 |
Center for Nonlinear Studies |
Suppose we monotonically load an elastic-plastic material from the zero stress level beyond the plastic limit. If the material is inhomogeneous, fractal patterns of plasticized grains are found to gradually form in the material domain and the sharp kink in the stress-strain curve is replaced by a smooth change. This is the case for a range of different elastic-plastic materials of metal or soil type, made of isotropic or anisotropic grains with random fluctuations in material properties (plastic limits, elastic and plastic moduli…). The set of plasticized grains has a fractal dimension growing from 0 to 2 in 2d (respectively, 0 to 3 in 3d), with the response under kinematic loading being stiffer than that under mixed-orthogonal loading, which in turn is stiffer than the traction controlled one. A qualitative explanation of the morphogenesis of fractal patterns is given from the standpoint of scaling analysis of phase transitions in condensed matter physics. While the foregoing provides one of the operating mechanisms of morphogenesis of fractals in nature, from a general perspective, we have to ask whether continuum mechanics can be generalized to handle field problems of materials with fractal geometries. At present, this appears possible for fractal porous media specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d, and a resolution length scale R. The focus is on fractal media with lower and upper cut-offs, through a theory based on dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. The theory depends on a product measure grasping the anisotropy of fractal dimensions and the ensuing lack of symmetry of the Cauchy stress, which then naturally leads to micropolar continuum mechanics.
Host: Ivan Christov