Renormalization analysis of continuum models on fractal domains

fractal;fractional calculus;product measure;renormalization;micropolar continuum;reciprocal;uniqueness and variational theorems

Li, Jun ; Ostoja-Starzewski, Martin ; Ostoja-Starzewski ; Martin

关键词: fractal;    fractional calculus;    product measure;    renormalization;    micropolar continuum;    reciprocal;    uniqueness and variational theorems;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/34287/Li_Jun.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This thesis builds on the recently begun extension of continuum thermomechanics to fractal media which are specified by a fractional mass scaling law of the resolution length scale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through renormalization analysis into continuum models, in which the fractal dimension D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D and R, as well as a surface fractal dimension d. While the original formulation was based on a Riesz measure—and thus more suited to isotropic media - the new model is based on a product measure capable of describing local material anisotropy. This measure allows one to grasp the anisotropy of fractal dimensions on a mesoscale and the ensuing lack of symmetry of the Cauchy stress. It allows a specification of geometry configuration of the continua by ‘fractal metric’ coefficients, on which the continuum mechanics is subsequently built. Finally, the reciprocity, uniqueness and variational theorems are established for development of approximate numerical solutions.