Mechanics of Materials - Rami Zakaria · Mechanical Properties of Materials Mechanics of Materials Dr. Rami Zakaria Reference: 1. ... strength of the material connected to its physical
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5. Mechanical Properties of Materials
Mechanics of Materials
Dr. Rami Zakaria
Reference:
1. Mechanics of Materials: R.C. Hibbeler, 9th ed, Pearson
2. Mechanics of Materials: J.M. Gere & B.J. Goodno, 8th ed, Cengage Learning
2
Term Definition Notes
Specimen Typical piece of the material under study/ Sample.
Elasticity Being able to recover original shape after deformation
To yield To give up/ to bend
Ultimate Maximum
To Fracture/ Fail To break
Permanent Cannot be recovered / everlasting
Ductile materials … deform (strain) under tensile forces / can be thinned out
Brittle materials … break suddenly / don’t deform before breaking
Stiffness Rigidity / doesn’t bend easily
Resilience Ability to recover to original form or shape/ Energy per unit
volume during the elastic deformation
To creep To act slowly
Fatigue Weariness / weakening of a material because of repeated stresses
Introduction
3
We will see how stress and strain are connected together.
In order to perform tension or compression test
on a material, we make a standard shape
specimen (constant circular cross-section) and
measure strain against stress.
We are interested in stress-strain diagram,
which can be obtained by experiment (as the
strength of the material connected to its physical properties).
For tension test, we apply force and measure the
change in length (using optical-extensometer for
example), or we can read the strain directly from
an electrical-resistance strain gauge.
The stress-Strain Diagram
4
1- Conventional Stress-Strain Diagram
00 LA
P
Let’s take an example, the Conventional Stress-Strain Diagram for steel:
We notice 4 regions depending on the material
behaviour:
Elastic Behavior Curve is a straight line (for most of the region)
until a proportional limit stress . Then the
curve bend and flatten out until the elastic limit,
just before the yield stress (usually difficult
to find as it is very close the proportional
limit) . If the load removed until this point, the
specimen will return back to its original shape.
pl
Y
5
Yielding This is when any increase in stress will cause a permanent deformation (we call it plastic
deformation). At this region the specimen will continue changing length (strain) without
any increase in load
Strain Hardening Yielding is ended, and an increase in load will
make the curve to rise until it reaches a
maximum stress, we call ultimate stress .
Up to this point the length is increasing and the
cross-section area is decreasing in a (kind of)
uniform manner.
u
Necking After the ultimate stress, the cross-section area
will decrease in a local region, making what
looks like a neck. Eventually the graph curve
downward until the specimen breaks at the
fracture stress . f
6
2- True Stress-Strain Diagram
Instead of calculating the (engineering) stress and strain using the original area
A0 and length L0 , we can calculate the true stress and true strain at the instant
when the load is measured.
The difference between the two graph become more obvious in the strain-
hardening region, and we can see in the true diagram that stress keeps
increasing (because the area is decreasing at the necking area) .
For most application we are interested in the elastic region, because the
deformation there is not sever and not permanent. The difference between the
two graphs in this region is very small.
Ductile and Brittle Materials
7
Ductile Materials:
They have 4 distinct behaviours (elastic, yielding, strain hardening,
and necking) when they are loaded.
This means they strain a lot before they fracture.
Examples: mild steel, brass, and zinc.
Some materials do not have a well-
defined yielding point, so we assume it
(graphically) using an offset method, and
call it a yield strength σYS (for example
aluminium alloy yield strength at 0.2%)
8
Brittle Materials
They have very little or no yielding before they fracture.
This means they fracture and fail suddenly.
When they are under tension, they
don’t have a well-defined tensile
fracture stress σtf (it takes place in a
random imperfection or microscopic crack,
and it is relatively small). What we can
calculate is the average of that stress
from many cases.
When they are under compression,
they show a much higher
compressive fracture stress σcf .
Examples: gray cast iron , glass, and concrete.
Notes:
1- Because the fracture point of concrete under tensile forces is small, we almost always
reinforce the concrete structures with steel bars.
2- In most cases materials can shift between the two behaviours (Ductile & Brittle)
depending on the material components (such as the carbon percentage), or the temperature
(high temperature make some materials softer – more ductile), etc…
10
Hooke’s Law
11
We saw that the relationship between stress and strain is linear within the elastic region.
We can describe this relationship using Hooke’s law
E
This relationship represents (only) the straight line at
the beginning of the σ-ε graph.
E is a constant called the modulus of elasticity, (it
represents the slope of the line), and it has the same
unit as stress (Pa, psi, …). It describes the stiffness of
a material (stiff materials have larger modulus of
elasticity E).
Notice in the example of steel alloys that all grades
have almost the same E=29(103) ksi.
Strain Hardening:
12
We saw that when a specimen is loaded in the plastic
region, and then unloaded, the elastic strain is recovered
but the plastic strain remains (permanent set).
In the graphs, notice that the slope OA is similar to the
slope O’A’.
What happened that the new yielding point now (A’) is
higher that the original yielding point (A), also the
elastic region is larger, while the plastic region is
smaller than originally.
13 [2] James M.Gere & Barry J.GooDNO
When a load is applied to a member, the deformations cause
strain energy to be stored in the material. The strain energy
per unit volume or strain energy density is equivalent to the
area under the stress–strain curve.
This area up to the yield point is called the modulus of
resilience, ur.
It represents the total energy per unit volume can be
absorbed without causing permanent damage (useful when
designing shock absorbers).
The entire area under the stress– strain diagram is called the
modulus of toughness, ut.
It represents the total energy per unit volume can be
absorbed before failing (useful when designing members
which could be overloaded accidently).
Strain Energy:
Eu
pl
plplr
2
2
1
2
1
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19
(Classroom Exercise) A bar having a length of 5 in. and cross-sectional area of 0.7 in2 is subjected
to an axial force of 8000 lb.
If the bar stretches 0.002 in., determine the modulus of elasticity of the material (the material
has linear-elastic behavior).
20
(Classroom Exercise) The stress–strain diagram for polyethylene, is determined from testing a
specimen that has a gauge length of 10 in. a) Determine the Modulus of Elasticity E
b) Determine the modulus of resilience, ur
c) If a load P on the specimen develops a strain of ε=0.024 . Determine the approximate length of the
specimen, when the load is removed. Assume the specimen recovers elastically.
σpl
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22
Poisson’s Ratio:
rLlatlong
,
Only within the elastic range of a homogenous and isotropic materials
23
When materials are in service for long periods of time, considerations of
creep become important. Creep is the time rate of deformation, which occurs
at high stress and/or high temperature. Design requires that the stress in the
material not exceed an allowable stress which is based on the material’s
creep strength.
Fatigue can occur when the material undergoes a large number of cycles of
loading. This effect will cause microscopic cracks to form, leading to a brittle
failure. To prevent fatigue, the stress in the material must not exceed a
specified endurance or fatigue limit.
Failure Due to Creep & Fatigue
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