Volumetric Strain - Definition, Formula

By Aina Parasher|Updated : August 7th, 2022

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

What is Volumetric Strain?

Volumetric strain is defined as the ratio of change in volume of a body to its original volume due to the application of some external deformation causing forces. It is also known as Dilation. The volumetric strain is generally denoted by EV and 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 referred to as Bulk modulus. It is generally denoted by K.

K =Direct Stress/Volumetric strain= σ/EV

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,

Exx/E-μσy/E-μσz/E

Eyy/E-μσx/E-μσz/Eand

Ezz/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)

EVx/E-μσy/E-μσz/E +σy/E-μσx/E-μσz/E +σz/E-μσx/E-μσz/E

EV= (1-2μ) (σxyz)/E

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

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FAQs on Volumetric Strain

  • Volumetric strain (also known as Dilation) is defined as the ratio of change in volume of a body to its original volume due to the application of some external deforming forces. The volumetric strain is generally denoted by EV.

  • We know that volumetric strain is a ratio of change in volume of a body to its original volume. SI unit for both change in volume and volume will be m3. So, the volumetric strain will be a unitless quantity.

  • Volumetric strain is defined as the ratio of change in volume of a body to its original volume due to the application of some external deforming forces. Therefore, the equation for volumetric strain will be EV=ΔV/V.

  • 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 referred to as Bulk modulus. It is generally denoted by K.

    K = Direct Stress/ Volumetric strain = σ/EV

  • The volumetric strain for a spherical body of diameter d is given as:

    EV=3δd/d

    EV=3Ed

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