Open Access
Issue
ESAIM: M2AN
Volume 50, Number 2, March-April 2016
Page(s) 593 - 631
DOI https://doi.org/10.1051/m2an/2015073
Published online 21 March 2016
  1. E. Acerbi, V. Chiado Piat, G. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains. Nonlin. Anal. Theory Methods Appl. 18 (1992) 481–496. [Google Scholar]
  2. N.D. Alikakos, Lp bounds of solutions of reaction-diffusion equations. Commun. Partial Differ. Eq. 4 (1976) 827–868. [Google Scholar]
  3. G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482–1518. [Google Scholar]
  4. G. Allaire, Shape Optimization by the Homogenization Method. Springer (2002). [Google Scholar]
  5. A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North Holland (1978). [Google Scholar]
  6. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer (2010). [Google Scholar]
  7. M.-A.-J. Chaplain, The strain energy function of an ideal plant cell wall. J. Theoret. Biol. 163 (1993) 77–97. [CrossRef] [Google Scholar]
  8. A. Chavarría-Krauser and M. Ptashnyk, Homogenization approach to water transport in plant tissues with periodic microstructures. Math. Model. Nat. Phenom. 8 (2013) 80–111. [CrossRef] [EDP Sciences] [Google Scholar]
  9. P.-G. Ciarlet, Mathematical elasticity. Volume I: Three-dimensional elasticity. North-Holland (1988). [Google Scholar]
  10. P.-G. Ciarlet and P. Ciarlet Jr., Another approach to linear elasticity and Korn’s inequality. C.R. Acad. Sci. Paris Ser. I 339 (2004) 307–312. [Google Scholar]
  11. D. Cioranescu and J. Saint Jean Paulin, Homogenization of reticulated structures. Springer (1999). [Google Scholar]
  12. D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40 (2008) 1585–1620. [CrossRef] [MathSciNet] [Google Scholar]
  13. D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 44 (2012) 718–760. [CrossRef] [MathSciNet] [Google Scholar]
  14. J.-R. Colvin, The size of the cellulose microfibril. J. Cell Biol. 17 (1963) 105–109. [CrossRef] [PubMed] [Google Scholar]
  15. D.-J. Cosgrove, Growth of the plant cell wall. Nat. Rev. Molec. Cell Biol. 6 (2005) 850–86. [CrossRef] [PubMed] [Google Scholar]
  16. I. Diddens, B. Murphy, M Krisch and M. Müller, Anisotropic elastic properties of cellulose measured using inelastic X-ray scattering. Macromolecules 41 (2008) 9755–9759. [CrossRef] [Google Scholar]
  17. J. Dumais, S.-L. Shaw, C.-R. Steele, S.-R. Long and P.-M. Ray, An anisotropic-viscoplastic model of plant cell morphogenesis by tip growth. Int. J. Developmental Biol. 50 (2006) 209–222. [CrossRef] [PubMed] [Google Scholar]
  18. R. Dutta and K.-R. Robinson, Identification and characterization of stretch-activated ion channels in pollen protoplasts. Plant Physiol. 135 (2004) 1398–1406. [CrossRef] [PubMed] [Google Scholar]
  19. R.-J. Dyson, O.-E. Jensen, A fibre-reinforced fluid model of anisotropic plant cell growth. J. Fluid Mech. 655 (2010) 472–503. [CrossRef] [Google Scholar]
  20. R.-J. Dyson, L.-R. Band and O.-E. Jensen, A model of crosslink kinetics in the expanding plant cell wall: Yield stress and enzyme action. J. Theoret. Biol. 307 (2012) 125–136. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  21. T. Fatima, A. Muntean and M. Ptashnyk, Error estimate and unfolding method for homogenization of a reaction-diffusion system modeling sulfate corrosion. Appl. Anal. 91 (2012) 1129–1154. [CrossRef] [Google Scholar]
  22. Y.C. Fung, Biomechanics: mechanical properties of living tissues. Springer (1993). [Google Scholar]
  23. R.-P. Gilbert and A. Mikelić, Homogenizing the acoustic properties of the seabed: Part I. Nonlin. Anal. 40 (2000) 185–212. [CrossRef] [MathSciNet] [Google Scholar]
  24. M.-E. Gurtin, E. Fried and L. Anand, The Mechanics and Thermodynamics of Continua. Cambridge University Press (2010). [Google Scholar]
  25. L. Haggerty, J.H. Sugarman, R.K. Prud’homme, Diffusion of polymers through polyacrylamide gels. Polymer 29 (1988) 1058–1063. [CrossRef] [Google Scholar]
  26. W. Jäger and U. Hornung, Diffusion, convection, adsorption, and reaction of chemicals in porous media. J. Differ. Eq. 92 (1991) 199–225. [Google Scholar]
  27. W. Jäger, A. Mikelić and M. Neuss-Radu, Homogenization limit of a model system for interaction of flow, chemical reactions, and mechanics in cell tissues. SIAM J. Math. Anal. 43 (2011) 1390–1435. [CrossRef] [MathSciNet] [Google Scholar]
  28. V.-V. Jikov, S.-M. Kozlov and O.-A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer (1994). [Google Scholar]
  29. C.-J. Jennedy, A. Šturcová, M.-C. Jarvis and T.-J. Wess, Hydration effects on spacing of primary-wall cellulose microfibrils: a small angle X-ray scattering study. Cellulose 14 (2007) 401–408. [CrossRef] [Google Scholar]
  30. A. Korn, Über einige ungleichungen, welche in der theorie del elastichen und elektrishen schwingungen eine rolle spielen. Bullettin Internationale, Cracovie Akademie Umiejet, Classe des sciences mathématiques et naturelles (1909) 705–724. [Google Scholar]
  31. J.-H. Kroeger, R. Zerzour and A. Geitmann, Regulator or driving force? The role of turgor pressure in oscillatory plant cell growth. PLoS One 6 (2011) e18549. [CrossRef] [PubMed] [Google Scholar]
  32. O. Ladyzhenskaya, V. Solonnikov and N. Ural’ceva, Linear and quasilinear equations of parabolic type. American Mathematical Society (1968). [Google Scholar]
  33. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod (1969). [Google Scholar]
  34. A. Logg and G.N. Wells, DOLFIN: automated finite element computing. ACM Trans. Math. Software 37 (2010). [CrossRef] [MathSciNet] [Google Scholar]
  35. A. Logg, K.-A. Mardal, G.N. Wells et al., Automated solution of differential equations by the finite element method. Springer (2012). [Google Scholar]
  36. A.J. Majda and A.L. Bertozzi, Vorticity and Incompressible Flow. Cambridge Texts Appl. Math. (2001). [Google Scholar]
  37. A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenisation techniques. SIAM J. Math. Anal. 40 (2008) 215–237. [CrossRef] [MathSciNet] [Google Scholar]
  38. M.-L. Mascarenhas, Homogenization of a viscoelastic equations with non-periodic coefficients. Proc. Roy. Soc. Edinburgh: Sect. A Math. 106 (1987) 143–160. [Google Scholar]
  39. A. Mikelić and M.-F. Wheeler, On the interface law between a deformable porous medium containing a viscous fluid and an elastic body. Math. Models Methods Appl. Sci. 22 (2012) 1250031. [CrossRef] [MathSciNet] [Google Scholar]
  40. F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials. Vol. 31 of Progr. Nonlin. Differ. Equ. Appl. Birkhäuser Boston, Boston, MA (1997) 21–43. [Google Scholar]
  41. J. Necas, Les méthodes directes en théorie des équations elliptiques. Academie, Prague (1967). [Google Scholar]
  42. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608–623. [CrossRef] [MathSciNet] [Google Scholar]
  43. O. Oleinik, A.-S. Shamaev and G.-A. Yosifian, Mathematical problems in Elasticity and Homogenization. North Holland (1992). [Google Scholar]
  44. K.B. Ølgaard and G.N. Wells, Optimisations for quadrature representations of finite element tensors through automated code generation. ACM Trans. Math. Software 37 (2010). [Google Scholar]
  45. J.-B. Passioura and S.-C. Fry, Turgor and cell expansion: beyond the Lockhart equation. Aust. J. Plant Physiol. 19 (1992) 565–576. [CrossRef] [Google Scholar]
  46. A. Peaucelle, S.A. Braybrook, L. Le Guillou, E. Bron, C. Kuhlemeier and H. Hofte, Pectin-induced changes in cell wall mechanics underlie organ initiation in Arabidopsis. Curr. Biol. 21 (2011) 1720–1726. [CrossRef] [PubMed] [Google Scholar]
  47. S. Pelletier, J. Van Orden, S. Wolf, K. Vissenberg, J. Delacourt, Y.-A. Ndong, J. Pelloux, V. Bischoff, A. Urbain, G. Mouille, G. Lemonnier, J.-P. Renou and H. Hofte, A role for pectin de-methylesterification in a developmentally regulated growth acceleration in dark-grown Arabidopsis hypocotyls. New Phytol. 188 (2010) 726–739. [CrossRef] [PubMed] [Google Scholar]
  48. T.-E. Proseus and J.-S. Boyer, Calcium deprivation disrupts enlargement of Chara corallina cells: further evidence for the calcium pectate cycle. J. Exp. Bot. 63 (2012) 1–6. [CrossRef] [PubMed] [Google Scholar]
  49. M. Ptashnyk, Derivation of a macroscopic model for nutrient uptake by a single branch of hairy-roots. Nonlin. Anal.: Real World Appl. 11 (2010) 4586–4596. [CrossRef] [Google Scholar]
  50. M. Ptashnyk and B. Seguin, Periodic homogenization and material symmetry in linear elasticity, 1504.08165 (2015). [Google Scholar]
  51. R. Redlinger, Invariant sets for strongly coupled reaction-diffusion systems under general boundary conditions. Arch. Rational Mech. Anal. 108 (1989) 281–291. [Google Scholar]
  52. E.-R. Rojas, S. Hotton and J. Dumais, Chemically mediated mechanical expansion of the pollen tube cell wall. Biophys. J. 101 (2011) 1844–1853. [CrossRef] [PubMed] [Google Scholar]
  53. E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory. Springer (1980). [Google Scholar]
  54. J. Simon, Compact sets in the space Lp(0,T;B). Ann. Mat. Pure Appl. (IV) CXLVI (1987) 65–96. [Google Scholar]
  55. J. Smoller, Shocke Waves and Reaction-Diffusion Equations. Springer (1994). [Google Scholar]
  56. C. Somerville, S. Bauer, G. Brininstool, M. Facette, T. Hamann, J. Milne, E. Osborne, A. Paredez, S. Persson, T. Raab, S. Vorwerk and H. Youngs, Toward a systems approach to understanding plant cell walls. Science 306 (2004) 2206. [CrossRef] [PubMed] [Google Scholar]
  57. L.-H. Thomas, V.-T. Forsyth, A. Šturcová, C.-J. Kennedy, R.-P. May, C.-M. Altaner, D.-C. Apperley, T.-J. Wess and M.-C. Jarvis, Structure of cellulose microfibrils in primary cell walls from collenchyma. Plant Physiol. 161 (2013) 465–476. [CrossRef] [PubMed] [Google Scholar]
  58. B.-A. Veytsman and D.-J. Cosgrove, A model of cell wall expansion based on thermodynamics of polymer networks. Biophys. J. 75 (1998) 2240–2250. [CrossRef] [PubMed] [Google Scholar]
  59. P.J. White, The pathways of calcium movement to the xylem. J. Exp. Bot. 52 (2001) 891–899. [CrossRef] [PubMed] [Google Scholar]
  60. S. Wolf and S. Greiner, Growth control by cell wall pectins. Protoplasma 249 (2012) 169–175. [CrossRef] [Google Scholar]
  61. S. Wolf, K. Hématy and H. Höfte, Growth control and cell wall signaling in plants. Ann. Rev. Plant Biol. 63 (2012) 381–407. [Google Scholar]
  62. S. Wolf, J. Mravec, S. Greiner, G. Mouille and H. Höfte, Plant cell wall homeostasis is mediated by Brassinosteroid feedback signaling. Curr. Biol. 22 (2012) 1732–1737. [CrossRef] [PubMed] [Google Scholar]
  63. U.-Z. Zimmermann, D. Hüs ken and E.-D. Schulze, Direction turgor pressure measurements in individual leave cells of Tradescantia virginiana. Planta 148 (1980) 445–453. [CrossRef] [Google Scholar]
  64. G. Zsivanovits, A.-J. MacDougall, A.-C. Smith and S.-G. Ring, Material properties of concentrated pectin networks. Carbohyd. Res. 339 (2004) 1317–1322. [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you