A Scaling Framework for Mechanical Memristance: Dimensionless Metrics and Material Design Maps

Abstract

Mechanical memristors are systems whose dissipative response depends on the history of previous loading through an evolving internal state. History-dependent forces and dissipation occur in a wide range of materials and devices, including viscoelastic polymers, shape-memory materials, piezoelectrics, granular media and field-responsive fluids. Determining which of these responses admits a mechanical-memristor representation requires a constitutive test, as well as a comparison of scales. In this work, a fractional-order mechanical memristor model is developed and cast into a nondimensional form to identify the governing parameters controlling memory-dependent dissipation. The formulation leads to a set of dimensionless groups that characterise dissipation magnitude, memory-state scale and memory transfer. These quantities are combined into an effective mechanical memristance screening index \(\Mh = βγ|\mathcal H_α(Ω)|\), which provides a conditional measure of local damping modulation at matched response amplitude, constitutive slope and reference scales. Illustrative parameter scenarios are then constructed for material classes including shape-memory polymers, shape-memory alloys, hydrogels, nanocellulose, lignin-rich materials, natural fibres, piezoelectric polymers, piezoelectric ceramics, electrorheological fluids, magnetorheological fluids and granular dampers. The framework establishes a common basis for comparing memory-dependent damping within the adopted constitutive description and identifies the calibration required for its application to candidate materials and devices