Lab Session No. 02
Objective:
“To verify the Hook’s Law using the
extension of wire apparatus and also determine the Young’s Modulus of
elasticity of the test material wire”
Apparatus:
Ø Young’s Modulus Apparatus
Ø Weights and hangers
Ø Test Material Wire
Ø Metre rod
Load Hanger 1N |
|
Load Hook |
|
Vernier |
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Young's Modulus Apparatus
Description
Stress
We begin by considering the sectioned area
to be subdivided into small areas, such as ∆A shown in Fig. As we reduce ∆A to
a smaller and smaller size, we will make two assumptions regarding the
properties of the material. We will consider the material to be continuous,
that is, to consist of a continuum or uniform distribution of matter
having no voids. Also, the material must be cohesive, meaning
that all portions of it are connected together, without having breaks, cracks,
or separations. A typical finite yet very small force ∆F, acting on ∆A,
is shown in Fig. This force, like all the others, will have a unique direction,
but to compare it with all the other forces, we will replace it by its three
components, namely, ∆Fx, ∆Fy, and ∆Fz.
As ∆A approaches zero, so do ∆F and its components; however, the
quotient of the force and area will approach a finite limit. This quotient is
called stress, and it describes the intensity of the internal
force acting on a specific plane (area) passing through a point.
Normal Stress
The intensity of the force acting normal to ∆A
is referred to as the normal
stress, s (sigma).
Since ∆Fz is normal to the area then
If the normal force or stress “pulls” on ∆A as shown in Fig it is tensile stress,
whereas if it “pushes” on ∆A it is compressive
stress.
Shear Stress
The intensity of force acting tangent to ∆A is called the shear stress,
τ(tau). Here we have two shear stress components,
The subscript notation z specifies the orientation of the area ∆A, Fig., and x and y indicate the axes along which each shear stress acts
Strain
The measure of deformation is called as
strain.
Normal Strain
If an axial load P is applied to the
bar in Fig., it will change the bar’s length L0 to a length L. We will define the average normal strain ε
(epsilon) of the bar as the change in its length d (delta) = L - L0 divided by its
original length, that is
Shear Strain
Deformations not only cause line segments to elongate or
contract, but they also cause them to change direction. If we select two line
segments that are originally perpendicular to one another, then the change in
angle that occurs between them is referred to as shear strain. This angle is
denoted by g (gamma) and is always measured in radians (rad),which are
dimensionless. For example, consider the two perpendicular line segments at a
point in the block shown in Fig. 2–3a. If an applied loading causes the block to
deform as shown in Fig. 2–3b, so that the angle between the line segments
becomes u, then the shear strain at the point becomes
Notice that if θ is smaller than π/2 then
the shear strain is positive, whereas if θ is larger than π/2, then the shear
strain is negative.
Conventional Stress–Strain Diagram
The nominal
or engineering stress is
determined by dividing the applied load P by the specimen’s original
cross-sectional area A0. This calculation assumes that the stress is constant
over the cross section and throughout the gage length. We have
Likewise, the nominal
or engineering strain is
found directly from the strain gage reading, or by dividing the change in the
specimen’s gage length, d, by the specimen’s original gage length L0.
Thus,
When these values of s and P are plotted,
where the vertical axis is the stress and the horizontal axis is the strain,
the resulting curve is called a conventional
stress–strain diagram. A typical example of this curve is shown in
Fig.. Realize, however, that two stress–strain diagrams for a particular
material will be quite similar, but will never be exactly the same. This is
because the results actually depend upon such variables as the material’s
composition, microscopic imperfections, the way the specimen is manufactured,
the rate of loading, and the temperature during the time of the test.From the
curve in Fig., we can identify four different regions in which the material
behaves in a unique way, depending on the amount of strain induced in the
material. Elastic Behavior. The initial region of the curve, indicated in light
orange, is referred to as the elastic region. Here the curve is a straight
line up to the point where the stress reaches the proportional limit, σpl. When the stress slightly
exceeds this value, the curve bends until the stress reaches an elastic limit.
For most materials, these points are very close, and therefore it becomes
rather difficult to distinguish their exact values. What makes the elastic
region unique, however, is that after reaching sY, if the load is
removed, the specimen will recover its original shape. In
other words, no damage will be done to the material.
Because the curve is a straight line up to spl, any
increase in stress will cause a proportional increase in strain. This fact was
discovered in 1676 by Robert Hooke, using springs, and is known as Hooke’s law. It is expressed
mathematically as
Here E represents the constant of
proportionality, which is called the modulus
of elasticity or Young’s
modulus, named after Thomas Young, who published an account of it in
1807.
Procedure:
1.
Place weight hanger on the load
hook.
2.
Apply initial load to remove
stiffness in wire.
3.
Find the least count of
measuring scale.
4.
Measure diameter and length of
the wire by using metre rod and Vernier calliper.
5.
Apply different loads on the
hanger, the wire extends and draw the table.
6.
Using data available in table
calculate the young’s modulus of specimen wire.
Observations and Calculations:
Formula
𝑆𝑡𝑟𝑒𝑠𝑠=𝐹𝑜𝑟𝑐𝑒 𝐴𝑟𝑒𝑎
𝑠𝑡𝑟𝑎𝑖𝑛= 𝐸𝑙𝑜𝑛𝑔𝑎𝑡𝑖𝑜𝑛𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ
𝑌𝑜𝑢𝑛𝑔′𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠=𝐸= 𝑠𝑡𝑟𝑒𝑠𝑠𝑆𝑡𝑟𝑎𝑖𝑛
𝑆𝑡𝑟𝑒𝑠𝑠=𝐹𝑜𝑟𝑐𝑒 𝐴𝑟𝑒𝑎
𝑠𝑡𝑟𝑎𝑖𝑛= 𝐸𝑙𝑜𝑛𝑔𝑎𝑡𝑖𝑜𝑛𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ
𝑌𝑜𝑢𝑛𝑔′𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠=𝐸= 𝑠𝑡𝑟𝑒𝑠𝑠𝑆𝑡𝑟𝑎𝑖𝑛
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