Mechanics of systems of affine bodies. Geometric foundations and applications in dynamics of structured media

Abstract

Discussed are geometric structures underlying analytical mechanics of systems of affine bodies. Presented is detailed algebraic and geometric analysis of concepts like mutual deformation tensors and their invariants. Problems of affine invariance and of its interplay with the usual Euclidean invariance are reviewed. This analysis was motivated by mechanics of affine (homogeneously deformable) bodies, nevertheless, it is also relevant for the theory of unconstrained continua and discrete media. Postulated are some models where the dynamics of elastic vibrations is encoded not only in potential energy (sometimes even not at all) but also (sometimes first of all) in appropriately chosen models of kinetic energy (metric tensor on the configuration space), like in Maupertuis principle. Physically the models may be applied in structured discrete media, molecular crystals, fullerens, and even in description of astrophysical objects. Continuous limit of our affine-multibody theory is expected to provide a new class of micromorphic media.

Constraints and symmetry in mechanics of affine motion

“…To be honest, some of them are also partially contained in Eringen's theory of micromorphic media, i.e., continua of infinitesimal affine bodies [8]. Later on, we developed the theory in various aspects [9,10,11,12,14,19,20,26,27,28,29,30,31,32,33,34,35,36,37,38,39,43] and some of our results were confirmed and developed by many people [15,16,17,21,22,40,41,42]. Let us also mention the papers like [3,4,5,6,18,44].…”

Multi-Value Microstructural Descriptors for Complex Materials: Analysis of Ground States

“…A unified view on the matter foresees that, besides the deformation through which we reach a placement taken as reference-say B, and other macroscopic shapes B a -we have a field, defined over B itself, which takes values on a differentiable manifold M (see [6,37,39,42]) that we call the manifold of microstructural shapes. The common assumption that M is finite-dimensional is sufficient to include in the framework, as special cases, models that we know in solid-state physics (example those for ferroelectrics, magnetoelastic materials, quasicrystals, elastomers) and also more abstract schemes such as the Cosserat one [10] (called also micropolar), used for models of beams and shells (among many, see the basic papers [21] and [62] on the matter, the first one being that opening the application of Cosserat ideas to the description of the elastic structural elements) or liquid crystals in smectic order, and the micromorphic one (either considering microstrain or deformable directors, see [22,29,45]), adopted for polymers or models of strain-gradient plasticity, which have also been the playground for several analytical and geometrical investigations (see, example, [11,30,47,51,64]). An exception is the choice to describe crack paths in a solid by means of Radon measures over the natural Grassmanian constructed over B, taking into account at every point the possibility that a crack could occur there along some direction (see [26])-here, in a sense, the manifold of microstructural shapes is infinite-dimensional.…”

Mechanics of infinitesimal gyroscopes on Mylar balloons and their action‐angle analysis

“…where I is the scalar moment of inertia of the plane rotator. Let us now rewrite the above kinetic energy (2) in the form where we have explicitly separated the mass factor, ie,…”