lPPEA
ll
l
EA
xdxl
EAPdxU
ll
02
02
00
2
1
2
1
2
1
Stress and strain are used instead of
load and elongation
VE
VEV
lAl
l
A
PU
22
2
1
2
1
2
1
2
1
where V=Al is volume of the bar. Strain energy density
x
P
Strain Energy
Elongation
Exte
rna
l fo
rce
P0
x0
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Mechanics of Materials 25
Strain Energy Density Function A
222
2222
2
2112
2
1
xyyzzx
zyxzyx
xyxyyzyzzxzxzzyyxx
G
E
A
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Mechanics of Materials 26
2.5 Fundamental Principle of Elasticity
2.5.1 Principle of Virtual Work
Assume that the mechanical system is in equilibrium under
applied forces and prescribed geometrical constraints. Then, the
sum of all the virtual work, denoted by ’W, done by the external
and internal forces existing in the system in any arbitrary
infinitesimal virtual displacements satisfying the prescribed
geometrical constraints is zero.
0
S
iiV
iiV
ijij dSuT- dVuX dVeW
0
S
iiV
iiV
dSuT- dVuXdVAW
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Mechanics of Materials 27
2.5.2 Principle of minimum potential energy
The structures will be in
equilibrium when the potential
energy is minimum.
Potential energy
=U+W
S
iV
i dSugdVuGW V
i dVuAU
V
=0
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Mechanics of Materials 28
2.5.3 Saint Venant’s Principle Statically equivalent force systems which act over a
given small portion S on the surface of a body
produce approximately the same stress and
displacement at a point in the body sufficiently far
removed from the region S over which the force
system act.
P
P
P
P
P
P
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Mechanics of Materials 29
A convenient theorem to obtain concentrated forces where the
displacement is prescribed or a displacement at the point where a
concentrated force is applied.
(1) Castigliano’s first theorem
When the total energy of the system A(ui) is expressed as a
function of the prescribed displacement ui, the reaction force Pi in the
direction of ui at the point is
(2) Castigliano’s second theorem
When the total complementary energy B(Pi) of the system is
expressed as a function of the applied load Pi, the displacement ui in
the direction of Pi at the point is
2.5.4 Castigliano’s theorem
ii
u
AP
ii
P
Bu
From Arches Higdon, Edward H. Ohlsen, william B.Stiles,
William F. Riley, Mechanics of Materials, John Wiley & Sons
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Mechanics of Materials 30
E,A,l,
T
Temperature rise → Thermal expansion of the material
(Thermal strain)
Constraint of the deformation → Thermal stress
Appendix Thermal stress
Example A bar of a length l is constrained
at its both ends. Obtain the stress when the
temperature of the bar is elevated by T.
(Solution)
When the bar is not constrained, the thermal strain of the bar is
= T
When the bar does not expand owing to the constraints, the strain
should be zero. Then, stress to cause the mechanical strain of
must occur, that is,
=ET
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