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Deformation of Solids
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Marline Kurishingal
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Deformation of Solids
Definitions:
• Stress: is a measure of the force required to cause a
particular deformation.
• Strain: is a measure of the degree of deformation.
• Elastic Modulus: the ratio of stress to strain
Elastic Modulus =
The elastic modulus determines the amount of force
required per unit deformation. A material with large
elastic modulus is difficult to deform, while one with
small elastic modulus is easier to deform.
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Deformation of Solids : Changes in Length
Changes in Length
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To stretch or compress something you must
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exert a force on it at either end.
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Tensile Stress is the force per unit cross-
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sectional area exerted on the ends.
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(Note the surface whose area we wish to measure is perpendicular to the
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force.)
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Changes in Length (continued)
• Tensile Strain is the fractional
Tensile Stress change in original length.
• Young's Modulus ( Y ) is the
ratio of tensile stress to tensile
strain:
Y= = =
where F is the applied force, LO is
the original length of the
object, A is the cross-sectional
area of the object, and ∆L is the
change in the length of the
object. Notice that Y has S.I. units
of N/m 2.
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Hooke’s Law
Hooke’s Law states that, for relatively
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small deformations of an object, the
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displacement of the deformation is directly
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proportional to the deforming force or load.
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Forces can cause objects to deform.
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The way in which an object deforms depends on
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its dimensions, the material it is made of, the size
of the force and direction of the force.
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If you measure how a spring stretches (extends its length) as you
apply increasing force and plot extension (e) against force (F);
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P – proportionality limit
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Calculating stress
• Stress
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Stress is a measure of how strong a material is. This is defined
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as how much force the material can stand without undergoing
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some sort of physical change.
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Hence, the formula for calculating stress is the same as the
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formula for calculating pressure:
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where σ is stress (in Newtons per square metre but usually
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Pascals, commonly abbreviated Pa).
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Calculating strain
Stress causes strain.
• Applying force on an object causes it to stretch. Strain is
a measure of how much an object is being stretched.
Strain is the ratio of extension to the original length.
• The formula for strain is:
• Where is the original length of some bar being
stretched, and l is its length after it has been stretched. Δl
is the extension of the bar, the difference between these
two lengths.
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Calculating Young's Modulus
• Young's Modulus is a measure of the stiffness
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of a material. It is defined as the ratio of stress
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to strain. It states how much a material will
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stretch (i.e., how much strain it will undergo)
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as a result of a given amount of stress.
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• The formula for calculating it is:
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• Strain is unit less so Young's Modulus has the same units
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as stress, i.e. N/m² or Pa.
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Tensile strength & Yield strength
Tensile Strength
Tensile strength which is also known as Ultimate tensile
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strength or ultimate strength is the maximum stress that a
material can withstand while being stretched or pulled before
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failing or breaking. Tensile strength is the opposite
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of compressive strength and the values can be quite different.
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Yield Stress or Yield strength or Yield point
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The yield stress is the level of stress at which a material will
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deform permanently. This is also known as Yield strength or
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Yield point. Prior to the yield point the material will
deform elastically and will return to its original shape when
the applied stress is removed.
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It can be experimentally determined from the slope of a stress-
strain curve created during tensile tests conducted on a sample
of the material.
• The value of the Young's Modulus is quoted for various
materials but the value is only approximate.
• This is because Young's Modulus can vary considerably
depending on the exact composition of the material.
• For example, the value for most metals can vary by 5%
or more, depending on the precise composition of the
alloy and any heat treatment applied during
manufacture.
• If a big force only produces a small extension then the
material is 'stiff' and E is a big value. If a force produces
a big extension then the material is not very stiff - it is
easier to stretch and the value of E will be smaller.
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An experiment to measure the Young's Modulus
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An experiment to measure the Young's Modulus
(continued)
• To minimize errors the control wire is the same length, diameter
and material as the test wire. This means that errors due to
expansion (from the surroundings) during the experiment are
avoided as the test wire and control wire would both expand by the
same amount and the scale would adjust position and eliminate the
error.
• The wire must have no kinks in it otherwise there will be big
extensions due to the wire straightening out rather than just
stretching.
• Care must be taken that the limit of proportionality is not
exceeded. This can be checked by removing the load after each
addition of the weight. If the limit has not been exceeded the wire
should return to the length it was before the weight was added.
• The wire is as long as possible (usually about 2m long) and it is
as thin as possible so that as big an extension as possible can be
recorded. (A typical extension for a 5N loading will be 1mm).
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An experiment to measure the Young's Modulus
(continued)
• The test wire is loaded with the weight hanger so
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that it is taut before readings are taken.
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• The vernier scale is read and the result recorded as
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addition of 0N.
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• Weights - usually starting at 0N and increasing in 5N
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increments to 100N - are then added and a reading
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of the vernier scale is taken at each addition.
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• The experiment should be repeated twice and any
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anomalous results repeated and checked.
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An experiment to measure the Young's Modulus
(continued)
• A graph of load against
extension is plotted. It
should be a straight line
through the origin
(provided measurements
are accurate).
• The gradient of that graph
will be F/e. Using that
value we can find the
value of Young's Modulus
for the wire. l/A X Gradient
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Proportionality limit and Yield strength
• Proportionality limit and Elastic limit
Maximum amount a material can be stretched by a force and still
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(or may) return to its original shape depends on the material.
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• Yield point or Yield strength
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The point where there is a large permanent change in length with
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no extra load force.
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yield point :- interface between elasticity and plasticity
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Elastic limit - up to which material can sustain the load and return
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back to its original position.
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Although these two points are so close to each other it can be
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treated as one, on a case to case basis.
It depends upon material whether it's brittle or ductile.
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Explaining Graph in the previous slide.....
• The stretching behavior is summarized in a stress-strain graph in the
previous slide. As the stress is increased initially Hooke's Law is
obeyed - the stress-strain relationship for the wire is linear & elastic.
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• Just before the plastic region is reached we get the limit of
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proportionality - beyond this for a small section we see non-linear
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behaviour but the stretching is still elastic.
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• After the yield strength, the material enters the plastic deformation
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region, which means that the stretch of the wire is permanent. (For
example, if the wire is stressed to point A on the graph and the stress
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is slowly decreased, the stress-strain curve follows the dotted line
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instead of the original curve to point B and there is a permanent
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extention when all stress is removed.) At the facture point the wire
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snaps.
• Differences in the shape and limits of the stress-strain diagram
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determines whether a material is considered ductile or brittle, elastic
or plastic.
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Polymeric materials
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Strength, Ductility & Toughness
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High Yield strength & Low yield strength
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Energy in deformations
• Whenever we apply force to an object, it will
cause deformation. If the deformation caused is
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within the elastic limit, the work done in
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deforming the object is stored within it as
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potential energy. We call this (elastic) 'strain
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energy'. It can be released from the object by
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removing the applied force.
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• The strain energy then performs work in un-
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deforming the object and returns to its original
state.
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Force-extension graphs for typical
ductile, brittle and polymeric
materials, including an
understanding of ultimate
tensile stress.
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