Normal Stresses & strains in three dimensions

It is also known as Concept of uni-axial loading.
To finding volumetric strain we have to consider three axis of body i.e. X, Y, Z. If we apply load in any one direction then such kind of loading is called as uni-axial loading. Here consider we apply load (P) in X-direction.
Volumetric strain for rectangular bar -

Consider a rectangular bar of having dimensions length(L), breadth(b) and thickness(t) is subjected to tensile load(P) in X-direction as shown in fig.
Stress in X-direction
σ_x = ^P / _A
σ_y = σ_z = 0 ⇒(∵Due to no stress(load) in Y and Z directions.)
Now, strain in X-direction
e_x = ^σ_x / _E ⇒(∵Young's modulus E = ^s/_e)
strain in Y and Z-directions are lateral strains,
e_y = e_z = -µ X Linear strain ⇒(∵Poisson's Ratio = ^{Lateral strain} / _{Linear strain})
     = -µ X e_x
     = -µ X ^σ_x / _E
∴ Volumetric strain of bar,
e_v = ^dV / _V
     = e_x + e_y + e_z
     = ^σ_x / _E - µ X ^σ_x / _E - µ X ^σ_x / _E
     = ^σ_x / _E (1 - µ - µ)
     = ^σ_x / _E (1 - 2µ)
     = e(1 - 2µ) ⇒(∵^σ_x / _E = e_x = linear strain i.e. e)

e_v = e(1 - 2µ)