Statistical physics of structural design

In this thesis, problems of structural optimisation are approached through analytic and computational techniques. A particular focus is the effect of hierarchical design. The first chapter forms an introduction for the reader. Chapter 2 investigates the optimisation of elastic support on a buckling rod. A cost function is associated with the strength of the total elastic support provided to a beam of uniform cross-section supporting a compressive load. Through a perturbative method, it is found that for a low cost of support, a single, centrally placed support is optimal; furthermore it is found, using simulational and analytic methods, that the optimal support placement undergoes a series of bifurcations as the cost increases. The nature of these bifurcations is non-trivial and, although the analogy is not complete, there exist similarities between the solution to this problem and Landau theory of second-order phase transitions. In Chapter 3, the theme of hierarchical design is introduced. By analysing all possible failure modes, it is shown that a hierarchical design is highly efficient for withstanding external pressure loading in the limit of low applied pressures. By changing the level of hierarchy, the scaling law for volume of material required for structural stability against the applied external pressure can be changed systematically. For a given applied pressure, a particular level of hierarchy is shown to be optimal. This optimal level of hierarchy increases without bound as the pressure decreases. The Hausdorff dimension of the optimal structure and its dependence on applied pressure is found. Two example structures are presented, although the design is applicable to any convex shape. The fourth chapter of this thesis investigates the use of hierarchical geometry for a highly efficient interface between two surfaces. It is proposed that for a given strength of surface interaction, alterations to the geometry of the interface play a strong role in determining the force that is required to separate the surfaces. In particular the case of two surfaces with one being very much more rigid than the second is investigated. Increasing the hierarchical order of the design is seen to change the scaling relationship between the interface interaction strength and failure load. In Chapter 5, a hierarchical design for high mechanical efficiency under compressive loading is fabricated and mechanically tested. The particular design has previously been shown to be highly efficient under compressive loading. The scaling of material required to build a stable structure against a specific loading has previously been shown to be dependent on the level of hierarchy. A second order design is fabricated using rapid prototyping techniques. Additionally, a similar design based on hollow tubes rather than solid beams is proposed and is shown to make further savings on volume when compared to the original design. The final investigation presented in this thesis focuses on the role of imperfections in determining the buckling load of a hierarchical design. A two-dimensional design is proposed before simple, single beam, imperfections are added to the structure. The dependence of the structure on the magnitude of the imperfections is calculated analytically for the generation-1 and 2 designs. In the generation-1 structure, the magnitude of the imperfection is related to the reduction in failure load by a one-half power-law. The behaviour of a generation-2 frame with a single beam perturbed in thickness is found to be dominated by the behaviour of the generation-1 subframe. The behaviour found analytically is confirmed with finite element simulations for the generation-1 structure.