Speaker: Martin Ostoja-Starzewski

Abstract:

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 (resp. 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. 
In both subject areas discussed here, the novel correlation function 
recently developed by E. Porcu et al. (2007, 2008) allow a further 
advancement of these models. In particular, the combination of stochastic 
local with global effects is crucial in realistic modeling of shock waves 
in man-made as well as geophysical media and, equally so, in 
physically-based stochastic finite element methods.