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In most contact theories, the most popular of which are the three models of Hertz, Derjaguin, Muller and Toporov (DMT) and Johnson, Kendall and Roberts (JKR), biological cells were considered as an elastic material which is not a proper assumption. The elastic assumption in the case of biological cells could lead to neglecting the loading history as a result of which the stresses and strains applied to the material would not be studied accurately. In this paper, developing the three mentioned elastic models into viscoelastic models, simulating and comparing them with empirical data obtained through the indentation test of the MCF-7 cancer cell showed that the viscoelastic state presents a better prediction of biological cell behavior compared to that of an elastic state. The selection of the suitable creep function for objects in contact is another issue that has a significant importance in the viscoelastic case and this was investigated. Different mechanical models of a cell were studied and simulated for all three named theories among which the creep function obtained from the Kelvin model, a parallel combination of spring-damper, simplified the simulation and gave more precise results for modeling due to the fact that the obtained results from this model are closer to experimental ones and simpler than other models. On the other hand, for a more exact prediction of cell behavior, this model was modified by an equivalent elasticity module which considered cell components instead of the cell cortex only. The results of the simulation confirmed that a new elasticity module can improve the accuracy of cell models. After choosing the suitable mechanical model for the cell, we scrutinized the capability of the developed theories in predicting the results for biological liquid environments. Although the results of the Hertz and DMT viscoelastic models are closer to experimental ones in comparison with viscoelastic JKR, neglecting adhesion makes their prediction in biological liquid environments weak and erroneous. Therefore, it can be concluded that the developed viscoelastic model of JKR is more accurate and has a better performance in different environments than the other mentioned models.

Developing viscoelastic contact models and selecting suitable creep function for spherical biological cells