  When elastic solids are sheared, a nonlinear effect named after Poynting
gives rise to normal stresses or changes in volume. We provide a novel relation
between the Poynting effect and the microscopic Gr\"uneisen parameter, which
quantifies how stretching shifts vibrational modes. By applying this relation
to random spring networks, a minimal model for, e.g., biopolymer gels and solid
foams, we find that networks contract or develop tension because they vibrate
faster when stretched. The amplitude of the Poynting effect is sensitive to the
network's linear elastic moduli, which can be tuned via its preparation
protocol and connectivity. Finally, we show that the Poynting effect can be
used to predict the finite strain scale where the material stiffens under
shear.
