ENGR 225
Section 3.1

3.8
Load increase
–
> stress increase.
Deformation increase
–
> strain increase.
Relation between Stress and Strain.
Tension Test
Standard test specimen
Universal Testing Machine (UTM)
Tension Test
http://www.youtube.com/watch?v=9suShuEwc7I&feature=related
Nominal Stress:
σ
= P/A
0
Where A
0
is original cross section
Nominal Strain:
ε
=
δ
/L
0
Where L
0
is the original gauge length and
δ
is the change in gauge length
No two stress

strain diagrams for a particular material will be
exactly the same since the results depend on:
material’s composition
microscopic imperfections,
way material is manufactured
rate of loading
temperature
Stress

Strain Diagrams
Stress

Strain Diagrams
Stress is proportional to strain
(within proportional limit)
in
most Solids
Not so for fluids !!
Mild Steel
y
= 25
pl
Ductility
•
The extent of plastic deformation that a
material undergoes before fracture.
Ductile Material
–
A material that can be subjected to large strains
before it ruptures.
–
Ductility can be measured by percent elongation or
percent reduction in area
at the time of fracture
.
–
Mild Steel : 38%
–
Mild Steel : 60%
%)
100
(
Elongation
Percent
o
o
f
L
L
L
%)
100
(
Area
in
Reduction
Percent
o
o
f
A
A
A
Ductility
•
Why use ductile materials?
–
Capable of absorbing shock or energy
–
When overloaded, usually exhibit large
deformation before failing.
Aluminum
Yield Strength
•
Yield Strength is not a physical property of
the material.
•
We will use approach that
–
Yield strength
–
Yield point
–
Elastic limit
–
Proportional limit
•
All coincide unless otherwise stated
Natural Rubber
(Nonlinear Elastic Behavior)
Brittle Material
–
A material that exhibits little of no yielding
before failure. It fractures suddenly under
tension.
Grey Cast Iron
Concrete
Modulus of Elasticity
E =
Stress / Strain
Hooke’s Law
= E
Modulus of Elasticity
•
Modulus of Elasticity (E) indicates stiffness
of a material.
If material is stiff, E is large
(for steel, E = 200 GPa)
If material is spongy, E is small
(for vulcanized rubber, E = 0.70 MPa)
•
Strength
•
Ductility
•
Brittleness
•
Stiffness
•
Resilience
•
Toughness
•
Endurance
•
Rigidity
Material Properties
Strain Hardening
•
Strain hardening is
used to establish a
higher yield point for a
material
•
The modulus of
elasticity stays the
same.
•
The ductility
decreases.
Chapter 3 Lecture Example 1
A tension test for a steel alloy results in the stress

strain diagram shown.
Calculate the modulus of elasticity and the yield strength based on a 0.2%
offset. Identify on the graph the ultimate stress and the fracture stress.
Strain Energy
•
Energy stored in a
material due to
deformation
Modulus of Resilience
Chapter 3
Lecture
Example 2
The stress

strain diagram for an aluminum alloy that is used for making aircraft
parts is shown. If a specimen of this material is stressed to 600 MPa, determine
the permanent strain that remains in the specimen when the load is released.
Also, find the modulus of resilience both before and after the load application.
Toughness
•
The area under the stress

strain curve
•
The amount of energy per unit volume that
the material dissipates prior to fracture.
Modulus of Toughness
Strength, Toughness
and Ductility
•
Strength ~ related to the height of the
curve
•
Ductility ~ related to the width of the curve
•
Toughness ~ related to area under the
curve
Poisson’s Ratio
Poisson’s Ratio
Poisson’s Ratio
•
When a deformable body is subjected to a
tensile force, not only does it elongate, but
it also contracts.
•
Longitudinal is in the direction of the tensile
force.
long
lat
French mathematician Simeon Denis Poisson
•
Value of
is positive
•
Value of
same in tension and compression
•
Range of
0.25 to 0.35
•
constant only in the elastic range
•
Maximum possible value of
is 0.5 (Section 10.6)
•
Only Longitudinal force is acting to cause the lateral
strain.
Poisson’s Ratio
Shear Stress
–
Strain Diagram
Shear Stress
–
Strain Diagram
G
Modulus of Rigidity
G
)
1
(
2
v
E
G
Derivation of this equation in section 10.6
Chapter 3 Lecture Example 3
This is a titanium alloy. Determine the shear modulus, G, the
proportional limit, and the ultimate shear stress. Find the
maximum d where the material behaves elastically. What is the
magnitude of V to cause d?
Failure of turbine blade due to creep
Failure of steam pipe due to creep
Creep
•
Creep is the time

related deformation of a
material for which temperature and stress play
an important role.
•
Creep results from sustained loading below the
measured yield point.
•
Members are designed to resist the effects of
creep based on their creep strength, which is the
highest amount of stress a member can
withstand during a specified amount of time
without experiencing creep strain.
Creep
Fatigue
•
Fatigue occurs in metals when stress or strain is
cycled. It causes a brittle fracture to occur.
•
Members are design to resist fatigue by
ensuring that the stress in the member does not
exceed its endurance limit.
•
This the maximum stress member can resist
when enduring a specified number of cycles.
Concept Questions
•
How would you explain strain to one of
your engineering classmates?
–
Strain is a linear deformation due to stress
–
Percent increase in length under tension or
compression
Concept Questions
•
How would you characterize a ductile
material? Give several examples.
–
A material that can be subjected to large
strains before it ruptures.
–
Steel, wood, natural rubber, brass, copper,
gold, aluminum
Concept Questions
•
How would you characterize a brittle
material? Give several examples.
–
A material that exhibits little or no yielding
before failure.
–
Cast iron, concrete, glass, ceramics
Concept Questions
•
What is the difference between
engineering stress and true stress?
–
Engineering stress assumes a constant cross
sectional area during elongation.
A tale of two cities
•
How can you explain Elasticity?
–
Property of material by which it returns to
original dimensions on unloading.
•
How can you explain Plasticity?
–
C
haracteristic
of material by which it cannot
return to original dimensions on unloading
and undergoes permanent deformation.
Strength
–
Capacity to resist loads
–
yield stress. Higher
y
, higher strength.
Resilience
–
Measure of energy absorbed without permanent damage .
Toughness
–
Measure of energy absorbed before fracturing.
Ductility
–
Property of material which allows large deformation before fracture.
Brittleness
–
Property of material which allows little or no yielding before fracture.
Endurance
–
Ability to sustain cyclic loads.
Stiffness
–
Mechanical property indicated by E. Higher E means stiff material. Steeper
slope in the Stress
–
Strain diagram.
Rigidity

Mechanical property of material indicated by G.
Material Properties
3.7
(a) A structural member in a nuclear reactor is made of Zirconium
alloy. If an axial load of 4 kip is to be supported by the member,
determine the required cross sectional area. Yield stress for Zirconium
is 57.5
ksi
. Factor of safety 3.
(b) What value of load would cause an elongation of 0.02 inch in
this member if length of member is 3 ft.
E
zr
= 14 x 10
3
ksi
.
3.26
A short cylindrical block of 2014

T6 Aluminum having an original
diameter of 0.5 in and an original length of 1.5 in is placed in the
smooth jaws of a vise and a compressive force of 800 lb is applied.
Determine
(a) The decrease in length.
(b) The new diameter.
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