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TRANSCRIPT
Chapter 6 - 1
ISSUES TO ADDRESS...
• Stress and strain: What are they and why arethey used instead of load and deformation?
• Elastic behavior: When loads are small, how much deformation occurs? What materials deform least?
• Plastic behavior: At what point does permanent deformation occur? What materials are most resistant to permanent deformation?
• Toughness and ductility: What are they and howdo we measure them?
Chapter 6: Mechanical Properties
Chapter 6 - 2
Stress and Strain
(a) Tensile Load – elongation and (+)strain
(b) Compressive load –contraction and (-)strain
(c) Shear stress – shear strain (γ)
(d) Torque - Torsional deformation
Chapter 6 - 3
• Simple tension: ex. cable
Common States of Stress
• Torsion(a form of shear): ex. drive shaft
Ski lift (photo courtesy P.M. Anderson)
FF
- Shear stress is a function the applied torque T
- Shear strain is related to the angle of twist, ϕ
Chapter 6 - 4
Common States of Stress
- Shear Force is when opposingforces are applied across the material, having the effect of cutting it in two. The material would require shear strength to resist this force.
- Scissors cut material by applying a local shear stress at the cutting location which exceeds the material's shear strength.
• Shear stress: Bench shears, Scissors,
Chapter 6 - 5
(photo courtesy P.M. Anderson)Canyon Bridge, Los Alamos, NM
• Simple compression:
Note: compressivestructure member( < 0 here).(photo courtesy P.M. Anderson)
OTHER COMMON STRESS STATES (i)
Balanced Rock, Arches National Park
Chapter 6 - 6
• Bi-axial tension: • Hydrostatic compression:
Pressurized tank
< 0h
(photo courtesyP.M. Anderson)
(photo courtesyP.M. Anderson)
OTHER COMMON STRESS STATES (ii)
Fish under water
z > 0
> 0
Chapter 6 - 7
Stress has units:N/m2 (Pa) or lbf /in2 (psi)
Engineering Stress• Shear stress, :
Area, Ao
Ft
Ft
Fs
F
F
Fs
= Fs
Ao
• Tensile stress, :
original area before loading
=Ft
Ao2f
2m
Nor
in
lb=
Area, Ao
Ft
Ft acts tangentialto the surface
acts perpendicularto the cross-section
Chapter 6 - 8
• Tensile strain (): • Lateral strain ( ):
Strain is alwaysdimensionless.
Engineering Strain
• Shear strain ():
90º
90º - y
x = x/y = tan
Lo
Adapted from Fig. 6.1(a) and (c), Callister & Rethwisch 8e.
/2
Lowo
L L
wo
L/2
L
Chapter 6 - 9
Stress-Strain Testing• Typical tensile test
machine
Adapted from Fig. 6.3, Callister & Rethwisch 8e. (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
Adapted from Fig. 6.2,Callister & Rethwisch 8e.
gauge length
→ to measure changes in the length of an object
Chapter 6 - 10
Stress-Strain Testing• Universal Testing Machine (UTM) : INSTRON
http://www.youtube.com/watch?v=CkSsRLUeRQIhttp://www.youtube.com/watch?v=qDqc-HAxNrQ
Chapter 6 - 11
Linear Elastic Properties
• Modulus of Elasticity, E: (also known as Young's modulus)the slope of the linear portion of the stress-strain curve, it is usually specific to each material; a constant, known value
• Hooke's Law:
= E
Linear-elastic
E
F
Fsimple tension test
SI unit for the E is Gpa :
E= F/(A
1GPa = 109N/m2
cf. psi (lb/ in2)kgf/cm2
Elastic deformation is not permanent; when the load is removed, the part returns to its original shape and dimensions.
Chapter 6 -
Shear Modulus
y
x
12
= x/y = tan Shear stress() to shear strain():
= G ,
G is Shear Modulus (Units: N/m2)
90º
90º - y
x
Chapter 6 -
- Ratio of lateral to axial strain calledPoisson's ratio, .
- (Tension) shrink laterally (Compression) bulge.
Poisson's ratio,
13
- dimensionless.
- Sign: lateral strain opposite to longitudinal strain
- Theoretical value: for isotropic material: 0.25
- (Typical value) 0.24 - 0.30 (Maximum value) 0.50
L
Lateral strain
Tensile strain
x
z
y
z
Chapter 6 - 14
Chapter 6 - 15
Poisson's ratio,
• Poisson's ratio, :
Units:E: [GPa] or [psi]: dimensionless
> 0.50 density increases
< 0.50 density decreases (voids form)
L
-
L
metals: ~ 0.33ceramics: ~ 0.25polymers: ~ 0.40
Lateral strain
Tensile strain
Chapter 6 -
Elastic Modulus, Poisson’s Ratioand Shear Modulus
- For isotropic material:
E = 2G(1+) G ~ 0.4E
- Single crystals are usually elastically anisotropic
- Elastic behavior varies with crystallographic direction.
16
Chapter 6 - 17
Non-linear Elastic Behavior
- Elastic portion of the stress-strain curve is not linear. ex. Gray cast iron, concrete
& many polymer
- It is not possible to determine a modulus of elasticity
- Either tangent or secant modulus is normally used
http://www.youtube.com/watch?v=clLEhdbQ6Ug
Chapter 6 - 18
Mechanical Properties : atomic scale
• Slope of stress strain plot (which is proportional to the elastic modulus) depends on bond strength of metal:
Adapted from Fig. 6.7,Callister & Rethwisch 8e.
→ elastic modulus is proportional to the slope of each curve at the equil. interaction separation
E∝
Chapter 6 - 19
Mechanical Properties- The modulus of elasticity diminishes with increasing temperature
Chapter 6 -20
• Have assumed elastic deformation is time independent
(applied stress produces instantaneous strain)
• Elastic deformation takes time; can continue even after loadrelease. This behavior is known as anelasticity.
(→ time dependence of elastic deformation)
• Small effect in metals; can be significant for polymers(visco-elastic behavior).
Anelasticity (visco-elastic behavior)
Chapter 6 - 21
MetalsAlloys
GraphiteCeramicsSemicond
PolymersComposites
/fibers
E(GPa)
Based on data in Table B.2,Callister & Rethwisch 8e.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
4
6
10
20
40
6080
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)*
Chapter 6 - 22
• Simple tension:
FLo
EAo
L Fw o
EAo
• Material, geometric, and loading parameters allcontribute 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
Chapter 6 - 23
Elastic means reversible!
Elastic Deformation2. Small load
F
bonds stretch
1. Initial 3. Unload
return to initial
- Reversible for small strains
- Stress removed → material returns tooriginal size
F
Linear- elastic
Non-Linear-elastic
Chapter 6 - 24
Plastic Deformation (Metals)1. Initial 2. Small load 3. Unload
planesstill sheared
F
elastic + plastic
bonds stretch & planes shear
plastic
F
linear elastic
linear elastic
plastic
• stress not proportional to strain
• deformation is not reversible : permanent/plastic
• deformation occurs by breaking and re-arrangement of atomic bonds (crystalline materials by motion of defects)
→ Permanent deformation for metals is accomplished by means of a process called slip, which involves the motion of dislocations.
Chapter 6 - 25
(at lower temperatures, i.e. T < Tmelt/3)Plastic (Permanent) Deformation
• Simple tension test:
Engineeringstress,
engineering strain,
Elastic+Plastic at larger stress
p
plastic strain
Elastic initially
Adapted from Fig. 6.10(a),Callister & Rethwisch 8e.
permanent (plastic) after load is removed
- After removal of the stress, the large number of atoms that have relocated, do not return to original position.
- plastic deformationcorresponds to the breaking of bonds with original atom neighbors and then reforming bondswith new neighbors.
Chapter 6 - 26
(at lower temperatures, i.e. T < Tmelt/3)Plastic (Permanent) Deformation
• Most structures are designed to ensure that only elastic deformationresults when stress is applied.
• A structure that has plastically deformed, or experienced a permanent change in shape, may not be capable of functioning as intended.
• Yield strength is usually more important than tensile strength. Once yield stress has been passed, structure has deformed beyond acceptable limits.
Necked region in a fractured sample
http://www.youtube.com/watch?v=hD_NJaZIpT0
Chapter 6 - 27
• Stress at which noticeable plastic deformation hasoccurred.
when p = 0.002
Yield Strength, y
y = yield strength
Adapted from Fig. 6.10(a),Callister & Rethwisch 8e.
tensile stress,
engineering strain,
y
p = 0.002
Elastic Plastic
Yield Point : strain deviates from being proportional to stress (Proportional Limit)
P - Yield point (P) is the gradual elastic to plastic transition.
- Yield strength is a measure of resistance to plastic deformation
Chapter 6 - 28
Yield Strength, y
→ σy = average stress at the lower yield point
Strain Strain
Str
ess
Str
ess
0.005
σy σy→ σy = Stress required to produce some amount of strain (e.g., ε= 0.005)
• Nonlinear elastic deformation • Some low-carbon steels with yield point phenomenon
Chapter 6 - 29
Room temperaturevalues
Based on data in Table B.4,Callister & Rethwisch 8e.a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & tempered
Yield Strength : ComparisonGraphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibers
PolymersY
ield
str
engt
h, y
(MP
a)
PVC
Har
d to
mea
sure
,
sinc
e in
tens
ion,
frac
ture
usu
ally
occ
urs
befo
re y
ield
.
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
¨
Chapter 6 - 30
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 & Rethwisch 8e.
y
strain
Typical response of a metal
F = fracture or
ultimate
strength
Neck – acts as stress concentrator
engi
neer
ing
TS
stre
ss
engineering strain
• Maximum stress on engineering stress-strain curve.
Chapter 6 - 31
• After yielding, the stress necessary to continue plastic deformation in metals increases to a maximum point (M) and then decreases to the eventual fracture point (F).• All deformation up to the maximum stress is uniform throughout the tensile sample. • However, at max stress, a small constriction or neck begins to form.• Subsequent deformation will be confined to this neck area.• Fracture strength corresponds to the stress at fracture.
Region between M and F:• Metals: occurs when noticeable necking starts.• Ceramics: occurs when crack propagation starts.• Polymers: occurs when polymer backbones are aligned and about to break.
Tensile Strength, TS
Chapter 6 - 32
Elastic Strain Recovery (during plastic deformation)
Adapted from Fig. 6.17, Callister & Rethwisch 8e.
Str
ess
Strain
3. Reapplyload
2. Unload
D
Elastic strainrecovery
1. Load
yo
yi
- Deformed plastically,stress released, materialhas permanent strain.
- If stress is reapplied,material again respondselastically at the beginningup to a new yield point thatis higher than the originalyield point.
(Strain hardening)
- Elastic strain beforereaching the yield point iscalled elastic strainrecovery.
Chapter 6 -
Stress-Strain Diagram
Strain ( ) (L/Lo)
41
2
3
5
Elastic Region
PlasticRegion
StrainHardening Fracture
ultimatetensile strength
Elastic regionslope =Young’s (elastic) modulusyield strength
Plastic regionultimate tensile strengthstrain hardeningfracture
necking
yieldstrength
UTS
y
εEσ
ε
σE
12
y
ε ε
σE
Chapter 6 -
Stress-Strain Diagram (cont)
• Elastic Region (Point 1 –2)- The material will return to itsoriginal shape after the material is unloaded(like a rubber band).
- The stress is linearly proportionalto the strain in this region.
εEσ σ : Stress(psi)E : Elastic modulus (Young’s Modulus) (psi)ε : Strain (in/in)
- Point 2 : Yield Strength : a point where permanent deformation occurs. (If it is passed, the material will no longer return to its original length.)
ε
σE or
Strain ( )
41
2
3
5
Elastic Region
PlasticRegion
StrainHardening
Fracture
UTS
y
Chapter 6 -
• Strain Hardening
- If the material is loaded again
from Point 4, the curve will
follow back to Point 3 with the
same Elastic Modulus (slope).
- The material now has a higher
yield strength of Point 3.
Stress-Strain Diagram (cont)
Strain ( )
41
2
3
5
Elastic Region
PlasticRegion
StrainHardening
Fracture
UTS
y
- Raising the yield strength by permanently straining the material is called Strain Hardening; an increase in σy due to plastic deformation.
Chapter 6 -
• Tensile Strength (Point 3)
- The largest value of stress on the diagram is called Tensile Strength(TS) or Ultimate TensileStrength(UTS)
- It is the maximum stress whichthe material can support without breaking.
Stress-Strain Diagram (cont)
Strain ( )
41
2
3
5
Elastic Region
PlasticRegion
StrainHardening
Fracture
UTS
y
• Fracture (Point 5)
- If the material is stretched beyond Point 3, the stress decreases as necking and non-uniform deformation occur.
- Fracture will finally occur at Point 5.
Chapter 6 - 37
Tensile Strength: Comparison
Si crystal<100>
Graphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibers
PolymersTe
nsile
stre
ngth
, TS
(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
PP
PC 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 fib
C fibersAramid fib
Based on data in Table B.4,Callister & Rethwisch 8e.a = annealedhr = hot rolledag = agedcd = cold drawncw = cold workedqt = quenched & temperedAFRE, GFRE, & CFRE =aramid, glass, & carbonfiber-reinforced epoxycomposites, with 60 vol%fibers.
Room temperaturevalues
Chapter 6 - 38
Ductility
- A measure of the degree of plastic deformation that has been sustained at fracture.
- A metal that experiences very little or no plastic deformation upon fracture is termed brittle.
- expressed quantitatively as either percent elongation(%EL) or percent reduction in area(%RA)
Chapter 6 - 39
• Plastic deformation that has been sustained at fracture
Ductility
• Another ductility measure: 100xA
AARA%
o
fo -=
x 100L
LLEL%
o
of
Lf
Ao AfLo
Adapted from Fig. 6.13, Callister & Rethwisch 8e.
Engineering tensile strain,
Engineering tensile stress,
smaller %EL(brittle if %EL<5% )
larger %EL(ductile if %EL>5%)
→ A material that suffers very little plastic deformation is brittle.
Chapter 6 - 40
Ductility
- The magnitudes of both yield and tensile strengths decline with increasing temperature
- Ductility increases with temperature. (ex. Fe)
Chapter 6 - 41
--brittle response (aligned chain, cross linked & networked case)--plastic response (semi-crystalline case)
Stress-Strain Behavior: Elastomers
3 different responses:
A – brittle failure
B – plastic failure
C - highly elastic (elastomer)
Chapter 6 - 42
• The ability to absorb energy up to fracture : Energy to break a unit volume of material (energy per unit volume of material, e.g. J/m3 ).
• Approximate by the area under the stress-strain curve.• A “tough” material has strength and ductility.
Toughness
Brittle fracture: elastic energyDuctile fracture: elastic + plastic energy
Adapted from Fig. 6.13, Callister & Rethwisch 8e.
small toughness (unreinforced polymers)
Engineering tensile strain,
Engineering tensile stress,
small toughness (ceramics)
large toughness (metals)
Chapter 6 - 43
Resilience, Ur• Ability of a material to store energy
– Energy stored best in elastic region
– Resilient materials have high yield strengths and low moduli of elasticity; ex. spring applications.
If we assume a linear stress-strain curve this simplifies to
Adapted from Fig. 6.15, Callister & Rethwisch 8e.
yyr 21U y2
1 y
E
y
2E2
y dUr 0
Chapter 6 - 44
True Stress & Strain- The decline in the stress
past the maximum(M) seems to indicate that the metal is becoming weaker. However, it is increasing in strength.
- However, cross-sectional area is decreasing rapidly within the neck region, where deformation is occurring.
iT AF
- It is more meaningful to use a true stress-true strain scheme.
True stress; the load divided by the actual cross-sectional area(Ai) of the specimen at that load;
- True strain; calculated using actual and not original dimensions, defined by
oiT ln
Chapter 6 - 45
True Stress & Strain• If no volume change occurs during deformation ;
• Engineering stress and strain are related according to :
1ln
1
T
T
Adapted from Fig. 6.16, Callister & Rethwisch 8e.
The true stress necessary to sustain increasing strain continues to rise past the tensile point M’
00 AA ii
iT AF
oiT ln
00 AA ii
Valid only to the onset of necking
Chapter 6 -46
In an undeformedthermoplastic polymer tensile sample,
(a) the polymer chains are randomly oriented.
(b)When a stress is applied, a neck develops as chains become aligned locally. The neck continues to grow until the chains in the entire gage length have aligned.
(c) The strength of the polymer is increased
Chapter 6 - 47
Stress-Strain Results for Steel Sample
Chapter 6 -
Example 1. Convert the change in length data in the table to engineering stress and strain and plot a stress-strain curve.
Chapter 6 -
Example 1 SOLUTION
Chapter 6 -
Example 2. Compare engineering stress and strain with true stress and strain for the aluminum alloy in Example 1 at (a) the maximum load. The diameter at maximum load is 0.497 in. and at fracture is 0.398 in.
Example 3 SOLUTION
Chapter 6 - 51
Hardness• Resistance to permanently indenting the surface.• Large hardness means:
-- resistance to plastic deformation or cracking incompression.
-- better wear properties.
e.g., 10 mm sphere
apply known force(1 to 1000g)
measure size of indent 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
Chapter 6 - 52
Hardness: MeasurementTable 6.5
Chapter 6 - 53
Conversion of Hardness Scales
Also see: ASTM E140 - 07 Volume 03.01Standard Hardness Conversion Tables for Metals Relationship Among Brinell Hardness, Vickers Hardness, Rockwell Hardness, Superficial Hardness, Knoop Hardness, and Scleroscope Hardness
Chapter 6 - 54
Correlation between Hardness and Tensile Strength
• Both hardness and tensile strength are indicators of a metal’s resistance to plastic deformation.
• For cast iron, steel and brass, the two are roughly proportional.
• Tensile strength (psi) = 500*BHR
Chapter 6 - 55
Variability in Material Properties
• Elastic modulus is material property
• Measured material properties are not exact quantities.
ex. A number of identical tensile samples give a variety of modulus, yield strength, and tensile strength.
• A number of factors lead to uncertainties in measure data;
ex. Test method, variation in specimen fabrication procedures, operator bias, apparatus calibration, inhomogeneities, and lot difference
• Scatter and variability of materials properties are inevitable and must be dealt with appropriately.
• “What is the probability of failure of this alloy under these given circumstances” instead of asking the question, “What is the fracture strength of this ally?”
Chapter 6 - 56
Variability in Material Properties
• Specify a typical value and degree of dispersion;
• Statistics
– Mean
– Standard Deviation
s n
xi x 2n1
1
2
n
xx n
n
where n is the number of data points
Chapter 6 - 57
• Design uncertainties mean we do not push the limit.• Factor of safety, N
Ny
working
Often N isbetween1.2 and 4
• Example: Calculate a diameter, d, to ensure that yield doesnot occur in the 1045 carbon steel rod below. Use a factor of safety of 5.
Design or Safety Factors
220,000N
d2 / 4 5
Ny
working
1045 plain
carbon steel: y = 310 MPa
TS = 565 MPa
F = 220,000N
d
Lo
d = 0.067 m = 6.7 cm
Chapter 6 - 58
• Stress and strain: These are size-independentmeasures of load and displacement, respectively.
• Elastic behavior: This reversible behavior oftenshows a linear relation between stress and strain.To minimize deformation, select a material with alarge elastic modulus (E or G).
• Toughness: The energy needed to break a unitvolume of material.
• Ductility: The plastic strain at failure.
Summary
• Plastic behavior: This permanent deformationbehavior occurs when the tensile (or compressive)uniaxial stress reaches y.
Chapter 6 - 59
• Concept Check : 6.1-6.4
• Example problem : 6.1~6.6
• Design example : 6.1
• Questions and Problems :6.4~6.8, 6.10, 6.11, 6.13, 6.15~6.18, 6.20, 6.22~6.28, 6.35, 6.36, 6.39, 6.49, 6.50, 6.54, 6.57, 6D1
Problems