Recently we reported on the development of a composite material exhibiting reversible shape memory effect. A Ti–25Ni–25Cu (at.%) alloy was obtained by the melt spinning technique as amorphous–crystalline ribbons with a thickness of approximately 40 μm. The thickness of the amorphous and crystalline layers (dа and dc, respectively) was varied by electrochemical polishing. It has been ascertained that with varying the relationship dc/dа the martensite transformation and shape-recovery temperatures do not actually change, while the minimum radius of the ribbon bending decreases from 8.0 mm to 2.4 mm with increasing the relationship dc/dа from 0.33 to 1.40. The maximum reversible strain comprises 0.4% at dc/dа = 0.82. On the basis of experimental data obtained the phenomenological description, providing an explanation for nature of the phenomena taking place in the rapidly quenched amorphous-crystalline ribbon composite, has been proposed.