A simple two-dimensional model for crack propagation

Topology and Toughening of Sparse Elastic Networks

The toughening of sparse elastic networks, such as hydrogels, foams, or meshes against fracture is one of the most important problems in materials science. However, the principles of toughening have not yet been established despite urgent engineering requirements and several efforts made by materials scientists. Here we address the above-mentioned problem by focusing on the topology of a network. We perform fracture experiments for two-dimensional periodic lattices fabricated from rubber strings and connecters with well-defined topological structures. We find that systematic increase in the largest coordination number while maintaining the average coordination number (=4) as constant leads to significant improvement in toughness. We reproduce the observed toughening behavior through numerical simulations and confirm that the stress concentration in the vicinity of a crack tip can be controlled by the topology of the network. This provides a new strategy for creating tough sparse elastic networks, especially hydrogels.

Force-induced breakdown of flexible polymerized membrane

We consider the fracture of a free-standing two-dimensional (2D) elastic-brittle network to be used as protective coating subject to constant tensile stress applied on its rim. Using a molecular-dynamics simulation with a Langevin thermostat, we investigate the scission and recombination of bonds, and the formation of cracks in the 2D graphenelike hexagonal sheet for different pulling force f and temperature T. We find that bond rupture occurs almost always at the sheet periphery, and the first mean breakage time <τ> of bonds decays with membrane size as <τ> ∝N(-β), where β≈0.50±0.03 and N denotes the number of atoms in the membrane. The probability distribution of bond scission times t is given by a Poisson function W(t)∝t(1/3)exp(-t/<τ>). The mean failure time <τ(r)> necessary to rip off the sheet declines with growing size N as a power law <τ(r)>∝N(-φ(f)). We also find <τ(r)>∝exp(ΔU(0)/k(B)T), where the nucleation barrier for crack formation ΔU(0)∝f(-2), in agreement with Griffith's theory. <τ(r)> displays an Arrhenian dependence of <τ(r)> on temperature T. Our results indicate a rapid increase in crack spreading velocity with growing external tension f.

Crossover from fingering to fracturing in deformable disordered media

We investigate the displacement of one fluid by another in a deformable medium with pore-scale disorder. We develop a model that captures the dynamic pressure redistribution at the invasion front and the feedback between fluid invasion and microstructure rearrangement. Our results suggest how to collapse the transition between invasion percolation and viscous fingering in the presence of quenched disorder. We predict the emergence of a fracturing pattern for sufficiently deformable media, in agreement with observations of drainage in granular material. We identify a dimensionless number that appears to govern the crossover from fingering to fracturing.

Gaps in the erythrocyte membrane skeleton: a stretched net model

The geometry of spectrin-free regions in the erythrocyte membrane skeleton is modeled using Monte Carlo calculations for an incomplete triangular lattice of entropy springs under tension. Intact springs correspond to normal spectrin molecules, and cut springs correspond to spectrin that is missing or unable to associate normally. As springs are cut and the network is allowed to relax to mechanical equilibrium, gaps in the network appear. Geometrical properties of these gaps are obtained as a function of the fraction of springs cut. The most important property modeled is the area of the largest spectrin-free region; this area increases approximately exponentially as the fraction of normal spectrin decreases from 100% to approximately 50%. The effect of these gaps on lateral diffusion and vesiculation is discussed.

Models for Material Failure and Deformation

Simple computer models have been used to investigate a variety of pattern formation processes associated with material failure and deformation. These models reproduce surprisingly well the characteristic morphologies observed in a wide range of real systems. They provide a sound basis for the development of more realistic models that can be used to develop a better understanding of the mechanical properties of real materials. The present algorithms are adequate for some purposes, but substantial improvements are needed if simulation results are to make a major contribution to our theoretical understanding of the asymptotic fractal scaling and universality properties of patterns generated by failure and deformation phenomena.