Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 80 / 1 / iii

In one dimension the periodic conductivity cell problem reduces toHence $a(y)(1+w^{\prime })=C$ is constant. Periodicity requires ${\int }_0^1{w}^{\prime }dy=0$, soThe effective flux average is $a^{\ast }={\int }_0^{1}a(1+w^{\prime })dy=C$, givingOne can exhibit the corrector explicitly: $w^{\prime }=a^{\ast }/a_1-1$ in phase 1 and $w^{\prime }=a^{\ast }/a_2-1$ in phase 2. Its integrated change over a period is zero, so a continuous piecewise affine periodic $w$ exists, and an additive constant fixes its mean.

This is precisely the normal component of the laminate conductivity in part (i). In one dimension there is no tangential direction: every heat-flow path crosses the phases in sequence, so only the harmonic-mean response remains.