# Volumetric Strain and Bulk Modulus

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## Volumetric Strain

When a body is subjected to a system of forces, it undergoes some changes in its dimensions. InÂ other words, the volume of the body is changed. The ratio of the change in volume to the originalÂ volume is known as volumetric strain. Mathematically, volumetric strain,

Îµv = Î´V / V

where,

Î´V = Change in volume,

and V = Original volume.

Notes :

1. Volumetric strain of a rectangular body subjected to an axial force is given as

Îµv = Î´V / V=Îµ{1-(2/m)}

where, Îµ = Linear strain

2. Volumetric strain of a rectangular body subjected to three mutually perpendicular forces is given by

Îµv = Îµx + Îµy + Îµz

where, Îµx, Îµy and Îµz are the strains in the directions x-axis, y-axis and z-axis respectively.

## Bulk Modulus

When a body is subjected to three mutually perpendicular stresses, of equal intensity, then theÂ ratio of the direct stress to the corresponding volumetric strain is known as bulk modulus. It isÂ usually denoted by K. Mathematically, bulk modulus,

K =Â Direct stress/Volumetric strain =Ïƒ /(Î´V/V)

1. Relation Between Bulk Modulus and Youngâ€™s Modulus

The bulk modulus (K) and Young’s modulus (E) are related by the following relation,

K =Â m.E/{3 (m-2)}=Â E / {3(1-2Î¼)}

2. Relation Between Youngâ€™s Modulus and Modulus of Rigidity

The Young’s modulus (E) and modulus of rigidity (G) are related by the following relation,

G =Â m.E / {2 (m+1)} = E / {2(1+Î¼)}

## ImpactÂ Stress

Sometimes, machine members are subjected to the loadÂ with impact. The stress produced in the member due to the fallingÂ load is known as impact stress.

Consider a bar carrying a load W at a height h and fallingÂ on the collar provided at the lower end, as shown in Fig. 1.

Let A = Cross-sectional area of the bar,
E = Young’s modulus of the material of the bar,
l = Length of the bar,
Î´l = Deformation of the bar,
P = Force at which the deflection Î´l is produced,
Ïƒi = Stress induced in the bar due to the applicationÂ of impact load, and
h = Height through which the load falls.

Note : When h = 0, then Ïƒi = 2W/A. This means that the stress in the bar when the load in applied suddenly is

ReferenceÂ A textbook of Machine Design by R.S.Khurmi and J.K.Gupta

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