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

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.