# Concept of Elasticity, Stress, Strain, Modulus of Elasticity

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### Introduction:

#### Deformation:

• The forces applied to a body can produce a change in shape or size or both shape and size of the body. Such a change is called deformation.

#### Deforming Force:

• The forces which produce deformation in the body are called deforming forces.

#### Notes:

• Deformation is always directly proportional to deforming force.
• Due to deforming force the length of a wire (chord) or volume of a body or shape of body changes.
• When deformation takes place, the body is said to be strained or in a deformed state.
• In the deformed state, the applied force is numerically equal to internal elastic restoring force within a body but in opposite direction.
• Deformation produced in a body is due to change in relative positions of the molecules within a body, due to applied deforming force.

#### 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.
• As solids have a definite shape and definite volume, therefore, they alone possess elasticity of shape as well as volume.  Liquids have a definite volume but an indefinite shape.  Hence, they possess volume elasticity.  Gases possess volume elasticity.
• 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 elastic property of the body.

#### Plasticity:

• When a body is acted upon by deforming forces the shape and/or size of the body changes.  But, if the deforming forces are removed, the body retains its new shape and size.  Such body is called as plastic body and the property is called as plasticity.
• e.g. Plaster of Paris, Clay, Mud, Plastic, etc. shows plasticity.
• A body which can easily be deformed when very small deforming force is applied and which does not regains its original shape or size or both but retains its new shape is called a perfectly plastic body.
• No body in the universe is a perfectly plastic body. Wall putty can be considered as a perfectly plastic body.
• In plastic bodies, no internal elastic force or restoring force is produced. If produced they are negligible. Hence they can not regain original shape but in absence of restoring force, they retain their new shape.

#### Rigidity:

• When a body is acted upon by deforming forces,  the shape and size of the body do not get altered, whatever may be the magnitude of deforming forces.  Such body is called as a rigid body and the property is known as rigidity.
• In practice, it is not possible for us to get a perfectly rigid body. But large blocks of metal, stones can be considered rigid bodies.
• In rigid bodies, internal force of attraction are so high that there is no relative motion between two particles of the body. Hence there is no change in the shape of the body.

### Elasticity in Detail:

#### Stress:

• The net internal elastic force (restoring force) acting per unit area of the surface which is subjected to deformation is called stress.

Stress = F / A

• Stress is denoted by letter ‘σ’. S.I. Unit of stress is N m-2  or Pa (pascal) and its dimensions are [L-1M1T-2].  Units and dimensions of stress are the same as that of pressure.

#### Strain:

• The change in dimension per unit original dimension of a body subjected to deforming forces is called as Strain.

Strain = Change in dimension / Original dimension

• A strain is denoted by letter ‘e’.  It is a pure ratio, (Ratio of two similar quantities) hence it is unitless, dimensionless quantity [L0M0T0] .
• Strain is tensor quantity. (Neither scalar nor vector).

Hooke’s Law of Elasticity:

• Statement: Within the elastic limit, the stress developed in the body is directly proportional to the strain produced in the body.
• Explanation :

By Hooke’s Law,

Stress      ∝  Strain

∴ Stress      =      Constant    Strain

∴ Stress / Strain  = Constant

• This constant of proportionality is called as the modulus of elasticity or coefficient of elasticity. Its units and dimensions are the same as that of stress.  Its S.I. unit is N m-2  or Pa (pascal) and its dimensions are [L-1M1T-2].
• Depending upon the nature of stress and strain these constants are called as (1) Young’s modulus (2) Bulk modulus and (3) Modulus of rigidity.

Graphical Representation:

#### Elastic Limit:

• It is the upper limit of deforming force up to which if the deforming force is removed, the body regains its original shape and size completely.
• If the deforming force is increased beyond this limit, there is permanent deformation in the body called permanent set.
• Elastic limit is a property of a body, while elasticity is the property of the material of the body.

#### Longitudinal stress:

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

Longitudinal stress = F / A

• Stress is denoted by letter ‘σ’. S.I. Unit of stress is N m-2  or Pa (pascal) and its dimensions are [L-1M1T-2].  Units and dimensions of stress are the same as that of pressure.

#### Tensile stress:

• When the deforming force is such that there is the increase in the length of the body, then the stress produced in the body is called tensile stress.

#### Compressive stress:

• When the deforming force is such that there is a decrease in the length of the body, then the stress produced in the body is called compressive stress.

#### Longitudinal strain:

• When the deforming forces are such that there is a change in the length of the body, then the strain produced in the body is called longitudinal strain. The longitudinal strain is further classified into two types. Tensile strain and tensile strain.
• Mathematically the longitudinal strain is given by

Longitudinal strain = Change in length(l) / Original length (L)

• The longitudinal strain has no unit and no dimensions.

#### Tensile 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.

#### Compressive strain:

• When the deforming force is such that there is the decrease in the length of the body, then the strain produced in the body is called compressive strain.

#### Young’s Modulus of Elasticity:

• Within elastic limit, the ratio of the longitudinal stress to the corresponding longitudinal strain in the body is always constant, which is called as Young’s modulus of elasticity.
• It is denoted by letter “Y” or “E”. Its S.I. Unit of stress is N m-2  or Pa (pascal) and its dimensions are [L-1M1T-2].
• Mathematically

Youn’s modulus of elasticity = Longitudinal stress / Longitudinal strain

• Young’s modulus of elasticity is not defined for liquids and gases.

#### Expression for Young’s Modulus of Elasticity of the Material of a Wire:

• 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.

Poisson’s Ratio:

• The concept of this constant was introduced by physicist Simeon Poisson.
• 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.
• It is denoted by ‘m’. It has no unit. It is dimensionless quantity.

Poisson’s Ratio = Lateral strain / Longitudinal strain

• 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.
• Poisson’s ratio of cork is zero, that of metal is 0.3 and that of rubber is 0.5
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