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#### 1.1 Concept of elasticity:

- The property by virtue of which material bodies regain their original dimensions (size, shape or both) after removal of deforming force is called elasticity. The material exhibiting elasticity is called elastic material and the body is called the elastic body.
- e.g. Rubber, Steel, Aluminum, Sponge etc.
- When the body is stretched the inter-atomic spacing increases and when it is compressed the inter-atomic spacing decreases. In both the cases, internal forces are created in the body which tend to restore the atoms back to their original positions. Such internal forces are called as internal elastic forces or restoring forces. The magnitude of restoring or internal elastic force is the same as applied force. These restoring forces are responsible for the elastic property of the body.
- Solids possess elasticity of shape as well as volume. Liquids and gases possess volume elasticity.

#### 2.1 Elastic Bodies:

- The property by virtue of which material bodies regain their original dimensions (size, shape or both) after removal of deforming force is called elasticity. The body exhibiting elasticity is called the elastic body.
- e.g. Rubber, Steel, Aluminum, Sponge etc.

#### 2.2 Plastic Bodies:

- Plasticity is the property of a material to undergo a permanent deformation even after removal of external deforming forces. The body exhibiting plasticity is called a plastic body.
- e.g. Plaster of Paris, Clay, Mud, Plastic, etc. shows plasticity.

#### 3.1 Expression for Young’s Modulus of Elasticity:

- Consider a wire of length ‘L’ and radius of cross-section ‘r’ is fixed at one end and stretched by suspending a load of ‘mg’ from the other end. Let ‘‘ be the extension produced in the wire when it is fully stretched.

- Now, by the definition of Yong’s modulus of elasticity we have This is an expression for Young’s modulus of elasticity of a material of a wire.

This is an expression for Young’s modulus of elasticity of a material of a wire.

#### 3.2 S.I. Unit and Dimensions of Youn’s Modulus of Elasticity (Y):

- S.I.unit of Young’s modulus of elasticity is N m
^{-2}or Pa (pascal) and its dimensions are [L^{-1}M^{1}T^{-2}].

#### 4.1 Expression for Bulk Modulus of Elasticity:

- When the deforming forces are such that there is a change in the volume of the body, then the stress produced in the body is called volume stress.

Volumetric Stress = Load / Area = Pressure Intensity = dP

- When the deforming forces are such that there is a change in the volume of the body, then the strain produced in the body is called volume strain. The volumetric strain is unitless and dimensionless quantity.

Volumetric strain = – Change in volume (dV)/ Original Volume (V)

- Within the elastic limit, the ratio of volumetric stress to the corresponding volumetric strain in a body is always constant, which is called as Bulk modulus of elasticity.
- It is denoted by letter ‘K’. Its S.I. Unit of stress is N m
^{-2}or Pa (pascal) and its dimensions are [L^{-1}M^{1}T^{-2}].

This is an expression for bulk modulus of elasticity of the material of the body

#### 4.2 S.I. Unit and Dimensions of Bulk Modulus of Elasticity (K):

- S.I. Unit of bulk modulus of elasticity is N m
^{-2}or Pa (pascal) and its dimensions are [L^{-1}M^{1}T^{-2}].

#### 5.1 Expression for Modulus of Rigidity:

- Within the elastic limit, the ratio of the shear stress to the corresponding shear strain in the body is always constant, which is called as modulus of rigidity.
- It is denoted by the letter ‘η’. Its S.I. Unit of stress is N m
^{-2}or Pa (pascal) and its dimensions are [L^{-1}M^{1}T^{-2}]. - Consider a rigid body as shown in the figure which is fixed along the surface CD. Let it be acted upon by tangential force F along surface AB as shown. Let lateral surface AD get deflected through angle θ as shown. The tangential force F per unit area of surface AB is called as shear stress.

#### 5.2 S.I. Unit and Dimensions of Modulus of Rigidity (η):

- S.I. Unit of the modulus of rigidity is N m
^{-2}or Pa (pascal) and its dimensions are [L^{-1}M^{1}T^{-2}].

#### 6.1 Stress Versus Strain Graph:

- The behaviour of wire under increasing load can be studied using Searle’s apparatus. The wire whose behaviour is to be studied is used in the apparatus, at the free end, increasing loads are applied. For each load, stress and strain are calculated. Then the behaviour of wire is studied by plotting a graph, stress versus strain.

- For ductile material, the graph is as shown. From O to A graph is a straight line which clearly indicates that the stress is directly proportional to strain, which indicates that Hooke’s Law is obeyed in this region. Point A is called the limit of proportionality.
- The elastic limit is the point up to which the Hooke’s law is applicable. Stress corresponding to this is called the elastic limit. If the load is removed before the elastic limit is crossed, then the wire will be able to recover its original length completely.
- If the load is further increased, we get curve AA’ which indicate that Hooke’s law is not obeyed. The extension starts increasing faster than the load, and the graph bends towards the strain axis. If the wire is strained up to a point A’ and then if the load is removed, the wire is not able to recover its original length. However, the wire still retains its elastic properties. We can see it by the fact, that when the load is steadily reduced, a new straight line graph such as A’O’ is obtained. In this case, the wire undergoes permanent deformation. The corresponding permanent strain OO’ is called permanent set or permanent strain or residual strain.
- If the load is increased further, a point B is reached, at which the tangent to the curve becomes parallel to strain axis. It indicates that there is the extension in the wire without an increase in the load. Here wire exhibit plastic flow. The point B is called as yield point and corresponding stress is called as yield stress.
- Initially, as wire elongates area of cross-section decreases uniformly, but if the wire is loaded beyond point B, stress at some local point starts increasing rapidly due to neck formation in that region and ultimately wire breaks. This point is called as the breaking point, and corresponding stress is called as breaking stress or ultimate stress or ultimate strength.
- For ductile material, there is neck formation at breaking point C. Before breaking ductile material always show plastic flow. For obtaining an appreciable extension of wire in Serle’s experiment, the specimen wire should be long and thin.

#### 7.1 Expression for Strain Energy in a Strained Wire:

- Consider a wire of length ‘L’ and area of cross-section ‘A’ be fixed at one end and stretched by suspending a load ‘M’ from the other end. The extension in the wire takes place so slowly that it can be treated as quasi-static change; because internal elastic force in the wire is balanced by the externally applied force and hence acceleration is zero. Let at some instant during stretching the internal elastic force be ‘f’ and the extension produced be ‘x’. Then,

- Since at any instant, the external applied force is equal and opposite to the internal elastic force, we can say that the work done by the external applied force in producing a further infinitesimal displacement dx is

- Let ‘
*l*‘ be the total extension produced in the wire, and work done during the total extension can be found by integrating the above equation.

This is an expression for the work done in stretching wire.

- The work done by the external applied force during stretching is stored as potential energy (U) in the wire and is called as strain energy in the wire.
- Thus the strain energy is given by

This is an expression for strain energy in a strained wire.

**8.1 Strain Energy Per Unit Volume of a Wire:**

- Write complete answer 7.1 and proceed further
- The work done by external applied force during stretching is stored as potential energy (U) in the wire and is called as strain energy in the wire.

- This is an expression for strain energy or potential energy per unit volume of stretched wire. This is also called as the energy density of the strained wire. Its S.I. unit is J m
^{-3}and its dimensions are [L^{-1}M^{1}T^{-2}].

#### 9.1 Poisson’s Ratio:

- When a rod or wire is subjected to tensile stress, its length increases in the direction of stress, but its transverse dimensions decrease and vice-versa. i.e. when the length increase, the thickness decreases and vice-versa. In other words, we can say that the longitudinal strain is always accompanied by a transverse strain.
- The ratio of transverse strain to the corresponding longitudinal strain is called Poisson’s ratio.

#### 9.2 Limiting Value of Poisson’s Ratio:

- For homogeneous isotropic medium -1 ≤ m ≤ 0.5
- In actual practice, Poisson’s ratio is always positive. There is no material with negative Poisson’s ratio.

#### 10.1 Definition of Longitudinal Stress:

- When the deforming forces are such that there is a change in the length of the body such as wire or beam, then the stress produced in the body is called longitudinal stress. Longitudinal stress is further classified into two types. Tensile stress and compressive stress.

#### 10.2 Definition of Longitudinal Strain:

- When the deforming force is such that there is an increase in the length of the body, then the strain produced in the body is called tensile strain.

#### 10.3 Definition of Volume Stress:

- When the deforming forces are such that there is a change in the volume of the body, then the stress produced in the body is called volume stress.

#### 10.4 Definition of Volume Strain:

- When the deforming forces are such that there is a change in the volume of the body, then the strain produced in the body is called volume strain.

#### 10.5 Definition of Shear Stress:

- When the deforming forces are such that there is a change in the shape of the body, then the stress produced is called shearing stress.

#### 10.6 Definition of Shear Strain:

- When the deforming forces are such that there is a change in the shape of the body, then the strain produced in the body is called shear strain.

#### 11.1 Method of Determination of Young’s Modulus of Elasticity of a Wire:

**Searle’s Experiment:**

#### Apparatus:

- Two identical wires A and B are suspended from a rigid support so that the points of suspension are very close to each other. Searle’s apparatus blocks are attached to the lower ends of the wires by means of chucks F
_{1}and F_{2}. Searle’s apparatus block consists of two metal frames P and Q. The two frames are loosely connected by cross strips in such a way that the frame Q can move relatively with respect to frame P. A spirit level S is hinged to the frame P and is rested on the tip of a micrometer screw M which can work in a nut fixed in the frame Q. At the lower end, each frame carries a hanger from which slotted weights can be suspended. Wire A is dummy wire from which a fixed load of about 1 kg (dead weight) is suspended.

**Procedure:**

- Initially, the length (L) of wire B is measured. Its mean radius (r) is found with the help of micrometer screw gauge. Micrometer screw is adjusted to bring the bubble in the spirit level at the centre and the reading is noted.
- The load suspended from wire B is then increased in equal steps of about 0.5 kg-wt. let ‘m’ be the mass in the hanger. Each time, after waiting for about two minutes, the bubble is brought to the centre by rotating the screw and micrometer reading is noted. This is extension or elongation (l
_{1}) in the wire. This way five to six readings are taken. - After loading procedure is complete the wire is unloaded in the same steps of 0.5 kg-wt and the readings ( l
_{2}) are noted again for each step.

**Calculations:**

- The mean of the readings for loading ( l
_{1}) and unloading ( l_{2}) is calculated and represented as (*l*) for each step. Then Young’s modulus of the material is calculated in each step using formula,

- The average value of Young’s Modulus (Y) is calculated. A care should be taken to avoid possible errors.

**Sources of Errors:**

- Error due to kinks in the wire.
- Errors due to a backlash of the screw.
- Error due to bending (yielding) of the support.
- Error due to thermal expansion or contraction.
- Error due to the crossing of the elastic limit and/or slipping of the wire from the chucks.

#### 12.1 Point of Proportionality:

- The point of proportionality is the point on the stress-strain curve above which the stress in a material is no longer linearly proportional to strain i.e. it is the point above which Hooke’s law is not obeyed. Point of proportionality is also called proportional limit.

#### 12.2 Elastic Limit:

- It is the maximum stress to which an elastic body can be subjected without causing permanent deformation is called the elastic limit.

#### 12.3Yield Point:

- The point on the stress-strain curve at which the strain begins to increase without any increase in the stress is called the yield point.

#### 12.4 Breaking Stress:

- The maximum stress which can be applied to a wire is called breaking stress.

#### 12.5 Perfectly Elastic Bodies:

- Those bodies which regain their original shape, size or both completely after removal of the deforming force are called perfectly elastic bodies.
- No perfectly elastic body exists in nature but for practical purpose quartz and phosphor bronze can be considered as perfectly elastic bodies.

#### 12.6 Perfectly Plastic Bodies:

- Those bodies which retain their new shape, size or both completely after removal of the deforming force are called perfectly plastic bodies.
- No perfectly plastic body exists in nature but for practical purpose wax, putty, and clay can be considered as perfectly plastic bodies.

#### 13.1 To prove that within elastic limit, Young’s modulus of the material of wire is the stress required to double the length of wire.

As the length is doubled, change in length = *l* = 2L – L = L

Strain = *l* / L = L/L = 1

Now, Young’s modulus of elasticity is given by

Y = Stress / Strain = Stress/1 = Stress

- Hence within elastic limit, Young’s modulus of the material of wire is the stress required to double the length of wire.

#### 14.1 Distinguishing Between Elasticity and Plasticity:

Elasticity |
Plasticity |

The body exhibiting elasticity regains its shape or size after removal of the external force. | The body exhibiting plasticity retains its new shape or size after removal of the external force. |

There is a temporary change in dimensions of the body on application of the deforming force. | There is a permanent change in dimensions of the body on application of the deforming force. |

Internal restoring force is set up inside the body. | No Internal restoring force is set up inside the body. Or they are very negligible. |

The ratio of stress to corresponding strain produced is constant. | The ratio of stress to corresponding strain produced is not constant. |

materials exhibiting elasticity: Steel, rubber, etc | Materials exhibiting plasticity: PVC, plaster of Paris, wax, etc. |

** 15.1 Distinguishing Between Young’s Modulus, Bulk Modulus and Modulus of Rigidity:**

Young’s Modulus of Elasticity |
Bulk Modulus of Elasticity |
Modulus of Rigidity |

Within elastic limit it is the ratio of longitudinal stress to longitudinal strain | Within elastic limit it is the ratio of volumetric stress to volumetric strain | Within elastic limit it is the ratio of shear stress to shear strain |

It is associated with the change in the length of a body. | It is associated with the change in the volume of a body. | It is associated with the change in the shape of a body. |

It exists in solid material bodies | It exists in solids, liquids, and gases. | It exists in solids only. |

It is a measure of the stiffness of a solid material | It determines how much the body will compress under a given amount of external pressure. | It describes an object’s tendency to shear |

Young’s modulus of the material of a wire is given by |
Bulk modulus of the material of a body is given by | Shear modulus of the material of a body is given by |

**16.1 Applications of Elastic Behaviour of Material:**

- The concept of elasticity is useful in the design of bridges, construction of homes, designs of structures, machinery, industry etc.
- The concept of elasticity is used in the design of crane ropes. The diameter of the ropes and the material can be calculated using the concept of stress. When designing some margins using the concept of a factor of safety is kept to avoid accidents.

Science > Physics > Elasticity > You are Here |

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It is so easy to understand! Very nice information! Thanks!