Davit, Y. and Osborne, J. M. and Byrne, H. M. and Gavaghan, D. J. and PittFrancis, J. M. (2012) Validity of the CauchyBorn rule applied to discrete cellularscale models of biological tissues. Physical Review E . (Submitted)

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Abstract
The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to PDEbased continuum methods, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discretetocontinuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the CauchyBorn rule (CBR). Although the CBR does not hold exactly for noncrystalline materials, it may still be used as a first order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to 2D cellularscale models by assessing the mechanical behaviour of model biological tissues, including crystalline (honeycomb) and noncrystalline reference states. The numerical procedure consists in precribing an affine deformation on the boundary cells and computing the position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For centrebased models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stressfree configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertexbased models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete/continuous modelling, adaptation of atomtocontinuum (AtC) techniques to biological tissues and model classification.
Item Type:  Article 

Subjects:  D  G > General 
Research Groups:  Oxford Centre for Collaborative Applied Mathematics 
ID Code:  1553 
Deposited By:  Peter Hudston 
Deposited On:  05 Jul 2012 07:50 
Last Modified:  29 May 2015 19:15 
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