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Transcript of ME 330 Engineering Materials
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ME 330 Engineering MaterialsLectures 2-3
Tensile Properties• Elastic properties• Yield-point behavior• Plastic deformation• True vs. Engineering stress • Stress-strain curves• Fracture surfaces• Hardness Testing
Please read chapters 1 (Lecture 1) & 6 (Lecture 2)Please read chapters 1 (Lecture 1) & 6 (Lecture 2)
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Where We Are Going...
• Engineers design products to carry loads, transmit forces, etc.
• Characterize a material’s behavior through properties– Measure properties in lab test … extrapolate behavior to
different scenario– Alternative is proof testing everything!
• Basic mechanical testing– Look for response to applied forces
• Apply load, measure deformation• Indent surface, measure hardness
– Quantify words like “strong”, “ductile”, “hard”, etc
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Basic Mechanical TestsTension
Most common mechanical testGage section reduced to ensure deflection hereLoad cell measures applied load Extensometer ensures l measured from gage region
CompressionSimilar to tensile testGood for brittle specimens … hard to gripOften much different properties in compression
TorsionTest of pure shearMember twisted by angle , calculate shear strainMeasure applied torque, calculate shear stress
Bending
In all cases, a displacement is applied and you measure loadCalculate stress from measured loadCalculate strain from change in gage length
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Tension Test
Measure load and displacement
Compute stress and strain
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Review of Stress and Strain
• Stress: force per unit area
• Traditional units: MPa or ksi• Ao is original area• A is instantaneous area
• Strain: “relative” change in length
• Dimensionless quantity• Lo is original length (“gage length”)• L is instantaneous length
Often interested in measuring force and deformation in a size independent manner
AreaForce
LengthLength
oAF:gEngineerin
o
oLLL:gEngineerin
Ao
Lo
: TFTrueA
oT L
Lln:True A
LFrom dT=dL/L
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Relation Between Stress & Strain
Tension (+) Compression (-)
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Typical Stress-Strain Curves
0.1 10 100
ceramics
metals
polymers
(M
Pa)
(%)
Stre
ngth
Ductility
Stiffness
Energy Absorption
Elastic Plastic
Yield Today, we’ll talk about the different: Regions in stress-strain spaceProperties important to design
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Elastic Region & Properties
~0.1 10 100
ceramics
metals
polymers
(M
Pa)
(%)
Stiffness
ElasticElastic region: proportional stress and strainStiffness = Modulus of Elasticity ductility
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Elastic Material BehaviorS
tress
(MP
a)
Strain (%)
Stre
ss (M
Pa)
Strain (%)
Linear Non-linear
1
2
secant modulus @ 1
tangent modulus @ 2E
Elastic region: strain returns to zero when stress removedElastic Modulus (E) - measure of stiffness
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Elastic BehaviorS
tress
(MP
a)
Strain (%)
Stre
ss (M
Pa)
Strain (%)
linear non-linear
E Secant Modulus
Tangent Modulus
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Atomic Level Effects on Modulus
F
F
Many metals
Most ceramics
F F Most polymers
• Strength of interatomic bonds: stiffness of springs• Atomic packing: springs per unit area
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Atomistic Origins of ElasticityForce
Atomic separation, r
Force
r
oo rr2
2
rr drd
drdFE
Strong bonding,stiff
Weak bonding,compliant
ro Energy(r)dr
d)r(F
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Final Notes on Stiffness• Interatomic bonding
– Ceramics - Ionic & Covalent– Metals - Metallic & Covalent– Polymers - Covalent &
Secondary• Packing
– Ceramics & Metals • Highly ordered crystals• Dense packing
– Polymers• Randomly oriented chains• Loosely packed
• Temperature effects– Effect depends on types of
bonds– As temperature increases,
modulus decreases
Material E (GPa)
Silicon Carbide 475Ceramics Alumina 375
Glass 70
Steel 210Metals Brass 97
Aluminum 69
PVC 3.3
Polymers Epoxy 2.4LDPE 0.23
(M
Pa)
(%)
Ceramics
Metals
E
Polymers
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Elastic Constitutive Relationfor 1-D Tensile Loading (linear materials)
• Hooke’s Law: Stress and strain are directly related by modulus of elasticity,
• Poisson’s ratio: Strain perpendicular to applied load is related to the axial strain,
– Maximum (constant volume) : = 0.50– Minimum: = 0– Look at change in volume in a cube of side length, L
– Volume increases during tensile, elastic deformation (if 0.50)
E
z
y
z
x
z
x
0 0 0{ (1 )} { (1 )} { (1 )}xx yy zzLxLxL L x L x L 2
0 0 0 0 0{ (1 )} { (1 )} { (1 )} { (1 )} { (1 )}zz zz zz zz zzL x L x L L x L 3 2 2 30{1 (1 2 ) ( 2) }zz zz zzL 30{1 (1 2 ) }zzL
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Elastic Behavior
Elastic Modulus
Elastic Modulus
Poisson’s Ratio
12
12
E
z
y
z
x
allongitudin
transverse
12GE
E
0rrdrdFE
Axial
Shear G
for isotropic material
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Elastic +Plastic Properties
0.1 10 100
ceramics
metals
polymers
Stre
ngth
Ductility
(M
Pa)
(%)
Stiffness
Energy Absorption
Elastic Plastic
Yield
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Elastic UnloadingS
tress
(MP
a)
Strain (%)plastic elastic
total strain = elastic + plastic
Stress – always elastic, no concept of plastic stress
p
pe
E
E E
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Review Stress and StrainEngineering Stress
Engineering Strain
True Strain
True Stress
oAF
1AF
oo
o
LL
LLL
1lnlnln
AA
LL o
oT
Constant Volume 00LAAL
Lo
do
Ao~ L
F
d
A~
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Modeling Plastic Deformation:True Stress and Strain
• True stress-strain values for plasticity … takes into account large area changes during plastic deformation
• Can relate true values to engineering values– Valid only for constant plastic deformation– Assuming constant volume, ,
)1ln()L/Lln( oT
L*AL*A oo
AAo
L Lo
* / * oo
o
L LL L
L
)1(*T
o
oT AA
APAP
oL*AL*A
o
o
oL/L*1
L/LL/L*
AAoo
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Elastic Constitutive Relationfor Simple Shear
F
F
F
F
Again, stress and strain are directly related, by shear modulus, GG: G
For isotropic materials, shear and elastic modulus are related by: 1G2E
Shear stress:oAF
Ao
Shear strain:
)tan(
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Stress & Strain in 3-Dimensions
x
z
yx
zx
z
yxy
zy
yx
yz
xz
x
y
z
xy
xz
yx
yz
zx
zy
x
y
z
xy
xz
yx
yz
zx
zy
Need to relate stress to strain
klijklij C
Originally 9 independent components Cijkl has 81 constants!!Equilibrium indicates ij = ji 6 components 36 constants (most general anisotropic matl)Elastic strain is reversible, so Ci j= Cji 21 constantsBased on crystal symmetry, for cubic crystals 3 constantsFor an isotropic crystal, need only 2 constants to describe 3-D responseRelate 1-D tests to complex loading
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1 0 0 0
1 0 0 0
1 0 0 0
2(1 )0 0 0 0 0
2(1 )0 0 0 0 0
2(1 )0 0 0 0 0
x x
y y
z z
xy xy
yz yz
xz xz
E E E
E E E
E E E
E
E
E
3-Dimensional Elastic Stress State
1 0 0 0
1 0 0 0
1 0 0 0
0 0 0 0 00 0 0 0 00 0 0 0 0
xy xz
x y z
x xyx yz
y yx y z
z zzyzx
xy xyx y z
yz yz
xyxz xz
yz
xz
E E E
E E E
E E E
GG
G
Isotropic Material
Orthotropic Material
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Yield Point
~0.1 10 100
ceramics
metals
polymers
(M
Pa)
(%)
Stiffness
Elastic
Yield Yield point marks the transition from elastic to plastic deformation
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(M
Pa)
(%)0.1
y
(M
Pa) (%)0.2
0.2%y
(M
Pa)
(%)
ly
uy
Yield Point Behavior
• Proportional limit marks the end of linearity• Yield point marks the beginning of plastic deformation
– Some materials show an obvious transition, y
– Often need to define 0.2% offset yield, 0.2%y
– Sometime see an upper (uy) and lower (ly) yield stresses occur
• Caused by significant dislocation-solute interaction
• Common in BCC iron based alloys
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Plastic Region
~0.1 10 100
ceramics
metals
polymers
(M
Pa)
(%)
Stiffness
Elastic Plastic
Yield Stress is no longer proportional to strainPlastic deformation is permanent, non-recoverable
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plastic elastic
E E
p
pe
E
No concept of “plastic stress”
Upon unloading, strain is partitioned between recovered and permanent.
Plastic Phenomena
Uniformdeformation
Necking begins:
Localizeddeformation
0dd
(MPa
)
(%)
y2
y1
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Upon reloading, stress-strain curvefollows the same path to failure.
Plastic Phenomena
(MPa
)
(%)
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True vs. Engineering - Curve
(M
Pa)
(%)
• Decreasing area in plastic regime higher “true” stresses• Once a neck forms,
– Equations are invalid– True curve overpredicts actual stress due to triaxial stress
state
Engineering
True
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True vs. Engineering - CurveCompression
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Plastic Constitutive Response
• Can approximate relation between true stress-strain curve in constant plastic deformation region by:
– K is the strength parameter– n is the strain-hardening exponent
• 0 n 1• if n = 0, elastic-perfectly plastic response• if n = 1, ideally elastic material• as n increases, achieve more strain hardening
– Typically valid only for some metals and alloys– Termed “power law hardening”
nTT K
(M
Pa)
(%)
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Measures of Energy Absorption: Toughness vs. Resilience
(M
Pa)
(%)
Resilience: Ability to absorb energy without
permanent deformation - (elastic only)
Toughness: Total energy absorption capability
of a material - (elastic + plastic)
•Units: Energy per unit volume•Define: Energy stored during deformation •Graphically: Area under - curve
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Stress-Strain Properties (cont.)
yyr
y
dU
21
0 Modulus of Resilience
nTT K Stress vs. Strain Eq. uTy for
EEU yy
yr 221 2
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Measures of Strength
(MPa
)
(%)
f
Fracture stress, f
0.2%
0.2%y
0.2% offset yield strength, 0.2%y
UTS
Ultimate Tensile Stress, UTS
f
Fracture strain, f (~Ductility)
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Measures of Ductility
Percent Elongation: Sensitive to gage length Does not account for necking
100*LLLEL%o
o
Lo L
Area Reduction: Insensitive to gage length Does account for necking Sensitive to cross-section
100*A
AAAR%o
o
AAo
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Stress-Strain Properties
% Elongation
% Reduction in Area
100xL
LLEL%o
o
100xA
AARA%o
o
Yield Strength y 0.2% offset or lower yield point
UTS u Highest stress on curve
Proportional limit = highest linear stress
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Material Deformation & Fracture
From Callister, p.126
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Brittle•Cleavage failure•Flat,rough fracture surface•No necking•Failure in tension
•Ductile•Completely ductile failure necks to a point•Cup-cone fracture surface
•Necking prior fracture•Cavities initiate in neck•Voids coalesce to form crack•Final failure in shear
•Discuss more completely in fracture
Fracture Surfaces
From Callister, p.187
Brittle Ductile
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Shear in Tension Test?’
’
’
’
’
’
’
’
’
’
2-D Mohr’s Circle
All stress states on a diameter of this circle are equivalent, just rotation of axes
’
(’/2, ’/2)
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Mohr’s CircleGeneralized 2-D Loading
• Stress state (tensor) depends on coordinate frame chosen
• Mathematical construct to ease coordinate transform
• Rotation of in material space is equivalent to 2* in Mohr space– Example: pure shear
• rotate 45º on material unit• rotate 90º on Mohr’s
circle
2xy
22x
yx
2R
2C
2
R
C
x
yxy
-/2/2
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Mohr’s Circle Examples
y
x
yx
xy
y
y
y
y
x x= -y
~ 20º
~ 70º
max
~ 10º
~ 35º
min max
= 0 (x ,xy)
(y ,yx)
(y ,0)
(y ,0)(x ,0)
max
45º
max
45º
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Failure mode - simple models
f
f
Ductile failure -Tresca criteria
f
f
Brittle failure- Maximum normal stress criteria
More complex failure theory - Von Mises (energy based)
21223
213
212e 2
2
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Hardness Testing• Scratch Test - very qualitative
– Mohs• Penetration Tests
– Brinell– Rockwell– Knoop– Vickers
• Hardness testing measures ability to resist plastic deformation– Need to eliminate effect of elastic deformation
• Brinell - load applied for 30 sec• Rockwell - initial preload and differential depth measurement
• To measure individual grain hardness, use Knoop or Vickers (lab #8)
Microhardness
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Brinell Hardness
D
d
F
22 dDD2D
FBHN
• Large, hard spherical indentor
• Relatively large loads (500-3000 kg)
• Hold load for 30 sec.• Leaves large indent in
specimen• Manually measure
indentation with calibrated microscope
• Single scale for all materials• Takes average hardness
over many grains
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Rockwell Hardness
d1
F1
d2
F2
d1
F1
d2
F2
Rockwell B
Rockwell C
• Most common hardness test method• Many scales: 2 important for us:
– Rockwell B- soft materials• Spherical indentor• Low loads (~100 kg)• small indention
– Rockwell C- hard materials• Conical indentor• Slightly higher loads (~150 kg)• Very small indention
• Measures differential penetration depth (initial preload, 10 kg)
• Machines are fully automated• Scale limits 20-100 (HRB, HRC, etc)
– if exceeded, switch test
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Conversions & Correlations
• Can convert from one scale to the other - approximately
• Brinell Hardness number (HB) is approximately related to tensile strength by:
• in steels only (empirical relation)
)ksi(HB*5.0)MPa(HB*45.3
UTS
UTS
From Callister, p.139
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• Scales are designed for flat specimens– Need “curvature correction” for round
specimens– Avoid specimen edges and other indents
• Specimen thickness must be at least 10x indention depth
Notes on Hardness Testing
Disadvantages“Relatively” nondestructive“Relatively” quantitative
AdvantagesCheapSimple test“Relatively” nondestructive“Relatively” quantitativeCorrelates with tensile strength
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Statistical Testing• When conducting experimental
testing, data will vary.• Be aware of your sources of
variability:– Specimen manufacture– Machine
variations/malfunctions– Environmental changes– Improper procedure– Random variables
• In lab, report your statistical differences, don’t hide them.
• For more in-depth analysis, look into IE230.
• Measure of average value:Mean Value
• Measure of scatter: Standard Deviation
• Relative measure of scatter:“Coefficient of variation”
n
xx
n
1ii
1n
xxs
n
1i
2i
xsCv
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Thermal Properties• Often design to utilize a material’s thermal properties
– Energy storage– Insulative or Conductive – Use thermally activated switches (beam expands and
closes switch)
• Properties we care most about– Heat Capacity (C)– Conduction (q) – Thermal Expansion (T)
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Heat Capacity & Conduction• Heat (Q) and Temperature (T) are related by
• Property can be measured at:– Constant volume, Cv
– Constant pressure, Cp
– Condensed phases (solid in our case) are more often at constant pressure
• Heat always flows from high energy to low
– qx is heat flux, k is thermal conductivity– Metals are excellent conductors due to free electrons– Ceramics and polymers are usually considered insulators
dTdQCCdTdQ
dxdTkqx
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Thermal Expansion• Temperature change will induce a change in dimensions
• If a bar is heated while physically constrained, induce a thermal stress
• Thermal expansion coefficient is strongly dependent on material (shape of force vs. atomic separation curve)– Polymers: ~100-200 x 10-6 C-1
– Metals: ~10-20 x 10-6 C-1
– Ceramics: ~1-10 x 10-6 C-1
oflT TTll
oflT
ofleT
TTEE
TT0
l = lo
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New Concepts & Terms• Elastic Properties
– Elastic (Young’s) Modulus• Secant Modulus• Tangent Modulus
– Poisson’s ratio– Linear vs. Nonlinear– Isotropic vs. orthotropic
• Yield-point behavior– Proportional limit– 0.2% offset yield strength– Upper & lower yield
• Plastic Deformation– Neck– Uniform vs. localized deformation– Mohr’s circle
• True vs. Engineering stress– Engineering: original area– True: instantaneous area
• Stress-strain curves– Yield strength– Ultimate Tensile Strength– Fracture Strength– Fracture Strain – Toughness, Resilience – Ductility (%AR, %EL)
• Fracture Surfaces– Cleavage– Cup-cone
• Hardness Testing– Rockwell– Brinell
• Statistics (mean, standard deviation)• Thermal Properties
– Heat Capacity– Thermal Expansion– Conduction
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