# What is Volumetric Strain? Definition & Formula

By BYJU'S Exam Prep

Updated on: September 25th, 2023

**The Volumetric Strain** is generally denoted by E_{V}. Strain in a body is the ratio of change in dimension to its original dimension when the body deforms under the application of load. Strain can be classified into 4 types: longitudinal strain, lateral strain, shear strain and volumetric strain. In this article, we will focus on the volumetric strain.

In this article, we will first define volumetric strain and then derive the volumetric strain formula for three different types of members. These members are listed below:

- Rectangular bar
- Cylindrical rod
- Spherical body

Table of content

**What is Volumetric Strain?**

Volumetric strain is defined as the ratio of change in the volume of a body to its original volume due to the application of some external deformation-causing forces. It is also known as Dilation and is important for the GATE exam. The general equation for volumetric strain is given as –

**E _{V }= ΔV/V**

where

- ΔV = change in volume
- V = original volume

**Bulk Modulus (K): **When a body is subjected to stresses of equal intensity in 3 mutually perpendicular directions, then the ratio of this direct stress to the volumetric strain is called Bulk modulus. It is generally denoted by K.

K =Direct Stress/Volumetric strain= σ/E_{V}

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**Volumetric Strain for Rectangular Bar**

This section will derive the volumetric strain formula for a rectangular bar. To define volumetric strain expression for a rectangular bar, let us assume a rectangular prismatic member of length L, width B, and depth D subjected to triaxial stresses, as shown in the figure below.

The initial volume of the rectangular bar,

**V = L×B×D**

The change in volume due to the applied stresses,

**ΔV = δL×B×D + L×δB×D + L×B×δD**

We know that volumetric strain,

E_{V}=ΔV/V

E_{V}=δL/L+δB/B+δD/D

We know that,

δL/L = E_{x} (strain in the x-direction)

δB/B = E_{y} (strain in the y-direction)

and δD/D = E_{z} (strain in the z-direction)

So,

E_{V }= E_{x}+E_{y}+E_{z}….(i)

We also know that,

E_{x }= σ_{x}/E-μσ_{y}/E-μσ_{z}/E

E_{y }= σ_{y}/E-μσ_{x}/E-μσ_{z}/Eand

E_{z }= σ_{z}/E-μσ_{x}/E-μσ_{z}/E

where

- μ = Poisson’s ratio
- E = Young’s modulus of elasticity

Putting the value of x, y and z in equation (i)

E_{V}=σ_{x}/E-μσ_{y}/E-μσ_{z}/E +σ_{y}/E-μσ_{x}/E-μσ_{z}/E +σ_{z}/E-μσ_{x}/E-μσ_{z}/E

E_{V}= (1-2μ) (σ_{x}+σ_{y}+σ_{z})/E

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**Volumetric Strain for Cylindrical Rod**

In this section, we will derive the volumetric strain formula for a cylindrical rod. To define volumetric strain expression for a cylindrical rod, let us assume a cylindrical rod of length L and diameter d as shown in the figure below

The initial volume of the cylindrical rod,

V=(π/4)d^{2}.L

The change in volume due to applied stresses

ΔV=(π/4)[d^{2}.δL+L.2dδd]

We know that volumetric strain,

E_{V}=ΔV/V

E_{V}=[δL/L+2. δd/d]

We know that,

δL/L=E_{L} (strain in the longitudinal direction)

δd/d=E_{d} (strain in the radial direction)

So,

E_{V}=E_{L}+2E_{d}

**Volumetric Strain for a Spherical Body**

In this section, we will derive the volumetric strain formula for a spherical body. To define volumetric strain expression for a spherical body, let us assume a sphere of diameter d, as shown in the figure below.

The initial volume of the sphere,

V=(π/6)d^{3}

The change in volume due to applied stresses

V=(π/6).3δd.d^{2}

We know that volumetric strain,

E_{V}=ΔV/V

E_{V}=3δd/d

E_{V}=3E_{d}