me 330 engineering materials
DESCRIPTION
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”, etcTRANSCRIPT
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)
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
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
Tension Test
Measure load and displacement
Compute stress and strain
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
Relation Between Stress & Strain
Tension (+) Compression (-)
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
Elastic Region & Properties
~0.1 10 100
ceramics
metals
polymers
(M
Pa)
(%)
Stiffness
ElasticElastic region: proportional stress and strainStiffness = Modulus of Elasticity ductility
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
Elastic BehaviorS
tress
(MP
a)
Strain (%)
Stre
ss (M
Pa)
Strain (%)
linear non-linear
E Secant Modulus
Tangent Modulus
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
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
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
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
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
Elastic +Plastic Properties
0.1 10 100
ceramics
metals
polymers
Stre
ngth
Ductility
(M
Pa)
(%)
Stiffness
Energy Absorption
Elastic Plastic
Yield
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
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~
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
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(
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
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
Yield Point
~0.1 10 100
ceramics
metals
polymers
(M
Pa)
(%)
Stiffness
Elastic
Yield Yield point marks the transition from elastic to plastic deformation
(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
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
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
Upon reloading, stress-strain curvefollows the same path to failure.
Plastic Phenomena
(MPa
)
(%)
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
True vs. Engineering - CurveCompression
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)
(%)
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
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
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)
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
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
Material Deformation & Fracture
From Callister, p.126
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
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)
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
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
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
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
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
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
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
• 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
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
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)
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
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
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
Next Lecture ...
• Please read chapters 2 & 3Please read chapters 2 & 3