1)Elasticity:
    The property of a body by virtue of which it tends to regain its original size and shape  after the removal of deforming forces is called elasticity.
2) Elastic body:
   When a body regains its original size and shape completely after the removal of deforming forces, then a body is said to be perfectly elastic.
The nearest approach to a perfectly elastic body is a quartz fibre.
3) Plastic Body:
    When a body does not have a tendency to recover its original size and shape even after the removal of deforming forces, then the body is said to be perfectly plastic. 
    The nearest approach to a perfectly plastic body is putty.
4) Rigid Body:
     When the external forces does not produce any deformation in the body, the body is called a rigid body.
The nearest approach to a rigid body is diamond.
5) Stress:
     When external forces are applied on a body, the body is distorted, i.e., the different portions of the body move relative to each other. Due to these displacements atomic forces (restoring forces) are set up inside the body to restore the original form.
     "The restoring force per unit area set up inside the body is called stress". This is measured by the magnitude of the deforming force acting per unit area of the body when the equilibrium is established.
a) Normal stress: If the stress is normal to the surface, it is called normal stress. The stress is always normal in the case of change in length of a wire or volume of a body. The normal stress can further be "compressive" or "tensile" according as it produce a decrease or 'increase' in length or volume.
                   Compressive stress                                          Tensile stress

b) Shearing or Tangential stress:
    When the stress is tangential to the surface it is called tangential or shearing stress.Due to this stress, the shape of the body changes or it gets twisted.
c) Units:
   →  C.G.S. Unit dyne / cm2
   →  M.K.S. Unit Newton/ m2
   →  Dimensional formula is M' L-1 T-2
d) Further it is not a scalar as at a given point for a given force, it has different values in different directions. It is not a vector as it has no fixed direction as at a point for different cross sections it has different directions.
      Hence stress is a Tensor quantity.
6) Difference between stress and pressure:
    Though both stress and pressure are force per unit area they are different as
i) Pressure is always normal to the area while stress can be either normal or tangential.
ii) Pressure on a body is always compressive while stress can be compressive or tensile
iii) Pressure is scalar while stress in a tensor
7) Strain:
    The external forces acting on a body cause a relative displacement of its various parts; a change in length, volume or shape takes place. The body is then said to be strained.
                    

    Strain has no dimensions as it is a pure number.
a) Longitudinal or Linear Strain:
    Suppose the length of a rod increases from its natural value L to L + e, when pulled by equal and opposite forces along the length. The fractional change  is called the longitudinal strain.

If the length increases from its natural length, the longitudinal strain is called tensile strain.
If the length decreases from its natural length, the longitudinal strain is called compressive strain.

b) Volume strain:
     When a body is subjected to volume stress, there occurs a change in its volume. If ∆v is the change in volume and 'v' is the original volume, then change in volume

     Volume strain is equal to three times its longitudinal strain
c) Shearing strain:
    When a deforming force is applied to a body parallel to its surface then its shape (not size) changes. The strain produced in this way is known as shearing strain.
    i) Shearing strain is measured by the angle through which a line originally normal to the fixed surface is turned. 
 ii) Shearing strain = 2 × longitudinal strain
iii) When a body is sheared, two mutually perpendicular strains are produced in it one extentional strain and the other compressional strain and Extentional strain =   

                           
8) Hooke's Law and Modulii of Elasticity:
a) Hooke's Law:
With in the elastic limit, stress is directly proportional to the strain.
       
 The constant of proportionality E is called Modules of elasticity of the material.
 Its value depends upon the nature of material and the manner in which the body is deformed.
 There are three modulii of elasticity corresponding to three types of the strain called Young's modulus, Bulk modulus and Rigidity modulus.
(b) Young Modulus (Y)
  i) When a Wire or rod is stretched by a longitudinal force. The ratio of the longitudinal stress to

                   
v) Young's modulus is defined for solids but not for liquids and gases.
C) Bulk Modulus (K)
  i) When a solid or fluid is subjected to uniform pressure all over the surface, such that the shape remains the same, then there is change in volume. Within in the elastic limit, the ratio of stress to volume strain is called bulk modulus.
  
ii) The negative sign shows that, with increase in pressure by ∆P, the Volume
decreases by ∆V.
iii) Liquids and gases have only volume elasticity.
iv) Gases have two types of elasticities.
Isothermal elasticity = P ; Adiabatic elasticity = γP
v) The reciprocal of bulk modulus is called compressibility.

The shearing strain is defined as the angle θ in radians through which a line normal to fixed surface has turned. For small values of angle, shearing strain

    Only solids can exhibit a shearing as they have definite shape.
9) Important points regarding the Modulii of elasticity (Y, K, η )
i) The value of modulii of elasticity is independent of the magnitude of the stress and strain.
ii) For a given material there can be different modulii of elasticity depending on the type of stress applied and strain resulting.
iii) The modulii of elasticity has same dimensional formula and units as that of stress since strain is dimension less.
i.e., Dimensional formula for Y, K and η is ML-1T-2 while units are dyne / Cm2 (or)  N/m2
iv) Greater the value of modulii of elasticity more elastic in the material.v) For a rigid body e, ∆V and θ=0, so Y,K or η will be infinity. i.e.Elasticity of a rigid body is infinite.
vi) Esolid > E liquid > Egas
vii) with rise in temperature Y, K and η decreases.
10) Poisson's Ratio:
When a force is applied on a wire then, its length increases but its diameter decreases.
Thus two strains are produced by a single force.
i) Longitudinal Strain =
ii) Lateral Strain = -∆D
                                   D
"With in the limit of proportionality, the ratio of lateral strain to longitudinal strain
is constant for a given material and this constant is called Poisson's ratio (σ)".
        

a) Negative sign indicates that if the length increases, then the radius decreases.
b) Since it is the ratio of two strains, it has no units and dimensions.
c) Theoretical Values of σ lies between -1 and + 1/2
d) Practical Values of σ lies between Zero and + 1/2
11. Strain Energy:
The work done in stretching a wire is stored in the wire in the form of potential energy
and this is called as strain energy.

The breaking stress is also called tensile strength.
       Awire of length L, is hanging without break. Then its breaking stress is L e g. Where e is the density of material. Breaking stress repairs on the material of the wire. It does not depend on the length of wire.
Thermal Stress:
   If a rod is fixed between two supports, due to change in temperature its length will change and so it will exert a stress on the supports. This stress is called thermal stress.
 ... Thermal Stress = Y ∆ t
Thermal Stress is independent if length of the rod.
If a gas is enclosed in a vessel of any rigid material, the change in pressure or thermal
∞stress is ∆P = γK ∆t
    where γ = coefficient of cubical expansion
              K  = Bulk modulus
             ∆t  = change in temperature