relation between young modulus, shear modulus and bulk modulus?

Question

can some one please give the derivation of this relationship with explanation and if you could draw some diagrams or find some on net please provide me the links.

Answer 1

E= young's modulus or modulus of elasticity;
υ = poisson' s ratio
G= shear modulus or modulus of rigidity
K = bulk modulus;

as you should know:
σx = Eεx;

when there are a tensile stress along x axis it also produces
a contraction in the transverse y and z directions. the transverse strain has benn found by experience to be a constant fraction of the strain in longitudinal direction. this is known as poisson's ratio (υ);
► εy = εz = -υεx = - υσx/E;
suppose the the stresses in three dimentions, by superposition the overall relations would be

► εx = 1/E [ σx - υ(σy + σz)];
► εz = 1/E [ σz - υ(σy + σx)];
► εy = 1/E [ σy - υ(σx + σz)];

adding up both sides of above three equations:we get

εx +εz+ εy= (1-2υ)/E [ σx + σy + σz];

εx +εz+ εy is called volume strain and σx + σy + σz = 3σm where σm is mean stress;
► volume strain = (1-2υ)/E [3σm];

K = bulk modulus = σm/volume strain = E/3(1-2υ)

also
τxy = G γxy , τzy = G γzy ,τxz = G γxz ,

►G = E/2(1+υ)

Answer 2

Shear Modulus Of Elasticity

Answer 3

Modulus Of Rigidity Equation

Answer 4

Equation 1


G =
E/2(1 + v)


Equation 2


K =
E/(1 - 2v)



where:
G = shear modulus,
K = bulk modulus,
E = Young's modulus, and
v = Poisson's ratio.

Answer 5

E,
G =E/2(1 + v),
K =E/3(1 - 2v),

where:
v = Poisson's ratio,
E = Young's modulus,
G = shear modulus,
K = bulk modulus,

Alireza Khorshidi ^_^

Answer 6

This Site Might Help You.

RE:
relation between young modulus, shear modulus and bulk modulus?
can some one please give the derivation of this relationship with explanation and if you could draw some diagrams or find some on net please provide me the links.

Answer 7

Y = G(1+n)

The analytical derivation can be obtained in books on Solid Mechanics.