Physics Complete: ELASTICITY

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Bulk Modulus

Bulk modulus is the ratio bulk stress to the bulk strain.Bulk stress is the force acting on a body per unit area in all directions.Generally this is also referred as pressure.Bulk strain is the ratio of change in the volume to its original volume.

When a body is submerged in a fluid, it undergoes a hydraulic stress (equal in magnitude to the hydraulic pressure). This leads to the decrease in the volume of the body thus producing a strain called volume strain . The ratio of hydraulic stress to the corresponding hydraulic strain is called bulk modulus. It is denoted by symbol B.

B = – p/(ΔV/V)

The negative sign indicates the fact that with an increase in pressure, a decrease in volume occurs. That is, if p is positive, ΔV is negative.Thus for a system in equilibrium, the value of bulk modulus B is always positive. SI unit of bulk modulus is the same as that of pressure i.e., N m–2 or Pa.

The reciprocal of the bulk modulus is called compressibility and is denoted by k. It is defined as the fractional change in volume per unit increase in pressure.

k = (1/B) = – (1/Δp) × (ΔV/V)

Bulk moduli for solids are much larger than for liquids, which are again much larger than the bulk modulus for gases (air).

Thus solids are least compressible whereas gases are most compressible. Gases are about a million times more compressible than solids! Gases have large compressibility, which vary with pressure and temperature.

The in compressibility of the solids is primarily due to the tight coupling between the neighboring atoms. The molecules in liquids are also bound with their neighbors but not as strong as in solids. Molecules in gases are very poorly coupled to their neighbors.

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Problems and Solutions on Elasticity

Shear modulus
Elastic behavior of Solids
Stress and strain
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Determination of Young's modules

Elastic Shear modulus

Shear Modulus is a measure of strength of the shape of a body. A rigid body's shearing modules is going to be infinite and it means its shape is very consistent and to change it we need infinite force.

The ratio of shearing stress to the corresponding shearing strain is called the shear modulus of the material and is represented by G. It is also called the modulus of rigidity.

Generally to test the strength of the shape of the body its lower surface is fixed and force is applied horizontal to the upper surface.

Shearing stress is the horizontal force applied per unit surface area.Shearing strain can be measured as the angle shifted by the upper surface when compared with the lower surface.

The ratio of shearing stress and strain is called shearing modulus.

Mechanical Properties of Solids and Young's Modulus

Problems and Solutions on Young's Modulus of a wire

Rigidity Modulus and Bulk Modulus

Behavior of a Wire under Increasing Load

Breaking Stress and Its Applications

Experimental determination of Young's modulus of the wire

Energy stored in the wire

Problems and Solutions on Elasticity

Elastic behavior of Solids
Stress and strain
Stress and Strain Curve
Determination of Young's modules

Elasticity Young’s Modulus Determination

A typical experimental arrangement to determine the Young’s modulus of a material of wire under tension is shown in Figure below. It consists of two long straight wires of same length and equal radius suspended side by side from a fixed rigid support.

The wire A (called the reference wire) carries a milli metre main scale M and a pan to place a weight. The wire B (called the experimental wire) of uniform area of cross section also carries a pan in which known weights can be placed.

A vernier scale V is attached to a pointer at the bottom of the experimental wire B, and the main scale M is fixed to the reference wire A. The weights placed in the pan exert a downward force and stretch the experimental wire under a tensile stress.

The elongation of the wire (increase in length) is measured by the vernier arrangement. The reference wire is used to compensate for any change in length that may occur due to change in room temperature, since any change in length of the reference wire due to temperature change will be accompanied by an equal change in experimental wire.

Both the reference and experimental wires are given an initial small load to keep the wires straight and the vernier reading is noted. Now the experimental wire is gradually loaded with more weights to bring it under a tensile stress and the vernier reading is noted again. The difference between two vernier readings gives the elongation produced in the wire.

Let r and L be the initial radius and length of the experimental wire, respectively. Then the area of cross-section of the wire would be πr². Let M be the mass that produced an elongation ΔL in the wire. Thus the applied force is equal to Mg, where g is the acceleration due to gravity. The Young’s modulus of the material of the experimental wire can be determined with

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Sress and Strain Curve

Stress is the force experienced by a body per unit area and strain is the ratio of change in the shape to its original dimensions.The ratio of stress and strain is called modules of elasticity and it depends on the nature of the solid materiel.

The relation between the stress and the strain for a given material under tensile stress can be found experimentally. In a standard test of tensile properties, a test cylinder or a wire is stretched by an applied force. The fractional change in length (the strain) and the applied force needed to cause the strain are recorded.

The stress-strain curves vary from material to material. These curves help us to understand how a given material deforms with increasing loads. From the graph, we can see that in the region between O to A, the curve is linear. In this region, Hooke’s law is obeyed.

The applied force is gradually increased in steps and the change in length is noted. A graph is plotted between the stress and the strain produced. A typical graph for a metal is shown in Figure. Analogous graphs The body regains its original dimensions when the applied force is removed. In this region, the solid behaves as an elastic body.

In the region from A to B, stress and strain are not proportional. Nevertheless, the body still returns to its original dimension when the load is removed. The point B in the curve is known as yield point (also known as elastic limit) and the corresponding stress is known as yield strength (Sy) of the material.

If the load is increased further, the stress developed exceeds the yield strength and strain increases rapidly even for a small change in the stress. The portion of the curve between B and D shows this. When the load is removed, say at some point C between B and D, the body does not regain its original dimension.

In this case, even when the stress is zero, the strain is not zero. The material is said to have a permanent set. The deformation is said to be plastic deformation. The point D on the graph is the
ultimate tensile strength (Su) of the material.

Beyond this point, additional strain is produced even by a reduced applied force and fracture occurs at point E. If the ultimate strength and fracture points D and E are close, the material is said to be brittle. If they are far apart, the material is said to be ductile.

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Elastic behavior of Solids
Stress and strain

Stress and Strain

When a force is applied on body, it is deformed to a small or large extent depending upon the nature of the material of the body and the magnitude of the deforming force. The deformation may not be noticeable visually in many materials but it is there. When a body is subjected to a deforming force, a restoring force is developed in the body.

This restoring force is equal in magnitude but opposite in direction to the applied force. The restoring force per unit area is known as stress. If F is the force applied and A is the area of cross section of the body, Magnitude of the stress = F/A .

The SI unit of stress is N m–2 or pascal (Pa) . There are three ways in which a solid may change its dimensions when an external force acts on it. These are shown in Figure below. In Fig(a), a cylinder is stretched by two equal forces applied normal to its cross-sectional area.

The restoring force per unit area in this case is called tensile stress. If the cylinder is compressed under the action of applied forces, the restoring force per unit area is known as compressive stress. Tensile or compressive stress can also be termed as longitudinal stress.

In both the cases, there is a change in the length of the cylinder. The change in the length ΔL to the original length L of the body (cylinder in this case) is known as longitudinal strain.

However, if two equal and opposite deforming forces are applied parallel to the cross-sectional area of the cylinder, as shown in Fig. (b), there is relative displacement between the opposite faces of the cylinder. The restoring force per unit area developed due to the applied tangential force is known as tangential or shearing stress.

As a result of applied tangential force, there is a relative displacement Δx between opposite faces of the cylinder as shown in the Fig. (b). The strain so produced is known as shearing strain and it is defined as the ratio of relative displacement of the faces Δx to the length of the cylinder L.
Shearing strain x L = tan θ .

where θ is the angular displacement of the cylinder from the vertical (original position of the cylinder). Usually θ is very small, tan θ is nearly equal to angle θ .

It can also be visualized, when a book is pressed with the hand and pushed horizontally, as shown in Fig. (c). Thus, shearing strain = tan θ ≈ θ .

In Fig. (d), a solid sphere placed in the fluid under high pressure is compressed uniformly on all sides. The force applied by the fluid acts in perpendicular direction at each point of the surface and the body is said to be under hydraulic compression. This leads to decrease in its volume without any change of its geometrical shape.

The body develops internal restoring forces that are equal and opposite to the forces applied by the fluid . The internal restoring force per unit area in this case is known as hydraulic stress and in magnitude is equal to the hydraulic pressure (applied force per unit area).

The strain produced by a hydraulic pressure is called volume strain and is defined as the ratio of change in volume (ΔV) to the original volume (V).

Since the strain is a ratio of change in dimension to the original dimension, it has no units or dimensional formula.

Elastic Behaviour of Solids

The property of the body because of which they are able to come back to their original status after removing the external force applied is called elasticity.

In a solid, each atom or molecules surrounded by neighboring atoms or molecules. These are bonded together by inter atomic or intermolecular forces and stay in a stable equilibrium position. When a solid is deformed, the atoms or molecules are displaced from their equilibrium positions causing a change in the inter atomic (or intermolecular) distances.

When the deforming force is removed, the inter atomic forces tend to drive them back to their original positions. Thus the body regains its original shape and size. The restoring mechanism can be visualized by taking a model of spring-ball system shown in the Figure below. Here the balls represent atoms and springs represent inter atomic forces.

If you try to displace any ball from its equilibrium position, the spring system tries to restore the ball back to its original position. Thus elastic behavior of solids can be explained in terms of microscopic nature of the solid. Robert Hooke, an English physicist performed experiments on springs and found that the elongation produced in a body is proportional to the applied force or load .

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