3.2 Fracture Formation in Growing Surfaces

Trees are growing bodies, and growth sometimes produces fracturing (Federl and Prusinkiewicz 2004). As was discussed in Chapter 1 regarding bark formation and the inner structure of a tree trunk, various types of fractures and cracks can be observed which occur in trees due to various reasons. Such reasons include but are not limited to; natural properties of the wood and external factors like wind or sudden temperature changes.

A very interesting approach was developed by Pavol Federl and Przemyslaw Prusinkiewicz regarding fracture formation in growing surfaces. In their paper “Finite Element Model of Fracture Formation on Growing Surfaces”(Federl and Prusinkiewicz. 2004), the authors present an interesting model to generate fractures. This model is based on bi-layered materials. The model is capable of simulating growth or shrinkage-resulted patterns of fractures. The top layer in this simulation is attached to the bottom layer. The top layer cracks as a result of the bottom layer’s growth or shrinkage. This directional growth process results in stress on the top layer, and when the stress value exceeds the threshold level in the top layer, the surface cracks. Continuing growth or shrinkage of the bottom layer produces a pattern of fractures.

The authors are using the finite element method (FEM), which is used to form computer simulations for material failure. FEM is a numerical way of solving partial differential equations. It is often used to analyze stresses caused to materials, and the material’s reaction to them in engineering. This method requires a description of the material structure, its boundaries and external forces in order to determine a change of material shape. The authors mainly dedicate their method to describing the shape of tree bark and drying mud.

Fig 22. Picture taken from: Federl and P. Prusinkiewicz. 2004. Finite element model of fracture formation on growing surfaces. Proceedings of Computational Science. ICCS 2004 (Krakow, Poland, June 6−9, 2004), Part II, Lecture Notes in Computer Science 3037, Springer, Berlin, pp. 138-145.

For these two purposes, different initial shapes are used. In the case of bark, the bottom layer expands, and fractures are simulated in the top layer. The general idea behind this method is based upon linear elastic fracture mechanics and an approximation of the stress field near the crack point using the theory of linear elasticity. A fracture appears where the highest level of stress exceeds the material’s threshold stress value. The direction of the crack is dictated by a maximum stress direction. The crack appears perpendicular to the direction of stress. The fracture propagates as long as the potential energy released by the fracture exceeds the amount of energy needed to create the fracture. This method generates very impressive fracture patterns from an aesthetic point of view. Unfortunately it is not implemented yet in any commercial tree generators to my knowledge. The reason might be that the simulation of such fracture patterns requires quite intensive calculations, and takes quite a long time. Yet the time it takes to perform these calculations is less than it would have taken to model it manually (Federl and Prusinkiewicz 2004).

The results of this method show that this is a viable tool for modeling patterns of fractures. The model can be applied to many different situations, but its results are very good to simulate tree surfaces.

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