Volumetric Strain and Bulk Modulus

Volumetric Strain

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

εv = δV / V

where,

δV = Change in volume,

and V = Original volume.

Notes :

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

εv = δV / V=ε{1-(2/m)}

where, ε = Linear strain

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

εv = εx + εy + εz

where, εx, εy and εz 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 the ratio of the direct stress to the corresponding volumetric strain is known as bulk modulus. It is usually denoted by K. Mathematically, bulk modulus,

K = Direct stress/Volumetric strain =σ /(δV/V)

1. Relation Between Bulk Modulus and Young’s Modulus

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

K = m.E/{3 (m-2)}= E / {3(1-2μ)}

2. Relation Between Young’s Modulus and Modulus of Rigidity

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

G = m.E / {2 (m+1)} = E / {2(1+μ)}

Impact Stress

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

Consider a bar carrying a load W at a height h and falling 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,
δl = Deformation of the bar,
P = Force at which the deflection δl is produced,
σi = Stress induced in the bar due to the application of impact load, and
h = Height through which the load falls.

Note : When h = 0, then σi = 2W/A. This means that the stress in the bar when the load in applied suddenly is
double of the stress induced due to gradually applied load.

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