Mechanical Behavior
Material Sciences and Engineering
MatE271
Week9
1
Mechanical Behavior
1. Support load
- Applied vs. dead weight
- Static vs. dynamic
2. Controlled deformation
-Small vs. large
3. Reliability
• How microstructure affects
mechanical properties
• Tailored microstructure for
mechanical properties
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Application
Week 9
2
1
Goals for this unit (Ch. 6)
Ø Detailed coverage of basic mechanical properties
- Describe the concepts of stress and strain
- Differentiate between elastic and plastic deformation
- Quantify elastic properties of materials
- Describe measures of hardness, ductility,
toughness and strength
- Understand fracture, fatigue and creep failures
Material Sciences and Engineering
MatE271
3
Week 9
6.1 Stress vs. Strain
Engineering Stress - load/original cross sectional area
Area
(Ao)
load
Displacement (DL)
Length
(Lo)
Stress s = P/ Ao
(N/m2 )
Strain e = DL/ Lo
For tensile or compressive stresses
There are also shear and torsional stresses
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
4
2
Stress vs. Strain: units
ØStress
s = F/Ao (where Ao is the original cross-sectional
area)
psi (pounds force per square inch)
MPa (Mega Pascals = 106 N/m2 )
ØStrain
e = DL/Lo (where Lo is the original length)
unitless-sometimes expressed as a
percentage
Material Sciences and Engineering
MatE271
Week 9
5
Shear Stress vs. Shear Strain
Dy
a
Lo
z
y
Ps
x
t = Ps / As
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
t
g = tan a = Dy /Lo
MatE271
Week 9
6
3
Application of Loads
Tension
Material Sciences and Engineering
Compression
MatE271
Week 9
7
Application of Loads
Shear
Torsional
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
8
4
Tensile testing
Load cell
• One of the most common stress-strain
Grip
tests performed is tensile testing
Specimen
Gage length
• There are standards for the shape and size
Grip
Crosshead
and finish of test specimens
• Tensile testing equipment elongates a
specimen at a constant rate and measures:
– Load (load cell)
– Elongation (extensometer)
Material Sciences and Engineering
MatE271
Tensile testing
Week 9
Tensile
strength
Yield
strength
Load cell
9
Fracture
Specimen
Gage length
Grip
Stress
Grip
Crosshead
Elastic
Plastic
Strain
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
10
5
Elastic Deformation
ØDefinition
• When stress and strain are proportional
• Non-permanent
• When stress is removed, strain disappears
• i.e. the sample returns to it’s original shape
ØWhat is happening?
• small changes in inter-atomic spacing
• bonds are stretching but not breaking
Material Sciences and Engineering
MatE271
Week 9
11
Modulus of elasticity depends on bond strength!
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
12
6
Modulus of Elasticity
• Slope of stress-strain curve in
elastic region
s = (E)(e) (Hooke’s Law)
E - modulus of elasticity
(Young’s modulus)
Material
E (GPa)
Steel
207
Aluminum
69
Al2O3
370
SiC
470
Material Sciences and Engineering
Elastic modulus is the
slope of the atom force
vs distance curve at
equilibrium spacing
MatE271
Week 9
13
Elastic Stress - Strain Behavior
ØShear stress and strain are also
proportional to each other in the elastic
region:
t = Gg
t = shear stress
g = shear strain
G = shear modulus
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
Compare to
s = Ee
MatE271
Week 9
14
7
Example Problem
ØA tensile force of 2000N is applied along
the axis of an aluminum cylindrical
specimen (E = 70 GPa, 1 m long, radius
0.01 m). Assuming the deformation is
elastic, estimate the elongation.
Material Sciences and Engineering
MatE271
Week 9
15
Poisson’s Ratio
ØQ. When a specimen is elongated in
one direction - what happens in the
other two directions?
ØA. Usually, they contract.
ØThe ratio of lateral to axial strains is
called Poisson’s ratio
ey
e
The - sign assures u
u=- x =will be positive
ez
ez
z
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
16
8
Poisson’s Ratio
Material Sciences and Engineering
MatE271
Week 9
17
Poisson’s Ratio
Ø Q. What is Poisson’s ratio for an isotropic
material?
Ø A. If the properties are the same in all
directions, then n = 0.25
Ø Most metals have a n = 0.25 to 0.35
Ø Admissible range
-1 £ n £ 0.5
• for no volume change n = 0.5
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
18
9
Poisson’s Ratio
ØShear and elastic moduli are related:
E = 2G(1+n)
ØMost materials are elastically anisotropic
• E varies with crystallographic direction
• most polycrystalline materials may be considered
to be isotropic
ØMost engineering materials are
polycrystalline
Material Sciences and Engineering
MatE271
Week 9
19
Tensile
strength
Plastic Deformation
Fracture
Stress
Ø There is a limit to
Yield
strength
how much a metal
can be deformed
before it will not
return to its original
shape when the stress
Plastic
is removed
Elastic
Ø After reaching elastic
Strain
limit, deformation is
plastic (in metals
dislocation movement).
(in ceramics
micro-cracking)
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
20
10
Plastic Deformation
Stress
Elastic Plastic
sy
Strain
Material Sciences and Engineering
Ø In metals:
Plastic deformation
corresponds to the breaking
of bonds with atom
neighbors and reforming
bonds with new neighbors
- (dislocation motion)
Ø Beyond Yield point,
stress is not proportional
to strain (Hooke’s law is
not valid)
MatE271
Week 9
21
Slip produces plastic deformation
• During plastic deformation,
shear stresses cause dislocation
movement resulting in slip.
• This deformation is permanent
(not recovered when stress is
removed.)
Check week 5 slides 20-33
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
22
11
Yielding and Yield Strength
Ø Most structures are designed such that only elastic
deformation occurs when a stress is applied
Ø The point at which plastic deformation occurs
must be known (what stress level will bend the
metal permanently?)
Ø Phenomenon is called yielding
Ø For metals that experience a gradual transition, the
point is called the proportional limit
Material Sciences and Engineering
MatE271
Week 9
23
Proportional Limit
Elastic Plastic
Stress
sy
P
ØHow do you know
where sy is?
ØBy convention, a
specified strain offset
of 0.002 is used to
identify the yield
strength, sy.
Strain
0.002
0.2%
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
24
12
Stress (MPa)
Elastic recovery after plastic deformation
Elastic Recovery
Strain (mm/mm)
Material Sciences and Engineering
MatE271
25
Week 9
Work Hardening (Strain Hardening)
Yield
point
multiplies the number of dislocations
• As each increment of plastic
Stress
• Process of plastic deformation (slip)
deformation occurs, dislocations find
it harder and harder to move because
of “entanglement” with ever increasing
Strain
number of dislocations
• Result is that yield strength increases after
plastic deformation (“strain hardening”)
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
26
13
Yield Point Phenomenon
Stress
Ø Some steels show a yield
point which occurs abruptly
Upper
Yield Pt.
sy
Lower
Yield Pt.
Strain
Material Sciences and Engineering
Ø Yield point is taken as the
average stress of the lower
yield point
Ø Yield points for steels vary
from 5,000 to 200,000 psi!
MatE271
27
Week 9
Tensile
strength
Tensile Strength
Yield
strength
Fracture
Stress
Ø After yielding, stress increases
to a maximum, then decreases,
and eventually the material
Elastic
fractures
Ø Tensile strength is the stress
at the maximum of the engineering
stress vs strain curve.
Ø Deformation up to this point is uniform
throughout the sample
Ø After maximum stress, necking occurs
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
Plastic
Strain
28
14
True vs. Engineering Stress and Strain
• Does material actually get
weaker after TS has been
exceeded?
• No, that is an “artifact” of
using engineering stress
instead of true stress in the plot.
• X-sectional area is decreasing,
and especially after necking
starts.
Material Sciences and Engineering
MatE271
Week 9
29
True Stress and Strain
True Stress = P/A
True Strain = DL/L
(A: is the current area)
(L:current length)
Vol= AL =Ao Lo
P L
strue=
Ao Lo
• When strength of a metal is cited, for design purposes, the
yield strength is used.
• The fracture strength is the stress at fracture
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
30
15
Definition - Ductility
Ø Measure of degree of plastic deformation that has
been sustained before fracture
Ø If there is little plastic deformation before fracture
--- called brittle
Ø Ductility = percent elongation
%EL =
Material Sciences and Engineering
(l f - lo )
lo
x 100
MatE271
Week 9
31
MatE271
Week 9
32
Ductility
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
16
Ductility
Why is ductility important?
Ø Specifies how much a structure will deform before
fracture
Ø Specifies how much deformation is allowable
during fabrication
Ø Ductility is strongly temperature dependent
– i.e., ductile-to-brittle transitions
Material Sciences and Engineering
MatE271
Week 9
33
Toughness
Ø Capacity to absorb energy when deformed, up to
fracture
Ø Given by area under curve
Ø Describes the combination of strength and ductility
tougher
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
34
17
Charpy Impact Test of Toughness
Seldom have complete stess-strain curve, so an impact
test is usually used to measure toughness
Material Sciences and Engineering
MatE271
Week 9
35
Comparison of Mechanical Characteristics
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
36
18
F
Hardness
Hardness: surface resistance to indentation
d
H= F/Aprojected
Ap
- Quantitative means use a small indenter
forced into the surface
- Indenter: round (ball)
pointed (cone or pyramid)
Material Sciences and Engineering
MatE271
Week 9
37
Hardness Tests
Ø There is a correlation between tensile strength
and hardness
Ø Hardness tests are simple and inexpensive
Ø Hardness tests are nondestructive (you still have a
usable sample when you are done)
Ø Other properties can be estimated from hardness
information.
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
38
19
Strength, MPa
Tensile Strength often scales with Hardness
Hardness, BHN
Material Sciences and Engineering
MatE271
Week 9
39
Hardness Tests
Ø Although the scales are quantitative, the numbers
are only relative (rather than absolute values)
Ø Only compare hardness values obtained using the
same method
Ø Methods of testing
• Rockwell Hardness
• Brinell Hardness
• Knoop and Vickers Microhardness
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
40
20
Rockwell Hardness
Ø Most common method
Ø Indenters are hardened steel balls of various
diameters
Ø The hardness is determined by the
difference in depth of the indentation of two
different loads
Ø Modern instruments are automated
Material Sciences and Engineering
MatE271
Week 9
41
Brinell Hardness
Ø Hard, spherical indenter is forced into the surface
(like for Rockwell)
Ø The indentor is steel or WC (tungsten carbide)
Ø Standard loads are used
Ø The load is maintained for a specified amount of
time
Ø The diameter of the indentation is measured with a
microscope
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
42
21
Knoop and Vickers
Ø Very small diamond indenter with a
pyramid geometry is forced into the
specimen.
Ø The resulting impression is measured
Ø Knoop is frequently used for ceramics
Material Sciences and Engineering
MatE271
Week 9
43
Summary of Standard Hardness Tests
Material Sciences and Engineering
Material Sciences and Engineering,
MatE271
MatE271
Week 9
44
22