What is Volumetric Strain? Definition & Formula
By BYJU'S Exam Prep
Updated on: September 25th, 2023
The Volumetric Strain is generally denoted by EV. 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 –
EV = 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/EV
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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,
EV=ΔV/V
EV=δL/L+δB/B+δD/D
We know that,
δL/L = Ex (strain in the x-direction)
δB/B = Ey (strain in the y-direction)
and δD/D = Ez (strain in the z-direction)
So,
EV = Ex+Ey+Ez….(i)
We also know that,
Ex = σx/E-μσy/E-μσz/E
Ey = σy/E-μσx/E-μσz/Eand
Ez = σz/E-μσx/E-μσz/E
where
- μ = Poisson’s ratio
- E = Young’s modulus of elasticity
Putting the value of x, y and z in equation (i)
EV=σx/E-μσy/E-μσz/E +σy/E-μσx/E-μσz/E +σz/E-μσx/E-μσz/E
EV= (1-2μ) (σx+σy+σz)/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)d2.L
The change in volume due to applied stresses
ΔV=(π/4)[d2.δL+L.2dδd]
We know that volumetric strain,
EV=ΔV/V
EV=[δL/L+2. δd/d]
We know that,
δL/L=EL (strain in the longitudinal direction)
δd/d=Ed (strain in the radial direction)
So,
EV=EL+2Ed
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)d3
The change in volume due to applied stresses
V=(π/6).3δd.d2
We know that volumetric strain,
EV=ΔV/V
EV=3δd/d
EV=3Ed