Final answer:
The Poisson's ratio of a material of a wire that maintains its volume constant under external normal stress is zero, because the transverse strain is zero in case of incompressible materials under such conditions.
Step-by-step explanation:
To determine the Poisson's ratio of a material for a wire that maintains constant volume under external normal stress, we must understand the definition of Poisson's ratio (ν). Poisson's ratio is the negative ratio of transverse to axial strain. When a material is stretched, it usually gets thinner. Conversely, when a material is compressed, it usually gets wider. However, if the volume of the material remains constant under stress, then the material does not expand or contract in the perpendicular direction when an axial load is applied.
Therefore, for a wire whose volume remains constant when a normal stress is applied, the change in length is completely compensated by a change in the other two dimensions, and there is no transverse strain. This means that for any axial strain, transverse strain is zero. By the definition of Poisson's ratio, which is ν = - (transverse strain)/(axial strain), and since the transverse strain is zero, the Poisson's ratio of such a material would be zero (ν = 0).