Fractal design for efficient brittle plates under gentle pressure loading

We present a scalable, parallel implementation of a solver for the solution of a phase-field model for quasi-static brittle fracture. The code is available as open source. Numerical solutions in 2d and 3d with adaptive mesh refinement show optimal scaling of the linear solver based on algebraic multigrid, and convergence of the phase-field model towards exact values of functionals of interests such as the crack opening displacement or the total crack volume. In contrast to un...

This work shows that fractals can be obtained from Mechanical Laws without being forced by any algorithm, closing the gap between the Platonic world of Mathematics and Nature. Fractal tree crown directly emerges when applying elasticity theory to branching stresses in a binary tree. Vertical displacements of nodes are given by the Takagi curve, while the horizontal ones are given by a linear combination of inverses of $\beta$-Cantor functions. In addition, both fractal dimens...

In this work we aim to develop a unified mathematical framework and a reliable computational approach to model the brittle fracture in heterogeneous materials with variability in material microstructures, and to provide statistic metrics for quantities of interest, such as the fracture toughness. To depict the material responses and naturally describe the nucleation and growth of fractures, we consider the peridynamics model. In particular, a stochastic state-based peridynami...

This paper proposes a methodology for architecting microstructures with extremal stiffness, yield, and buckling strength using topology optimization. The optimized microstructures reveal an interesting transition from simple lattice like structures for yield-dominated situations to hierarchical lattice structures for buckling-dominated situations. The transition from simple to hierarchical is governed by the relative yield strength of the constituent base material as well as ...

We have developed a simulation technique that uses non-linear finite element analysis and elastic fracture mechanics to compute physically plausible motion for three-dimensional, solid objects as they break, crack, or tear. When these objects deform beyond their mechanical limits, the system automatically determines where fractures should begin and in what directions they should propagate. The system allows fractures to propagate in arbitrary directions by dynamically restruc...

The purpose of the present paper is to present the main applications of a new method for the determination of the fractal structure of plane curves. It is focused on the inverse problem, that is, given a curve in the plane, find its fractal dimension. It is shown that the dynamical approach extends the characterization of a curve as a fractal object introducing the effects of mass density, elastic properties, and transverse geometry. The dynamical dimension characterizes mate...

A progress report on two recent theoretical approaches proposed to understand the physics of irreversible fractal aggregates showing up a structural transition from a rather dense to a more multibranched growth is presented. In the first approach the transition is understood by solving the Poisson equation on a squared lattice. The second approach is based on the discretization of the Biharmonic equation. Within these models the transition appears when the growth velocity at ...

In this paper we demonstrate the influence of the pore pressure to the development of a hydraulically-driven fracture in a poroelastic medium. We present a novel numerical model for propagation of a planar hydraulic fracture and prove its correctness by demonstration of the numerical convergence and by comparison with known solutions. The advantage of the algorithm is that it does not require the distinguishing of the fracture's tips and reconstruction of the numerical mesh a...

We have recently developed some simple continuum models of static granular media which display "fragile" behaviour: they predict that the medium is unable to support certain types of infinitesimal load (which we call "incompatible" loads) without plastic rearrangement. We argue that a fragile description may be appropriate when the mechanical integrity of the medium arises adaptively, in response to a load, through an internal jamming process. We hypothesize that a network of...

Drawing a direct analogy with the well-studied vibration or elastic modes, we introduce an object's fracture modes, which constitute its preferred or most natural ways of breaking. We formulate a sparsified eigenvalue problem, which we solve iteratively to obtain the n lowest-energy modes. These can be precomputed for a given shape to obtain a prefracture pattern that can substitute the state of the art for realtime applications at no runtime cost but significantly greater re...