An Analytical Technique for Transient Heat Conduction in Multilayer Structures With Intralayer Linearly Graded Properties

Abstract

Abstract. This study presents an analytical solution for one-dimensional transient heat conduction in multilayer slabs with linearly varying thermal conductivity and diffusivity within each layer. The multilayer system is reformulated as an equivalent single-layer slab with piecewise-linear property variations, and the solution is derived by extending a previously established orthogonal function expansion technique based on Heaviside functions. The resulting temperature expression includes modified time-dependent coefficients involving sine and cosine integrals. Verification against a benchmark solution and finite element simulations confirms the accuracy of the proposed solution across both transient and steady-state regimes. The framework is also applied to functionally graded materials (FGMs), where a piecewise-linear approximation of continuously varying properties enables accurate thermal analysis. In an illustrative FGM case, the proposed solution achieves comparable accuracy to conventional piecewise-constant models while requiring significantly fewer layers—offering improved physical consistency and computational efficiency. This makes the method highly suitable for thermal analysis of graded systems.