NONLOCAL MATHEMATICAL MODEL OF PROCESSES IN DISPERSED MATERIALS

Abstract

The paper presents a nonlocal physico-mathematical model describing two interrelated phenomena in dispersed materials: the emergence of a quasiperiodic cluster mesostructure and the setting of the material during hardening. The model is based on a nonlocal free energy functional with a competing Yukawa-type interaction potential combining long-range attraction and shortrange repulsion. The variational stationarity condition reduces to a nonlinear Hammerstein integral equation for the order parameter spatial distribution, taken as local particle concentration or bond density. Both formulations share the same mathematical framework; the bond density choice establishes a direct link with percolation theory. Stability analysis of the homogeneous state in Fourier space shows that when λ exceeds a critical value λc, the system loses stability at wavenumber kc with cluster scale Lc = 2πlₐ/kc. Analytical expressions for kc and λc are derived. Setting is described as the moment ξ* at which λ(ξ) crosses λc(ξ) during hydration. A two-stage mechanism is established: structural mode at ξ*, percolation cluster at ξ** when the bond amplitude reaches A₀perc. A spectral FFT Picard algorithm confirms the analytical predictions and yields a structural regime diagram.