Modèles mécaniques de réseaux de fibres 2D et de textiles, Mechanical models for 2D fibber networks and textiles

Thesee - Giuliana Indelicato

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134 pages

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Sous la direction de Jean-François Ganghoffer, Franco Pastrone
Thèse soutenue le 25 février 2008: Université de Turin - Italie, INPL
Ce travail aborde trois problèmes fondamentaux liés au comportement mécanique de matériaux tissés. Dans une première partie, le modèle de Wang et Pipkin pour des tissés, décrits comme des réseaux de fibres inextensibles comportant une résistance au cisaillement et à la flexion, est généralisé en un modèle prenant en compte la résistance à la torsion des fibres. Une application au comportement d’une coque cylindrique constituée de fibres hélicoïdales est traitée. Dans une deuxième partie, nous analysons l’impact de la géométrie de l’armure du tissé sur les propriétés de symétrie de l’énergie de déformation. Pour des réseaux constitués de deux familles de fibres, quatre configurations distinctes d’armure existent, selon l’angle entre les fibres et les propriétés mécaniques des fibres. Les propriétés de symétrie de l’armure déterminent le groupe de symétrie matérielle du réseau, sous l’action duquel la densité d’énergie est invariante. Dans ce contexte, des représentations des énergies de déformation d’un tissé invariantes par le groupe de symétrie matérielle du réseau sont établies. La relation entre les invariants du groupe et la courbure des fibres est analysée. Dans une troisième partie, des modèles de textiles considérés comme des surfaces dotées d’une microstructure sont élaborés, à partir d’une modification des modèles classiques de coques de Cosserat, dans lesquels la microstructure décrit les ondulations des fils à l’échelle microscopique. A partir d’une représentation du fil comme un elastica d’Euler, une expression explicite de l’énergie élastique microscopique est obtenue, qui permet d’établir un modèle simple du comportement mécanique macroscopique de tissés
-Structures tissées
-Modèles de microstructure pour des tissés
-Surfaces renforcées par des fibres
-Invariants anisotropes
-Symétries matérielles
In this work, we discuss three basic problems related to the mechanical behavior of textile materials. First, we extend the model of Wang and Pipkin for textiles, described as networks of inextensible fibers with resistance to shear and bending, to a model in which resistance to twist of the individual fibers is taken into account, by including torsion contributions in the elastic stored energy. As an example, we study the behaviour of a cylindrical shell made of helical fibers. Second, we study how the geometry of the weave pattern affects the symmetry properties of the deformation energy of a woven fabric. For networks made by two families of fibers, four basic types of weave patterns are possible, depending on the angle between the fibers and on their material properties. The symmetry properties of the pattern determine the material symmetry group of the network, under which the stored energy is invariant. In this context, we derive representations for the deformation energy of a woven fabric that are invariant under the symmetry group of the network, and discuss the relation of the resulting group invariants with the curvature of the fibers. Third, we develop a model for textiles viewed as surfaces with microstructure, using a modification of the classical Cosserat model for shells, in which the microstructure accounts for the undulations of the threads at the microscopic scale. Describing the threads as Euler's elastica, we derive an explicit expression for the microscopic elastic energy that allows to set up a simple model for the macroscopic mechanical behavior of textiles
-Woven fabric
-Microstructure models for woven fabrics
-Fibber reinforced surface
-Anisotropic invariants
-Material symmetry
Source: http://www.theses.fr/2008INPL009N/document

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Publié par	Thesee
Nombre de lectures	34
Langue	English
Poids de l'ouvrage	7 Mo

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LIENS

Code de la propriété intellectuelle. Articles L 122.4
Code de la propriété intellectuelle. Articles L 335.2 – L 335.10
http://www.cfcopies.com/V2/leg/leg_droi.php
http://www.culture.gouv.fr/culture/infos-pratiques/droits/protection.htm
`UNIVERSITA DEGLI STUDI DI TORINO
Dipartimento di Matematica – Dottorato di Ricerca in Matematica – XIX ciclo
Coordinatore del Dottorato Prof. F. Arzarello – a.a.2003/2007 – settore: MAT 07
INSTITUT NATIONAL POLYTECHNIQUE DE LORRAINE
´ ´Ecole Doctorale: EMMA – Laboratoire: LEMTA – Sp´ecialit´e: M´ecanique et Energ´etique
Mechanical models for 2D ﬁber networks and textiles
Mod`eles m´ecaniques de r´eseaux de ﬁbres 2D et de textiles
Giuliana Indelicato
Jury
Prof. Franco Pastrone, Directeur de th`ese
Prof. Jean-Franc¸ois Ganghoﬀer, Codirecteur de th`ese
Prof. Philippe Boisse, Rapporteur
Prof. Mario Pitteri, Rapporteur
Torino – 25 February 2008iiIntroduction
Motivation and main results
The mechanics of textile materials has been extensively studied in the past 50 years, but
it still provides a wealth of interesting problems both for modelization and mathematical
analysis. The availability of nonstandard materials, such as smart materials, the increased
computationalcapabilities,alsorelatedtocomputergraphics,haverecentlyspurredarenewed
interest in basic theoretical research in this area. Moreover, the explosive increase of research
on biological tissues and materials has provided a new ﬁeld to which the mechanics of ﬁber-
reinforced materials, initially developed for textiles, can be successfully applied.
In this work we discuss three basic problems related to the mechanical behavior of textile
materials.
First, we extend the model of Wang and Pipkin for textiles, described as networks of
inextensible ﬁbers with resistance to shear and bending, to a model in which resistance to
twist of the individual ﬁbers is taken into account, by including torsion terms in the elastic
stored energy.
Second, we study how the geometry of the weave pattern aﬀects the symmetry properties
of the elastic and bending energy of a woven fabric. For networks made by two families of
ﬁbers, four basic types of weave patterns are possible, in dependence of the angle between
the ﬁbers and their material properties. The symmetry properties of the pattern determine
the material symmetry group of the network, under which the stored energy is invariant. In
this context, we derive representations for the elastic and bending energy of a woven fabric
that are invariant under the symmetry group of the network, and discuss the relation of the
resulting group invariants with the curvature of the ﬁbers.
iiiiv Introduction
Third, we develop a model for textiles viewed as surfaces with microstructure, using a
modiﬁcation of the classical Cosserat model for shells, in which the microstructure accounts
for the undulations of the threads at the microscopic scale. Describing the threads as Euler’s
elasticas, we derive an explicit expression for the microscopic elastic energy that allows to
derive a simple model for the macroscopic mechanical behavior of textiles.
Textiles
Intextilefabricsdiﬀerentscalescanbeidentiﬁed: themacroscopicscale,correspondingto
the textile proper; an intermediate scale, at which the undulation of the yarn (or thread) can
bedistinguished; andthemicroscopicscale,thatcorrespondstotheyarnanditsconstituents:
the ﬁbers. We remark that, in most of this work, the term ﬁber is used as a synonym of yarn
or thread, while in the technical literature the ﬁber is a microscopic constituent of a thread.
The techniques used to manufacture a textile can produce diﬀerent structures, such as
knitted materials or woven textiles. A knitted textile is made of a single yarn that forms a
loop into a previous loop formed by the yarn itself. A woven textile is formed by the interlace
of two families of yarns: the weft and the warp.
Basic mathematical models for textiles
The ﬁrst mathematical model for a woven material was proposed by Peirce [49] at the
beginningofthetwentiethcentury. Thismodelessentiallydescribesthegeometricalstructure
of a plain weave, namely a tissue in which warp and weft interlace alternately. The yarns
formingthetextileweredescribedasﬂexible, circularcylinders, interlacedtogetherinregular
patterns to form the fabric.
Another model that focuses on the geometrical description of the structure of the textile,
describing some of its mechanical features, is due to Kawabata [32]. In order to simplify the
description of the geometrical structure of the weave, he neglects the undulation of the yarns.
This model takes into account both the biaxial and uniaxial tensile properties, such as the
shear properties of a plain weave. The compressibility of the yarn under the action of lateral
compressive forces is discussed, and it is shown that the compressive properties of yarn have
a great inﬂuence on the tensile properties of the fabric.TEXTILES v
In order to model tissues from a mechanical point of view, several attempts have been
madetoinvestigatethemechanicalpropertiesofwovenfabricsandtodescribetherelationship
between the force exerted on a textile material and its deformation. One of the ﬁrst attempts
was due to Olofsson in 1964 [44], who proposed a mechanical model that extends the model
of Peirce, incorporating the bending stiﬀness of the yarns.
AnotherimportantmodelwasdevisedbyGrosbergandKedia[19], whostudiedtheload-
extension modulus of a cloth, showing that it depends not only on the bending modulus of
the yarn and its geometry in the cloth, but also on the history of deformation of the fabric.
Theyfoundthatfabricswhichstillretaintheirstressedcondition, whicharosewhenthecloth
was made, have a much higher modulus. Many other models have been proposed after them,
and the literature in this ﬁeld is still growing.
For the sake of completeness, we mention a third family of models, still diﬀerent from the
above, the so-called energetic models, of which we only discuss the one due to De Jong and
Postle [14], see also Aimene et al. [2]. According to these authors, the uniaxial extension
curve of the crimped thread is related to three distinct deformation mechanisms:
I- the loss of textile weave crimp or yarn undulation (at the macrolevel);
II- the loss of the undulations of the threads inside the fabric (at the mesoscopic level);
III- the extension of the yarn,
and they were able to develop a formula for the energy that accounts for this behavior.
We ﬁnally mention the so-called microstructural models: these constitute a macroscopic
approach to textiles that treats the tissue as a deformable surface as a whole, and accounts
fortheinﬂuenceoftheﬁnestructurethroughadditional,microstructuralﬁelds. Theresulting
modelsareabletocapturethosecharacteristicsoftheﬁnestructureofatextilethatdetermine
itsresponse, forinstancethestress-straincurve. Forexample, themodelproposedbyMagno,
Ganghoﬀer and others [39], [38], is principally related with the second of the extension
mechanisms discussed by De Jong and Postle, and allows to determine the axial deformation
and the axial stress.
The existing models for inextensible networks
We brieﬂy review below some popular theories that describe the mechanical behavior of
cloth and cable networks as networks of inextensible ﬁbers.vi Introduction
In 1955, Rivlin [59] proposed a theory of networks formed by inextensible cords. He
considered the mechanics of a plane net made by two families of parallel inextensible cords,
and obtained general solutions of traction boundary value problems.
Later, between 1980 and 1986, in a series of papers, Pipkin, [54] [52] [51] [50], further
developed the theory of inextensible networks.
In1986,WangandPipkin,[66][67],formulatedatheoryofinextensiblenetswithbending
stiﬀness. The resulting continuum theory is a special case of ﬁnite-deformation plate theory,
in which each ﬁber has a bending couple proportional to its curvature.
In 2001, a theory of bending and twisting eﬀects in three-dimensional deformations of
an inextensible network was presented by Luo and Steigmann [37]. They derived the Euler-
Lagrange equations and boundary conditions by a minimum-energy principle.
Many other models have been presented in the literature, among which we only mention
Boisse et al. [9].
Some open problems
The mechanics of textile materials is an active and fast growing research ﬁeld, and some
of the open problems in this area are related to the predictive capabilities of the existing
models, a problem that can actually be addressed nowadays due to the increased eﬃciency of
numerical methods, speciﬁcally ﬁnite element methods, for