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
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