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This work investigates the mechano-biological features of cells cultured in monolayers in response to different osmotic conditions. In-vitro experiments have been performed to quantify the long-term effects of prolonged osmotic stresses on the morphology and proliferation capacity of glioblastoma cells. The experimental results highlight that both hypotonic and hypertonic conditions affect the proliferative rate of glioblastoma cells on different cell cycle phases. Moreover, glioblastoma cells in hypertonic conditions display a flattened and elongated shape. The latter effect is explained using a nonlinear elastic model for the single cell. Due to a crossover between the free energy contributions related to the cytosol and the cytoskeletal fibers, a critical osmotic stress determines a morphological transition from a uniformly compressed to an elongated shape.