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Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Lecture Note for Solid Mechanics- Stress-Strain Relationships and Material Properties -
Prof. Sung Ho YoonDepartment of Mechanical EngineeringKumoh National Institute of Technology
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Text book : Mechanics of Materials, 6th ed.,
W.F. Riley, L.D. Sturges, and D.H. Morris, 2007.
Prerequisite : Knowledge of Statics, Basic Physics, Mathematics, etc.
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Stress-strain diagrams
It is necessary to relate loads and temperature changes on the structure to the deformations produced by the loads and temperature changes
Relationship between load and deformation depends on dimension of member and type of material
Stress-Strain Relations and Material Properties
Dependent of cross sectional area & length
Independent of cross sectional area
Independent of length cross sectional area and length
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
The tensile test
Data for stress-strain curves are obtained by applying an axial load to a specimen and measuring the load and deformation simultaneously
Stress is obtained by dividing the load by the initial area (nominal stress) or the actual area (true stress)
Extensometer
Loadcell
Grip or fixture
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Strain measurement
Instruments for measuring strain : strain gages or extensometers Nominal strain is computed on the initial length of the specimen True strain is computed on the actual length of the specimen
Example of stress-strain curves
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Modulus of elasticity (Young’s modulus)
The slope of straight line portion of stress-strain curve is used to measure the stiffness of material Young’s modulus (Modulus of elasticity) : Shear modulus (Modulus of rigidity) :
E
G
Proportional limit (A)The maximum stress for which stress and strain are proportional
Young’s modulus (E)The slope of straight line portion of stress-strain curve
Tangent modulus (Et)The slope of stress-strain curve at a particular point beyond the proportional limit
Secant modulus (Es)The slope of stress-strain curve at any point
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Elastic limit Elastic if strain resulting from loading disappears when the load is removed Elastic limit (point D) is the maximum stress for which material acts elastically Line BC is parallel to line OA Precise value is difficult to obtain
Work hardening (or strain hardening)
If loading again, stress-strain diagram follows the unloading curve until it reaches a stress a little less than the maximum stress during initial loading
The proportional limit (point B) is greater than that for the initial loading
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Yield point
The stress at which there is an appreciable increase in strain with no increase in stress
Low carbon steels
Yield strength
The stress that induce a specified permanent set with 0.2 percent (0.002)
Stress indicated by intersection of CB and stress-strain diagram
Ultimate strength
The maximum stress based on the original area developed in a material before rupture
The nominal stress decreases beyond the ultimate strength
The true stress continues to increase until rupture
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Elastoplastic Materials
When the stress exceeds the proportional limit, empirical equations have been proposed relating between stress and strain
Plastic strain is 16-20 times the elastic strain at the proportional limit
Poisson’s ratio
Ratio of the lateral strain to longitudinal strain
//
a
l
long
lat
GE )( 12
Ductility
Capacity for plastic deformation
Controls the amount of cold forming to which a material may be subjected
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example 4-1) A 100-kip axial load is applied to a 1x4x90-in rectangular bar. When loaded, the 4-in side measures 3.9986-in and the length has increased 0.09-in. determine Poisson’s ratio, the modulus of elasticity, and the modulus of rigidity of the material.
(Sol)
ksiEGrigidityofModulus
ksiEelasticityofModulus
ratiosPoisson
ksiAP
L
L
in
long
lat
longlong
latlat
lat
260935012
0002512
0002500100025
350001000000350
2514
100
00100090090
000350400140
00140499863
,).(
,)(
:
,.
:
...:'
..
....
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Principle of superposition
If the stresses do not exceed the proportional limit and the deformations are small, the effects of separate loadings can be added algebraically
Generalized Hooke’s law
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
dyE
dyE
dyd
dxE
dxE
dxd
xyyy
yxxx
yxz
xyy
yxx
E
E
E
1
1
Extension to the state of tri-axial stresses
yxzz
xzyy
zyxx
E
E
E
1
1
1
yxzz
xzyy
zyxx
E
E
E
)())((
)())((
)())((
1211
1211
1211
For the case of plane stress
yxzzE
)(
))((1
2110 yxz
1
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
2
0)45cos()45sin(0
sincoscossin 22
x
x
xyyxnt
)1(2
EG
zxzxzx
yzyzyz
xyxyxy
EG
EG
EG
)(
)(
)(
12
12
12
1
0)45cos()45sin(2
sincoscossin2 22
x
xx
yx
yx
xyyxnt
xy
x
y
a
t
12
1
EG
GG
xx
xntnt
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example 4-2) When the machine part of alloy steel with E=210GPa and =0.3 was subjected to a biaxial stress, the measured strains were x=+1394m/m, y=-660m/m, and xy=2054m/m. Determine (a) stress components at the point (b) principal stresses and maximum stress at the point. Locate the planes on which these stresses act.
(Sol)
(a)
MPamN
E
CMPamN
E
TMPamN
E
xyxy
xyy
yxx
9165109165
1020543012
1021012
85510855
1013943066030110210
1
27610270
1066030139430110210
1
26
69
26
62
9
2
26
62
9
2
.)(.
))(().()(
)(.)(.
)())(.().(
)()(
)())(.().(
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(b)
0
51245124
3457344
62341110
91652
8552762
855276
22
3
2
1
22
221
zp
p
p
xyyxyx
pp
CMPaMPaTMPaMPa
)(..)(.
..
).(..
,
522
018555276
9165222
62345124734421
21
21
.
.).(.
).(tan
.).(.
max
p
yx
xyp
ppp
MPa
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example 4-3) When the machine part of alloy steel with E=30,000ksi and =0.3 was subjected to a biaxial stress, the measured strains were a=+650in/in, b=+475in/in, and c=-250in/in. Determine (a) stress components at the point (b) principal strains and maximum strain at the point. Locate the planes on which these stresses act (c) principal stresses and maximum stress at the point. Locate the planes on which these stresses act.
(Sol) (a)
ksiksiE
CksiksiE
TksiksiE
xyxy
xyy
yxx
xyxyyxb
ycbxa
356356105503012
0003012
813181311065030250301
000301
961896181025030650301
000301
550475
250475650
6
622
622
22
..))(().(
,)(
)(..)())(.().(
,
)(..)())(.().(
,
cossinsincos
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(b)
711543312
611110250650
5502
1055
41711
3274527200
72745272004527200
222
21
3
2
1
22
21
..
.)(
tan
.)(
.
..
,
max
pp
yx
xyp
ppp
yxzp
p
p
xyyxyxpp
ksiE 171210810543012
0003012
6 .))(.().(
,)( maxmax
(c)
06035993
104727304327301
000301
72074220
104327304727301
000301
3
62
1222
62
2121
zp
ppp
ppp
Cksiksi
ETksiksi
E
)(..
).(...
,
)(..
).(...
,
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Thermal strain
Most materials when unconstrained expand under heating and contract under cooling
Coefficient of thermal expansion : thermal strain due to one degree change in temperature
TT
Total strains
Strains due to temperature change and strains due to applied loads are independent
Most materials expand uniformly in all directions when heated
Neither the shape of the body nor the shear stresses and shear strains are affected by temperature changes
TETtotal
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) The aluminum [E=70GPa and =22.5(10-6)/oC ] block rests on a smooth, horizontal surface. When the body is subjected to a temperature change of ∆T=20 oC. Determine (a) The thermal strains (b) The deformations in the coordinate directions (c) The shearing strain
(Sol) (a)
mmmm
TTzTyTxT
/450/)10(450
)20)(10(5.226
6
Thermal strain :
(b)
TTtotal
),,( TzTyTx
),,( zyx
)( xy
Since the block is not constrained, there are no applied forces and there are no stresses. The load portion of the strain is zero, .
mmL
mmL
mmL
Tzz
Tyy
Txx
0338.0)75)(10(450
0225.0)50)(10(450
0450.0)100)(10(450
6
6
6
(c) Since the block is not constrained, there are no stresses and the total strains the same as the thermal strain. The thermal strain is the same in all directions. The original angle do not change. That is, there are no shearing strain, . 0xy
Stress-Strain Relations and Material Properties
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) Consider a 1.5m long brass [E=100GPa and =17.6(10-6)/oC ] strap. Determine (a) how much the strap would have to be heated to lengthen the strap enough to insert the pin through the eyelets. (b) stress existing in the strap after the strap cooed back to room temperature.
(Sol) (a) TEL
Total strain :
If no stress is applied to the strap and the temperature is raised to stretch the strap by 1mm (0.001m)
CT
TLTE
o937
001051106170 6
.
.).())(.(
(b)
)(.)(.
.).()(
TMPamN
LTE
766106766
001051010100
26
9
If the strap stays stretched by 1mm after it cools down
Stress-Strain Relations and Material Properties