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
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.
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