  We report high magnetic field thermal-expansion and magnetostriction results
on CeRhIn$_5$ single crystals. Several transitions, both first and second
order, are observed when the field is applied perpendicular to the
crystallographic c-axis. The magnetic field dependence of the thermal-expansion
coefficient above 15 K, where the magnetic correlations are negligible, can be
explained supposing an almost pure $| \pm 5/2>$ ground state doublet, in
apparent contradiction with neutron scattering experiments. Although the
spin-lattice interaction is relevant in this compound, the effect of the
magnetic correlations on the elastic properties is relatively weak, as revealed
by resonant ultrasound spectroscopy experiments.
