Materials Tutorial 3 - Stress, Strain, and the Young Modulus
Stress
and Strain
Wires obey Hooke's Law just like a spring. This is because bonds between atoms stretch just like springs:
If we stretch a
wire, the amount it stretches by depends on:
its length
its diameter
the material
it’s made of.
If we have two of the same material and length but of different thicknesses, it is clear that the thicker wire will stretch less for a given load. We make this a fair test by using the term tensile stress which is defined as the tension per unit area normal to that area. The term normal means at 90^{o} to the area.
We can also talk of the compression force per unit area, i.e. the pressure.
Stress
= Load (N) =
F
area (m^{2}) A
In some text books you may see stress given the code s (sigma, a Greek letter 's'). Therefore:
You
will have met the expression
F/A before.
It is, of course, pressure, which implies a
squashing force. A stretching force gives an expression of the same kind.
Units are newtons per square metre (N/m2) or Pascals
(Pa).
1 Pa = 1 N
m^{-2}
You must always convert areas to square metres for this equation to work. Remember that radii will often be given in mm or cm. This is a common bear trap.
1 mm^{2} = 1 x 10^{-6} m^{2}
Therefore if you get an area of 10^{-2} m^{2},^{ } you probably have forgotten to do the conversion to square metres.
Find the area of a wire of diameter 0.75 mm in m^{2}. |
If we have a wire
of the same material and the same diameter, it doesn’t take a genius to see
that the wire will stretch more for a given load if it is longer.
To take this into account, we express the extension as a ratio of the
original length. We call this the tensile
strain which we define as the extension per
unit length.
Strain = extension (m)
original length (m)
There are no units for strain; it’s just a number. It can sometimes be expressed as a percentage.
You will find that the same is true for when we compress a material.
What is the strain of a 1.5 m wire that stretches by 2 mm if a load is applied? |
Stress-Strain
Curves
Stress-strain graphs are really a development of force-extension graphs, simply taking into account the factors needed to ensure a fair test. A typical stress-strain graph looks like this:
We
can describe the details of the graph as:
P
is the limit of proportionality,
where the linear relationship between stress and strain finishes.
E
is the elastic limit.
Below the elastic limit, the wire will return to its original shape.
Y is the yield point, where plastic deformation begins. A large increase in strain is seen for a small increase in stress.
UTS
is the ultimate tensile stress,
the maximum stress that is applied to a wire without its snapping.
It is sometimes called the breaking
stress. Notice that beyond
the UTS, the force required to snap the wire is less.
S
is the point where the wire snaps.
We can draw stress-strain graphs of materials that show other properties.
Curve
A shows a brittle
material. This material is
also strong because there is little strain for a high stress.
The fracture of a brittle material is sudden and catastrophic, with
little or no plastic deformation. Brittle
materials crack under tension and the stress increases around the cracks.
Cracks propagate less under compression.
Curve
B is a strong
material which is not ductile. Steel wires stretch very little,
and break suddenly. There can be a lot of elastic strain energy in a
steel wire under tension and it will “whiplash” if it breaks. The
ends are razor sharp and such a failure is very dangerous indeed.
Curve
C is a ductile
material
Curve
D is a plastic
material. Notice a very large strain for a small stress. The
material will not go back to its original length.
The
Young Modulus
The
Young Modulus is defined as the
ratio of the tensile stress and the tensile strain.
Young modulus = tensile stress
tensile strain
tensile stress = force =
F
area A
tensile strain = ___extension __ = Dl
original length l
Young Modulus has the physics code E, so we can write:
which becomes:
Units
for the Young Modulus are Pascals (Pa) or newtons
per square metre (Nm^{-2}).
The
Young Modulus describes pulling forces.
We can link the Young Modulus to a stress strain graph.
The Young Modulus is the gradient of the stress-strain graph for the region that obeys Hooke’s Law. This is why we have the stress on the vertical axis when we would expect the stress to be on the horizontal axis.
The area under the stress strain graph is the strain energy per unit volume (joules per metre^{3}).
Strain energy per unit volume = 1/2 stress x strain.
The units arise because stress is in N m^{-2} and strain is m m^{-1 } (NOTE: This unit here is not "millimetres to the minus one", but metres per metre which mean no units).
N m^{-2} x m m^{-1} = N m m^{-3}. N m is joules, hence Jm^{-3}
Area is the strain energy per unit volume. So we can write this equation:
Area = ½ × stress × strain
In Physics code:
The term E' is pronounced "E-prime" or "E-dashed" and is being used as the code for the elastic strain energy per unit volume.
Al is area × length = volume
A wire made of a particular material is loaded with a load of 500 N. The diameter of the wire is 1.0 mm. The length of the wire is 2.5 m, and it stretches 8 mm when under load. What is the Young Modulus of this material? | ||
What is the elastic strain energy per unit volume for the wire in question 3? |
Measuring Stress and Strain
We can measure the Young Modulus doing a simple experiment like this:
We need to measure the diameter of the wire using a micrometer. A video tutorial in how to do this is in one of the links. Then we measure the extension as we increase the load, recording the observations. We need to convert the force into stress, and extension into strain. From that we plot a stress-strain graph, using the gradient to calculate the Young Modulus.
The value we get for E is often rather low. This is because the wire might suffer the following defects:
It might not be of uniform diameter. A smaller diameter will cause a greater stress.
The crystal structure of the wire may be degraded by the wire drawing process. The wire is made by pulling the metal through a small hole called a die.
There are also many uncertainties. The greatest uncertainty comes in measuring the diameter.