Invariants of Mesoscale Thermal Conductivity and Resistivity Tensors in Planar Random Checkerboards
Dr. Shivakumar Ranganathan (American University of Sharjah), Ahmed Dalaq (American University of Sharjah)
Multiscale Mechanics of Particulate Media
Mon 4:20 - 5:40
Sayles 105
We consider Fourier type heat conduction in two-phase planar random checkerboards at finite mesoscales. Although the microstructure under consideration is of 50 % volume fraction of each phase at infinite mesoscale, we admit randomness in the volume fraction at finite mesoscales by considering arbitrary volume fractions with a defined probability. Boundary conditions that stem from the Hill-Mandel homogenization condition are then applied to obtain rigorous bounds on the mesoscale resistivity and conductivity tensors. Several material combinations are chosen in order to determine the statistics of the invariants of these tensors as a function of contrast and the mesoscale. Finally, a dimensionless scaling function is defined and its form is established using which one can estimate the approach from a Statistical Volume Element to the Representative Volume Element.