Abstract
A two-dimensional elastic fractal tree model is proposed considering the trunk clamped to the ground and extremities under the action of moments. For the proposed problem the normal stress is lower in the trunk and higher in the extremities. The limiting case in which stress is constant in all orders corresponds to the physiological principle of minimum work proposed by Murray (Proc Natl Acad Sci USA 12:207–214, 1926). When longitudinal and transverse proportions coincide the tree is said to be geometrically similar. Geometric similarity, constancy of stress, uniform distribution of elastic energy and of material quantity among orders occur in group. These four states are correlated to each other: the occurrence of two implies that the other two also occur. The proposed approach considers both individual and systemic properties and highlights the inherent connection among them. Flexibility is shown to increase from the trunk to the extremities. Considering a stochastic distribution of external moments the branching design is shown to minimize relative dispersion of stress while concentrating forces through the trunk.
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This work was partially supported by the CNPq (National Research Council) and FAPERJ (Rio de Janeiro Research Foundation) through research projects.
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Communicated by Fernando Alves Rochinha.
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Barros, M.M., Bevilacqua, L. Elastic fractal trees: a correspondence among geometry, stress, resilience and material quantity. J Braz. Soc. Mech. Sci. Eng. 37, 1479–1483 (2015). https://doi.org/10.1007/s40430-014-0288-y
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DOI: https://doi.org/10.1007/s40430-014-0288-y