Line defect evolution in finite-dimensional manifolds

Paolo Maria Mariano

doi:10.3934/dcdsb.2012.17.575

Article Contents

2012, Volume 17, Issue 2: 575-596. Doi: 10.3934/dcdsb.2012.17.575

This issue Previous Article Adhesive flexible material structures Next Article Maps into projective spaces: Liquid crystal and conformal energies

Line defect evolution in finite-dimensional manifolds

Paolo Maria Mariano^1,

1.
DICeA, University of Florence, via Santa Marta 3, I-50139 Firenze

Received: September 30, 2011

Published: February 2012

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Abstract

A fiber bundle $Y$ having as a prototype fiber a finite-dimensional, differentiable manifold, not embedded in any linear space, and as a base a space-time tube constructed on the $n-$dimensional point space is the geometric environment considered here. After assigning a Lagrangian to its first jet bundle, the notion of defect and the representation of its evolution in this setting is discussed first. Then the geometry is restricted to a base involving fit regions of a three-dimensional Euclidean space, and the balances of actions on defects in the abstract fiber which can be pictured on the basis of $Y$ by a smooth space-type line $l$ are derived in a non-purely conservative setting (a d'Alembert-Lagrange principle is involved). The main result is related to the case involving non-constant peculiar energy along $l$: the covariance of the action balances along $l$, including the configurational ones, is related to the validity of an integral Nöther-like relation which has the meaning of action power relative to the defect motion.

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Mathematics Subject Classification: Primary: 74A, 70S05; Secondary: 74A30, 74A99.

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