A uniform cylindrical wire of a given isotropic material is subjected to a longitudinal tensile stress S . If the bulk modulus of the material is K , the fractional change in the volume of the wire is:

Step-by-step solution

Let the Young's modulus of the material be Y and its Poisson's ratio be . When a longitudinal tensile stress S is applied, the longitudinal strain is: _ l = S Y The lateral strain is: _ r = - _ l = - S Y The fractional change in volume (volumetric strain) for a cylinder is given by: V V = _ l + 2 _ r Substituting the strains: V V = S Y - 2 S Y = S Y (1 - 2 ) We know the relation between Young's modulus Y , bulk modulus K , and Poisson's ratio is: Y = 3K(1 - 2 ) Rearranging this relation gives: 1 - 2 Y = 1 3K Substi