Statistical admissibility and long-wavelength structural convergence determine representative support in granular materials

Abstract

Representative elementary areas and volumes provide the scale bridge between resolved granular microstructure and continuum-scale material properties, yet they are still commonly selected from the apparent stabilization of a single scalar observable. Such plateaus can be false indicators of representativeness when statistically incompatible regions are averaged together or when long-wavelength structural correlations remain unresolved. We introduce a general representative-support framework for granular materials based on three sequential requirements: statistical admissibility of the sampled domain, persistent convergence of the apparent field mean and the low-wavenumber covariance spectrum, and validation against independent effective properties. The concept is tested on seven large-area backscattered-electron images of Arabian dune sands and mineral-resolved QEMSCAN maps. The structure-only criterion identifies a representative elementary area of $1536$ pixels, corresponding to approximately $2.01\mathrm{mm}$. Without property-specific fitting or recalibration, the same physical scale transferred to the QEMSCAN grid $(\approx 2.04\mathrm{mm})$ coincides with the onset of weak size dependence in apparent thermal conductivity, elastic stiffness, and directional Young's moduli. Thus, the representative scale inferred from long-wavelength granular organization independently predicts the convergence of distinct constitutive responses. The results show that representative support should be assigned only after both statistical compatibility and long-range structural convergence have been established. Although demonstrated here on two-dimensional dune-sand sections, the criterion is formulated for granular materials generally and has a direct mathematical extension to three-dimensional voxelized volumes.