  We obtain stronger laboratory constraints on the coupling constants of
axion-like particles to nucleons from measurements of the normal and lateral
Casimir forces between sinusoidally corrugated surfaces of a sphere and a
plate. For this purpose, the normal and lateral additional force arising in the
experimental configurations due to two-axion exchange between protons and
neutrons are calculated. Our constraints following from measurements of the
normal and lateral Casimir forces are stronger than the laboratory constraints
reported so far for masses of axion-like particles larger than 11eV and 8eV,
respectively. A comparison between various laboratory constraints on the
coupling constants of axion-like particles to nucleons obtained from the
magnetometer measurements, Eotvos- and Cavendish-type experiments, and from the
Casimir effect is performed over the wide range of masses of axion-like
particles from 10^{-10}eV to 20eV.
