Isomorphic Memory-Dissipation Dynamics in Granular Consolidation and Polymer Swelling
Abstract
1. Summary This paper bridges the gap between macroscopic geotechnical engineering (soil consolidation and secondary creep) and microscopic pharmaceutical science (controlled-release polymer hydrogels). By replacing abstract phenomenological parameters with rigorous physical units, the framework demonstrates that both systems share an identical differential dissipation topology governed by coupled deformation and stress gradients. 2. Key Formulas & Equations Coupled State-Space Memory Equation: $$\frac{d}{dt} \begin{bmatrix} x(t) \\ p(t) \end{bmatrix} = \begin{bmatrix} 0 & \frac{1}{m_{eff}} \\ -k & -\gamma \end{bmatrix} \begin{bmatrix} x(t) \\ p(t) \end{bmatrix} - \int_{0}^{t} M(t-t') \begin{bmatrix} 0 \\ v(t') \end{bmatrix} dt'$$ (Where $x(t)$ is displacement/strain, $p(t)$ is momentum/stress, $k$ is structural stiffness, $\gamma$ is instantaneous damping, and $M(t-t')$ is the historical memory kernel). 3. Key Vocabulary & Keywords Isomorphic Topology: Identical mathematical structure governing disparate physical systems. Memory Kernel ($M(t)$): Integral term capturing historical relaxation and delayed energy dissipation over time. Non-Fickian Transport: Deviations from standard diffusion caused by polymer chain relaxation and swelling stress. Darcy Flow / Pore Pressure: Macroscopic hydraulic gradients driving fluid expulsion in granular media. 4. Physical Significance Eliminates the need for separate, disconnected empirical models for soil compaction and hydrogel drug delivery. Proves that apparent behavioral complexity across physical scales stems from parameter variation ($G, \eta, k_B$) rather than structural mathematical novelty. Provides a predictive pathway to fit experimental laboratory trial data directly into a unified differential framework. 5. License & Archival Metadata Repository Target: Zenodo Preprint Repository. License Recommendation:Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) Attribution: Independent Research Conspectus, under the Conserved Informational Modulation (CIM) and Systemic Relaxation Tensors framework.
