#### Vol. 6, No. 1-4, 2011

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Fractals in thermoelastoplastic materials

### Jun Li and Martin Ostoja-Starzewski

Vol. 6 (2011), No. 1-4, 351–359
##### Abstract

Fractal patterns are observed in computer simulations of elastic-plastic transitions in linear, locally isotropic thermoelastic-hardening plastic heterogeneous materials. The models involve 2D aggregates of homogeneous grains with weak random fluctuations in thermal expansion coefficients, equivalent to modeling the effects of random residual strains. The spatial assignment of material randomness is a nonfractal, strict-white-noise random field on a $256×256$ square lattice. The flow rule of each grain follows associated plasticity with loading applied through either one of three macroscopically uniform boundary conditions admitted by the Hill–Mandel condition. Upon following the evolution of a set of grains that become plastic (plastic set), we find that it has a fractal dimension increasing smoothly from 0 towards 2. Transitions under various types of model randomness and combinations of material constants are examined. While the grains possess sharp elastic-plastic stress-strain curves, the overall stress-strain responses are smoothly curved and asymptote toward plastic flows of reference homogeneous media. As the randomness decreases to zero, they turn into conventional curves with sharp kinks of homogeneous materials. Overall, the fractal dimension $D$ of the plastic set is a readily accessible parameter to investigate transition patterns in many materials.

##### Keywords
random heterogeneous materials, thermoelastoplasticity, elastic-plastic transition, fractal pattern