Heat conduction in periodic laminates with probabilistic distribution of material properties

Abstract

This contribution deals with a problem of heat conduction in a two-phase laminate made of periodically distributed micro-laminas along one direction. In general, the Fourier’s Law describing the heat conduction in a considered composite has highly oscillating and discontinuous coefficients. Therefore, the tolerance averaging technique (cf. Woźniak et al. in Thermomechanics of microheterogeneous solids and structures. Monografie - Politechnika Łodzka, Wydawnictwo Politechniki Łodzkiej, Łodź, 2008) is applied. Based on this technique, the averaged differential equations for a tolerance–asymptotic model are derived and solved analytically for given initial-boundary conditions. The second part of this contribution is an investigation of the effect of material properties ratio \(\omega\) of two components on the total temperature field \(\theta\), by the assumption that conductivities of micro-laminas are not necessary uniquely described. Numerical experiments (Monte Carlo simulation) are executed under assumption that \(\omega\) is a random variable with a fixed probability distribution. At the end, based on the obtained results, a crucial hypothesis is formulated.