Volumetric Strain: Definition, Formula with Derivation

Hello, friends in this article we are going to discuss the volumetric strain in which you will learn what is volumetric strain and also know the formula of volumetric strain for rectangular bar, thin cylinder, sphere, and cube with their derivation.

But before knowing volumetric strain, first, we need to know what is strain and its types such that we can understand volumetric strain better.

So without wasting time let's get started.

What is Strain?

When a body of an elastic material is subjected to an axial force it undergoes a change in dimensions. The change in dimension with respect to its original dimension is known as strain. 

Mathematically, It can be expressed as

Strain = Change in Dimension/Original Dimension 

It has no unit.

There are following types of strain are found:
  1. Longitudinal Strain Or Linear Strain 
  2. Lateral Strain 
  3. Shear Strain
  4. Volumetric Strain

Here we only discuss volumetric strain in a detail.

What is Volumetric strain?

When an elastic body is subjected to external deforming forces such that there is a change in volume of the body, then the ratio of change in volume to original volume is called volumetric strain.

volumetric strain

Mathematically it can be represented as,

eV = ΔV/V

Volumetric Strain of Rectangular Bar or Prismatic Member

The initial volume of the rectangular bar,

V = L x B x D 


Change in volume of the rectangular bar,

ΔV = (Δl × Bx D) + (Δb x L x D) + (Δd x L x B) 


The volumetric strain of rectangular bar ( eV ),

ΔV/V = (Δl/L) + (Δb/B) + (Δd /D)

So it can be written as,

eV = Σxx + Σyy + Σzz


Σxx = Strain in x-x direction

Σyy = Strain in y-y direction

Σzz = Strain in z-z direction

As we know,

Σxx = (σx/E) + (σy/E) - μ((σz/E)

Σyy = (σy/E) - μ((σx/E) - μ((σz/E)

Σzz = (σz/E) - μ((σx/E) - μ((σy/E)

Now putting these value,

eV = {(σx/E)+(σy/E) - μ((σz/E)} +{(σy/E) - μ(σx/E) - μ(σz/E)} +{(σz/E) - μ((σx/E) - μ((σy/E})

After calculating these values the final result will be obtained.

eV = {(σx + σy + σz)/E} × (1- 2μ)

If loading is triaxial with equal and alike stresses then,

σx = σy = σz = σ


eV = {(3σ)/E} × (1- 2μ)

As we know,

E = 3K × (1- 2μ)


eV = σ/K

Volumetric Strain of thin Cylinder or Cylindrical Rod 

The initial volume of the thin cylinder,

V = (π/4) × d^2 × l

Change in volume of the thin cylinder,

ΔV = π/4(d^2 × Δl + l × 2d × Δd)


Volumetric strain for cylindrical rod or thin cylinder will be,

eV = (Δl/l) + (2 × Δd/d)

So, it can be written as,

eV = eL + 2 × eD

Volumetric Strain for Spherical Body

The initial volume of spherical,

V = 4/3 × π × R^3


R  = d/2


The initial volume of the sphere,

V = π /6 × d^3

Change in volume of the sphere,

ΔV = π /6 × 3d^2 × Δd

Then volumetric strain of sphere body will,

eV = ΔV/V = 3Δd/d = 3eD

Volumetric Strain for Cube

Initial volume of cube,

V = L^3

Change in volume of the cube,

ΔV = 3L^2 ΔL


Volumetric strain of cube will be,

eV = ΔV/V = (3L^2 ΔL)/L^3

After solving this the final result will be,

eV = ΔV/V = 3ΔL/L

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So here you have to know the volumetric strain and their formula for rectangular bar, thin cylinder, sphere, and cube with their derivation.

I hope you will learn all aspects related to the volumetric strain if you have any doubt then you are free to ask me by mail or on the contact us page.

Thank You.

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