Fractals in elastic-hardening plastic materials

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Plastic grains are found to form fractal patterns in elastic-hardening plastic materials in two dimensions, made of locally isotropic grains with random fluctuations in plastic limits or elastic/plastic moduli. The spatial assignment of randomness follows a strict-whitenoise random field on a square lattice aggregate of square-shaped grains, whereby the flow rule of each grain follows associated plasticity. Square-shaped domains (comprising 256 × 256 grains) are loaded through either one of three macroscopically uniform boundary conditions admitted by the Hill-Mandel condition. Following an evolution of a set of grains that have become plastic, we find that it is monotonically plane filling with an increasing macroscopic load. The set's fractal dimension increases from 0 to 2, with the response under kinematic loading being stiffer than that under mixed-orthogonal loading, which, in turn, is stiffer than the traction controlled one. All these responses display smooth transitions but, as the randomness decreases to zero, they turn into the sharp response of an idealized homogeneous material. The randomness in yield limits has a stronger effect than that in elastic/plastic moduli. On the practical side, the curves of fractal dimension versus applied stress-which indeed display a universal character for a range of different materials-offer a simple method of assessing the inelastic state of the material. A qualitative explanation of the morphogenesis of fractal patterns is given from the standpoint of a correlated percolation on a Markov field on a graph network of grains.

Original languageEnglish (US)
Pages (from-to)603-621
Number of pages19
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2114
StatePublished - Feb 8 2010


  • Elastic-plastic transition
  • Fractal pattern
  • Markov random field
  • Random heterogeneous materials

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering
  • General Physics and Astronomy


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