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arXiv · 2608.01996

Geometric theory of generalised continua using moving frames

Boris Kolev·Cl{é}ment Ecker
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Abstract

Generalised continuum theories couple the macroscopic deformation and the micro-/meso-scopic deformation of an underlying micro-structure. They can account for internal length-scale effects and higher-order mechanical loadings absent from classical Cauchy elasticity and has shown its efficiency in modelling metamaterials. This has led to a proliferation of higher-grade and higher-order models (e.g. strain-gradient, micromorphic, micro-polar) whose underlying kinematic reduction strategies, in particular to reduce the number of material parameters, are rarely identifiable from the free energy alone. It leaves two open issues: the lack of a systematic criterion for selecting an appropriate model, and a persistent ambiguity regarding the physical status of the local frames (''directors of matter'') used in their kinematic description. Are they physical quantities describing the change of state of the given micro-structure or arbitrary kinematic descriptors of its deformation\,? This work addresses both questions through a gauge-theoretic formulation of generalised continua in finite strains. While relying on tools and modelling choices consistent with the existing geometric literature on continuum mechanics, the present approach departs from it in its objectives, aiming at a unifying classification of available mechanical models rather than the description of a specific microstructural phenomenon such as defects. In this work, generalised configurations are defined as moving frames over classical configurations, and invariance with respect to the reference generalised configuration is shown to be a necessary and sufficient condition for a gauge invariance, recovering the micromorphic theory as the general-purpose theory of the deformation of arbitrary directors of matter. A systematic classification of first-order generalised media follows from structural group reduction, while strain-gradient continua are recovered through convected frames, and finally constrained media (e.g. couple-stress) are addressed.

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Boris Kolev, Cl{é}ment Ecker. 2026-08-03. Geometric theory of generalised continua using moving frames. https://arxiv.org/abs/2608.01996

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