Poisson's Ratio: The three stresses and strains do not operate independently. Stresses produce strains in lateral directions as the solid tries to retain its original volume. Poisson's ratio is a measure of how successful this is.
When an axial force is applied along the longitudinal axis of a bar, the length of a bar will increase but at the same time its lateral dimension (width) will be decreased so, it is called as Poisson' ratio.
Value of Poisson's ratio is same in tension and compression
Under uniaxial loading0≤ μ ≤ 0.5μ = 0 for corkμ = 0.5 For perfectly plastic body(Rubber)μ = 0.25 to 0.42 for elastic metalsμ = 0.1 to 0.2 for concreteμ = 0.286 mild steelμ is greater for ductile metals than for brittle metals. Volumetric Strain
It is defined as the ratio of change in volume to the initial volume. Mathematically
Volumetric strain,
Volumetric Strain Due to Single Direct Stress
The ratio of change in volume to original volume is called volumetric strain.
ev = e1 + e2 + e3
Volumetric strain:
For the circular bar of diameter d:
Volumetric Strain due to Three Mutually Perpendicular Stress System: When a body is subjected to identical pressure in three mutually perpendicular direction, then the body undergoes uniform changes in three directions without undergoing distortion of shape.
or 
Shear Modulus or Modulus of Rigidity
Modulus of rigidity :
* At principal planes, shear stress is always zero.
* Planes of maximum shear stress also contains normal stress.
Relationship between E, G, K and μ :
Modulus of rigidity:
Bulk modulus:
Analysis of Stress and Strain
We will derive some mathematical expressions for plains stresses and will study their graphical significance in 2D and 3D
Stress on Inclined Section PQ due to Uniaxial Stress
Consider a rectangular cross-section and we have to calculate the stress on an inclined section as shown in figure.
Normal stress :
Stress on an inclined section
Tangential stress
Resultant stress
Stress Induced by State Simple Shear
Induced stress is divided into two components which are given as
Normal stress: 
Tangential stress:
Stress Induced by Axial Stress and Simple Shear
Normal stress
Tangential stress
Principal Stresses and Principal Planes
The plane carrying the maximum normal stress is called the major principal plane and normal stress is called major principal stress. The plane carrying the minimum normal stress is known as minor principal stress.
Major principal stress 
:
Minor principal stress 
:
Across maximum normal stresses acting in plane shear stresses are zero.
Computation of Principal Stress from Principal Strain
The three stresses normal to shear principal planes are called principal stress, while a plane at which shear strain is zero is called principal strain.
For two dimensional stress system, σ3 = 0
Maximum Shear Stress
The maximum shear stress or maximum principal stress is equal of one half the difference between the largest and smallest principal stresses and acts on the plane that bisects the angle between the directions of the largest and smallest principal stress, i.e., the plane of the maximum shear stress is oriented 45° from the principal stress planes.
Principal Strain
For two dimensional strain system,
Where, e1 = Strain in x-direction
e2 = Strain in y-direction
φ = Shearing strain relative to OX and OY
Maximum Shear Strain:
The maximum shear strain also contains normal strain which is given as
45° Strain Rosette or Rectangular Strain Rosette
Rectangular strains Rosette are inclined 45° to each other
Principal strains:
