CHE 333 Class 11. Mechanical Behavior of Materials. Elastic Deformation. Consider a metal rod fixed at one end. At the other end a load can be applied by some manner. When a small amount of load is applied, if the length of the metal rod was measured it would be longer. If the
CHE 333 Class 11
Mechanical Behavior of Materials
Consider a metal rod fixed at one end.
At the other end a load can be applied
by some manner. When a small amount
of load is applied, if the length of the metal
rod was measured it would be longer. If the
load is removed, and the rod measured
again, it would return to the original
length. It is said that the deformation
was recovered. This type of deformation
is ELASTIC, that is all recovered on load removal.
It was also found that the extension of the rod
was directly proportional to the load applied.
Load extension data would be as shown
in the diagram.
Service loads should be ELASTIC
Following elastic deformation, the load
extension curve is no longer linear,
as shown in the diagram. After the linear
elastic portion, a non linear region starts
which indicates the start of PLASTIC
deformation. If the load is removed at
a point after plastic deformation is initiated
the metal rod will not return to the same
length as the initial length. It will be longer
by the amount of plastic deformation. The
new increase length is the plastic
deformation. In this case all the deformation
was not recovered. The elastic portion is
recovered but not the plastic deformation.
The load removal curve decreases parallel
to the elastic deformation line.
after load removal
The load extension data can be transformed
into Stress Strain data by normalising
with respect to material dimensions.
The stress is the load divided by the
original cross sectional area.
s = L/A
s – stress , units MPa, or psi or ksi
L – load applied
A – original cross sectional area
The strain is the increase in length
normalised by the original length.
e = Dl/l
e – strain – dimensionless (in/in)
Dl – increase in length
l – original length
Strain is often given in percent so x100
As the normalisations are by constants
the shapes of the curves stays the same.
Strain rate is e/t. Most materials are strain
rate sensitive that is their mechanical behavior
depends on the rate of deformation.
Hooke’s Law is concerned with
s = Ee
Stress is proportional to strain,
But only in the elastic region.
This is the “elasticity” or elastic
Modulus of materials, sometimes
Called “Young’s Modulus”.
Metal Youngs Mod
Yield Stress Ultimate Tensile Stress
The Yield Stress is at the
onset of plastic deformation.
The Ultimate Tensile Stress is
the maximum stress during
the stress strain test.
Manufacturing between YS and UTS
The strain to failure can be
measured from the stress strain
The 0.2% yield stress is used for
materials such as steel as the
yield point is sometimes difficult
to determine. At 0.2% strain a line
is drawn parallel to the elastic
portion of the data until it intersects
the plastic portion of the data. The
stress level at this point is the 0.2%
yield stress. (0.002 strain)
Ultimate Tensile Stress
Strain at Failure
Failure at this stress
Brittle materials exhibit little on
no plastic deformation region.
Only elastic deformation is found.
The energy of failure is then the
area under the stress stain curve,
which for a brittle material is the
area of a right angel triangle,
or half base multiplied by the height.
Or half the strain at failure multiplied
by the stress at failure.
Plastic deformation adds a considerable
amount of energy to the failure process.
Ceramics and martensitic steels show
Energy of failure is the area under the
stress strain curve. For brittle materials
it is half the strain multiplied by the
At the UTS, for metals local deformation starts, and thereafter the deformation is concentrated
locally. This causes a “NECK” to occur shown above along with the crack at failure.The
cross section is reduced at the failure point compared to the region outside the neck. One
measure of “DUCTILITY” besides elongation at failure is “reduction of area”
ROA = final cross sectional area/ original cross sectional area
Final failure in round bar is often
characterized for a ductile material
as a “Cup and Cone” failure. An
example is shown. The fracture starts
in the interior of the material and spreads
internally until only a small annulus of material
remains. This then shears at 45o to the
applied stress. The more ductile the material
the larger the shear lip.
A sheet material tensile sample is shown above. ASTM has standard dimensions. At either
end is a grip area, and in the center is the gauge length which is a narrower section to ensure
failure outside the grip area effects. The thickness and width of the sample need to be known
to calculate the stress data and the original length to calculate the strain at failure.
A failed sample is compared to a new untested sample. Note the failure is at 45o to the
applied stress. The local deformation in this case is very near the failure point. ROA
Data would be very difficult in this case. Elongation at failure would be more useful
A failed polymer sample has a large elongation at failure in comparison to the metal sample.
Sample is 0.5 in wide to provide a scale.
Polymers generally have low elastic modulus and long elongations to failure compared to