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

Handle URI:
http://hdl.handle.net/10754/600159
Title:
Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues
Authors:
Davit, Y.; Osborne, J. M.; Byrne, H. M.; Gavaghan, D.; Pitt-Francis, J.
Abstract:
The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to continuum methods based on partial differential equations, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born 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 two-dimensional cellular-scale models by assessing the mechanical behavior of model biological tissues, including crystalline (honeycomb) and noncrystalline reference states. The numerical procedure involves applying an affine deformation to the boundary cells and computing the quasistatic 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 center-based cell models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free 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 vertex-based cell 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 and continuous modeling, adaptation of atom-to-continuum techniques to biological tissues, and model classification. © 2013 American Physical Society.
Citation:
Davit Y, Osborne JM, Byrne HM, Gavaghan D, Pitt-Francis J (2013) Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues. Phys Rev E 87. Available: http://dx.doi.org/10.1103/PhysRevE.87.042724.
Publisher:
American Physical Society (APS)
Journal:
Physical Review E
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
30-Apr-2013
DOI:
10.1103/PhysRevE.87.042724
PubMed ID:
23679466
Type:
Article
ISSN:
1539-3755; 1550-2376
Sponsors:
This work was based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorDavit, Y.en
dc.contributor.authorOsborne, J. M.en
dc.contributor.authorByrne, H. M.en
dc.contributor.authorGavaghan, D.en
dc.contributor.authorPitt-Francis, J.en
dc.date.accessioned2016-02-28T06:43:59Zen
dc.date.available2016-02-28T06:43:59Zen
dc.date.issued2013-04-30en
dc.identifier.citationDavit Y, Osborne JM, Byrne HM, Gavaghan D, Pitt-Francis J (2013) Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues. Phys Rev E 87. Available: http://dx.doi.org/10.1103/PhysRevE.87.042724.en
dc.identifier.issn1539-3755en
dc.identifier.issn1550-2376en
dc.identifier.pmid23679466en
dc.identifier.doi10.1103/PhysRevE.87.042724en
dc.identifier.urihttp://hdl.handle.net/10754/600159en
dc.description.abstractThe development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to continuum methods based on partial differential equations, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born 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 two-dimensional cellular-scale models by assessing the mechanical behavior of model biological tissues, including crystalline (honeycomb) and noncrystalline reference states. The numerical procedure involves applying an affine deformation to the boundary cells and computing the quasistatic 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 center-based cell models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free 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 vertex-based cell 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 and continuous modeling, adaptation of atom-to-continuum techniques to biological tissues, and model classification. © 2013 American Physical Society.en
dc.description.sponsorshipThis work was based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherAmerican Physical Society (APS)en
dc.titleValidity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissuesen
dc.typeArticleen
dc.identifier.journalPhysical Review Een
dc.contributor.institutionUniversity of Oxford, Oxford, United Kingdomen
dc.contributor.institutionUniversite de Toulouse, Toulouse, Franceen
dc.contributor.institutionIMFT Institut de Mecaniques des Fluides, Toulouse, Franceen
kaust.grant.numberKUK-C1-013-04en

Related articles on PubMed

All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.