Question

define  young modulus 2 bulk modulus 3rigidity modulus state relation between them

Answer 1

Youngs modulus of elasticity Y is defined for solids.  It is the ratio of stress σ to strain ε. Stress is the Compressive or tensile force per unit area of cross section exerted on a solid object.  The strain is the ratio of elongation or extension in the size of object (in the direction of force) to the original total length of the object.
 
        Y = Stress / Strain = σ/ε = (F/A ) / (ΔL / L)  = F L / (A ΔL)   N/m²

Bulk modulus of elasticity is the change in the pressure ΔP in the substance to the ratio of volume strain.  The change in pressure causes a change in the volume of fluid.

        B =  - ΔP / (ΔV / V) = - V ΔP / ΔV     N/m²

Rigidity modulus G or (1/K ) of elasticity is the ratio of lateral (shearing) stress to the lateral (shearing) strain or the angle of bend in the substance.  A tangential stress is applied on the substance which is fixed at one end, and the angle bent at the free end is measured.

        G = shearing stress / shearing strain  =  (F / A) / θ
 
    Poissons ratio = σ_P = (Δd/d) / (ΔL/L)
          = lateral strain (in diameter d of cross section) / longitudinal strain (in length L along tensile force)   
    
relations:
               for most substances is  Y  = 3 G
                   Y = 2 G ( 1 + σ_P ) 
                    Y < 3 B  
                    Y = 3 B ( 1 - σ_P)  
 
                      1/Y = 1/(4G) + 1/(6B)