Fractal pattern formation at elastic-plastic transition in heterogeneous materials

Fractal patterns are observed in computational mechanics of elastic-plastic transitions in two models of linear elastic/perfectly-plastic random heterogeneous materials: (1) a composite made of locally isotropic grains with weak random fluctuations in elastic moduli and/or yield limits; and (2) a polycrystal made of randomly oriented anisotropic grains. In each case, the spatial assignment of material randomness is a non-fractal, strict-white-noise field on a 256 x 256 square lattice of homogeneous, square-shaped grains; the flow rule in each grain follows associated plasticity. These lattices are subjected to simple shear loading increasing through either one of three macroscopically uniform boundary conditions (kinematic, mixed-orthogonal or traction), admitted by the Hill-Mandel condition. Upon following the evolution of a set of grains that become plastic, we find that it has a fractal dimension increasing from 0 towards 2 as the material transitions from elastic to perfectly-plastic. While the grains possess sharp elastic-plastic stress-strain curves, the overall stress-strain responses are smooth and asymptote toward perfectly-plastic flows; these responses and the fractal dimension-strain curves are almost identical for three different loadings. The randomness in elastic moduli alone is sufficient to generate fractal patterns at the transition, but has a weaker effect than the randomness in yield limits. In the model with isotropic grains, as the random fluctuations vanish (i.e. the composite becomes a homogeneous body), a sharp elastic-plastic transition is recovered.Comment: paper is in pres

Fractals at elastic–plastic transitions in metals, soils, and rocks

A range of studies indicate that a fractal growth of plastic domains is characteristic of elastic–plastic transitions in metallic, soil-like, and rock-like materials where elastic moduli and/or coefficients of friction, cohesion, and dilatation [1-5]. More specifically, all these material parameters are taken as nonfractal random fields in 2D or 3D, with weak noise-to-signal ratios, in a statistical continuum models. Statistical analysis is used to assess the anisotropy of those shear bands. All the macroscopic responses display smooth transitions but, as the randomness vanishes, they turn into a sharp response of an idealized homogeneous material. Notably, increasing hardening modulus and friction makes the transition more rapid, randomness in cohesion has a stronger effect than randomness in friction, but dilatation has practically no influence. Adapting the concept of scaling functions (first introduced for metals), we link the elastic–plastic transition in random Mohr–Coulomb media to phase transitions in condensed matter physics: the fully plastic state is a critical point and, with three order parameters (the “reduced Mohr–Coulomb stress,” “reduced plastic volume fraction,” and “reduced fractal dimension”), three scaling functions are introduced to unify the responses of different materials. The critical exponents are universal regardless of randomness in various constitutive properties and their random noise levels. REFERENCES [1] Li, J., Ostoja-Starzewski, M. Fractal pattern formation at elastic–plastic transition in heterogeneous materials. ASME J. Appl. Mech. 2010, 77, 021005-1-7. [2] Li, J., Ostoja-Starzewski, M. Fractals in elastic-hardening plastic materials. Proc. R. Soc. Lond. A. 2010, 466, 603–621. [3] Li, J., Ostoja-Starzewski, M. Fractals in thermo-elasto-plastic materials. J. Mech. Mater. Struct. 2011, 6(1–4), 351–359. [4] Li, J., Saharan, A., Koric, S. and Ostoja-Starzewski, M. Elastic–plastic transition in three-dimensional random materials: massively parallel simulations, fractal morphogenesis and scaling function. Phil. Mag. 2012, 92(22), 2733–2758. [5] Li, J., Ostoja-Starzewski, M. Fractal shear bands at elastic–plastic transitions in random Mohr–Coulomb materials. ASCE J. Eng. Mech. 2014, online

Edges of Saturn’s rings are fractal

The images recently sent by the Cassini spacecraft mission (on the NASA website http://saturn.jpl.nasa.gov/photos/halloffame/) show the complex and beautiful rings of Saturn. Over the past few decades, various conjectures were advanced that Saturn’s rings are Cantor-like sets, although no convincing fractal analysis of actual images has ever appeared. Here we focus on four images sent by the Cassini spacecraft mission (slide #42 “Mapping Clumps in Saturn’s Rings”, slide #54 “Scattered Sunshine”, slide #66 taken two weeks before the planet’s August 2009 equinox, and slide #68 showing edge waves raised by Daphnis on the Keeler Gap) and one image from the Voyager 2 mission in 1981. Using three box-counting methods, we determine the fractal dimension of edges of rings seen here to be consistently about 1.63 ~ 1.78. This clarifies in what sense Saturn’s rings are fractal

Electric-field-induced displacement of a charged spherical colloid embedded in an elastic Brinkman medium

When an electric field is applied to an electrolyte-saturated polymer gel embedded with charged colloidal particles, the force that must be exerted by the hydrogel on each particle reflects a delicate balance of electrical, hydrodynamic and elastic stresses. This paper examines the displacement of a single charged spherical inclusion embedded in an uncharged hydrogel. We present numerically exact solutions of coupled electrokinetic transport and elastic-deformation equations, where the gel is treated as an incompressible, elastic Brinkman medium. This model problem demonstrates how the displacement depends on the particle size and charge, the electrolyte ionic strength, and Young's modulus of the polymer skeleton. The numerics are verified, in part, with an analytical (boundary-layer) theory valid when the Debye length is much smaller than the particle radius. Further, we identify a close connection between the displacement when a colloid is immobilized in a gel and its velocity when dispersed in a Newtonian electrolyte. Finally, we describe an experiment where nanometer-scale displacements might be accurately measured using back-focal-plane interferometry. The purpose of such an experiment is to probe physicochemical and rheological characteristics of hydrogel composites, possibly during gelation

Spatial behaviour of solutions of the Moore-Gibson-Thompson equation

In this note we study the spatial behaviour of the Moore-Gibson-Thompson equation. As it is a hyperbolic equation, we prove that the solutions do not grow along certain spatial-time lines. Given the presence of dissipation, we show that the solutions also decay exponentially in certain directions.Peer ReviewedPostprint (author's final draft

Comment on "Hydrodynamics of fractal continuum flow" and "Map of fluid flow in fractal porous medium into fractal continuum flow"

In two recent papers [Phys. Rev. E 85, 025302(R) (2012) and Phys. Rev. E 85, 056314 (2012)], the authors proposed fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. While in general providing a certain advancement of continuum mechanics modeling of fractal media to fluid flows, some results and statements to previous works need clarification. We first show that the nonlocal character those authors alleged in our paper [Proc. R. Soc. A 465, 2521 (2009)] actually does not exist; instead, all those works are in the same general representation of derivative operators differing by specific forms of the line coefficient c_1. Next, the claimed generalization of the volumetric coefficient c_3 is, in fact, equivalent to previously proposed product measures when considering together the separate decomposition of c_3 on each coordinate. Furthermore, the modified Jacobian proposed in the two commented papers does not relate the volume element between the current and initial configurations, which henceforth leads to a correction of the Reynolds’ transport theorem. Finally, we point out that the asymmetry of the Cauchy stress tensor resulting from the conservation of the angular momentum must not be ignored; this aspect motivates a more complete formulation of fractal continuum models within a micropolar framework