On the distribution of tree growth stresses — Part I: An anisotropic plane strain theory

Summary

Growth stress distributions in trees are derived using the hypothesis that longitudinal and circumferential growth strains are continuously induced at the periphery of the growing stem. An axisymmetric plane deformation model serves as a basis for the cylindrically orthotropic analysis. By accounting for the influence on the total stress distribution of the strains in each new growth increment, a self-equilibriating distribution of growth stresses develops in the stem. In regions away from the central core of the stem, a simple closed form solution of the general anisotropic problem is given. The growth stress results of Kübler can be recovered from the present results by assuming isotropy in the cross section of the stem and avoiding the central region near the pith. The transition point where the peripheral longitudinal tension zone ends is shown to remain essentially unchanged in the more general theory from that given by Kübler. However, the point where the compression circumferential stress zone ends is shown to be a function of the ratio of the modulii in the radial and circumferential directions whereas in Küblers' theory it was independent of the elastic constants. Detailed results are given for the development of growth stresses as the stem grows and comparisons with the earlier theory noted.