When we calculate the energy stored in the rod after elongation by an external force, we only consider longitudinal expansion. My question is that according to Poisson, there is also a sideways/transverse contraction in the rod. If any, how significant is the error in calculating the potential energy of the rod(if we consider it also)?
The contraction occurs with zero transverse force. So the transverse force times the transverse displacement is zero. No work is done and, thus, this contributes nothing to the stored elastic energy.