High magnetic field thermal-expansion and magnetostriction of URu$_2$Si$_2$

We present high magnetic field (up to 45 T) thermal-expansion and magnetostriction results on URu$_2$Si$_2$ single crystals. The volume change associated with the transition to the ``hidden'' order phase becomes increasingly discontinous as the magnetic field is raised above 30 T. This confirms recent thermal conductivity and specific heat experiments indicating a strong coupling between the ``hidden'' order parameter and the lattice which suggest some sort of charge ordered state. Several other transitions are observed at higher fields, many of them showing hysteresis, while a change in the sign of the magnetostriction coefficient is observed at the metamagnetic transition ($B_M$ $\sim$ 38 T).