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Biomechanics of growing trees: mathematical model, numerical resolution and perspectives

Fourcaud T., Guillon T., Dumont Y.. 2011. In : Simos Theodore E. (ed.), Psihoyios George (ed.), Tsitouras C. (ed.). Numerical analysis and applied mathematics : International Conference on Numerical Analysis and Applied Mathematics 2011, Halkidiki, Crete, Greece, 19-25 September 2011. Melville : AIP, p. 734-737. (AIP Conference proceedings, 1389). International Conference on Numerical Analysis and Applied Mathematics 2011, 2011-09-19/2011-09-25, Halkidiki (Grèce).

The growth of trees is characterized by the elongation and thickening of its axes. New cells are formed at the periphery of the existing body, the properties of the older inner material being unchanged. The calculation of the progressive deflection of a growing stem is not a classical problem in mechanics for three main reasons: 1- the hypothesis of mass conservation is not valid; 2- the new material added at the periphery of the existing and deformed structure does not participate retroactively to the total equilibrium and tends to "fix" the actual shape; 3- an initial reference configuration corresponding to the unloaded structure cannot be classically defined to formulate the equilibrium equations. This paper proposes a theoretical framework that allows bypassing these difficulties. Equations adapted from the beam theory and considering the strong dependencies between space and time are given. A numerical scheme based on the finite element method is proposed to solve these equations. The model opens new research perspectives both in mathematics and plant biology.

Mots-clés : modèle mathématique; modèle de simulation; mécanique; croissance; arbre

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