Fractal design for efficient brittle plates under gentle pressure loading

Because of Euler buckling, a simple strut of length $L$ and Young modulus $Y$ requires a volume of material proportional to $L^3 f^{1/2}$ in order to support a compressive force $F$, where $f=F/YL^2$ and $f\ll 1$. By taking into account both Euler and local buckling, we provide a hierarchical design for such a strut consisting of intersecting curved shells, which requires a volume of material proportional to the much smaller quantity $L^3 f\exp[2\sqrt{(\ln 3)(\ln f^{-1})}]$.

The principle of hierarchical design is a prominent theme in many natural systems where mechanical efficiency is of importance. Here we establish the properties of a particular hierarchical structure, showing that high mechanical efficiency is found in certain loading regimes. We show that in the limit of gentle loading, the optimal hierarchical order increases without bound. We show that the scaling of material required for stability against loading to be withstood can be al...

A solid slender beam of length $L$, made from a material of Young's modulus $Y$ and subject to a gentle compressive force $F$, requires a volume of material proportional to $L^{3}f^{1/2}$ [where $f\equiv F/(YL^{2})\ll 1$] in order to be stable against Euler buckling. By constructing a hierarchical space frame, we are able to systematically change the scaling of required material with $f$ so that it is proportional to $L^{3}f^{(G+1)/(G+2)}$, through changing the number of hier...

The fracture resistance of structures is optimised using the level-set method. Fracture resistance is assumed to be related to the elastic energy released by a crack propagating in a normal direction from parts of the boundary which are in tension, and is calculated using the virtual crack extension technique. The shape derivative of the fracture-resistance objective function is derived. Two illustrative two-dimensional case studies are presented: a hole in a plate subjected ...

In this paper we study the diffusely observed occurrence of Fractality and Self-organized Criticality in mechanical systems. We analytically show, based on a prototypical compressed tensegrity structure, that these phenomena can be viewed as the result of the contemporary attainment of mass minimization and global stability in elastic systems.

Using a two dimensional lattice model we investigate the crack growth under the influence of remote tensile forces as well as due to an internally applied pressure (hydraulic fracturing). For homogeneous elastic properties we present numerical finite-size scalings for the breaking stresses and pressures in terms of crack lengths and lattice sizes. Continuum theory predicts for the tensile and for the pressure problem identical scaling functions. Our findings for the tensile p...

We propose a new conceptual approach to reach unattained dissipative properties based on the friction of slender concentric sliding columns. We begin by searching for the optimal topology in the simplest telescopic system of two concentric columns. Interestingly, we obtain that the optimal shape parameters are material independent and scale invariant. Based on a multiscale self-similar reconstruction, we end-up with a theoretical optimal fractal limit system whose cross secti...

We present a theory for the toughness, damage tolerance, and tensile strength of a class of hierarchical, fractal, metamaterials. We show that the even though the absolute toughness and damage tolerance decrease with increasing number of hierarchical scales, the specific toughness (toughness per-unit density) grows with increasing number of hierarchical scales in the material, while the specific tensile strength and damage tolerance remain constant.

Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of elasticity equations for non-integer dimensional space, and its solutions for equilibrium case of fractal materials are suggested. Elasticity problems for fractal hollow ball and cylindrical fractal elastic pipe with inside and outside pre...

The thesis consists of four main chapters. In Ch.2 we discuss experimental results concerning the scaling behavior and fractality of fracture surfaces. In Ch.3 continuum and discrete models for fracture mechanics are reviewed and partially extended. In Ch.4 we present numerical results for a finite size scaling of the macroscopic fracture stress in the absence of any disorder in the material. We discuss in Ch.5, the main chapter, the technological important problem of hydraul...