ME 330 Engineering Materials
Lectures 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)
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 Tests
Tension
Most common mechanical test
Gage section reduced to ensure deflection here
Load cell measures applied load
Extensometer ensures l measured from gage region
Compression
Similar to tensile test
Good for brittle specimens … hard to grip
Often much different properties in compression
Torsion
Test of pure shear
Member twisted by angle , calculate shear strain
Measure applied torque, calculate shear stress
Bending
In all cases, a displacement is applied and you measure load
Calculate stress from measured load
Calculate strain from change in gage length
Tension Test
Measure load and displacement
Compute stress and strain
Review of Stress and Strain
Often interested in measuring force and deformation in a size independent manner
•
Stress: force per unit area
•
Ao
Force

Area
F
Engineering :  
Ao
True :  T 
F
A
L  Lo
Engineering :  
Lo
 L
True :  T  ln 
 Lo 
Lo
A
From dT=dL/L
L
•
•
Traditional units: MPa or ksi
Ao is original area
•
A is instantaneous area
Strain: “relative” change in length
Length

Length
•
•
•
Dimensionless quantity
Lo is original length (“gage
length”)
L is instantaneous length
Relation Between Stress & Strain
Tension (+)
Compression (-)
Typical Stress-Strain Curves
Yield
Elastic
Plastic
 (MPa)
Strength 
ceramics
Today, we’ll talk about the different:
Regions in stress-strain space
Properties important to design
metals
Energy Absorption
Stiffness
polymers
0.1
10
100
Ductility 
 (%)
Elastic Region & Properties
Elastic region: proportional stress and strain
Stiffness = Modulus of Elasticity  ductility
Elastic
 (MPa)
ceramics
metals
Stiffness
polymers
~0.1
10
100
 (%)
Elastic Material Behavior
Elastic region: strain returns to zero when stress removed
Elastic Modulus (E) - measure of stiffness
Non-linear

E

Strain (%)
Stress (MPa)
Stress (MPa)
Linear
2
tangent modulus @ 2
1
secant modulus @ 1
Strain (%)
Elastic Behavior
linear
non-linear
E
Strain (%)
Stress (MPa)
Stress (MPa)
Tangent Modulus
Secant Modulus
Strain (%)
Atomic Level Effects on Modulus
• Strength of interatomic bonds: stiffness of springs
• Atomic packing: springs per unit area
F
Many metals
Most ceramics
F
Most polymers
F
F
Atomistic Origins of Elasticity
Force
d
F( r ) 
dr
ro
Energy
(r)
Atomic separation, r
Force
d  
dF 

E 
  2 

 dr  r  ro  dr  r  r
o
2
Strong bonding,
stiff
Weak bonding,
compliant
r
Final Notes on Stiffness
 (MPa)
• Interatomic bonding
– Ceramics - Ionic & Covalent
– Metals - Metallic & Covalent
– Polymers - Covalent &
Secondary
Ceramics
Metals
E
Polymers
• 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
Ceramics
Metals
Polymers
Material
E (GPa)
Silicon Carbide
475
Alumina
375
Glass
70
Steel
Brass
210
97
Aluminum
69
PVC
3.3
Epoxy
2.4
LDPE
0.23
Elastic Constitutive Relation
for 1-D Tensile Loading (linear materials)
•
Hooke’s Law: Stress and strain are directly related by
modulus of elasticity,
  E
•
z
Poisson’s ratio: Strain perpendicular to applied load is
related to the axial strain,
y
x
 
z

x
z
– Maximum (constant volume) :  = 0.50
– Minimum:  = 0
– Look at change in volume in a cube of side length, L
LxLxL  {L0 (1   xx )}x{L0 (1   yy )}x{L0 (1   zz )}
 {L0 (1   zz )}x{L0 (1   zz )}x{L0 (1   zz )}  {L0 (1   zz )}x{L0 (1   zz )}2
 L30{1  (1  2 ) zz   (  2) zz2   2 zz3 }
 L30{1  (1  2 ) zz }
– Volume increases during tensile, elastic deformation
(if   0.50)
Elastic Behavior

  1
Elastic
E 
 2
Modulus

 2  1
Axial
  E
Shear
  G
Poisson’s Ratio
Elastic Modulus
 
  transverse
 longitudinal
E  2G1  
 dF 
E 

 dr  r r 0

x
z

y
z
for isotropic material
Elastic +Plastic Properties
Yield
Elastic
Plastic
 (MPa)
Strength 
ceramics
metals
Energy Absorption
Stiffness
polymers
0.1
10
100
Ductility 
 (%)
Elastic Unloading
total strain = elastic + plastic
Stress (MPa)
Stress – always elastic,
no concept of plastic stress
E
E
   e  p

   p
E
Strain (%)
plastic
elastic
Review Stress and Strain
Engineering   F
Ao
Stress
Engineering   L  Lo  L
Lo
Lo
Strain
Constant
Volume
F
Lo ~
AL  A0 L0
F
True Stress      1   
A
L
A
True Strain  T  ln    ln  o   ln 1   
L 
 A
 o
Ao
do
A
L
~
d
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,
T 
Ao * Lo  A * L
P P Ao

A A Ao
,
A
L / Lo  1 


*

 *  L / Lo 
A Ao L / Lo  AA** LL 
 o o
 L  L 
  *  L / Lo    * o
Lo
T   *(1 )
 T  ln( L / Lo )  ln( 1   )
L
Ao
Lo
Elastic Constitutive Relation
for Simple Shear

Ao
F
F

F
F
Shear stress:  
Ao
Shear strain:   tan(  )
Again, stress and strain are directly related, by
shear modulus, G:   G
For isotropic materials, shear and elastic modulus are related by:
E  2G 1   
Stress & Strain in 3-Dimensions
z
z
zx
xz
x
xy
zy
yz
yx
y
y
x
x
y
z
xy
xz
yx
yz
zx
zy
x
y
z
xy
xz
yx
yz
zx
zy
Need to relate stress to strain
 ij  Cijkl kl
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 constants
Based on crystal symmetry, for cubic crystals  3 constants
For an isotropic crystal, need only 2 constants to describe 3-D response
Relate 1-D tests to complex loading
3-Dimensional Elastic Stress State
 1
 E

 

 x  E
  
 y   
 z   E
 
 xy   0
 yz  
  
 xz 
 0

 0



E
1
E



E

0
0
0
0
0
0
0
E
E
1
E
0
0
2(1   )
E
0
0
0
2(1   )
E
0
0
0
0





0   x 

  y 
0  
  z 
  xy 
0  
  yz 
 
0   xz 

2(1   ) 
E 
0
Orthotropic Material
Isotropic Material
 1
 E
 x

 x 
      yx
 y   Ex
 z  
     zx
 xy    E
x
 yz  
   0
 xz 
 0

 0

 xy
 xz
0
0
0
0
Ey
1
Ez
0
0
0
0
0
0
0
0
Gxy
0
0
0
G yz
0
Ey
1
Ey


 zy

Ez
 yz
Ez

0 
 
 x
0   y 
 
  z 
 
0   xy 

  yz 
0  
 xz 

0

Gxz 
Yield Point
Yield
Yield point marks the transition from elastic to
plastic deformation
Elastic
 (MPa)
ceramics
metals
Stiffness
polymers
~0.1
10
100
 (%)
Yield Point Behavior
uy
0.2%y
ly
 (%)
0.1
 (MPa)
 (MPa)
 (MPa)
y
0.2
 (%)
•
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
Yield
Elastic
Plastic
Stress is no longer proportional to strain
Plastic deformation is permanent, non-recoverable
 (MPa)
ceramics
metals
Stiffness
polymers
~0.1
10
100
 (%)
Plastic Phenomena
 (MPa)
Uniform
deformation
Localized
deformation
Necking begins:
E
E
d
0
d
y2 Upon unloading, strain is partitioned
between recovered and permanent.
y1
  e   p
 

E
 p
No concept of
“plastic stress”
 (%)
plastic
elastic
 (MPa)
Plastic Phenomena
Upon reloading, stress-strain curve
follows the same path to failure.
 (%)
 (MPa)
True vs. Engineering - Curve
True
Engineering
 (%)
•
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
True vs. Engineering - Curve
Compression
Plastic Constitutive Response
• Can approximate relation between true stress-strain
curve in constant plastic deformation region by:
 (MPa)
 T  K T
n
– 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”
 (%)
Measures of Energy Absorption:
Toughness vs. Resilience
Resilience: Ability to absorb energy without
permanent deformation - (elastic only)
 (MPa)
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.)
y
Modulus of Resilience
1
U r   d   y y
2
0
1  y   y
U r   y   
2  E  2E
2
Stress vs. Strain Eq.
 T  K Tn
for  y   T   u
Measures of Strength
f
Fracture stress, f
UTS
0.2%y
 (MPa)
Ultimate Tensile Stress, UTS
0.2% offset yield strength, 0.2%y
Fracture strain, f (~Ductility)
0.2%
 (%)
f
Measures of Ductility
Lo
Ao
L
Percent Elongation: % EL   L  Lo  * 100
 Lo 
Area Reduction: % AR   Ao  A  * 100

Ao

A
Sensitive to gage length
Does not account for necking
Insensitive to gage length
Does account for necking
Sensitive to cross-section
Stress-Strain Properties
Proportional limit
Yield Strength
UTS
% Elongation
% Reduction in Area
= highest linear stress
 y  0.2% offset or lower yield point
 u  Highest stress on curve
L  Lo
% EL 
x 100
Lo
% RA 
Ao  A
x 100
Ao
Material Deformation & Fracture
From Callister, p.126
Fracture Surfaces
Brittle
Ductile
Brittle
•Cleavage failure
•Flat,rough fracture surface
•No necking
•Failure in tension
•Ductile
From Callister, p.187
•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
Shear in Tension Test?
’
’
2-D Mohr’s Circle
(’/2, ’/2)
’
’

’
’
’

’
’
’
’
All stress states on a diameter of this
circle are equivalent, just rotation of axes
Mohr’s Circle
y
Generalized 2-D Loading

•
x
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
/2
-/2



xy
C
2

R

C
 x  y
2
  2 
2
R   x



xy
 2 
2
Mohr’s Circle Examples
(y ,yx)
y

yx
xy
x
min
max
~ 35º
 ~ 10º
y

max
max
=0
~ 70º

~ 20º



y
y
max
45º
(y ,0)
(x ,0)

x= -y
y
(x ,xy)
(y ,0)
45º
x


Failure mode - simple models
Brittle failure-
Ductile failure -
Maximum normal stress criteria
Tresca criteria


f
f

f

f
More complex failure theory - Von Mises (energy based)

2
 2   1 2   3   1 2   3   2 2
e 
2

1
2
Hardness Testing
• Scratch Test - very qualitative
– Mohs
• Penetration Tests
–
–
–
–
Brinell
Rockwell
Knoop
Vickers
Microhardness
• 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)
Brinell Hardness
F
D
d
BHN 

D
2
D
F
D2  d 2

• 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
•
F1
Rockwell B
F2
•
Most common hardness test
method
Many scales: 2 important for us:
– Rockwell B- soft materials
• Spherical indentor
• Low loads (~100 kg)
• small indention
d1
F1
Rockwell C
– Rockwell C- hard materials
d2
• Conical indentor
• Slightly higher loads (~150
kg)
• Very small indention
F2
•
•
•
d1
d2
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:
 UTS  3.45 * HB ( MPa )
 UTS  0.5 * HB ( ksi )
•
From Callister, p.139
in steels only (empirical
relation)
Notes on Hardness Testing
• 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
Advantages
Cheap
Simple test
“Relatively” nondestructive
“Relatively” quantitative
Correlates with tensile strength
Disadvantages
“Relatively” nondestructive
“Relatively” quantitative
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
n
 xi
•
x  i 1
n
Measure of scatter:
Standard Deviation
n
s
•
  xi  x 
i 1
2
n 1
Relative measure of scatter:
“Coefficient of
variation”
s
Cv 
x
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
dQ  CdT
•
•
C
dQ
dT
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
q x  k
dT
dx
– qx is heat flux, k is thermal conductivity
– Metals are excellent conductors due to free electrons
– Ceramics and polymers are usually considered insulators
Thermal Expansion
•
Temperature change will induce a change in
dimensions
l
l
•
  T   l T f  To 
If a bar is heated while physically constrained,
induce a thermal stress
T   e  0
l = lo
•
 l T f  To   
 T   E l T f  To 

E
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
New Concepts & Terms
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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
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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 & 3