2026-09-21T14:59:19Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/34287 2023-07-10 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Ostoja-Starzewski, Martin Ostoja-Starzewski, Martin Li, Jun 2012-09-18T21:09:39Z 2012-09-18T21:09:39Z 2012-08 2012-09-18T21:09:39Z 2012-08 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. Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-07-19T15:18:55Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 2 Math MS_Jun Li.doc: 734208 bytes, checksum: db2240c353f55f809ee35c5cbcd17959 (MD5) Li_Jun.pdf: 358359 bytes, checksum: 0ccbfca9191e6d6e718df50c7e7ccdae (MD5) Made available in DSpace on 2012-09-18T21:09:39Z (GMT). No. of bitstreams: 3 Li_Jun.pdf: 378676 bytes, checksum: 4862777de37b48d5b89bae44604745e4 (MD5) Math MS_Jun Li.doc: 734208 bytes, checksum: db2240c353f55f809ee35c5cbcd17959 (MD5) license.txt: 4054 bytes, checksum: cabd2a8615925aff682adebc5a69a2b3 (MD5) http://hdl.handle.net/2142/34287 en Copyright 2012 Jun Li fractal fractional calculus product measure renormalization micropolar continuum reciprocal uniqueness and variational theorems Renormalization analysis of continuum models on fractal domains Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Thesis M.S. MS:Mathematics -UIUC 10KS0439MS