Fractal equilibrium configuration of a mechanically loaded binary tree

https://doi.org/10.1016/j.chaos.2021.111415Get rights and content

Highlights

  • •

    We find the equilibrium configuration of a loaded fractal binary tree.

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    Takagi curve and β-Cantor function emerge as fractals conforming the canopy.

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    Vertical and horizontal displacements are given by these emerging fractals.

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    Dimensions of emerging fractals are related, showing a connection between them.

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    Emerging fractal parameters depend on the tree mechanical and geometrical properties.

In this paper we study the equilibrium mechanics problem that originates in a binary tree with infinite levels subjected to loads on its topmost branches. The application of the laws of mechanics to find the equilibrium configuration shows that the functional forms of the vertical and horizontal displacements of its end nodes converge to a Takagi curve and a linear combination of inverses of β-Cantor functions respectively as the number of levels tend to infinity. As a consequence, the shape of the canopy results from the combination of these two emerging fractals that were not present in the unloaded tree. Besides, our study also shows that the analytical expressions of the emerging fractals depend on the mechanical properties of the binary tree, indicating that the binary tree is a link between these two emerging fractals. In addition, we prove that the fractal dimensions of Takagi and β-Cantor are related in this model.

Introduction

Trees are an integral and fundamental part of our environment for ecological and economic reasons. Such a vital element needs to be understood and studied not only from a biological point of view, but also from the point of view of its mechanical behavior. From the mechanical perspective, it is essential to determine what shape the tree structure acquires when it reaches equilibrium after being loaded on its canopy. This raises the question of whether the deformed tree structure generates or contributes to the fractal structure of the canopies, and if so, what are the emerging fractals.

To provide answers to the questions posed, in this paper, we will consider the deformations of a loaded binary tree [1]. A symmetrical binary tree is a good and simple approximation of the Dracaena draco tree (see Fig. 1), which is a prototypical fractal-shaped tree that will serve us as a model. Note that the canopy of a symmetrical binary tree is absolutely flat, but this is not what we observe in Nature. By applying the laws of mechanics to the trivial binary tree, more complex fractals that model the silhouette of the canopy will emerge.

Mandelbrot and Frame [2], [3] were the first ones to study two-dimensional self-similar binary trees, by using Iterated Function Systems (IFS) and geometric algorithms; the shape of the canopy was mathematically imposed. In this paper, we will mathematically prove that fractal structures emerge in a tree crown directly from a flat canopy, when branching deformations under stress are taken into account, and for this we will simply use the continuum elasticity theory, instead of IFS; that is, fractal structures in the tree will emerge from the application of mechanical principles to a simple pre-existing fractal, without resorting to mathematical algorithms or biological causes hidden in the tree’s biology.

Note that the fact of obtaining fractals by applying the laws of mechanics to other fractals makes it possible to know their analytical expression, something of not only scientific but also practical interest. In the particular case of our study, characterizing the canopy in terms of the “structure parameters” has many different applications, some of them with important economic implications. In architecture, the design of architectural canopies [4], [5], [6] requires the computation of the dimensions of the beams according to the characteristics of the materials, which is the essential condition for their construction with an optimal use of materials; among the architectural structures, it is worth mentioning the novel fractal tree-like photobioreactors [7], [8]. In the ecological context, having the parameters that characterize the fractal curve of the canopy makes possible to model how the shape of the canopy affects these processes: humidity and temperature in the canopy [9], [10], the study of the effect of the canopy on rainwater [11], [12] and also new simulations of processes in forest canopy [13] by simulating a forest as a set of tree canopies subject to winds [14], aimed at understanding pollen dispersal, seed dispersal or fire spread.

The Principle of Virtual Work [15], [16], [17], [18] will be the tool that we will use in our work, to see if a fractal subject to mechanical loads generates one or more fractals and in particular the canopy shape of the trees. This is not the only tool available; the mechanical study of fractals is very varied and an open subject, given the conceptual and technical problems involved, in which both general approaches and solutions to particular problems are sought for. Among the first methods proposed are: renormalization group theory [16], functional analysis [19], fractional calculus [20], geometrical theory of differential spaces [15], postulates [21] or dimensional regularization [22]. All these methods have dealt with the problem of not having surfaces and volumes with integer dimensions and therefore to define density, which is the variable that relates mass to volume in continuum mechanics, it is essential to use the fractal dimension of the object under study [15]. Among the particular cases are the study of vibrating fractal rods that take the form of well-known fractals such as the Cantor set [23] or the Koch curve [24], [25]. In these works we find the ideas of using an approximation to the fractal and developing recurrence equations and taking limits, which we will use in this paper. Note that, although standard definitions of stress and strain cannot be extended to fractals for obvious reasons, they can be readily used in finite approximations to fractals; that is one of the reasons for using the approximations. Another important reason is that the variational principles of mechanics can be applied to these approximations.

On this basis, we study (Section 2) the well-known binary tree [1] (see Fig. 2). We will assume that the cross-section of the branches are smaller and smaller as we ascend in the successive levels of the tree as the loads decrease. By applying the Principle of Virtual Work, we will calculate the vertical (Section 3) and horizontal (Section 4) displacements in the ith level of the tree. When the limit to infinity is taken, the Takagi curve (Section 3) and a linear combination of inverses of β-Cantor functions (Section 4) give the vertical and horizontal displacements respectively. These two fractals, Takagi and β-Cantor functions, not only appear related through the fractal tree, but also their fractal dimensions are related through the tree mechanical parameters, as we will prove in Section 5. Finally, we give the conclusions in Section 6.

Section snippets

The model: a structure of P levels

Let us consider a unloaded binary tree structure of P levels (see Fig. 2) with rigid bifurcation (branching) points. At those points, the branches depart with an angle θ with respect to the horizontal axis. All branches are assumed to have the same Young’s modulus E and shear modulus G regardless of the level at which they are located. On the other hand, their length Li is determined by the level i at which they are located and it is given by Li=L·21−i,i=1,..,P, being L the length of the

Vertical displacements in a structure of finite levels

As a previous step to calculate the vertical displacements in the structure of infinite levels, it is necessary to calculate the vertical displacements of the end nodes in a finite structure. In particular, we will calculate the displacement VδP,wR of an arbitrary end node w in a structure of P levels by using the Eq. (2) on a virtual load system consisting of a downward vertical load of value 1 on the end node w. Recall that VδP,wR are the vertical displacements per unit load, as the real load

Horizontal displacements in a structure of finite levels

To calculate the horizontal displacements HδP,wR of the end nodes using Eq. (2), we proceed in a similar way to that used in Section 3. Note that the structure and the real load system are symmetrical with respect to an axis perpendicular to the ground passing through the trunk junction point. Thus, due to the symmetry of the system, only the left half of the structure will be considered. Let us take a virtual load system consisting of a leftward horizontal load of value 1 on the end node w

The combination of Takagi and β-Cantor on the canopy and their relationship via the fractal dimension

The final shape of the canopy of a loaded binary tree of infinite levels is given by the combination of vertical displacements following a Takagi curve (see Section 3.2) and horizontal displacements following a linear combination of inverses of β-Cantor functions (see Section 4.2). Fig. 11a shows a deformed binary tree (which has a finite number of levels, but the analysis performed is equivalent), whose canopy is extracted in Fig. 11b and c. In Fig. 11b vertical displacements of the end nodes

Discussion and conclusions

We have proven that a simple fractal subject to the laws of mechanics simultaneously generates several more complicated and interrelated fractals. The application of continuum elasticity theory to a loaded binary tree shows that the vertical and horizontal displacements of its nodes follow two fractals. In the first case, the Takagi curve appears, and in the second case, it is a linear combination of inverses of β-Cantor functions. The parameters that determine the Takagi Curve and the β-Cantor

CRediT authorship contribution statement

Javier Rodríguez-Cuadrado: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing – original draft, Writing – review & editing. Jesús San Martín: Conceptualization, Methodology, Formal analysis, Writing – original draft, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank the referees for their helpful comments and insights.

This action is financed by Universidad Politécnica de Madrid as part of the UPM-Funded Research, Development and Innovation Programme, specifically targeting Funding for predoctoral contracts for the completion of doctoral degrees at UPM schools, faculties, and R&D centres and institutes including a period of at least three months as a visiting researcher outside Spain (international doctoral mention).

References (36)

  • A. Carpinteri et al.

    Static–kinematic duality and the principle of virtual work in the mechanics of fractal media

    Comput Methods Appl MechEng

    (2001)
  • I. Giorgio et al.

    Viscous second gradient porous materials for bones reconstructed with bio-resorbable grafts

    Extreme Mech Lett

    (2017)
  • P. Gudmundson et al.

    An analytic model for effective mechanical properties and local contact stresses in lithium-ion porous electrodes

    Extreme Mech Lett

    (2021)
  • A. Carpinteri et al.

    The elastic problem for fractal media: basic theory and finite element formulation

    Comput Struct

    (2004)
  • M. Epstein et al.

    The stiffness of self-similar fractals

    Int J Solids Struct

    (2008)
  • K. Davey et al.

    Analytical solutions for vibrating fractal composite rods and beams

    Appl Math Model

    (2011)
  • A. Carpinteri et al.

    Free vibration analysis of a von koch beam

    Int J Solids Struct

    (2010)
  • A. Carpinteri et al.

    Asymptotic analysis of a von koch beam

    Chaos Solitons Fractals

    (2009)

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