A new mathematical framework for modelling the biomechanics of growing trees with rod theory

Autor: Thomas Guillon, Thierry Fourcaud, Yves Dumont
Přispěvatelé: Botanique et Modélisation de l'Architecture des Plantes et des Végétations (UMR AMAP), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Institut National de la Recherche Agronomique (INRA)-Centre de Coopération Internationale en Recherche Agronomique pour le Développement (Cirad)-Institut de Recherche pour le Développement (IRD [France-Sud]), Université Montpellier 2 - Sciences et Techniques (UM2), Centre de Coopération Internationale en Recherche Agronomique pour le Développement (Cirad)-Institut National de la Recherche Agronomique (INRA)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD [France-Sud])
Jazyk: angličtina
Rok vydání: 2012
Předmět:
0106 biological sciences
Surface (mathematics)
Dependency (UML)
Constitutive equation
F62 - Physiologie végétale - Croissance et développement
Arbre
01 natural sciences
F50 - Anatomie et morphologie des plantes
Cell maturation strain
Gravitropism
03 medical and health sciences
Modelling and Simulation
Tropisme
Applied mathematics
Croissance
030304 developmental biology
Mathematics
Nonlinear partial differential equations
0303 health sciences
Continuous modelling
Partial differential equation
Spacetime
Basis (linear algebra)
U10 - Informatique
mathématiques et statistiques
Linear elasticity
Modèle de simulation
Mécanique
Cambial growth
Computer Science Applications
Tree (data structure)
Modeling and Simulation
[SDE.BE]Environmental Sciences/Biodiversity and Ecology
Surface growth
Algorithm
Modèle mathématique
010606 plant biology & botany
Zdroj: Mathematical and Computer Modelling
Mathematical and Computer Modelling, Elsevier, 2012, 55 (9-10), pp.2061-2077. ⟨10.1016/j.mcm.2011.12.024⟩
ISSN: 0895-7177
DOI: 10.1016/j.mcm.2011.12.024⟩
Popis: ACL-12-24; International audience; The analysis of the shape evolution of growing trees requires an accurate modelling of the interaction between growth and biomechanics, including both static and adaptive responses. However, this coupling is a problematic issue since the progressive addition of a new material on an existing deformed body makes the definition of a reference configuration unclear. This article presents a new mathematical framework for rod theory that allows overcoming this difficulty in the case of slender structures that grow both in length and diameter like tree branches. A key point in surface growth problems is the strong dependency between space and time. On this basis, the virtual reference configuration was defined as the set of initial geometric properties of the cross-sections at their date of appearance. The classical balance equations of the rod theory were then reformulated with respect to this evolving reference configuration. This new continuous formulation leads to an evolution equation of the relaxed configuration that takes into account changes in material and geometrical properties of the growing rod. Primary (linked to growth in length) and secondary (linked to growth in diameter) tropisms, i.e. the adaptive biomechanical response of growing trees to the local environment, were also considered as a component of remodelling in tree growth, which modifies the relaxed configuration. Analytical solutions of our growth model was found in simple cases, i.e. assuming planar and small deflections and considering a linear elastic constitutive law. Corresponding motion results were compared with results provided by the classical rod theory and analysed with regards to growth strategies involved in gravitropic responses. These first qualitative results show that the proposed mathematical model was able to simulate the main processes involved in tree growth. This mathematical formalism is particularly suited to study the biomechanical response of trees subjected to quasi-static loads. This contribution also provides new insight into a more general three-dimensional theory of surface growth and raises new mathematical challenges about the analysis of this original system of partial differential equations.
Databáze: OpenAIRE
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