Predicting of oak wood properties using X-ray inspection: representation, homogenisation and localisation. Part II: computation of macroscopic properties and microscopic stress fields

This paper is the second part of a comprehensive approach that allows the elastic and shrinkage properties of oak to be determined from the properties of the anatomical tissues and the spatial distribution of those tissues, obtained by a high resolution digital X-ray imaging device. The computation of the macroscopic properties employs a 2-D numerical tool, called MorphoPore, that utilises periodic homogenisation over the representative volume. The paper starts with the derivation of the mechanical formulation and analyses the effect of mesh refinement on the overall solution quality. Thereafter, a first example is presented that depicts the effect of fibre proportion and fibre zone shape on the macroscopic properties using a virtual anatomical pattern and a second example uses finite element meshes built from actual patterns coming from two positions of the same tree, just before and just after thinning the forest stand. The approach presented in this research allows the increase of rigidity and shrinkage coefficients due to the increase of the annual ring width to be quantified. Finally, two localisation problems are proposed in which the micro-stress fields are computed from the macroscopic loading. The first case assumes that no macroscopic stress field exists (free shrinkage configuration), whereas the second assumes that the macroscopic displacement is constrained along one direction (drying configuration)