1. Stress & Strain
Ductile materials
Fig 2. A stress–strain curve typical of structural steel
1. Ultimate Strength
2. Yield Strength
3. Rupture
4. Strain hardening region
5. Necking region.
A: Apparent stress (F/A0)
B: Actual stress (F/A)
Steel generally exhibits a very linear stress–strain
relationship up to a well defined yield point (Fig.2).
The linear portion of the curve is the elastic region
and the slope is the modulus of elasticity or Young's
Modulus. After the yield point, the curve typically
decreases slightly because of dislocations escaping
from Cottrell atmospheres. As deformation continues,
the stress increases on account of strain hardening
until it reaches the ultimate strength. Until this point,
the cross-sectional area decreases uniformly because

2. of Poisson contractions. The actual rupture point is in
the same vertical line as the visual rupture point.
However, beyond this point a neck forms where the
local cross-sectional area decreases more quickly than
the rest of the sample resulting in an increase in the
true stress. As shown in Fig.2, On an engineering
stress–strain curve this is seen as a decrease in the
apparent stress. However if the curve is plotted in
terms of true stress and true strain the stress will
continue to rise until failure. Eventually the neck
becomes unstable and the specimen ruptures
(fractures).
Less ductile materials such as aluminum and medium
to high carbon steels do not have a well-defined yield
point. For these materials the yield strength is
typically determined by the "offset yield method", by
which a line is drawn parallel to the linear elastic
portion of the curve and intersecting the abscissa at
some arbitrary value (most commonly 0.2%). The
intersection of this line and the stress–strain curve is
reported as the yield point. Plastic region is the point
where the material will stay deformed, The elastic
region is the point where the material can stretch no
further. Failure point is when the object breaks.

3. Brittle materials
Fig.3 Stress Strain Curve for Brittle materials
Brittle materials such as concrete and carbon fiber do
not have a yield point, and do not strain-harden.
Therefore the ultimate strength and breaking strength
are the same. A most unusual stress-strain curve is
shown in Fig.3. Typical brittle materials like glass do
not show any plastic deformation but fail while the
deformation is elastic. One of the characteristics of a
brittle failure is that the two broken parts can be
reassembled to produce the same shape as the original
component as there will not be a neck formation like
in the case of ductile materials. A typical stress strain
curve for a brittle material will be linear. Testing of
several identical specimen, cast iron, or soil, tensile
strength is negligible compared to the compressive
strength and it is assumed zero for many engineering
applications. Glass fibers have a tensile strength

4. stronger than steel, but bulk glass usually does not.
This is because of the stress intensity factor associated
with defects in the material. As the size of the sample
gets larger, the size of defects also grows. In general,
the tensile strength of a rope is always less than sum
of the tensile strength of its individual fibers.(less than
2%)

5. A universal testing machine
1.tensile machine:
Tensile testing, also known as tension testing,[1]
is a
fundamental materials science test in which a sample
is subjected to uniaxial tension until failure. The
results from the test are commonly used to select a
material for an application, for quality control, and to
predict how a material will react under other types of
forces. Properties that are directly measured via a
tensile test are ultimate tensile strength, maximum
elongation and reduction in area.[2]
From these
measurements the following properties can also be
determined: Young's modulus, Poisson's ratio, yield
strength, and strain-hardening characteristics.[3]

6. 2.three point pending machine :
Poisson's ratio
Figure 1: A cube with sides of length L of an
isotropic linearly elastic material subject to tension
along the x axis, with a Poisson's ratio of 0.5. The

7. green cube is unstressed, the red is expanded in the
x direction by ΔL due to tension, and contracted in
the y and z directions by ΔL'.
Poisson's ratio (ν), named after Siméon Poisson, is
the ratio, when a sample object is stretched, of the
contraction or transverse strain (perpendicular to
the applied load), to the extension or axial strain (in
the direction of the applied load).
When a material is compressed in one direction, it
usually tends to expand in the other two directions
perpendicular to the direction of compression. This
phenomenon is called the Poisson effect. Poisson's
ratio ν (nu) is a measure of the Poisson effect. The
Poisson ratio is the ratio of the fraction (or percent)
of expansion divided by the fraction (or percent) of
compression, for small values of these changes.
Conversely, if the material is stretched rather than
compressed, it usually tends to contract in the
directions transverse to the direction of stretching.
Again, the Poisson ratio will be the ratio of relative
contraction to relative stretching, and will have the
same value as above. In certain rare cases, a
material will actually shrink in the transverse
direction when compressed (or expand when
stretched) which will yield a negative value of the
Poisson ratio.

8. The Poisson's ratio of a stable, isotropic, linear
elastic material cannot be less than −1.0 nor greater
than 0.5 due to the requirement that Young's
modulus, the shear modulus and bulk modulus
have positive values [1]
. Most materials have
Poisson's ratio values ranging between 0.0 and 0.5.
A perfectly incompressible material deformed
elastically at small strains would have a Poisson's
ratio of exactly 0.5. Most steels and rigid polymers
when used within their design limits (before yield)
exhibit values of about 0.3, increasing to 0.5 for
post-yield deformation (which occurs largely at
constant volume.) Rubber has a Poisson ratio of
nearly 0.5. Cork's Poisson ratio is close to 0:
showing very little lateral expansion when
compressed. Some materials, mostly polymer
foams, have a negative Poisson's ratio; if these
auxetic materials are stretched in one direction,
they become thicker in perpendicular directions.
Anisotropic materials can have Poisson ratios
above 0.5 in some directions.
Assuming that the material is stretched or
compressed along the axial direction (the x axis in
the diagram):
where
ν is the resulting Poisson's ratio,

9. is transverse strain (negative for axial
tension (stretching), positive for axial
compression)
is axial strain (positive for axial tension,
negative for axial compression
Isotropic materials
For a linear isotropic material subjected only to
compressive (i.e. normal) forces, the deformation
of a material in the direction of one axis will
produce a deformation of the material along the
other axes in three dimensions. Thus it is possible
to generalize Hooke's Law (for compressive forces)
into three dimensions:
or
where

10. , and are strain in the direction of x, y and
z axis
σx , σy and are stress in the direction of x, y
and z axis
E is Young's modulus (the same in all
directions: x, y and z for isotropic materials)
ν is Poisson's ratio (the same in all directions:
x, y and z for isotropic materials)
These equations will hold in the general case which
includes shear forces as well as compressive forces,
and the full generalization of Hooke's law is given
by:
where δij is the Kronecker delta and
Negative Poisson's ratio materials(important)
Some materials known as auxetic materials display a
negative Poisson’s ratio. When subjected to positive
strain in a longitudinal axis, the transverse strain in the
material will actually be positive (i.e. it would increase
the cross sectional area). For these materials, it is usually
due to uniquely oriented, hinged molecular bonds. In
order for these bonds to stretch in the longitudinal

11. direction, the hinges must ‘open’ in the transverse
direction, effectively exhibiting a positive strain.[6]
Applications of Poisson's effect
One area in which Poisson's effect has a considerable
influence is in pressurized pipe flow. When the air or
liquid inside a pipe is highly pressurized it exerts a
uniform force on the inside of the pipe, resulting in a
radial stress within the pipe material. Due to Poisson's
effect, this radial stress will cause the pipe to slightly
increase in diameter and decrease in length. The decrease
in length, in particular, can have a noticeable effect upon
the pipe joints, as the effect will accumulate for each
section of pipe joined in series. A restrained joint may be
pulled apart or otherwise prone to failure.[7]
Another area of application for Poisson's effect is in the
realm of structural geology. Rocks, like most materials,
are subject to Poisson's effect while under stress. In a
geological timescale, excessive erosion or sedimentation
of Earth's crust can either create or remove large vertical
stresses upon the underlying rock. This rock will expand
or contract in the vertical direction as a direct result of
the applied stress, and it will also deform in the
horizontal direction as a result of Poisson's effect. This
change in strain in the horizontal direction can affect or
form joints and dormant stresses in the rock.[8]
.
The use of cork as a stopper for wine bottles is the result
of the fact that cork has a Poisson ratio of practically
zero. This means that, as the cork is inserted into the
bottle, the upper part which is not yet inserted will not
expand as the lower part is compressed. The force
needed to insert a cork into a bottle arises only from the

12. compression of the cork and the friction between the
cork and the bottle. If the stopper were made of rubber,
for example, (with a Poisson ratio of about 1/2), there
would be a relatively large additional force required to
overcome the expansion of the upper part of the rubber
stopper.
Temperature and Its Effects
* 2 basic effects:
• expansion / contraction
• change of material properties
Look, first, at the former:
Concept of Thermal Stresses and Strains
Materials and structures expand
and contract as the temperature
changes. Thus: (the thermal
strain will be discussed in the pdf
that had been attached with this
chapter) "Thermal strain.pdf"
Kites team