Numerical modelling of shape regulation and growth stresses in trees I. An incremental static finite element formulation

This paper is the first of a series focusing on the biomechanical analysis of live trees. The finite element method (fem) is the most common method used for the analysis of complex mechanical structures. Several fem industrial codes exist, but they need to be adapted to calculate the mechanical behaviour of growing trees. A general incremental model has, been developed for this specific application. In this model, time was discretised and for any developmental stage, a new equilibrium was written considering the increment of weight due to the mass of new wood layers and new vegetative elements being added. Maturation strains of new-formed cells were also considered for the simulation of the shoot reorientation process. This model was intended for use at the whole plant level. A multi-layer beam finite element is presented, which is well adapted to discretise tree limbs. The shape evolution of the structure was represented at each time step by the nodal displacement vector. The mechanical stresses induced as a result of growth were determined within the stem using a cumulative process taking into account the past history of each growth ring. The first basic results of growth stresses and shape evolution were compared with already published results at the branch level.