Enveloped viruses and nanosized biomimetic particles for drug and gene delivery enter target cells mainly through receptor-mediated endocytosis. A few models have been presented to elucidate the mechanics of particle engulfment by the cell membrane, showing how size and surface chemico-physical properties favor or oppose internalization. In this work, the effect of particle nonsphericity is addressed considering elliptical cylindrical particles with aspect ratio Gamma. Using a continuum energetic approach, three different conditions have been identified: for sufficiently small Gamma, the particle is not even wrapped by the cell membrane; for sufficiently large Gamma, the particle is partially wrapped ("frustrated endocytosis"); and for intermediate values of Gamma, the particle is fully wrapped and eventually internalized. Given the pleomorphism of viruses and the broad spectrum of shapes for nanosized biomimetic particles, the results presented may be of interest to virologists, pharmacologists, toxicologists, and nanotechnologists.