Tip growth in morpho-elasticity

Martine Ben Amar; Julien Dervaux

doi:10.5802/crmeca.27

Morphogenesis, elasticity

Tip growth in morpho-elasticity

Martine Ben Amar ^1,² ; Julien Dervaux ³

¹ Institut Universitaire de Cancérologie, Faculté de médecine, Sorbonne Université, 91 Bd de l’Hôpital, 75013 Paris, France
² Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
³ Laboratoire Matière et Systèmes Complexes, UMR 7057, CNRS and Université de Paris, 75013 Paris, France

Comptes Rendus. Mécanique, Volume 348 (2020) no. 6-7, pp. 613-625.

Résumé
Abstract

La croissance biologique génère des contraintes mécaniques qui contribuent à façonner la forme des tissus, des organes et des organismes vivants. En raison de l’extrême complexité des phénomènes de croissance biologique, il est en général impossible de prédire ces formes. Dans certains cas géométriquement simples, par exemple des tissus biologiques minces en croissance quasi-planaire tels que des feuilles, les lois de la mécanique contraignent les formes possibles. Toutefois, l’espace des formes atteignables reste particulièrement vaste. Dans ce compte-rendu, nous nous intéressons au cas particulier des pointes en croissance, que nous décrivons dans le cadre de la théorie de la morpho-élasticité et de la poro-élasticité non-linéaire, et qui partage des similarités frappantes avec deux sujets d’étude classiques en physique : la croissance dendritique et la digitation visqueuse. Les outils de l’analyse complexe sont mobilisés pour montrer qu’une parabole en croissance homogène est stable et ne développe pas de contrainte mécanique. En revanche, la forme de la pointe est fortement affectée par les perturbations du champ de croissance.

Growth of living species generates stresses which ultimately design their shapes. As a consequence, complex shapes, that everybody can observe, remain difficult to predict, even when the growth biology is over-simplified. One way to tackle this question consists in limiting ourselves to quasi-planar objects like leaves in the spring. However, even in this case the diversity of shapes is really vast. Here, we focus on growing tips with the aim to compare their role in elastic growth to classical viscous fingering and dendritic growth. With the help of complex analysis, we show that a parabola under constant growth is free of stress while growing but any growth perturbation will strongly affect its final shape. Two models of finite elasticity are considered: the Neo-Hookean and the poro-elastic model with incompressibility.

Publié le : 2020-11-16

DOI : 10.5802/crmeca.27

Mots clés : Nonlinear elasticity, Biological growth, Instabilities, Morphogenesis, Meristem growth

Affiliations des auteurs :

Martine Ben Amar ^{1, 2} ; Julien Dervaux ³

¹ Institut Universitaire de Cancérologie, Faculté de médecine, Sorbonne Université, 91 Bd de l’Hôpital, 75013 Paris, France
² Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
³ Laboratoire Matière et Systèmes Complexes, UMR 7057, CNRS and Université de Paris, 75013 Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Martine Ben Amar and Julien Dervaux},
     title = {Tip growth in morpho-elasticity},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {613--625},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {348},
     number = {6-7},
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     doi = {10.5802/crmeca.27},
     language = {en},
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Martine Ben Amar; Julien Dervaux. Tip growth in morpho-elasticity. Comptes Rendus. Mécanique, Volume 348 (2020) no. 6-7, pp. 613-625. doi : 10.5802/crmeca.27. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.27/

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