The temperature dependence of the dynamic Young’s modulus E, the elastic energy dissipation coefficient Q−1 and the heat flow (DSC) has been studied between 90 and 370 K in an Ni40Ti50Cu10 alloy containing various amounts nH of H (nH=H/Me=0; 0.004; 0.008; 0.013 and 0.018 at.). The Young’s modulus exhibits softening when the start temperature Ms of the B2→B19 martensitic transition is approached on cooling and a much steeper modulus decrease between Ms and Mf. This steep decrease appears to be associated with stress-induced motions of twin boundaries within the B19 martensite as it is drastically reduced by H pinning of these boundaries. No internal friction (IF) peak occurs at the B2→B19 transition and the values of Q−1 are high in the B19 martensite (≅100×10−4). Two IF peaks, PH and PTWH, occur below Ms in the H-doped material; the first is likely due to stress-assisted reordering of H elastic dipoles within a hydride phase, the second to H dragging processes by twin boundaries.