Chapter 6: Behavior Of Material Under Mechanical Loads = Mechanical Properties.
• Stress and strain: • What are they and why are they used instead of load and
deformation
• Elastic behavior: • Recoverable Deformation of small magnitude
• Plastic behavior: • Permanent deformation We must consider which materials are most
resistant to permanent deformation?
• Toughness and ductility: • Defining how much energy that a material can take before failure.
How do we measure them?
• Hardness:• How we measure hardness and its relationship to material strength
Elastic means reversible!
Elastic Deformation1. Initial 2. Small load 3. Unload
F
bonds stretch
return to initial
F
Linear- elastic
Non-Linear-elastic
Plastic means permanent!
Plastic Deformation (Metals)
F
linear elastic
linear elastic
plastic
1. Initial 2. Small load 3. Unload
planes still sheared
F
elastic + plastic
bonds stretch & planes shear
plastic
Comparison of Units: SI and Engineering Common
Unit SI Eng. CommonForce Newton (N) Pound-force (lbf)
Area mm2 or m2 in2
Stress Pascal (N/m2) or MPa (106 pascals)
psi (lbf/in2) or Ksi (1000 lbf/in2)
Strain (Unitless!) mm/mm or m/m in/in
Conversion Factors
SI to Eng. Common Eng. Common to SI
Force N*4.448 = lbf Lbf*0.2248 = N
Area I mm2*645.16 = in2 in2 *1.55x10-3 = mm2
Area II m2 *1550 = in2 in2* 6.452x10-4 = m2
Stress I - a Pascal * 1.450x10-4 = psi psi * 6894.76 = Pascal
Stress I - b Pascal * 1.450x10-7 = Ksi Ksi * 6.894 x106 = Pascal
Stress II - a MPa * 145.03 = psi psi * 6.89x 10-3 = MPa
Stress II - b MPa * 1.4503 x 10-1= Ksi Ksi * 6.89 = MPa
One other conversion: 1 GPa = 103 MPa
Stress has units: N/m2 (Mpa) or lbf/in2
Engineering Stress:
• Shear stress, :
Area, A
Ft
Ft
Fs
F
F
Fs
= FsAo
• Tensile stress, :
original area before loading
Area, A
Ft
Ft
= FtAo
2f
2mNor
inlb=
we can also see the symbol ‘s’ used for engineering stress
Geometric Considerations of the Stress State
The four types of forces act either parallel or perpendicular to the planar faces of the bodies.
Stress state is a function of the orientations of the planes upon which the stresses are taken to act.
• Simple tension: cable
Note: = M/AcR here. Where M is the “Moment” Ac shaft area & R shaft radius
Common States of Stress
Ao = cross sectional area (when unloaded)
FF
o F
A
o
FsA
M
M Ao
2R
FsAc
• Torsion (a form of shear): drive shaft Ski lift (photo courtesy P.M. Anderson)
(photo courtesy P.M. Anderson)Canyon Bridge, Los Alamos, NM
o F
A
• Simple compression:
Note: compressivestructure member( < 0 here).(photo courtesy P.M. Anderson)
OTHER COMMON STRESS STATES (1)
Ao
Balanced Rock, Arches National Park
• Bi-axial tension: • Hydrostatic compression:
Pressurized tank
< 0h
(photo courtesyP.M. Anderson)
(photo courtesyP.M. Anderson)
OTHER COMMON STRESS STATES (2)
Fish under water
z > 0 > 0
• Tensile strain: • Lateral strain:
• Shear strain:
Strain is alwaysDimensionless!
Engineering Strain:
90º
90º - y
x = x/y = tan
Lo
L L
wo
Adapted from Fig. 6.1 (a) and (c), Callister 7e.
/2
L/2
Lowo
We often see the symbol ‘e’ used for engineering strain
Here: The Black Outline is Original, Green is after
application of load
Stress-Strain: Testing Uses Standardized methods developed by ASTM for Tensile Tests it is ASTM E8
• Typical tensile test machine
Adapted from Fig. 6.3, Callister 7e. (Fig. 6.3 is taken from H.W. Hayden, W.G. Moffatt, and J. Wulff, The Structure and Properties of Materials, Vol. III, Mechanical Behavior, p. 2, John Wiley and Sons, New York, 1965.)
specimenextensometer
• Typical tensile specimen (ASTM A-bar)
Adapted from Fig. 6.2,Callister 7e.
gauge length
- During Tensile Testing,Instantaneous load and displacement is measured
These load / extension graphs depend on the size of the specimen. E.g. if we carry out a tensile test on a specimen having a cross-sectional area twice that of another, you will require twice the load to produce the same elongation.
LOAD vs. EXTENSION PLOTS
The Force .vs. Displacement plot will be the same shape as the Eng. Stress vs. Eng. Strain plot
The Engineering Stress - Strain curveDivided into 2 regions
ELASTIC PLASTIC
Linear: Elastic Properties• Modulus of Elasticity, E: (also known as Young's modulus)
• Hooke's Law: = E
Linear- elastic
E
Units:E: [GPa] or [psi]: in [Mpa] or [psi]: [m/m or mm/mm] or [in/in]
F
Ao/2
L/2
Lowo
Here: The Black Outline is Original,
Green is after application of load
• Typical Engineering Issue in Mechanical Testing:– Aluminum tensile specimen, square X-Section
(16.5 mm on a side) and 150 mm long– Pulled in tension to a load of 66700 N– Experiences elongation of: 0.43 mm
• Determine Young’s Modulus if all the deformation is recoverable
230
0
66700 244.99516.5*10
0.43 0.00344125Because we are to assume all deformation is recoverable, Hooke's Law can be assumed:
244.9950.00344
71219.6 71.2
NF MPaA
mmLL mm
MPaE E
E MPa GPa
Solving:
Poisson's ratio, • Poisson's ratio, :
Units:: dimensionless
> 0.50 density increases
< 0.50 density decreases (voids form)
L
-
L
metals: ~ 0.33ceramics: ~ 0.25polymers: ~ 0.40
• Elastic Shear modulus, G:
G
= G
Other Elastic Properties
simpletorsiontest
M
M
• Special relations for isotropic materials:
2(1 )EG
3(1 2)EK
• Elastic Bulk modulus, K:
pressuretest: Init.vol =Vo. Vol chg. = V
PP P
P = -K VVo
PV
K Vo
E is Modulus of Elasticity is Poisson’s Ratio
Looking at Aluminum and the earlier problem:
71.22 1 2 1 0.33
26.871.2
3 1 2 3 1 2 0.33
69.8
HB
HB
GPaEG
G GPaGPaEK
K GPa
MetalsAlloys
GraphiteCeramicsSemicond
Polymers Composites/fibers
E(GPa)
Based on data in Table B2,Callister 7e.Composite data based onreinforced epoxy with 60 vol%of alignedcarbon (CFRE),aramid (AFRE), orglass (GFRE)fibers.
Young’s Moduli: Comparison
109 Pa
0.2
8
0.6
1
Magnesium,Aluminum
Platinum
Silver, Gold
Tantalum
Zinc, Ti
Steel, NiMolybdenum
Graphite
Si crystal
Glass -soda
Concrete
Si nitrideAl oxide
PC
Wood( grain)
AFRE( fibers) *
CFRE*GFRE*
Glass fibers only
Carbon fibers only
Aramid fibers only
Epoxy only
0.4
0.8
2
46
10
20
406080
100
200
600800
10001200
400
Tin
Cu alloys
Tungsten
<100>
<111>
Si carbide
Diamond
PTFE
HDPE
LDPE
PP
Polyester
PSPET
CFRE( fibers) *
GFRE( fibers)*
GFRE(|| fibers)*
AFRE(|| fibers)*
CFRE(|| fibers)*
• Simple tension:
FLoEAo
L
Fw oEAo
• Material, geometric, and loading parameters all contribute to deflection.• Larger elastic moduli minimize elastic deflection.
Useful Linear Elastic Relationships
F
Ao/2
L/2
Lowo
• Simple torsion:
2MLo
ro4G
M = moment = angle of twist
2ro
Lo
Resilience, Ur
• Ability of a material to store (elastic) energy – Energy stored best in elastic region
If we assume a linear stress-strain curve this simplifies to
Adapted from Fig. 6.15, Callister 7e.
yyr 21U
y dUr 0
(at lower temperatures, i.e. T < Tmelt/3)Plastic (Permanent) Deformation
• Simple tension test:
engineering stress,
engineering strain,
Elastic+Plastic at larger stress
permanent (plastic) after load is removed
p
plastic strain
Elastic initially
Adapted from Fig. 6.10 (a), Callister 7e.
• Stress at which noticeable plastic deformation has occurred.
when p 0.002
Yield Strength, y
y = yield strength
Note: for 2 inch sample
= 0.002 = z/z
z = 0.004 in
Adapted from Fig. 6.10 (a), Callister 7e.
tensile stress,
engineering strain,
y
p = 0.002
Tensile properties Yielding Strength
Most structures operate in elastic region, therefore need to know when it ends
Some steels (lo-C)
Yield strength
-- Generally quoted
Proof stress
Proportional Limit
Room Temp. values
Based on data in Table B4,Callister 7e.a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & tempered
Yield Strength : ComparisonGraphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibersPolymers
Yie
ld s
treng
th,
y (M
Pa)
PVC
Har
d to
mea
sure
, si
nce
in te
nsio
n, fr
actu
re u
sual
ly o
ccur
s be
fore
yie
ld.
Nylon 6,6
LDPE
70
20
40
6050
100
10
30
200
300
400500600700
1000
2000
Tin (pure)
Al (6061) a
Al (6061) ag
Cu (71500) hrTa (pure)Ti (pure) aSteel (1020) hr
Steel (1020) cdSteel (4140) a
Steel (4140) qt
Ti (5Al-2.5Sn) aW (pure)
Mo (pure)Cu (71500) cw
Har
d to
mea
sure
, in
cer
amic
mat
rix a
nd e
poxy
mat
rix c
ompo
site
s, s
ince
in te
nsio
n, fr
actu
re u
sual
ly o
ccur
s be
fore
yie
ld.
HDPEPP
humid
dryPC
PET
¨
Tensile Strength, TS
• Metals: occurs when noticeable necking starts.• Polymers: occurs when polymer backbone chains are aligned and about to break.
Adapted from Fig. 6.11, Callister 7e.
y
strain
Typical response of a metal
F = fracture or ultimate strength
Neck – acts as stress concentrator e
ngin
eerin
g TS
stre
ss
engineering strain
• TS is Maximum stress on engineering stress-strain curve.
Tensile Strength : Comparison
Si crystal<100>
Graphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibersPolymers
Tens
ile st
reng
th, T
S (
MP
a)
PVC
Nylon 6,6
10
100
200300
1000
Al (6061) a
Al (6061) agCu (71500) hr
Ta (pure)Ti (pure) aSteel (1020)
Steel (4140) a
Steel (4140) qt
Ti (5Al-2.5Sn) aW (pure)
Cu (71500) cw
LDPE
PPPC PET
20
3040
20003000
5000
Graphite
Al oxide
Concrete
Diamond
Glass-soda
Si nitride
HDPE
wood ( fiber)
wood(|| fiber)
1
GFRE(|| fiber)
GFRE( fiber)
CFRE(|| fiber)
CFRE( fiber)
AFRE(|| fiber)
AFRE( fiber)
E-glass fibC fibers
Aramid fib
Room Temp. valuesBased on data in Table B4,Callister 7e.a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & temperedAFRE, GFRE, & CFRE =aramid, glass, & carbonfiber-reinforced epoxycomposites, with 60 vol%fibers.
• Plastic tensile strain at failure:
Adapted from Fig. 6.13, Callister 7e.
Ductility
• Another ductility measure: 100xA
AARA%o
fo -=
x 100L
LLEL%o
of
Engineering tensile strain,
Engineering tensile stress,
smaller %EL
larger %ELLf
Ao AfLo
Lets Try one (like Problem 6.29)Load (N) len. (mm) len. (m) l
0 50.8 0.0508 0
12700 50.825 0.050825 2.5E-05
25400 50.851 0.050851 5.1E-05
38100 50.876 0.050876 7.6E-05
50800 50.902 0.050902 0.000102
76200 50.952 0.050952 0.000152
89100 51.003 0.051003 0.000203
92700 51.054 0.051054 0.000254
102500 51.181 0.051181 0.000381
107800 51.308 0.051308 0.000508
119400 51.562 0.051562 0.000762
128300 51.816 0.051816 0.001016
149700 52.832 0.052832 0.002032
159000 53.848 0.053848 0.003048
160400 54.356 0.054356 0.003556
159500 54.864 0.054864 0.004064
151500 55.88 0.05588 0.00508
124700 56.642 0.056642 0.005842
GIVENS:
Leads to the following computed Stress/Strains:
e stress (Pa) e str (MPa) e. strain
0 0 0
98694715.7 98.694716 0.000492
197389431 197.38943 0.001004
296084147 296.08415 0.001496
394778863 394.77886 0.002008
592168294 592.16829 0.002992
692417257 692.41726 0.003996
720393712 720.39371 0.005
796551839 796.55184 0.0075
837739398 837.7394 0.01
927885752 927.88575 0.015
997049766 997.04977 0.02
1163354247 1163.3542 0.04
1235626755 1235.6268 0.06
1246506488 1246.5065 0.07
1239512374 1239.5124 0.08
1177342475 1177.3425 0.1
969073311 969.07331 0.115
0
2 20
0
use m if F in Newtons; in if F in lb
results in Pa (MPa) or psi (ksi)f
FA
A
andll
Leads to the Eng. Stress/Strain Curve:Engineering Stress Strain
0
200
400
600
800
1000
1200
1400
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Strain (m/m)
Stre
ss (M
Pa)
Magenta Line Model:
.002*.0021 to .0065
m Em E
T. Str. 1245 MPa
Y. Str. 742 MPa%el 11.5%
F. Str 970 MPa
E 195 GPa (by regression)
TOUGHNESS
High toughness = High yield strength and ductility
Dynamic (high strain rate) loading condition (Impact test)1. Specimen with notch- Notch toughness2. Specimen with crack- Fracture toughness
Is a measure of the ability of a material to absorb energy up to fracture
Important Factors in determining Toughness:
1. Specimen Geometry & 2. Method of load application
Static (low strain rate) loading condition (tensile stress-strain test)1. Area under stress vs strain curve up to the point of fracture.
• Energy to break a unit volume of material• Approximate by the area under the stress-strain curve.
Toughness
Brittle fracture: elastic energyDuctile fracture: elastic + plastic energy
very small toughness (unreinforced polymers)
Engineering tensile strain,
Engineering tensile stress,
small toughness (ceramics)
large toughness (metals)
Adapted from Fig. 6.13, Callister 7e.
Elastic Strain Recovery – After Plastic Deformation
Adapted from Fig. 6.17, Callister 7e.
It is an important factor in forming products (especially sheet metal and spring making)
True Stress & StrainNote: Stressed Area changes when sample is
deformed (stretched)• True stress
• True Strain
iT AF
oiT ln
1ln1
T
T
Adapted from Fig. 6.16, Callister 7e.
Strain Hardening
• Curve fit to the stress-strain response:
T K T n
“true” stress (F/Ai) “true” strain: ln(Li /Lo)
hardening exponent:n = 0.15 (some steels) n = 0.5 (some coppers)
• An increase in y due to continuing plastic deformation.
large Strain hardening
small Strain hardeningy 0
y 1
Earlier Example: True Properties
T. Stress T. Strain
0 0
98.74328592 0.000492
197.5875979 0.001003
296.5271076 0.001495
395.571529 0.002006
593.9401363 0.002988
695.1842003 0.003988
723.9956807 0.004988
802.525978 0.007472
846.1167917 0.00995
941.8040385 0.014889
1016.990761 0.019803
1209.888417 0.039221
1309.764361 0.058269
1333.761942 0.067659
1338.673364 0.076961
1295.076722 0.09531
1080.516741 0.108854
Necking Began
We Compute: T = KT
n to complete our True Stress vs. True Strain plot (plastic data to necking)
• Take Logs of both T and T
• ‘Regress’ the values from Yielding to Necking
• Gives a value for n (slope of line) and K (its T when T = 1)
• Plot as a continuation line beyond necking start
Stress Strain Plot w/ True ValuesEngineering Stress Strain
& True Stress Stain
0
200
400
600
800
1000
1200
1400
1600
1800
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Strain (m/m)
Stre
ss (M
Pa)
n 0.242 K 2617 MPa
Variability in Material PropertiesCritical properties depend largely on sample flaws
(defects, etc.). Most can exhibit Large sample to sample variability.
• Real (reported) Values, thus, are Statistical measures usually mean values.
– Mean
– Standard Deviation 21
2
1
n
xxs i
n
nxx n
n
where n is the number of data points (tests) that are performed
Ny
working
Ny
working
• Design uncertainties mean we do not push the limit!• Typically as engineering designers, we introduce a Factor of safety, N Often N is
between1.5 and 4
• Example: Calculate a diameter, d, to ensure that yield does not occur in the 1045 carbon steel rod below subjected to a
working load of 220,000 N. Use a factor of safety of 5.
Design or Safety Factors
1045 plain carbon steel: y = 310 MPa TS = 565 MPa
F = 220,000N
d
Lo
Solving:
2
20
2
2
2 26
2 3 2
2
3105
220000
4
310220000
54
220000 4 5 m310 104.52 10 m
6.72 10 m 6.72 cm
Yworking
working
Nm
NNF
A D
MNmN
D
D
D x
D x D
Hardness• Resistance to permanently (plastically) indenting the surface of a product.• Large hardness means: --resistance to plastic deformation or cracking in compression. --better wear properties.
e.g., Hardened 10 mm sphere
apply known force measure size of indentation after removing load
dDSmaller indents mean larger hardness.
increasing hardness
most plastics
brasses Al alloys
easy to machine steels file hard
cutting tools
nitrided steels diamond
Hardness: Common Measurement Systems
Callister Table 6.5
Comparing Hardness Scales:
Inaccuracies in Rockwell (Brinell) hardness measurements may occur due to: An indentation is made too near a specimen edge.
Two indentations are made too close to one another.
Specimen thickness should be at least ten times the indentation depth.
Allowance of at least three indentation diameters between the center on one indentation and the specimen edge, or to the center of a second indentation.
Testing of specimens stacked one on top of another is not recommended.
Indentation should be made into a smooth flat surface.
Correlation Between Hardness and Tensile Strength
Both measures the resistance to plastic deformation of a material.
HB = Brinell HardnessTS (psia) = 500 x HBTS (MPa) = 3.45 x HB
• Stress and strain: These are size-independent measures of load and displacement, respectively.• Elastic behavior: This reversible behavior often shows a linear relation between stress and strain. To minimize deformation, select a material with a large elastic modulus (E or G).
• Toughness: The energy needed to break a unit volume of material.• Ductility: The plastic strain at failure.
Summary
• Plastic behavior: This permanent deformation behavior occurs when the tensile (or compressive) uniaxial stress reaches y.