Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues

摘要

The development of new models of biological tissues that consider cells in a discrete manner is becomingnincreasingly popular as an alternative to continuum methods based on partial differential equations, althoughnformal relationships between the discrete and continuum frameworks remain to be established. For crystalnmechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can benmapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Bornnrule (CBR). Although the CBR does not hold exactly for noncrystalline materials, it may still be used as anfirst-order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal isnto investigate numerically the applicability of the CBR to two-dimensional cellular-scale models by assessing thenmechanical behavior of model biological tissues, including crystalline (honeycomb) and noncrystalline referencenstates. The numerical procedure involves applying an affine deformation to the boundary cells and computingnthe quasistatic position of internal cells. The position of internal cells is then compared with the prediction of thenCBR and an average deviation is calculated in the strain domain. For center-based cell models, we show that thenCBR holds exactly when the deformation gradient is relatively small and the reference stress-free configurationnis defined by a honeycomb lattice.We show further that the CBR may be used approximately when the referencenstate is perturbed from the honeycomb configuration. By contrast, for vertex-based cell models, a similar analysisnreveals that the CBR does not provide a good representation of the tissue mechanics, even when the referencenconfiguration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications ofnthese results for concurrent discrete and continuous modeling, adaptation of atom-to-continuum techniques tonbiological tissues, and model classification.