Higher-Gradient Thermoelastic Theory: A homogenization approach

Abstract

A unified periodic homogenization framework is developed for the construction of higher-gradient thermoelastic effective continua for heterogeneous materials. Five sequentially solved unit-cell boundary value problems (BVPs) deliver the complete set of effective moduli governing size- dependent elastic, hyperbolic thermal, non-local heat diffusion, and thermoelastic coupling effects. On the mechanical side, an extended Hill–Mandel lemma for strain gradient continua is established, and two BVPs for the classical displacement localizator and the strain-gradient localizator yield closed-form expressions for the Cauchy moduli, the coupling tensor, and the sixth-order strain- gradient tensor. An intrinsic correction is introduced that removes the spurious dependence on the choice and size of the representative unit cell and vanishes identically for homogeneous media. On the thermal side, upscaling of classical Fourier conduction via an extended Hill–Mandel thermal lemma shows that the Cattaneo–Vernotte hyperbolic constitutive relation emerges as a genuine upscaling effect. A unit cell BVP for the first-order thermal localizator delivers the effective conductivity tensor and the relaxation-time tensor. A unit cell BVP for the second-order localizator yields the higher-order conductivity, the thermal diffusivity, and the microstructural length scale that governs spatial non-locality in heat transport. A Bloch wave analysis of the periodic cell problem rigorously identifies τ as the sixth-order coefficient of the thermal dispersion relation contracted with the inverse squared effective diffusivity, establishing that τ ∝ ℓ²/αᴴ without any phenomenological assumption. Mechanical and thermal theories are unified via a Hill–Mandel lemma for fully coupled thermomechanics. BVP UCμ for the thermoelastic coupling localizator delivers the effective Cauchy coupling tensor, the hyperstress–temperature-gradient coupling tensor and the Grüneisen-type tensor, closing the two-way thermomechanical coupling loop at the macroscale. The framework is illustrated analytically for a periodically stratified composite under modulated laser heating. The second-gradient Cattaneo–Vernotte governing equation τT¨ + Ṫ = αᴴTxx − ℓ²αᴴTxxxx admits a biquadratic dispersion relation yielding two distinct thermal modes with separate length scales: a macroscopic diffusion mode δ₁ ∼ δ₀ and a microstructural boundary-layer mode δ₂ ∼ ℓ. This two-mode signature is the hallmark of higher-gradient thermal transport and provides an experimental protocol for simultaneous identification of αᴴ, τ, and ℓ from photothermal measurements. Explicit closed-form numerical results for a copper/polymer bilayer (period d = 50 μm, equal volume fractions) confirm the theory. A metamaterial design study demonstrates how the relaxation time can be tuned across orders of magnitude by adjusting the unit-cell period and phase contrast. The theory simultaneously resolves the infinite heat-wave speed paradox of Fourier’s law and the spurious unit-cell dependence of strain-gradient moduli, with applications to ultrafast laser processing, nanoscale thermal management, and advanced composite design.