Growth of green wood based on a phase field model

Abstract

Tree engineering is a young discipline utilizing trees as structural elements, where the determination of limit loads in tree trunks is of great importance. Simple numerical models underestimate the load-bearing capacity of green wood in contrast to experimental bending tests. A well-known reason for this is the residual stress state of the living tree lowering compressive stress towards the trunks surface. This results in an overall stress state, which increases the load capacity, since the tensile strength of wood is commonly higher than its compressive strength. By determining the residual growth stress, a more accurate evaluation of the load-bearing capacity of a living tree is possible. The residual stress state is a non-linear and time dependent function in thickness direction of the trunk. In order to simulate growth and growth stress, a phase field model is employed.

The morphology of a tree is the result of innumerable and often temporary environmental stimuli, which also change and interact with the genetically predisposed growth tropisms. Therefore, we use image processing to capture the individual tree morphology of an existing tree, which is based within the phase field model as predefined growth direction. This is the basis for primary growth in the model. Additionally the model simulates the secondary growth, which corresponds to the thickness of the trunk. Except in tropical areas, this growth is associated with growth rings, which we assign as an attribute to the modelled material. While in the branch structure several tropisms (e.g. gravitropism) are responsible for the off-centre accumulation of woody material, in the stem region we only follow the stress-induced growth. This mechanism can respond to either the principal tensile stress or the principal compressive stress in our model, as this difference is observed in hardwoods and softwoods.

Since the wood matrix represents an anisotropic material with a distinct fiber direction, we approach it in our model by a transversely isotropic constitutive law, whose principal direction coincides with the growth direction.