Ondas

Int J Fatigue 12 No 3 (1990) pp 185-190
C. Mattheck and S. Burkhardt
A new method has been developed which allows the reduction of localized notch stresses in two dimensional (2D) and three-dimensional (3D) elastic structures in a very effective way, with only a commercial finite-element code (the authors used ABAQUS) required. The method simulates on a computer the mechanism of tree growthcopying the self-optimization of living trees which always try to grow into a shape of constant surface stress. The success and efficiency of the method is demonstrated by 2D and 3D examples.
Key words: elastic structures; notches; biological structures; self-optimized shapes; finite-element code


The failure of mechanical structures under service conditions is most frequently due to fatigueloading and occurs preferentially at sites of h:gh local stresses often caused by the presence of notches. A design which is characterized by such a locally inhomogeneous stress distribution is always dangerous and therefore far from being an optimized design. However, for functional reasons it often cannot be avoided that edges, perforations, narrowing cross-sections, etc. are part of the design.The art of the designer must be to optimize the shape of the notches in such a way that they no longer act as stress risers.
Twenty years ago this would have been called an 'academic problem' because of the lack of manufacturing possibilities. Today just the opposite is true: an increasing number of high-performance CAM tools are available to machine even complex-shaped components. It is now up tothe designers to offer the optimum design to the manufacturer who wants to sell lightweight and fatigue-resistant products. To the authors' knowledge there are a number of methods on the market for optimizing frameworks, etc., consisting only of beams and trusses. However, no satisfactory, easy to apply and general method seems to be available for shape optimization of 2D and especially 3Dstructures. In Refs 1-4 it has been shown that biological structures self-optimize their shapes by growth with respect to the natural loading applied. The word optimum in this case means that in all the structures considered (tree butts, branch joints, deer antlers, antelope horns, tiger claws, camel thorns, etc.) a state of constant stress at the surface of the biological 'component' was always givenfor the natural loading case applied. The idea proposed in Refs 5-7 is to copy the mechanism of the tree growth in order to optimize mechanical engineering structures. It is based on the unproved but plausible assumption that biological load carriers can survive only if they are mechanically optimized, lightweight and fatigue resistant. It also relies on the law that not only the full grownbiological specimen is optimized but also the mechanism by which this optimum is reached: the mechanism of biological growth. Because the growth behaviour of trees is easier to simulate on a computer than the growth of bones, the method copies tree growth in order to provide the manufacturer with lightweight constructions with minimized notch stresses to guarantee a long fatigue life.
Computer-aidedshape optimization (CAO) by simulated tree growth
In trees only the outermost growth ring adapts to external loading. (With this knowledge not only the 'rain history' of a tree but also the load history can be seen looking on the growth rings in a tree stump!) The optimized shape of a tree stem is in this sense also documents its load history.1,8 This growth mechanism will be copied now by thefollowing numerical steps in the finite-element method (FEM) procedure:
(i) Generate an FEM structure with a reasonable design proposal. (If this is an extremely poor misconstruction it docs not matter. It will only require more computer time to improve!)
(ii) This FEM mesh is covered with a thin layer of finite elements having a much smaller Young's modulus than the material below. (This...