Percent elongation in a standard tension test specimen
Reduction in area that occurs in standard tension test
specimen in case of ductile materials
Percent reduction in area that occurs in standard
tension test specimen in case of ductile materials
For standard tensile test specimen subject to various
loads
The standard gauge length of tensile test specimen
The volume of material of tensile test specimen
remains constant during the plastic range which is
verified by experiments and is given by
Therefore the true strain from Eqs. (1-7) and (1-15)
The true strain at rupture, which is also known as the
true fracture strain or ductility
"
100
¼
l
f
À l
0
l
0
ð100Þð1-11Þ
A
r
¼
A
0
À A
f
A
0
ð1-12Þ
A
r100
¼
A
0
À A
f
A
0
ð100Þð1-13Þ
Refer to Fig. 1-3.
FIGURE 1-3 A standard tensile specimen subject to various
loads.
l
0
¼ 6:56
ffiffiffi
a
p
ð1-14Þ
A
0
l
0
¼ A
f
l
f
or
l
f
l
0
¼
A
0
A
f
¼
d
2
0
d
2
f
ð1-15Þ
"
tru
¼ ln
A
0
A
f
¼ ln
l
f
l
0
¼ 2ln
d
0
d
f
ð1-16Þ
where d
f
¼ minimum diameter in the gauge length
l
f
of specimen under load at that
instant,
A
r
¼ minimum area of cross section of
specimen under load at that instant.
"
ftru
¼ ln
1
1 À A
r
ð1-17Þ
where A
f
is the area of cross-section of specimen at
fracture.
Particular Formula
PROPERTIES OF ENGINEERING MATERIALS
1.5
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PROPERTIES OF ENGINEERING MATERIALS
From Eqs. (1-9) and (1-16)
Substituting Eq. (1-18) in Eq. (1-4) and using Eq. (1-3)
the true stress
From experimental results plotting true-stress versus
true-strain, it was found that the equation for plastic
stress-strain line, which is also called the strain-
strengthening equation, the true stress is given by
The load at any point along the stress-strain curve
(Fig 1-1)
The load-strain relation from Eqs. (1-20) and (1-2)
Differentiating Eq. (1-22) and equating the results to
zero yields the true strain equals to the strain harden-
ing exponent which is the instability point
The stress on the specimen which causes a given
amount of cold work W
The approximate yield strength of the previously
cold-worked specimen
The approximate yield strength since A
0
w
¼ A
w
By substituting Eq. (1-26) into Eq. (1-24)
The tensile strength of a cold worked material
The percent cold work associated with the deforma-
tion of the specimen from A
0
to A
0
w
Refer to Table 1-1A for values of "
ftru
of steel and
aluminum.
A
0
A
f
¼ 1 þ" or A
f
¼
A
0
1 þ "
ð1-18Þ
tru
¼ ð1 þ"Þ¼e
"
tru
ð1-19Þ
tru
¼
0
"
n
trup
ð1-20Þ
where
0
¼ strength coefficient,
n ¼ strain hardening or strain
strengthening exponent,
"
trup
¼ true plastic strain.
Refer to Table 1-1A for
0
and n values for steels and
other materials.
F ¼
s
A
0
ð1-21Þ
F ¼
0
A
0
"
n
tru
e
À"
tru
ð1-22Þ
"
u
¼ n ð1-23Þ
w
¼
0
ð"
w
Þ
n
¼
F
w
A
w
ð1-24Þ
where A
w
¼ actual cross-sectional area of the
specimen,
F
w
¼ applied load.
ð
sy
Þ
w
¼
F
w
A
0
w
ð1-25Þ
where A
w
¼ A
0
w
¼ the increased cross-sectional
area of specimen because of the elastic recovery
that occurs when the load is removed.
ð
sy
Þ
w
¼
F
w
A
0
w
%
w
ð1-26Þ
ð
sy
Þ
w
¼
0
ð"
w
Þ
n
ð1-27Þ
ð
su
Þ
w
¼
F
u
A
0
w
ð1-28Þ
where A
w
¼ A
u
, F
u
¼ A
0
ð
su
Þ
0
,
su
¼ tensile strength of the original
non-cold worked specimen,
A
0
¼ original area of the specimen.
W ¼
A
0
À A
0
w
A
0
ð100Þ or w ¼
A
0
À A
0
w
A
0
ð1-29Þ
where w ¼
W
100
Particular Formula
1.6 CHAPTER ONE
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PROPERTIES OF ENGINEERING MATERIALS
For standard tensile specimen at stages of loading A
0
w
is given by equation
Expression for ð
su
Þ
w
after substituting Eq. (1-28)
Eq. (1-31) can also be expressed as
The modulus of toughness
HARDNESS
The Vicker’s hardness number (H
V
) or the diamond
pyramid hardness number (H
p
)
The Knoop hardness number
The Meyer hardness number, H
M
The Brinell hardness number H
B
The Meyer’s strain hardening equation for a given
diameter of ball
A
0
w
¼ A
0
ð1 À wÞð1-30Þ
ð
su
Þ
w
¼
ð
su
Þ
0
1 À w
ð1-31Þ
ð
su
Þ
w
¼ð
su
Þ
0
e
"
tru
ð1-32Þ
Valid for A
w
A
u
or "
w
"
u
.
T
m
¼
ð
"
r
0
s
d" ð1-33aÞ
%
s
þ
su
2
"
r
ð1-34bÞ
where "
r
¼ "
u
¼ strain associated with incipient
fracture.
H
V
¼
2F sinð=2Þ
d
2
¼
1:8544F
d
2
ð1-35Þ
where F ¼ load applied, kgf,
¼ face angle of the pyramid, 1368,
d ¼ diagonal of the indentation, mm,
H
V
in kgf/mm
2
.
H
K
¼
F
0:07028d
2
ð1-36Þ
where d ¼ length of long diagonal of the projected
area of the indentation, mm,
F ¼ load applied, kgf,
0:07028 ¼ a constant which depends on one of
angles between the intersections of the
four faces of a special rhombic-based
pyramid industrial diamond indenter
172.58 and the other angle is 1308,
H
K
in kgf/mm
2
.
H
M
¼
4F
d
2
=4
ð1-37Þ
where F ¼ applied load, kgf,
d ¼ diameter of indentation, mm,
H
M
in kgf/mm
2
.
H
B
¼
2F
D½D À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D
2
À d
2
p
ð1-38Þ
where F in kgf, d and D in mm, H
B
in kgf/mm
2
.
F ¼ Ad
p
ð1-39Þ
where F ¼ applied load on a spherical indenter,
kgf,
d ¼ diameter of indentation, mm,
p ¼ Meyer strain-hardening exponent.
Particular Formula
PROPERTIES OF ENGINEERING MATERIALS
1.7
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PROPERTIES OF ENGINEERING MATERIALS
The relation between the diameter of indentation d
and the load F according to Datsko
1;2
The relation between Meyer strain-hardening expo-
nent p in Eq. (1-39) and the strain-hardening exponent
n in the tensile stress-strain Eq. ¼
0
"
n
The ratio of the tensile strength (
su
) of a material to
its Brinell hardness number (H
B
) as per experimental
results conducted by Datsko
1;2
For the plot of ratio of (
su
=H
B
Þ¼K
B
against the
strain-strengthening exponent n
Ã
(1)
The relationship between the Brinell hardness number
H
B
and Rockwell C number R
C
The relationship between the Brinell hardness number
H
B
and Rockwell B number R
B
F ¼ 18:8d
2:53
ð1-40Þ
p À 2 ¼ n ð1-41Þ
where p ¼ 2.25 for both annealed pure aluminum
and annealed 1020 steel,
p ¼ 2 for low work hardening materials such
as pH stainless steels and all cold rolled
metals,
p ¼ 2.53 experimentally determined value of
70-30 brass.
K
B
¼
su
H
B
ð1-42Þ
Refer to Fig. 1-4 for K
B
vs n for various ratios of
ðd=DÞ.
FIGURE 1-4 Ratio of ð
su
=H
B
Þ¼K
B
vs strain strengthen-
ing exponent n.
R
C
¼ 88H
0:162
B
À 192 ð1-43Þ
R
B
¼
H
B
À 47
0:0074H
B
þ 0:154
ð1-44Þ
Particular Formula
Ã
Courtesy: Datsko, J., Materials in Design and Manufacture, J. Datsko Consultants, Ann Arbor, Michigan, 1978, and Standard
Handbook of Machine Design, McGraw-Hill Book Company, New York, 1996.
1.8 CHAPTER ONE
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PROPERTIES OF ENGINEERING MATERIALS
The approximate relationship between ultimate tensile
strength and Brinell hardness number of carbon and
alloy steels which can be applied to steels with a Brinell
hardness number between 200H
B
and 350H
B
only
1;2
The relationship between the minimum ultimate
strength and the Brinell hardness number for steels
as per ASTM
The relationship between the minimum ultimate
strength and the Brinell hardness number for cast
iron as per ASTM
The relationship between the minimum ultimate
strength and the Brinell hardness number as per
SAE minimum strength
In case of stochastic results the relation between H
B
and
sut
for steel based on Eqs. (1-45a) and (1-45b)
In case of stochastic results the relation between
H
B
and
sut
for cast iron based on Eqs. (1-47a) and
(1-47b)
Relationships between hardness number and tensile
strength of steel in SI and US Customary units [7]
The approximate relationship between ultimate
shear stress and ultimate tensile strength for various
materials
The tensile yield strength of stress-relieved (not cold-
worked) steels according to Datsko
1;2
The equation for tensile yield strength of stress-
relieved (not cold-worked) steels in terms of Brinell
hardness number H
B
according to Datsko (2)
The approximate relationship between shear yield
strength ð
sy
Þ and yield strength (tensile)
sy
sut
¼ 3:45H
B
MPa SI ð1-45aÞ
¼ 500H
B
psi USCS ð1-45bÞ
sut
¼ 3:10H
B
MPa SI ð1-46aÞ
¼ 450H
B
psi USCS ð1-46bÞ
sut
¼ 1:58H
B
À 86:2MPa SI ð1-47aÞ
¼ 230H
B
À 12500 psi USCS ð1-47bÞ
sut
¼ 2:60H
B
À 110 MPa SI ð1-48aÞ
¼ 237:5H
B
À 16000 psi USCS ð1-48bÞ
sut
¼ð3:45; 0:152ÞH
B
MPa SI ð1-49aÞ
¼ð500; 22ÞH
B
psi USCS ð1-49bÞ
sut
¼ 1:58H
B
À 62 þð0; 10:3Þ MPa SI ð1-50aÞ
¼ 230H
B
À 9000 þð0; 1500Þ psi
USCS ð1-50bÞ
Refer to Fig. 1.5.
su
¼ 0:82
sut
for wrought steel ð1-51aÞ
su
¼ 0:90
sut
for malleable iron ð1-51bÞ
su
¼ 1:30
sut
for cast iron ð1-51cÞ
su
¼ 0:90
sut
for copper and copper alloy ð1-51dÞ
su
¼ 0:65
sut
for aluminum and aluminum alloys
ð1-51eÞ
sy
¼ð0:072
sut
À 205Þ MPa SI ð1-52aÞ
¼ 1:05
sut
À 30 kpi USCS ð1-52bÞ
sy
¼ð3:62H
B
À 205Þ MPa SI ð1-53aÞ
¼ 525H
B
À 30 kpi USCS ð1-53bÞ
sy
¼ 0:55
sy
for aluminum and aluminum alloys
ð1-54aÞ
sy
¼ 0:58
sy
for wrought steel ð1-54bÞ
Particular Formula
PROPERTIES OF ENGINEERING MATERIALS
1.9
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PROPERTIES OF ENGINEERING MATERIALS
The approximate relationship between endurance
limit (also called fatigue limit) for reversed bending
polished specimen based on 50 percent survival rate
and ultimate strength for nonferrous and ferrous
materials
FIGURE 1-5 Conversion of hardness number to ultimate
tensile strength of steel
sut
, MPa (kpsi). (Technical Editor
Speaks, courtesy of International Nickel Co., Inc., 1943.)
For students’ use
0
sfb
¼ 0:50
sut
for wrought steel having
sut
< 1380 MPa ð200 kpsiÞð1-55Þ
0
sfb
¼ 690 MPa for wrought steel having
sut
> 1380 MPa ð1-56aÞ
0
sfb
¼ 100 kpsi for wrought steel having
sut
> 200 kpsi USCS ð1-56bÞ
For practicing engineers’ use
0
sfb
¼ 0:35
sut
for wrought steel having
sut
< 1380 MPa ð200 kpsiÞð1-57Þ
0
sfb
¼ 550 MPa for wrought steel having
sut
> 1380 MPa SI ð1-58aÞ
0
sfb
¼ 80 kpsi for wrought steel having
sut
> 200 kpsi USCS ð1-58bÞ
0
sfb
¼ 0:45
sut
for cast iron and cast steel when
sut
600 MPa ð88 kpsiÞð1-59aÞ
0
sfb
¼ 275 MPa for cast iron and cast steel when
sut
> 600 MPa SI ð1-60aÞ
0
sfb
¼ 40 kpsi for cast iron and cast steel when
sut
> 88 kpsi USCS ð1-60bÞ
0
sfb
¼ 0:45
sut
for copper-based alloys
and nickel-based alloys ð1-61Þ
0
sfb
¼ 0:36
sut
for wrought aluminum alloys up toa
tensile strength of 275 MPa (40 kpsi)
based on 5 Â 10
8
cycle life ð1-62Þ
0
sfb
¼ 0:16
sut
for cast aluminum alloys
up to tensile strength of
300 MPa ð50 kpsiÞ based
on 5 Â10
8
cycle life ð1-63Þ
0
sfb
¼ 0:38
sut
for magnesium casting alloys
and magnesium wrought alloys
based on 10
6
cyclic life ð1-64Þ
Particular Formula
1.10 CHAPTER ONE
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PROPERTIES OF ENGINEERING MATERIALS
The relationship between the endurance limit for
reversed axial loading of a polished, unnotched speci-
men and the reversed bending for steel specimens
The relationship between the torsional endurance
limit and the reversed bending for reversed torsional
tested polished unnotched specimens for various
materials
For additional information or data on properties of
engineering materials
WOOD
Specific gravity, G
m
, of wood at a given moisture
condition, m, is given by
The weight density of wood, D (unit weight) at any
given moisture content
Equation for converting of weight density D
1
from
one moisture condition to another moisture condition
D
2
For typical properties of wood of clear material as per
ASTM D 143
0
sfa
¼ 0:85
0
sfb
¼ 0:43
sut
ð1-65Þ
0
sf
¼ 0:58
0
sfb
¼ 0:29
sut
for steel ð1-66aÞ
0
sf
% 0:8
0
sfb
% 0:32
sut
for cast iron ð1-66bÞ
0
sf
% 0:48
0
sfb
% 0:22
sut
for copper ð1-66cÞ
Refer to Tables 1-1 to 1-48
G
m
¼
W
0
W
m
ð1-67Þ
where W
0
¼ weight of the ovendry wood; N ðlbfÞ;
W
m
¼ weight of water displaced by the
sample at the given moisture
condition, N (lbf ).
W ¼
weight of ovendry wood and the contained water
volume of the piece at the same moisture content
ð1-68Þ
D
2
¼ D
1
100 þ M
2
100 þ M
1
þ 0:0135D
1
ðM
2
À M
1
Þ
ð1-69Þ
where D
1
¼ known weight density for same
moisture condition M
1
,kN/m
2
(lbf/ft
2
),
D
2
¼ desired weight density at a moisture
condition M
2
,kN/m
2
(lbf/ft
2
). M
1
and
M
2
are expressed in percent.
Refer to Table 1-47.
Particular Formula
PROPERTIES OF ENGINEERING MATERIALS
1.11
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PROPERTIES OF ENGINEERING MATERIALS
TABLE 1-1
Hardness conversion (approximate)
Brinell
29.42 kN (3000kgf ) load
Rockwell hardness number
10 mm ball Vickers A scale B scale C scale 15-N scale Shore Tensile strength,
sut
or Firth 0.588 kN 0.98 kN 1.47 kN 0.147 kN scleroscope approximate
Diameter Hardness hardness (60 kgf ) (100 kgf ) (150 kgf ) (15 kgf ) hardness
(mm) number number load load load load number MPa kpsi
2.25 745 840 84 65 92 91 2570 373
2.30 712 783 83 64 92 87 2455 356
2.35 682 737 82 62 91 84 2350 341
2.40 653 697 81 60 90 81 2275 330
2.45 627 667 81 59 90 79 2227 323
2.50 601 640 80 58 89 77 2192 318
2.55 578 615 79 57 88 75 2124 309
2.60 555 591 78 55 88 73 2020 293
2.65 534 569 78 54 87 71 1924 279
2.70 514 547 77 52 87 70 1834 266
2.75 495 528 76 51 86 68 1750 254
2.80 477 508 76 50 85 66 1675 243
2.85 461 491 75 49 85 65 1620 235
2.90 444 472 74 47 84 63 1532 222
2.95 429 455 73 46 83 61 1482 215
3.00 415 440 73 45 83 59 1434 208
3.05 401 425 72 43 82 58 1380 200
3.10 388 410 71 42 81 56 1338 194
3.15 375 396 71 40 81 54 1296- 188
3.20 363 383 70 39 80 52 1255 182
3.25 352 372 69 110 38 79 51 1214 176
3.30 341 360 69 109 37 79 50 1172 170
3.35 331 350 68 109 36 78 48 1145 166
3.40 321 339 68 108 34 77 47 1103 160
3.45 311 328 67 108 33 77 46 1069 155
3.50 302 319 66 107 32 76 45 1042 151
3 55 293 309 66 106 31 76 43 1010 146
3.60 285 301 65 106 30 75 42 983 142
3.65 277 292 65 105 29 74 41 955 138
3.70 269 284 64 104 28 74 40 928 134
3.75 262 276 64 103 27 73 39 904 131
3.80 255 269 63 102 25 73 38 875 127
3.85 248 261 63 101 24 72 37 855 124
3.90 241 253 62 100 23 71 36 832 120
3.95 235 247 61 99 22 70 35 810 117
4.00 229 241 61 98 21 70 34 790 114
4.05 223 234 97 19 770 111
4.10 217 228 96 18 33 748 108
4.15 212 222 96 16 32 730 106
4.20 207 218 95 15 31 714 103
4.25 201 212 94 14 690 100
4.30 197 207 93 13 30 680 98
4.35 192 202 92 12 29 662 96
4.40 187 196 91 10 645 93
1.12 CHAPTER ONE
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PROPERTIES OF ENGINEERING MATERIALS
TABLE 1-1
Hardness conversion (approximate) (Cont.)
Brinell
29.42 kN (3000kgf ) load
Rockwell hardness number
10 mm ball Vickers A scale B scale C scale 15-N scale Shore Tensile strength,
sut
or Firth 0.588 kN 0.98 kN 1.47 kN 0.147 kN scleroscope approximate
Diameter Hardness hardness (60 kgf ) (100 kgf) (150 kgf ) (15 kgf ) hardness
(mm) number number load load load load number MPa kpsi
4.45 183 192 90 9 28 631 91
4.50 179 188 89 8 27 617 89
4.55 174 182 88 7 600 87
4.60 170 178 87 5 26 585 85
4.65 167 175 86 4 576 83
470 163 171 85 3 25 562 81
4.80 156 163 83 1 24 538 78
4.90 149 156 81 23 514 74
5.00 143 150 79 22 493 71
5.10 137 143 76 21 472 68
5.20 131 137 74 451 65
5.30 126 132 72 20 435 63
5.40 121 127 70 19 417 60
5.50 116 122 68 18 400 58
5.60 111 117 65 17 383 55
PROPERTIES OF ENGINEERING MATERIALS
1.13
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PROPERTIES OF ENGINEERING MATERIALS
TABLE 1-1A
Mechanical properties of some metallic materials
Brinell
hardness Process/
Ultimate
strength,
sut
Yield
strength,
sy
Stress at
fracture,
f
Reduction
in area, A
f
True strain
at fracture
Strain hard-
ing exponent
Strength
coefficient,
0
Material H
B
Condition MPa kpsi MPa kpsi MPa kpsi % "
f
n MPa kpsi
Steel
RQC-100
a
290 HR
b
Plate 931 135 883 128 1331 193 67 1.02 0.06 1172 170
1005-1009 125 CD
c
Sheet 414 60 400 58 841 122 64 1.02 0.05 524 76
1005-1009 90 HR Sheet 345 50 262 38 848 123 80 1.60 0.16 531 77
1015 80 Normalized 414 60 228 33 724 105 68 1.14 0.26
1020
d
108 HR Plate 441 64 262 38 710 103 62 0.96 0.19 738 107
1045
e
225 Q and T
f
724 105 634 92 1227 178 65 1.04 0.13 1145 166
1045
e
410 Q and T 1448 210 1365 198 1862 270 51 0.72 0.08 2082 302
5160 430 Q and T 1669 242 1531 222 1931 280 42 0.87 0.06 2124 308
9262 260 Annealed 924 134 455 66 1041 151 14 0.16 0.22 1744 253
9262 410 Q and T 1565 227 1379 200 1855 269 32 0.38 0.06
950 150 HR Plate 531 77 311 48 1000 145 72 1.24 0.19 903 131
Aluminum:
2024-T351 ST, SH
g
469 68 379 55 558 81 25 0.28 0.03 455 66
2024-T4 ST and RT age
h
476 69 303 44 636 92 35 0.43 0.20 807 117
7075-T6 ST and AA
i
579 84 469 68 745 108 33 0.41 0.11 827 120
a
Tradename, Bethlehem steel Corp. Rolled quenched and tempered carbon steel. Used in structural, heavy applications machinery.
b
Hot-rolled.
c
cold-rolled.
d
low carbon, common machining
steels.
e
Bar stock, medium carbon high-strength machining steel.
f
Quenched and tempered.
g
Solution treated, strain hardened.
h
Solution treated and RT age.
i
Solution treated and artificially aged.
Source: SAE j1099, Technical Report on Fatigue properties, 1975.
1.14
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PROPERTIES OF ENGINEERING MATERIALS
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