  We present a new stochastic analysis for steady and transient one-dimensional
heat conduction problem based on the homogenization approach. Thermal
conductivity is assumed to be a random field K consisting of random variables
of a total large number N. Both steady and transient solutions T are expressed
in terms of the homogenized solution and its spatial derivatives, where
homogenized solution is obtained by solving the homogenized equation with
effective thermal conductivity. Both mean and variance of stochastic solutions
can be obtained analytically for K field consisting of identically distributed
(i.i.d) random variables. The mean and variance of T are shown to be dependent
only on the mean and variance of these i.i.d variables, not the particular form
of probability distribution function of i.i.d variables. Variance of
temperature field T can be separated into two contributions: the ensemble
contribution (through the homogenized temperature ); and the configurational
contribution (through different configrations) . The configurational
contribution is shown to be proportional to the local gradient of homogenized
solution. Large uncertainty of T field was found at locations with large
gradient of homogenized solution due to the significant configurational
contributions at these locations. Numerical simulations were implemented based
on a direct Monte Carlo method and good agreement is obtained between numerical
Monte Carlo results and the proposed stochastic analysis.
