# ENGR 225 Section 3.1 - 3.8

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29 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

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ENGR 225

Section 3.1
-

3.8

> stress increase.

Deformation increase

> strain increase.

Relation between Stress and Strain.

Tension Test

Standard test specimen

Universal Testing Machine (UTM)

Tension Test

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

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

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.

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

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

How can you explain Plasticity?

C
haracteristic

of material by which it cannot
and undergoes permanent deformation.

Strength

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

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.