  Shape memory effect associated with martensitic transformations is of the
rapidly developing field in nanotechnologies, where industrial use of systems
established on that effect provide greater flexibility on the nano-devices
fabrication of various kind. And therefore it addresses questions to the phase
transition phenomena at low-scale and its limitations and control. In this
report, we studied the crystal structure of tapered plates of Ti$_2$NiCu alloy
and the temperature $T_c$ at which the martensitic transition occurs. We
demonstrated that $T_c$ has a strongly descending character as a function of
the plate thickness $h$. The critical thickness value at which the transition
completely suppressed is 20 nm. Moreover, the obtained results for $T_c(h)$
curves indicate the hysteretic nature of the transition. These findings open
the pathway for size limits indication and regimes modulation where the
alloy-based nano-mechanical devices can be tuned to operate more efficiently.
