Apparent thermal conductivity of periodic two-dimensional composites

Effects of scale and boundary conditions, as well as mismatch between component phases, on apparent thermal conductivity of periodic, square-array, two-dimensional composites are investigated in this paper. The apparent conductivities are defined as those under either essential, natural, mixed or periodic boundary conditions applied to finite size material domains. It is shown that apparent conductivities obtained under mixed and periodic conditions are the same within numerical accuracy, and they are bounded by those obtained under essential and natural boundary conditions. Bounds are very sensitive to the mismatch of phase conductivities: the higher the mismatch, the wider the bounds. Bounds tighten as the window size increases. It is also observed that the window's location strongly affects the bounds' sharpness.