The Principle Bundle Structure of Continuum Mechanics

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arXiv:2210.11537v1 [physics.flu-dyn] 20 Oct 2022

The Principle Bundle Structure

Continuum Mechanics

Stefano Stramigioli,

University of Twente, The Netherlands

Abstract

In this paper it is shown that the structure of the conﬁguration space of any

continua is what is called in diﬀerential geometry a principle bundle Frankel

(2011). A principal bundle is a structure in which all points of the manifold

(each conﬁguration in this case) can be naturally projected to a manifold

called the base manifold, which in our case represents pure deformations.

All conﬁgurations projecting to the same point on the base manifold (same

deformation) are called ﬁbers. Each of these ﬁbers is then isomorphic to

the Lie group se(3) representing pure rigid body motions. Furthermore,

it is possible to deﬁne what is called a connection and this allows to split

any continua motion in a rigid body sub-motion and a deformable one in a

completely coordinate free way. As a consequence of that it is then possible

to properly deﬁne a pure deformation space on which an elastic energy can

be deﬁned. This will be shown using screw theory Ball (1900), which is vastly

used in the analysis of rigid body mechanisms but is not normally used to

analyse continua. Beside the just mentioned result, screw theory will also be

used to relate concepts like helicity and enstrophy to screw theory concepts.

Preprint submitted to Journal of Geometry and Physics October 24, 2022

1. Introduction

The study of continuous matter is of fundamental importance in all

branches of engineering mechanics Abraham and Marsden (1994), spanning

from solid deformations Jerrold E. Marsden (1983) to ﬂuids Chorin and Marsden

(1993). A lot of interest is present also about proper coordinate free anal-

ysis descriptions of continua, like for example the excellent recent work of

Kolev and Desmorat (2021).

On the other hand in rigid body mechanics, the theory of screws, Ball

(1900) is vastly used to describe the motion of mechanisms of interconnected

rigid bodies. A nice example of its use can for example be found in the

analysis of dynamic balance by de Jong et al. (2019). The theory of screws is

based on two theorems: Mozzi’s theorem, for which an hystorical perspective

can be found in Ceccarelli (2000)1and Poinsot’s theorem. Mozzi’s theorem

states that any instantaneous rigid body motion can be interpreted as the

action of a rotation around an axis in space superimposed to a possible motion

along the same axis, proportional to a scalar called the pitch which indicates

the length traveled along the line for a full rotation along the line, like the

pitch of a “workshop” screw. Poinsot’s theorem is a dual theorem which

basically says that any systems of forces applied to a rigid body, would have

the same eﬀect of a single force applied along a line in space and superimposed

to a possible torque around the same line, also relating the two by a scalar

pitch. These two theorems where then used in Ball (1900) to create a theory

based on the geometry of lines, or better screws, for the analysis of rigid

body kinematics. An extensive nice treatment of the mathematics and its

relation to Lie groups and Cliﬀord’s algebras can be found in Selig (2005),

and speciﬁcally to Lie algebras in Stramigioli et al. (2002). In this work it

will be shown that it is possible to associate to each velocity and vorticity in

a certain point of a continua, an inﬁnitesimal (associated to the inﬁnitesimal

volume at the point) screw describing them at the same time. Such an

inﬁnitesimal screw (relating to the inﬁnitesimal volume element with velocity

vand vorticity dv), as in the rigid body case, will be characterised by an axis

and a pitch, but in this case the motion of the point will be considered

acted upon by the inﬁnitesimal screw. It will be shown that the concepts of

helicity and enstrophy, well known in ﬂuid dynamics, will get a very geometric

1In the literature of screw theory Chasles theory (1830) is often cited but Mozzi’s

presented the same result in 1763.

interpretation in the context of inﬁnitesimal screws.

Considering that each of these screws are equivalent to elements of a Lie

algebra Stramigioli et al. (2002), they can be summed or integrated. Such an

operation will reveal the superimposed motion that each point will generate:

if all motions would follow the same rigid body motion, the resultant would

be such a motion, but in case of non rigid motions, the resultant will show

the “main rigid body motion” of the continuum. This construction will

be presented precisely and it will be shown that, thanks to this, it will be

possible to uniquely decompose, in a completely coordinate free way, the rigid

component of a continuum motion and the resting pure deformation. This

will be done by the introduction of what is called a diﬀerential geometric

connection Frankel (2011) in the principal bundle of motions.

During the paper, to stress the coordinate invariance of the approach,

diﬀerent terms will be used as synonymous like geometric or intrinsic.

The paper will start in Sec.2 with the general framework formulation of

the analysis of a continua. In Sec.3 it will be shown that the conﬁguration

space of a continuum has indeed a principle bundle structure whose ﬁbers

are isomorphic to SE(3). In Sec. 4 the geometry and basics of screws theory

will be reviewed to then in Sec.5 use this insight to deﬁne the connection on

the principle bundle introduced in Sec.3. Finally, in Sec.6 the results will be

framed in the context of a geometric description of potential energy of the

deformation, only using an energy formulation to then draw some concluding

remarks in Sec.7.

Notation. A compact, orientable, n-dimensional Riemannian manifold M

with (possibly empty) boundary ∂M models the spatial container of a non

relativistic continuum mechanical system. It possesses a metric ﬁeld gand

induced Levi-Civita connection ∇. The space of vector ﬁelds on Mis de-

ﬁned as the space of sections of the tangent bundle T M, that will be de-

noted by Γ(T M). The space of diﬀerential p-forms is denoted by Ωp(M)

and we also refer to 0-forms as functions, 1-forms as co-vector ﬁelds and

n-forms as top-forms. For any v∈Γ(T M), we use the standard deﬁnitions

for the interior product by ιv: Ωp(M)→Ωp−1(M) and the Lie derivative

operator Lvacting on tensor ﬁelds of any valence. The Hodge star operator

⋆: Ωp(M)→Ωn−p(M), the volume form µ=⋆1, as well as the ”musical” op-

erators ♭: Γ(T M)→Ω1(M) and # : Ω1(M)→Γ(T M), which respectively

transform vector ﬁelds to 1-forms and vice versa, are all uniquely induced

by the Riemannian metric in the standard way. To complement a purely

coordinate-free notation, we will represent tensorial quantities of interest in

local coordinates, denoted by x, when considered insightful, using Einstein

summation convention.

2. The Continuum Mechanics Setting

The goal of this paper is to show the intrinsic structures which appear in

the description of the kinematics of a continua. For this reason, there will be

no use of coordinates. The setting which will be used, using the approach of

Noll (1978) is the one of a continua moving in space and the various entities

of the problem will be introduced next.

2.1. Matter and Space

We consider the motion of matter in space where matter is mathematically

modeled as a manifold Mof dimension nto which some properties may be

associated via sections of a proper bundle. For example, in the case of a

purely mechanical description, we can consider µm∈Ωn(M) as the mass top

form describing the mass distribution, where we have indicated with Ωn(M)

the sections of the nalternating bundle to which n-forms on Mbelong. Any

other property could be deﬁned in the same way. What is important to

realise is that such properties are all those properties which are conserved

for physical reasons and will be advected by the motion because strictly

connected to the matter and matter only, independently of any motion it

takes in the space. This could be for example also electrical charge. On the

other hand the scalar density ρis a quantity which is not conserved/advected

because its deﬁnition does not depend only on the matter, but on the relation

between the mass topform µmand a representation of the volume form of

the space.

We consider space as an ndimensional Riemannian manifold (S, g) where

with g∈sym+(T0

2S) we indicate the positive deﬁnite symmetric metric ﬁeld

on Swith induced volume form µ∈Ωn(S).

2.2. The conﬁguration space

We can now put matter into space with an embedding of the form:

e:M → S

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摘要：

arXiv:2210.11537v1[physics.flu-dyn]20Oct2022ThePrincipleBundleStructureofContinuumMechanicsStefanoStramigioli,UniversityofTwente,TheNetherlandsAbstractInthispaperitisshownthatthestructureoftheconﬁgurationspaceofanycontinuaiswhatiscalledindiﬀerentialgeometryaprinciplebundleFrankel(2011).Aprincipalbun...

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