Modeling of Tensile Properties of Woven Fabrics and Auxetic Braided Structures by 
Multi-Scale Finite Element Method 
 
by 
 
Yang Shen 
 
 
 
 
A thesis submitted to the Graduate Faculty of 
Auburn University 
in partial fulfillment of the 
requirements for the Degree of 
Master of Science 
 
Auburn, Alabama 
December 14, 2013 
 
 
 
 
Keywords: multi-scale mechanical model, finite element method, woven fabric, 
helical auxetic yarn, biaxial tubular structure, analysis of mechanical property 
 
 
Copyright 2013 by Yang Shen 
 
 
Approved by 
 
Sabit Adanur, Chair, Professor of Polymer and Fiber Engineering 
Amnon Meir, Professor of Mathematics and Statistics 
Yanzhao Cao, Professor of Mathematics and Statistics 
Xinyu Zhang, Associate Professor of Polymer and Fiber Engineering 
ii 
 
 
 
 
 
 
Abstract 
 
 
This study is undertaken to build an appropriate mechanical model for the analysis of 
mechanical properties of forming fabrics by incorporation of multi-scale modeling concept 
comprising micro-mechanical model, meso-mechanical model and macro-mechanical model. 
The fabric is analyzed at three different scales: yarn, unit cell and fabric sheet. The 
micro-mechanical model is to obtain the overall properties of yarn structure. The 
meso-mechanical model concentrates on the mechanical analysis of the unit cell of the fabric 
with the periodic boundary conditions by inputting yarns? properties from the 
micro-mechanical model. The macro-mechanical model is to get the overall properties of 
desired fabric structure by extracting the results from meso-mechanical model. In the first 
part, six single-layer woven fabric samples, five two-layer fabric samples are analyzed. In the 
second part, a tubular braiding fabric structure made of helical auxetic yarns is manufactured 
and its properties are investigated by a similar multi-scale model. The models for both parts 
are validated by the comparison of simulation data and experimental data, which had good 
agreement. Detailed stress-strain field throughout the entire unit cell is determined, which can 
be further used in the analysis of fatigue and fracture properties. 
 
 
iii 
 
 
 
 
 
 
Acknowledgments 
 
 
I would like to give my great gratitude to my academic advisor, Dr Sabit Adanur, for his 
excellent guidance of research as well as courses during my entire academic studies at 
Auburn. I also would like to thank Dr Ramsis Farag for his help with the test experiments in 
the lab. My thanks go to Dr. Amnon J. Meir and Dr. Yanzhao Cao from the Department of 
Mathematics and Statistics for their advice in research.  
The financial support from Auburn University Intramural Grants Program is appreciated.  
I also would like to appreciate my friends and colleagues at Auburn for their kind help 
and support at Auburn. Special appreciations go to Song Chen because of his help of finite 
element method in the research. 
Finally, I dedicated this thesis to my loving wife Zi Yan for taking care of me in life and 
her unconditional love for me.  
 
 
 
 
 
 
 
 
iv 
 
 
 
 
 
Table of Contents 
 
                                                                           
Abstract ................................................................................................................................... ii 
Acknowledgments  ................................................................................................................. iii 
List of Tables ......................................................................................................................... vii 
List of Figures ....................................................................................................................... viii 
Chapter 1 Introduction ............................................................................................................. 1 
Chapter 2 Mechanical Model of Fabric Structure ..................................................................... 5 
 2.1 Literature Review ................................................................................................... 5 
 2.1.1 Fibrous Structure and Mechanical Behavior ............................................... 5 
2.1.2 Large Deflection and Nonlinearity .............................................................. 6 
2.1.3 Deformation in Textile Structure ................................................................ 6 
2.1.4 Background ................................................................................................ 7 
2.1.5 Classification of Mechanical Models of Fibrous Structures ........................ 8 
2.1.6 Analytical Modeling ................................................................................. 10 
2.1.6.1 Micro-Mechanical Model by Analytical Method .......................... 10 
2.1.6.2 Meso-Mechanical Model by Analytical Method ........................... 11 
2.1.6.3 Macro-Mechanical Model by Analytical Method ......................... 14 
2.1.7 Numerical Modeling................................................................................. 15 
2.1.7.1 Micro-Mechanical Model by Numerical Method .......................... 15 
v 
2.1.7.2 Meso-Mechanical Model by Numerical Method .......................... 17 
2.1.7.3 Macro-Mechanical Model by Numerical Method ......................... 19 
2.1.8 Evaluation ................................................................................................ 20 
 2.2 Experimental ........................................................................................................ 21 
2.2.1 Material Preparation ................................................................................. 21 
2.2.2 Geometry of Unit Cell .............................................................................. 21 
2.2.3 Measurement of Geometry ....................................................................... 25 
2.2.4 Mechanical Property Tests ....................................................................... 28 
2.2.5 Friction Coefficient Tests ......................................................................... 30 
 2.3 Modeling Approach .............................................................................................. 33 
2.3.1 Micro-Mechanical Modeling .................................................................... 33 
2.3.2 Meso-Mechanical Modeling ..................................................................... 36 
2.3.2.1 3D Solid Modeling of Unit Cell Geometry ................................... 37 
2.3.2.2 Material Property Definition ........................................................ 46 
2.3.2.3 Meshing ....................................................................................... 46 
2.3.2.4 Contact Definition ....................................................................... 49 
2.3.2.5 Periodic Boundary Conditions ..................................................... 51 
2.3.2.6 Derivation of Periodic Boundary Conditions ................................ 52 
2.3.2.7 Implementation of Boundary Conditions ...................................... 53 
2.3.3 Macro-Mechanical Modeling ................................................................... 54 
 2.4 Results and Analysis ............................................................................................. 55 
 2.5 Conclusions .......................................................................................................... 73 
vi 
Chapter 3 Mechanical Analysis of Auextic Behavior of a Novel Braiding Tubular Structure 
by FEM ...................................................................................................................... 74 
3.1 Literature Review ................................................................................................. 74 
 3.2 Material Preparation ............................................................................................. 75 
 3.3 Approach .............................................................................................................. 76 
 3.4 Analysis of the Properties by FEM ........................................................................ 80 
3.4.1 Micro-Mechanical Model ......................................................................... 80 
3.4.2 Meso-Mechanical Model .......................................................................... 85 
 3.5 Results and Discussion ......................................................................................... 87 
 3.6 Conclusions .......................................................................................................... 90 
References ............................................................................................................................. 91 
 
 
 
 
 
 
 
 
 
 
 
 
vii 
 
 
 
 
 
 
List of Tables 
 
 
Table 1 Geometric properties of six single-layer fabric samples .............................................. 22 
Table 2 Spacing for five two-layer fabric samples .................................................................. 27 
Table 3 Major and minor radius for two-layer fabrics ............................................................. 27 
Table 4 Mechanical properties of yarns for single-layer fabrics .............................................. 29 
Table 5 Mechanical properties of yarns for two-layer fabrics .................................................. 30 
Table 6 Friction coefficients of single-layer fabrics ................................................................ 32 
Table 7 Friction coefficients of two-layer fabrics.................................................................... 32 
Table 8 Mesh statistics ........................................................................................................... 48 
Table 9 Mechanical properties of wrap and core yarns as component of the helical auxetic 
yarn ............................................................................................................................ 74 
Table 10 Geometric parameters for the auxetic yarn ............................................................... 78 
Table 11 Geometry of the biaxial tubular structure ................................................................. 79 
 
 
 
 
 
 
 
viii 
 
 
 
 
 
List of Figures 
 
 
Figure 1 Integrated model of micro, meso and macro mechanical model .................................. 9 
Figure 2 Geometric model of plain woven fabric by Peirce .................................................... 12 
Figure 3 3D geometric model of plain woven unit cell by Peirce ............................................ 12 
Figure 4 3D representation of wove model proposed by Kemp ............................................... 12 
Figure 5 3D representation of elliptic model proposed by Olofsson ........................................ 13 
Figure 6 Beam model of filaments for loose structure ............................................................. 16 
Figure 7 Beam FE model of 50-filament twisted yarn ............................................................. 16 
Figure 8 Solid FE model of unit cell of twill (left) and satin (right) woven structure ............... 17 
Figure 9 Cross section of a typical two-layer woven fabric structure with unit cell of 8     
columns of weft yarns ............................................................................................. 22 
Figure 10 Geometry of single-layer fabrics in top view (left), right view (middle) and left     
view (right) ............................................................................................................. 24 
Figure 11 Geometry of two-layer fabrics in top view (left), right view (middle) and left view 
(right) ..................................................................................................................... 25 
Figure 12 Peirce?s model of simple woven fabric structure ..................................................... 25 
Figure 13 Illustration for measurement of centerline of warp yarn for two-layer fabric 2 ........ 26 
Figure 14 Universal mechanical testing machine with a camera device................................... 28 
Figure 15 Stress strain curves for yarns in six single-layer fabric samples .............................. 29 
Figure 16 Schematic of the friction coefficient testing device ................................................. 31 
Figure 17 Schematic of lenticular cross sections ..................................................................... 41 
Figure 18 3D solid model of unit cell of all fabric samples ..................................................... 46 
ix 
Figure 19 Schematic of the meshing of cross section by triangles and rectangles .................... 47 
Figure 20 Coarse mesh of unit cell of single-layer fabric sample 1 ......................................... 47 
Figure 21 Medium mesh of unit cell of single-layer fabric sample 1 ....................................... 48 
Figure 22 Fine mesh of unit cell of single-layer fabric sample 1 ............................................. 48 
Figure 23 A contact pair ......................................................................................................... 49 
Figure 24 Periodicity of boundary of a yarn............................................................................ 51 
Figure 25 Displacement constraints on the edge ..................................................................... 53 
Figure 26 Periodic boundary conditions on the single-layer fabric sample 5 ........................... 54 
Figure 27 The contour of stress distribution throughout the unit cell of single-layer 
fabric sample 1 in undeformed state (top) and 10% deformed state 
(bottom) in the warp direction ................................................................................. 57 
Figure 28 The contour of stress distribution throughout the unit cell of single-layer 
fabric 2 in undeformed state (top) and 10% deformed state (bottom) in 
the warp direction ................................................................................................... 58 
Figure 29 The contour of stress distribution throughout the unit cell of single-layer 
fabric 3 in undeformed state (top) and 10% deformed state (bottom) in 
the warp direction ................................................................................................... 59 
Figure 30 The contour of stress distribution throughout the unit cell of single-layer 
fabric 4 in undeformed state (top) and 10% deformed state (bottom) in 
the weft direction .................................................................................................... 60 
Figure 31 The contour of stress distribution throughout the unit cell of single-layer 
fabric 5 in undeformed state (top) and 10% deformed state (bottom) in 
the warp direction ................................................................................................... 61 
Figure 32 The contour of stress distribution throughout the unit cell of single-layer 
fabric 6 in undeformed state (top) and 10% deformed state (bottom) in 
the weft direction .................................................................................................... 62 
Figure 33 The contour of stress distribution throughout the unit cell of two-layer 
fabric 1 in undeformed state (top) and 10% deformed state (bottom) 
warp direction ......................................................................................................... 63 
Figure 34 The contour of stress distribution throughout the unit cell of two-layer 
fabric 2 in undeformed state (top) and 10% deformed state (bottom) 
warp direction ......................................................................................................... 64 
x 
Figure 35 The contour of stress distribution throughout the unit cell of two-layer 
fabric 3 in undeformed state (top) and 10% deformed state (bottom) 
warp direction ......................................................................................................... 65 
Figure 36 The contour of stress distribution throughout the unit cell of two-layer 
fabric 4 in undeformed state (top) and 10% deformed state (bottom) 
warp direction ......................................................................................................... 66 
Figure 37 The contour of stress distribution throughout the unit cell of two-layer 
fabric 5 in undeformed state (top) and 10% deformed state (bottom) 
warp direction ......................................................................................................... 67 
Figure 38 Comparison between experimental data and simulation data of 
single-layer fabric sample 1 .................................................................................... 68 
Figure 39 Comparison between experimental data and simulation data of 
single-layer fabric sample 2 .................................................................................... 68 
Figure 40 Comparison between experimental data and simulation data of 
single-layer fabric sample 3 .................................................................................... 69 
Figure 41 Comparison between experimental data and simulation data of 
single-layer fabric sample 4 .................................................................................... 69 
Figure 42 Comparison between experimental data and simulation data of 
single-layer fabric sample 5 .................................................................................... 70 
Figure 43 Comparison between experimental data and simulation data of 
single-layer fabric sample 6 .................................................................................... 70 
Figure 44 Comparison between experimental data and simulation data of two-layer 
fabric sample 1 ....................................................................................................... 71 
Figure 45 Comparison between experimental data and simulation data of two-layer 
fabric sample 2 ....................................................................................................... 71 
Figure 46 Comparison between experimental data and simulation data of two-layer 
fabric sample 3 ....................................................................................................... 72 
Figure 47 Comparison between experimental data and simulation data of two-layer 
fabric sample 4 ....................................................................................................... 72 
Figure 48 Comparison between experimental data and simulation data of two-layer 
fabric sample 5 ....................................................................................................... 73 
Figure 49 Schematic of manufacture of helical auxetic yarn ................................................... 76 
Figure 50 The cross sectional view of rotating surface............................................................ 76 
Figure 51 The braiding machine used ..................................................................................... 77 
xi 
Figure 52 Geometry of a helical auxetic yarn ......................................................................... 77 
Figure 53 Geometrical parameters .......................................................................................... 77 
Figure 54 Schematic of the biaxial tubular structure ............................................................... 79 
Figure 55 The biaxial tubular structure ................................................................................... 79 
Figure 56 3D solid model of the helical auxetic yarn .............................................................. 81 
Figure 57 Application of boundary conditions ........................................................................ 82 
Figure 58 The contour of stress distribution before (up) and after (down) deformation ........... 82 
Figure 59 Comparison between simulation data and experimental data for tensile property .... 82 
Figure 60 Curve fitting with bilinear function ......................................................................... 83 
Figure 61 Representation of helical auxetic yarn .................................................................... 84 
Figure 62 Poisson?s ratio versus longitudinal strain ................................................................ 85 
Figure 63 3D solid model of the unit cell ................................................................................ 86 
Figure 64 Illustration of tension and shear .............................................................................. 86 
Figure 65 Application of boundary conditions ........................................................................ 87 
Figure 66 Geometry of the unit cell ........................................................................................ 88 
Figure 67 The contour of stress distribution before deformation (left) and after 
deformation (right) ................................................................................................. 89 
Figure 68 Experimental data and simulation results of each parameter ................................... 89 
 
 
 
 
 
1 
Chapter 1 Introduction 
Computer aided designs for conventional engineering materials such as metal and 
concrete which are widely used in bridges, airplanes and other structures are well established 
to work out the effects of all the likely forces and aerodynamic properties using computer 
simulations. In contrast, interlaced fibrous structures and products are still produced largely 
on the basis of experience, intuition and trial-and-error due to their unique properties of 
strength, flexibility, inhomogeneous structure and non-linearity. In recent years, the 
development of new materials and new manufacturing techniques allowed a variety of 
interlaced fibrous structures to penetrate into high performance areas, from medical to civil 
engineering, from automotive to military, from space to undersea applications. On the one 
hand, these ever-expanding new application areas require interlaced structures to be 
engineered very carefully and precisely since their failure could be devastating. On the other 
hand, appropriate computer aided design for interlaced fibrous structure and fibrous 
reinforced composites could maximally utilize overall properties of structures to result in 
high quality to cost ratio product tailored for very specialized purposes. Such demands 
considerably stimulate the development of geometric and mechanical modeling of fibrous 
materials.  
Many former researches built analytical models to predict mechanical properties of 
fibrous materials based on assumptions and simplifications. The main drawbacks of 
analytical models are that most of them are quite complicated, time consuming and difficult 
to control the balance between necessary simplifications and accuracy. Application of finite 
element method emerges with rapid development of computer hardware and CAD programs 
2 
and it is gradually accepted as the first priority method to analyze properties of fibrous 
materials due to its advantages of accuracy, time saving and presentation of detailed stress 
distribution. After finite element analysis based on numerical method is well established and 
developed, a multi-level modeling concept by means of finite element method was also 
proposed and applied into all sorts of traditional materials analysis. Multi-scale concept 
divides materials into different scale. At each scale, properties data from previous scale is 
considered as inputs to generate overall properties of a larger unit by finite element method 
and the latter is then transferred into next scale as inputs to finally get the overall properties 
of the analyzed object. This concept could be suitably applied for mechanical analysis of 
interlaced fibrous structure. 
The fundamental theory of this multi-level model is that the fabric is considered to be 
made of many identical repeating units which have exactly the same properties. Each 
repeating unit comprises continuous and elastic yarns interlaced at a certain structure. The 
micro scale stage is to calculate the overall properties of each yarn or obtained directly from 
mechanical tests if the yarn structure is as simple as monofilament.  
The overall properties of one single yarn are approached by the homogenization process 
which connects the micro and meso-modeling. The meso scale stage is to calculate overall 
properties of repeating unit from yarn properties data of micro-stage. Finally the 
macro-mechanical modeling stage based on the generation of simplified structure predicts the 
mechanical performance of extended fabric pieces. The macro-mechanical data are collected 
from the homogenization process which transfers property of unit cell into property of overall 
fabric with desired geometry.   
3 
The first part of this research is to propose a multi-level model consisting of 
micro-mechanical model, meso-mechanical model and macro-mechanical model to 
efficiently and effectively analyze tensile properties of interlaced fibrous structure by finite 
element method. The multi-level model is constructed in detail so that it could contain most 
possible aspects from properties of yarn assemble to contact frictional coefficient which 
could affect the property of final product. Six single-layer woven fabrics and five different 
two-layer woven fabrics were investigated. Micro-mechanical properties and 3D geometric 
information both serve as input data for meso-mechanical modeling of repeating unit of each 
sample. Micro-mechanical properties including modulus, Poisson?s ratio and friction 
coefficient are obtained from test of each yarn. 3D geometric information of one repeating 
unit containing cross section, spacing and crimp height are directly measured using 
microscope with CellSens Standard 1.8.1 software (Olympus Corporation). A generative 
model is used in this research to accurately generate 3D solid geometry of repeating unit of 
each sample. The meso-mechanical model containing material definition, contact 
specification and periodic boundary conditions is built in Static Structure module of 
Ansysworkbench. Finally, homogenization process of stress-strain data obtained from 
meso-mechanical model extrapolates the macroscopic tensile properties of tested fabrics 
which are compared with experimental data. Tensile properties of samples in both warp and 
weft directions were simulated by this model and then compared with experimental data.  
The second part of the research introduces a novel auxetic tubular fabric structure by 
biaxial braiding and its mechanical analysis by FEM. 
Considerable research about auxetic materials has been undertaken due to their counter 
4 
intuitive mechanical behavior or negative Poisson?s ratio. In other words, auxetic materials 
are expanding in transverse direction while under longitudinal tension. Auxetic materials 
exhibit various potential applications because the negative Poisson?s ratio could improve 
fundamental mechanical properties such as shear modulus, indentation resistance and 
toughness. The fundamental mechanism of auxetic behavior ranges from the rotation of the 
bonds at the molecular level to a honey comb structure at a macroscopic level. Among all 
auxetic materials, the synthesized auxetics are highly promising because it is possible to 
carefully tailor mechanical properties and control the structure of synthesized auxetics for 
specialized applications or purposes. Auxetic fibers also provide a new route to fabricate 
large textile structures or composites through textile methods such as weaving, knitting and 
braiding. Under such circumstance, a helical auxetic yarn is proposed by Hook et al. through 
winding a high modulus yarn helically onto a low modulus yarn [1]. In this research, we use 
the helical auxetic yarns as basis to fabricate a biaxial tubular structure and study its 
mechanical properties and variable permeability attributed to helical auxetic yarns. 
The whole mechanical model was established on Ansysworkbench 14.0 which is a 
commercial finite element analysis program. The generative model as the 3D modeling 
technique was implemented in the modeling module of Ansys.   
The literature review gives an introduction of characteristics of mechanical behavior of 
fabric structures under certain deformations and applications of analytical models and 
numerical models in mechanical analysis of fabric structures. It is followed by the 
descriptions of the model. Then results and discussion follow. Finally, conclusions and 
suggestions are presented. 
5 
 
Chapter 2 Mechanical Model of Fabric Structure 
2.1 Literature Review 
2.1.1 Fibrous Structure and Mechanical Behavior 
Usually, a textile fabric is made of interlaced yarns which consist of elementary 
units?fibers. Textile structures could be classified according to the manufacturing process 
such as knitted, woven, braided and non-woven. The major difficulty of computational 
modeling of textiles is probably the geometrical complexity of the textiles as well as the 
manufacture pattern. For instance, a plain woven fabric structure could have complicated 
deformation even under simple loading. A tensile deformation of a spun yarn could result 
from the combination of bending, tensile, compression and even frictional sliding of the 
constituent arranged fibers. Furthermore, contact phenomenon such as bonding or frictional 
contact should also be taken into account as a necessary parameter in the mechanical model. 
The determination of contact region in the fabric and implementation of contact algorithm 
undoubtedly increase the complexity of the whole model. 
Therefore, many mechanical models or analysis of fabric structures are based on some 
assumptions and simplifications so that the model is accurate and not costly. For example, the 
yarn is usually assumed to be an elastic homogeneous material. The elastic property of 
homogeneous yarn could be either linear or non-linear. A non-linear elastic behavior could 
come from synergy of mechanical behavior of assemble of micro fibers if the yarn is 
multifilament. If the yarn is monofilament made of one certain polymer, the yarn would 
probably exhibit a viscoelastic behavior which is also a non-linear behavior.  
6 
As it is mentioned previously, a suitable contact model at where yarns contact with each 
other must be established in order to effectively analyze mechanics of fabric structure. 
Actually, the friction effect supports the stability of the whole structure for some certain 
multifilament fabrics. Frictional contact has great influence on stress and strain distribution in 
a fabric subjected to deformation because force would be transferred from one yarn to the 
contacting adjacent yarn.  
2.1.2 Large Deflection and Nonlinearity 
The complexity of the mechanical behavior of fabric structure could also introduce the 
large deflection and non-linearity. The relative large deformation is due to the flexibility of 
yarns and geometries structure of fabric. The yarn is a flexible material which allows large 
deformation before breakage because of low value of packing factor (the ratio of fibers 
volume to the total volume of the yarn). The air space left between fibers gives high mobility 
to fibers resulting high deformability and could be removed by exterior loading. This could 
also happen in the overall fabric since the fabric structure has air space between yarns [2]. 
The non-linearity behavior of fabric could be explained by synthesis of straightening of bent 
yarns and their subsequent elongation [3]. The straightening of yarns usually occurs at the 
initial stage of loading. Longitudinal elongation of yarns appears with the increase of load. 
Therefore, the stress-strain curve of a textile structure subjected to tensile deformation is 
nonlinear. 
2.1.3 Deformation in Textile Structure 
As discussed in section 2.1.1, even a simple loading condition applied onto a piece of 
fabric could lead to a complex deformation of straightening, bending, compression and 
7 
sliding of yarns in the meso-scale stage and respective deformations of fibers in micro-scale 
stage.  
In addition, huge deformation could occur through fibers in micro-mechanical scale 
when fabric is subjected to small deformation of macro-mechanical scale. Therefore, a simple 
deformation behavior happened in macro-mechanical behavior corresponds to a complex 
deformation in micro-mechanical scale. According to the macro-mechanical scale, 
deformation can be classified into simple deformations which are tensile, shear, bending and 
compression of the fabric sheet and complex deformation which is mainly the drape 
deformation. The drape property is very important characteristic which determines the 
formability of fabric especially when fabrics are used as reinforcement in composites. This 
research mainly focuses on the tensile properties of woven fabric structure. 
2.1.4 Background 
Composite materials exhibit a variety of advantages over conventional materials and 
have been under rapid development in terms of their design, manufacture and mathematical 
modeling. Under such circumstances, polymer materials are fabricated to form a tailored 
structure for a specialized purpose through textile manufacturing methods such as multi-layer 
weaving, multi-axial braiding and knitting due to their good quality to cost ratio [4]. The 
prediction of final product properties before the product is actually fabricated is extremely 
important for design and manufacture process in order to produce high quality to cost ratio 
product [5-6]. However, mathematical modeling of such fabric reinforced composites or 
novel fabric structures is complicated due to the complex structure. Fibrous materials present 
very different properties compared to other engineering materials because of their unique 
8 
properties such as strength and flexibility, inhomogeneous structure and visco-elasticity. 
Besides, the complicated crossover structure in fabric is also difficult to model [7]. The 
prerequisite of conducting a finite element analysis is to build an accurate geometric model of 
the object. A generative model which could generate all sorts of 3D fibrous structures has 
been proposed [8-9]. 
2.1.5 Classification of Mechanical Models of Fibrous Structures 
Analysis of mechanical properties of textile structures has been done during the last 
decades. Generally speaking, the models can be classified by the scale of the model and the 
modeling method used. According to the modeling method, the model of textile structure 
could be divided into analytical model and numerical model. In most cases, the analytical 
model is to figure out the relationship between the desired properties and parameters as 
variables. The conceptualization, which is the process from physical reality of textile 
structure to mathematical model, is of essence. In conceptualization process, some necessary 
assumptions and simplifications have to be made. The major characteristic of analytical 
model is the balance between the simplifications introduced and the accuracy of the model. 
With respect to the numerical model, the first step, conceptualization, is still the same as the 
analytical model. The entire solution domain could be subdivided into piecewise finite 
elements and the mathematical model is solved in each finite element. The final solution is 
approximated by integration of solutions over all finite elements. With rapid development of 
computer hardware and 3D model programs, now the numerical method is dominating in the 
field of mechanical modeling of fibrous structures and its reinforced composites. Another 
classification is based on the scale of the model, namely micro-mechanical model, 
9 
meso-mechanical model and macro-mechanical model. The micro-mechanical stage studies 
the property of yarns by taking into account the orientation, structure and properties of 
constituent fibers. This stage is actually the process of transferring from properties of 
constituent fibers to overall property of the yarn assemble. The meso-mechanical model 
focuses on the study of overall property of fabric unit cell by introduction of geometry of unit 
cell and considering yarns as homogeneous material. The overall property of homogeneous 
yarn is calculated from micro-mechanical model. Finally, the macro-mechanical model is to 
predict complicated deformation of overall fabric such as drape by considering fabric as 
continuum material [10]. Even though these three model stages were proposed as distinct 
analysis approaches, there have been efforts for the integration of these three models.    
The textile society proposed a modeling hierarchy based on three modeling scales: the 
micromechanical modeling of yarns, the meso-mechanical modeling of the fabric unit cell 
and the macro-mechanical modeling of the fabric sheet as shown in Figure 1 [11-12]. 
 
 
Figure 1 Integrated model of micro, meso and macro mechanical model 
 
According to the integrated textile modeling concept, the only input data over the entire 
model procedure are the fiber properties, the yarn structure and the fabric structure. In the 
10 
first modeling stage, the fiber properties and the yarn structure are introduced as input 
parameters for the mechanical analysis of the yarn and the calculation of the yarn properties. 
Then, the yarn properties are transferred to the meso-mechanical stage as inputs. In the 
meso-mechanical model, the yarn is considered to be homogeneous and the yarns? properties 
correspond to a homogenization process which connects the first and second model stages. 
Furthermore, the geometry of fabric unit cell is introduced in this stage to build accurate 3D 
solid model of the unit cell. The analysis is limited to the fabric unit cell. A second 
homogenization process is required to connect the second and third stages by calculation of 
property of unit cell and its attribution in the fabric sheet. 
Finally, the macro-mechanical modeling stage based on the assumption of continuum 
materials predicts the mechanical performance of extended fabric pieces in complex 
deformations. Each individual modeling procedure presents significant obstacles [12]. 
2.1.6 Analytical Modeling 
The first mechanical analysis of fabric structure was probably proposed by Peirce [13]. 
Most of the analytical models are based on the elastica theory and energy minimization. 
2.1.6.1 Micro-Mechanical Model by Analytical Method 
Some research concentrates on the study of multifilament twisted yarn in terms of 
micro-mechanical model. The objective of such research is to investigate the response of 
twisted yarn subjected to certain deformations. In most cases, yarn is considered to be 
homogeneous continuum of constant cross section and density along the length. And all 
fibers are assumed to have identical properties and perfectly elastic. Stress-strain curve of 
fibers, twist density and the initial specific volume are taken into account as input parameters. 
11 
The cylindrical- helix model by Hearle et al and the conical-helix model by Onder and Ba?er 
were established from the point of view of arrangement of fibers and geometry of fibers? 
assemble [14-15]. Detailed descriptions of mechanical behavior of yarns were studied. 
Tension, bending and torsion behavior of yarns were approached by means of force, 
stress-analysis and energy methods [16-20]. Komori employed a statistical approach by 
introduction of large quantity of data to analyze mechanical properties of yarn assemble [21]. 
The major challenge of multifilament yarn model is the balance between physical reality and 
idealistic condition of yarn assemble.  
2.1.6.2 Meso-Mechanical Model by Analytical Method 
The first attempt of analytical modeling of fabric unit cell is to predict the uniaxial or 
biaxial deformation of plain woven structure. Perice proposed three principal fundamental 
assumptions of geometrical model of plain woven fabric [13]. The flexible thread is assumed 
to be infinitely flexible, incompressible and inextensible with circular cross-sections as 
shown in Figure 2 and Figure 3 [13] [22].  This model has one main drawback which is the 
assumption of circular cross section shape because in most cases cross section shape is not 
perfectly circular due to exterior forces in manufacture processes. Thus, later, the Peirce?s 
geometry model was improved by changing the cross sectional shape for a better 
representation of interlaced structure. The racetrack and elliptical cross sectional shapes were 
employed into Peirce?s model as shown in Figure 4 and Figure 5 [23-24].  
 
12 
 
Figure 2 Geometric model of plain woven fabric by Peirce [13] 
 
 
Figure 3 3D geometric model of plain woven unit cell by Peirce [13] 
 
 
Figure 4 3D representation of woven model proposed by Kemp [23] 
 
13 
 
Figure 5 3D representation of elliptic model proposed by Olofsson [24] 
 
In Peirce?s model, the theory of elastica was proposed to treat the yarns as elastic slender 
rods. The shape of the yarn axis can be determined by the energy minimization of elastic 
slender while subjected to transverse forces. In short, the Peirce?s model and its 
corresponding models were all developed according to the equilibrium, energy or elastica 
method for the mechanical analysis. 
Another flexible thread model including the consideration of crimp and yarn extension 
was proposed by Freeston et al. [25]. This flexible thread model was based on the assumption 
of homogeneous, linear elasticity as well as linear work-hardening. Olofsson improved the 
Peirce?s model by changing the yarn cross section shape into a function of the forces acting 
on them [24]. Mathematical equations were derived in terms of equilibrium conditions, 
stress-strain relationship of tension and compression and energy in bending. Grosberg and 
Kedia employed an energy method on the analysis of initial load extension modulus of 
completely relaxed woven fabric with the assumptions of inextensible and incompressible 
yarn [26]. Dastoor et al also proposed a similar elastica method taking into account linear 
extensibility of the yarns by regarding the yarns as homogeneous, weightless slender rods, 
frictionless material [27]. In this analytical approach, a computational algorithm was 
14 
implemented for the solution of the equilibrium equations. Another analysis of the large 
deformation of partially and completely set plain woven fabrics was performed by Huang 
[28-29]. Both effects of extension and bending of yarns were considered based on the elastic 
model. His model introduces the bilinear moment ?curvature relation to explain the sliding of 
the fibers within the yarn, increasing the reliability of the study. The ?sawtooth? geometrical 
model which contains the force equilibrium and the displacement of the warp and weft yarns 
in thickness direction at contact points of crossing threads was proposed to elaborate finite 
deformation and biaxial deformation theory [30]. Dolatabadi and Kova? suggested an 
alternative geometrical model of woven fabric considering the yarn?s packing density as well 
as general fabric data as parameters [31-32]. 
2.1.6.3 Macro-Mechanical Model by Analytical Method 
An improvement of elastic theory based on the concept of planar and spatial elastica was 
developed to study complex deformation of fibers and fiber assembly by Konopasek [33-34]. 
This model includes considerations of nonlinear behavior of material, friction elasticity, 
elastic-plasticity and visco-elasticity. 
To be more specific, the planar elastica theory was proposed to study the large 
deflections of yarn in a plane and the bending behavior of fabric sheet. The spatial elastic was 
proposed to analyze fiber buckling and crimp. Computational tools were utilized in order to 
solve the nonlinear differential equations. Lloyd et al. published an alternative method to the 
theoretical mechanics of static drape of fabrics by solving the differential geometry of 
surfaces [35]. In addition, they developed a convenient implementation of the theoretical 
mechanics of fabrics since the fabric was considered as 2D continuum represented by a 
15 
surface with very small thickness embedded in the 3D Euclidean space. The dynamic 
deformation of fabrics was studied by means of differential geometry of surfaces [36]. A 
model was established to generate the surfaces made of a series of twisted curves. The 
relationship between the curvature of surface and the mechanical properties of material was 
investigated by checking differential geometry parameters. The concept of homotopy was 
applied in the mathematical model of deformation of the surface from the initial state to the 
final state. 
2.1.7 Numerical Modeling 
The expansion of powerful commercial software codes considerably stimulates the 
application of numerical method into research of textile structures. Two dominant methods in 
mechanical modeling of textile structure and its respective reinforced composite are Finite 
Element Method (FEM) and Boundary Element Method (BEM) [37-38]. 2D and 3D 
representations for the plain woven fabric structure were the first attempts to build 
mechanical model by numerical method. Keefe et al. proposed a 3D geometrical model with 
various compactions and angles based on the geometry of woven fabric [39]. Later, these 
models were validated for numerical methods [40-41]. The application of numerical methods 
such as FEM and BEM guides the direction of the approaches for textile micro and 
meso-mechanical models in the future. 
2.1.7.1 Micro-Mechanical Model by Numerical Method 
Munro et al [42] suggested a new approach using FEM to analyze the properties of 
aligned fiber assembly. The element used in the model is a cuboid cell with 8 nodes and each 
node contains 6 degrees of freedom. The model could correlate the energy contributions to 
16 
element stiffness with various user defined properties in which nonlinear behavior is 
successfully applied. The yarn models were discretized by dividing them into layers which 
was again split into finite element ranging from 1 to 22. The fibers were initially arranged to 
form the idealized helical-yarn geometry. The result from this model fit well with the 
expectation. The necking phenomenon of the yarn assembles appeared under tension and 
elements of the model were opened significantly under axial compression. 
The fast development of CAD tools makes the construction of 3D solid models of even 
complex textile structure possible. An FEM modeling of yarn of the loose condition and 
dense condition supported by the assumption of helicoidal filaments with a constant helix 
radius are given in Figures 6 and 7 [43]. 
 
 
Figure 6 Beam model of filaments for loose structure [43] 
 
 
Figure 7 Beam FE model of 50-filament twisted yarn [43] 
 
 
 
17 
2.1.7.2 Meso-Mechanical Model by Numerical Method 
Parametric solid modeling software programs have been developed to generate 3D solid 
model of complex woven structures with weave pattern, yarn size or spacing as parameters. 
In the unit cell, the yarn is still assumed to be homogeneous material in order to reduce the 
cost and time consumption during calculation [44]. The meso-mechanical model was greatly 
improved by various types of finite elements and implementation of material property pool 
which includes linear, nonlinear, elastic, plastic, viscoelastic, isotropic, orthotropic, 
anisotropic options etc. In addition, contact algorithms and large strain effects were also 
taken into account in the mechanical model of unit cell since they have great influence on the 
final result. Durville [45] proposed a 3D beam model to simulate mechanical behavior of 
woven fabrics in fibers scale. Tarfaoui and Akesbi [46] presented a more complicated 
structure of twill woven fabric whose unit cell comprises of three warp and three weft yarns. 
B-spline function was used in order to generate more precise 3D solid model of yarn in FEM 
[47]. Vassiliadis et al built a 3D model of complex interlaced woven fabric for the 
mechanical properties analysis as shown in Figure 8 [48].  
 
 
Figure 8 Solid FE model of unit cell of twill (left) and satin (right) woven structure [48] 
 
 
18 
Fibrous structure reinforced composites have been widely used in fields of automotive, 
marine and aerospace industry due to their high ratio of stiffness and strength to weight and 
multiple functions. Zhang and Harding used strain energy method of FEM to predict the 
elastic properties of plain woven composite structures [49]. The drawback of this approach 
pointed out by the authors was that the two undulations in one direction were not realistic for 
woven fabric. Naik improved this approach by considering the strand cross section geometry, 
possible gap between two adjacent yarns and the two direction undulation [50]. Whitcomb 
studied the effect of quadrature order, mesh density and material degradation on the predicted 
failure from in-plain loading [51]. The 3D solid model of composite is made of the volumes 
of woven unit cell and the volumes of matrix. The volume of matrix could be obtained by 
subtracting the volumes of fabric unit cell from the overall volume of composite unit cell. 
Some approaches were undertaken to investigate the homogenized elastic properties of fabric 
composites using the unit cell of composite structure. Circular, elliptic, compressed 
hexagonal and lenticular cross section shapes were assumed for the 3D construction of tows. 
The material properties of tows in the composite were defined as transverse isotropic and the 
matrix as isotropic. An approach proposed by Ng et al. [52] predicted the inplane elastic 
properties of a single layer 2/2 twill woven fabric composite. The mechanical model was 
programmed by the ANSYS Parametric Design Language (APDL). The 8-node solid 
elements with 3 degrees of freedom per node were used in the model. The whole model 
contains approximately 52000 finite elements and 12000 nodes. The contact areas were 
generated by subtracting the volume of fabric unit cell from the overall volume of composite 
unit cell and were then defined to be a shared entity for both the yarn and the matrix volumes. 
19 
Choi and Tamma [53] successfully predicted the in-plane elastic properties of a composite 
structure reinforced with plain woven fabric. The predicted elastic properties were then used 
for the damage analysis of laminated composite structures. The superposition principle was 
used for the prediction of homogenized properties of woven fabric reinforced composites. 
The overall model contains 520 wedge elements for the yarns and 256 brick elements for the 
matrix. In-plane tensile and shear deformation was obtained with a gradually incrementing 
load. A novel research in terms of generalized description of the internal structure of textile 
reinforcement was conducted by Leuven. Lomov et al. proposed a model for the internal 
geometry of 2D and 3D woven structure based on the minimization of total energy 
throughout unit cell and yarn mechanical properties. And this method could be extended to 
all sorts of fibrous structures such as woven, braided, knitted structures [54-56]. They also 
established a three-level hierarchy: the micro-, meso- and macro-level. The micro-level deals 
with the arrangement of fibers in the representative volume of yarn. The meso-level defines 
the internal structure of the reinforcement and direction and volume fraction within the yarn. 
Finally, the macro-level prescribes the 3D geometry of composite unit cell and distribution of 
the reinforcement properties. 
2.1.7.3 Macro-Mechanical Model by Numerical Method 
Macro-mechanical modeling of fabric or cloth mainly concentrates on the prediction of 
drape behavior and the virtual representation. According to the purpose and application of 
fabric, different techniques have been developed in the last decades. Numerical model such 
as the particle-model, the deformable node-bar model and the FEM were proposed for the 
purpose of better understanding of fabric behavior under deformation. Most 
20 
macro-mechanical models by numerical method mainly focus on drape properties of fabric, 
whereas meso-mechanical models with appropriate boundary condition are accurate enough 
to solve simple macro-deformation of tension, compression and shearing. Since this research 
only concentrates on the prediction of tensile property of fabric structure, there is no further 
introduction of macro-mechanical model of drape properties of fabric.  
2.1.8 Evaluation 
The review of literature of analytical models for the mechanical analysis of woven 
fabrics indicates that a globally accepted analytical model in terms of accuracy still remains 
as a goal. Even though there are so many different analytical models, most of which are based 
on totally distinct theories or principles, all of them were established on simplifications and 
assumptions in order to decrease the complexity of the model. The major drawback in 
analytical models is the balance between simplifications and the accuracy of the results. The 
two-dimensional approach and the attribution of linear isotropic properties defined for 
materials would result in inaccuracy. In addition, the analytical models have the difficulty of 
integrating of micro-, meso- and macro-level models. Analytical models have been replaced 
by or incorporated into the numerical models. The numerical methods have unique 
advantages to overcome obstacles such as geometrical representation, complex deformations, 
particular material properties, contact phenomena and large deflection effects. Moreover, 
numerical approach tools are able to generate good virtual representation of fabric under 
deformation. Most numerical methods are based on the Finite Element Method which could 
simplify the mechanical analysis by avoiding calculation of the bending rigidity of the yarn 
and its attribution at the modeled yarn. The homogenization process could be achieved 
21 
between each model level by application of boundary conditions so that the micro-, meso- 
and macro-level models are well connected. However, homogenization process may become 
problematic if the interpretation of data from previous stage is difficult for the next stage. 
There is still much work to collaborate each stage in order to generate realistic model and 
equivalent properties.   
2.2 Experimental 
2.2.1 Material Preparation 
Six single-layer woven fabrics and five two-layer woven fabrics were used for this 
research. Each yarn in the fabric is polyester monofilament with different diameter and 
mechanical properties. The detailed information of unit cell of each woven fabric was 
examined by observation under microscope.  
2.2.2 Geometry of Unit Cell  
The geometry of unit cell of each fabric samples was examined using computer software 
associated with a microscope (cellSens Standard 1.8.1 by Olympus Corporation). The 
CellSens Standard program is capable to measure the length and angle up to magnitude of 
micrometer. The yarns in the fabric structure are classified into warp yarn and weft yarn. The 
warp yarn is along the machine direction, while weft yarn is perpendicular to the warp yarn. 
For the two-layer woven fabrics, the top layer and bottom layer are made of weft yarns, while 
a warp yarn crosses over each weft yarn as shown in Figure 9.  
The unit cells geometry of each fabric sample are presented in Figures 10 and 11. The 
top view, right view (weft yarn direction) and left view (machine direction) of single-layer 
woven fabrics and two-layer fabrics are shown in the figures. The properties of single-layer 
22 
woven fabrics are given in Table 1. The patterns for two-layer fabrics are little bit 
complicated. The two-layer fabric samples 1, 4 and 5 have extra yarns in the top weft layer.  
 
Table 1 Geometric properties of six single-layer fabric samples (w: warp, f: filling) 
Fabric 1w 1f 2w 2f 3w 3 f 4w 4f 5w 5f 6 w 6 f 
Major 
radius 
(mm) 
0.575 0.525 0.555 0.555 0.445 0.355 0.435 0.255 0.25 0.21 0.22 0.175 
Minor 
radius 
(mm) 
0.51 0.485 0.45 0.45 0.415 0.305 0.405 0.195 0.195 0.195 0.195 0.15 
Spacing 
(mm) 
3.57 3.57 1.89 1.89 1.29 1.06 1.13 1.09 1.28 1.58 0.90 0.50 
Crimp 
height 
(mm) 
1.65 1.94 0.9 1 0.9 0.65 
Structure 1/1 2/1 4/1 3/1 3/1 3/1 
 
 
 
Figure 9 Cross section of a typical two-layer woven fabric structure with unit cell of 8 
columns of weft yarns 
 
23 
 
Single-layer fabric sample 1 
 
Single-layer fabric sample 2 
 
Single-layer fabric sample 3 
 
Single-layer fabric sample 4 
 
Single-layer fabric sample 5 
24 
 
Single-layer fabric sample 6 
Figure 10 Geometry of single-layer fabrics in top view (left), right view (middle) and left 
view (right) 
 
 
Two-layer fabric sample 1 
 
Two-layer fabric sample 2 
 
Two-layer fabric sample 3 
 
Two-layer fabric sample 4 
25 
 
Two-layer fabric sample 5 
Figure 11 Geometry of two-layer fabrics in top view (left), right view (middle) and left (right) 
view 
 
 
2.2.3 Measurement of Geometry 
The 3D model construction depends on the measurement of geometry of each unit cell. 
What geometrical parameters need to be measured matters. According to the Peirce?s model, 
the intact geometry of the unit cell of fabric can be determined as long as the spacing between 
yarns (p in Figure 12), diameter and crimp height (h in Figure 12) are given. Then, the crimp 
angle, ? in Figure 12, can be calculated so that the contact area at cross over point is 
determined. 
 
 
Figure 12 Peirce?s model of simple woven fabric structure 
 
CellSens Standard 1.8.1 can measure fine distances between two points. The spacing is 
measured through the distance between centroids of two adjacent yarns. It was found that the 
26 
cross sectional shape of yarns is not perfectly circular. To build a more accurate model, 
elliptic shape is used to describe cross section of each yarn. The major radius and minor 
radius of each yarn are measured. Finally, the crimp is measured through the distance 
between the highest point and lowest point over an undulated yarn. Table 1 summarizes the 
geometrical information of 6 samples of single-layer fabrics  
However, this principle is only suitable for simple woven structures which, in this case, 
are single-layer fabrics. It does not work for the more complicated woven structures such as 2 
layer fabrics. A generative model which contains an interpolated spline function and elliptical 
surface is applied into generation of 3D solid geometry of unit cell for 2 layer fabrics. This 
technique will be explained later in the modeling section. An undulated yarn could be 
described by the path of the centerline of its cross section. Besides major and minor radius 
and spacing, the coordinates of the nodes along the centerline are also measured. In order to 
draw an intact curve for the centerline of the warp yarn for a two-layer structure as shown in 
Figure 9, at least 8 nodes which are exactly above the centroids of bottom weft yarns have to 
be determined. Figure 13 shows the 8 centerline nodes (black dots) for the unit cell of 
two-layer fabric 2. This method for generating 3D solid model of fabrics could be extended to 
all sorts of fabric structures 
 
 
Figure 13 Illustration for measurement of centerline of warp yarn for two-layer fabric 2 
 
27 
Table 2 presents the spacing information for five two-layer fabric samples. Since spacing 
between top weft, bottom weft and warps could be different, each of them has to be measured 
separately. Table 3 shows the major and minor radius for two-layer fabrics. 
 
Table 2 Spacing for five two-layer fabric samples 
 spacing 
between top 
and bottom 
layer (mm) 
spacing 
between warp 
yarns (mm) 
spacing 
between top 
weft yarns 
(mm) 
spacing 
between 
bottom weft 
yarns (mm) 
Sample 1 0.234 0.187 0.280 0.560 
Sample 2 0.786 0.475 0.930 0.930 
Sample 3 0.881 0.384 0.877 1.755 
Sample 4 0.390 0.241 0.434 0.868 
Sample 5 0.303 0.222 0.377 0.755 
 
Table 3 Major and minor radius for two-layer fabrics 
  Major radius(mm)  Minor radius(mm) 
Sample 1 Warp yarn 0.072 0.058 
Weft bottom 0.091 0.080 
Weft top 0.066 0.053 
Sample 2 Warp yarn 0.208 0.175 
Weft bottom 0.221 0.173 
Weft top 0.221 0.173 
Sample 3 Warp yarn 0.258 0.202 
Weft bottom 0.284 0.272 
Weft top 0.227 0.165 
Sample 4 Warp yarn 0.115 0.090 
Weft bottom 0.175 0.091 
Weft top 0.175 0.091 
Top extra 0.056 0.044 
Sample 5 Warp yarn 0.098 0.069 
Weft bottom 0.120 0.095 
Weft top 0.120 0.095 
Weft extra 0.078 0.063 
28 
 
2.2.4 Mechanical Property Tests 
Since this model assumes that the yarn is homogeneous and isotropic, the linear isotropic 
material model is used to define elastic properties of yarns. A linear relationship is applied to 
fit the stress-strain curve of each yarn to describe mechanical properties. Thus, both 
stress-strain curve and Poisson?s ratio are measured by universal mechanical testing machine 
(INSTRON) according to the ASTM D3822 ? 07. A camera device associated with the 
mechanical test machine can capture the image of yarns under tension test (Figure 14). The 
width of each yarn can be measured under different tensile deformation from which the 
Poisson?s ratio can be calculated.  
 
Figure 14 Universal mechanical testing machine with a camera device (INSTRON 
Technology) 
The stress-strain curves of yarns in 6 single-layer fabric samples are given in Figure 15, 
which shows that some of them exhibit non-linear behavior. In an ideal realization of the 
model, the non-linear behavior should be taken into account. However, in this research, a 
simplified linear model is used.  
29 
 
Figure 15 Stress strain curves for yarns in six single-layer fabric samples 
 
The mechanical properties of each yarn are summarized in Tables 4 and 5. For 
single-layer fabric structure, both warp yarn and weft yarn have different modulus and 
Poisson?s ratio. Usually, the modulus of warp yarn is higher than that of weft yarn. With 
respect to two-layer fabric structure, the unit cell of two-layer structure contains top weft, 
bottom weft, warp yarn and even extra yarn. Each of them may have different mechanical 
properties.  
Table 4 Mechanical properties of yarns for single-layer fabrics 
Samples Direction Modulus (MPa) Poisson?s ratio 
single-layer 
sample 1 
Warp yarn 3990 0.39 
Weft yarn 1260 0.32 
single-layer 
sample 2 
Warp yarn 1080 0.35 
Weft yarn 870 0.36 
single-layer 
sample 3 
Warp yarn 1110 0.42 
Weft yarn 870 0.35 
single-layer 
sample 4 
Warp yarn 1440 0.4 
Weft yarn 890 0.37 
single-layer 
sample 5 
Warp yarn 610 0.41 
Weft yarn 74.1 0.34 
single-layer 
sample 6 
Warp yarn 1541 0.39 
Weft yarn 88.4 0.44 
30 
 
Table 5 Mechanical properties of yarns for two-layer fabrics 
 
 
 
 
 
 
 
 
 
 
 
 
 
2.2.5 Friction Coefficient Tests 
Previous research indicated that friction coefficient between fibers does not have much 
effect on mechanical properties while fabric is under axial or biaxial loading because relative 
sliding can be neglected when fabric is subjected to axial or biaxial loading. However, the 
friction coefficient turns out to be a very important aspect in terms of manufacturing process 
and behavior under shear loading or cyclic loading [56]. In addition, friction between yarns at 
crossover points directly supports the geometric integrity. For the completeness of the 
mechanical model, the friction coefficient has been taken into account in the FEA by 
  Modulus(MPa)  Poisson?s 
ratio 
Sample 1 Warp yarn 3320 0.34 
Weft bottom 788 0.32 
Weft top 1240 0.34 
Sample 2 Warp yarn 3510 0.33 
Weft bottom 931 0.33 
Weft top 382 0.35 
Sample 3 Warp yarn 2080 0.37 
Weft bottom 3452 0.36 
Weft top 2219 0.37 
Sample 4 Warp yarn 2637 0.43 
Weft bottom 453 0.41 
Weft top 194.6 0.43 
Top extra 1699 0.41 
Sample 5 Warp yarn 3599 0.39 
Weft bottom 703.2 0.39 
Weft top 333.5 0.38 
Weft extra 2469 0.40 
31 
incorporating a friction model at the contact zone. The friction coefficient was experimentally 
obtained using a device whose schematic is shown in Figure 16. Five samples were tested for 
each pair of yarns contacting to each other and the final value of friction coefficient was 
obtained by taking the average. The friction coefficient is calculated using: 
                                                              (1) 
where   is the number of turns per unit length,    is the initial tension on the lower free 
ends,   is the measured friction force, and   is the twist angle which can be calculated from 
the following equation: 
                                  
    
                             (2) 
where L, L1 and L2 are distances indicated in Figure 16. Friction coefficients of each fabric 
measured from experiments are shown in Tables 6 and 7. The single-layer fabrics only have 
one type of contacting condition which is the contact between the warp and weft yarn at 
crossover point, whereas the two-layer fabrics have more than one contacting condition since 
there are two or three different types of weft yarns including the extra yarn in the two-layer 
structure.  
 
Figure 16 Schematic of the friction coefficient testing device 
 
32 
Table 6 Friction coefficients of single-layer fabrics 
Fabric 1 2 3 4 5 6 
Yarn to yarn average dynamic 
friction coefficient 
0.102 0.112 0.104 0.109 0.112 0.109 
Coefficient of variation 2.2% 3.2% 2.9% 2.5% 2.4% 2.4% 
 
Table 7 Friction coefficients of two-layer fabrics 
Sample  Yarn to yarn 
average dynamic 
friction coefficient 
Coefficient of 
variation 
Sample 1 Warp yarn to 
top weft 
0.117 3.0% 
Warp yarn to 
bottom weft 
0.116 2.9% 
Sample 2 Warp yarn to 
top weft 
0.103 2.1% 
Warp yarn to 
bottom weft  
0.103 2.2% 
Sample 3 Warp yarn to 
top weft 
0.114 1.9% 
Warp yarn to 
bottom weft 
0.116 1.7% 
Sample 4 Warp yarn to 
top weft 
0.115 1.9% 
Warp yarn to 
bottom weft 
0.114 2.0% 
Warp yarn to 
extra weft 
0.111 2.1% 
Sample 5 Warp yarn to 
top weft 
0.112 2.3% 
Warp yarn to 
bottom weft 
0.112 2.9% 
Warp yarn to 
extra weft 
0.109 2.9% 
 
 
 
33 
2.3 Modeling Approach 
2.3.1 Micro-Mechanical Modeling  
The objective of micro-mechanical model is to describe the stress-strain relationship of 
the yarn and to ultimately obtain what displacements would result from given forces in 
different directions. The properties of yarn calculated from this stage will be further used in 
the meso-mechanical modeling.   
To begin with, the strain represents the deformation of a body in a certain direction. 
Engineering strain, is defined by the following equation. 
                                   
 
                                 (3) 
where    is the body length before deformation,    is the body length after deformation in 
the same direction. This is the simple definition of engineering strain in the uniform 
deformation scenario. If the deformation of the body is no longer uniform, the engineering 
strain can be defined by a more generic form: 
                                                               (4) 
where u is the displacement vector. So this expression can be extended to three orthogonal 
directions: 
          
                                                                (5) 
          
According to the mechanics of materials, the shear strains are defined in the following 
equations. 
 
34 
                         
                                                                 (6) 
                         
The deformation at any point in a body can be defined by tensile and shear 
strain                     . The strain occurred in the body would cause the internal force 
within the body to resist the deformation. This internal force can be expressed by stress. The 
stress is the internal distribution of force per unit area. The stress is usually calculated as the 
function of the strain.  
                                                                      (7) 
This function describing the relationship between stress and strain is usually called the 
material model or constitutive equation, which is what the micro-mechanical model seeks for. 
If the stress-strain relationship of the material is the same regardless of directions, then it is 
called isotropic material. The stress-strain relationships in different directions might not be 
identical for the same body. If so, the material is then named anisotropic material. Typically, 
the yarn material is an anisotropic material; however, it is not completely anisotropic. The 
stress response in the cross section plane is isotropic. So the yarn material is considered to be 
transverse isotropic material. If the material is assumed to be linear elastic, the stress can be 
defined as a function of strain with 5 constants. The constants are longitudinal modulus EL, 
transverse modulus ET, transverse Poisson?s ratio ?TT, transverse-longitudinal Poisson?s ratio 
?TL and transverse-longitudinal shear modulus GTL. From these 5 constants, the 
longitudinal-transverse Poisson?s ratio ?LT and transverse shear modulus GTT can be 
calculated by the following equations: 
35 
                                                                           
 
                            (8) 
                                                                            
   
                         (9) 
In this case, the stiffness matrix equation of transverse isotropic property can be established 
assuming that z axis is the yarn path direction and x and y axes are two orthogonal directions 
perpendicular to the z axis. The overall stiffness matrix can be expressed by the following 
equation: 
(10) 
 
However, all the yarns in this research are monofilaments which mean that each yarn is 
formed evenly by extruding and polyester is uniformly distributed throughout the entire yarn. 
Therefore it is quite reasonable to assume that the yarn is an isotropic homogeneous material. 
In order to have a simplified model for the description of mechanical properties of yarn, the 
isotropic model is defined for the homogeneous yarn.  The isotropic mechanical property 
has only two independent variables (E for modulus and ? for Poisson?s ratio). The expression 
of relationship between the stress and strain is simplified from equation (10) to equation (11). 
The modulus and Poisson?s ratio are experimentally measured using universal mechanical 
testing with a camera device. The data of modulus and Poisson?s ratio of each yarn are given 
in Section 3 experimental part.  
36 
              (11) 
Usually, a homogenization process is required as the last step in this stage to obtain the 
overall property of a yarn if the yarn is multifilament assembled by many fibers. However, 
the overall property of yarn is self-explained by the modulus and Poisson?s ratio because the 
yarn is homogeneous monofilament with isotropic properties.  
2.3.2 Meso-Mechanical Modeling 
The meso-mechanical modeling stage is to analyze mechanical properties of the unit cell. 
According to the multi-scale model concept, the behavior of a micro heterogeneous medium 
can be described by that of an equivalent homogeneous medium with appropriate boundary 
conditions according to the macroscopic geometry and boundary. In particular, the fabric is a 
heterogeneous material. However, one can consider fabric structure to be homogeneous 
material made of identical repeating unit cells which have exactly same properties. In this 
case, the behavior of fabric structure can be described by repeating unit cell structure with 
appropriate boundary conditions. The overall fabric?s simple deformation behavior such as 
unidirectional tension and shearing can be retrieved as long as the coresponding mechanical 
properties of unit cell are calculated. In this research, the tensile properties in longitudinal and 
transverse direction of unit cells are obtained through FEM in which the geometry of unit cell, 
modulus and Poisson?s ratio are introduced as input data. An appropriate boundary condition 
is applied onto the unit cell to simulate the real physical conditions over the unit cell and 
37 
more importantly to homogenize the mechanical properties throughout the unit cell. 
2.3.2.1 3D Solid Modeling of Unit Cell Geometry 
3D solid model of a single yarn can be described by its 2D shape of cross section and 
yarn path, which is similarly defined by an array of nodes of centroids. This kind of modeling 
is called generative model, which is formed by transformation of a lower dimensional shape 
to a higher dimensional shape [57].  
Consider a cross sectiona shape:  
z=f1(x, y), x? R1, y? R2                                    (12) 
And a curve in 3D space: 
z=f2 (?), ?? R3                                 (13)                                                           
Now the cross sectional shape (12) is translated along the tangent direction of the 3D 
curve (13) so that the new coordinate z is expressed as function of ? for every point (x, y) ?  
[R1, R2] given a ?. Then, a 3D solid structure V(x, y, ?) may be generated. In particular, the 
cross sectional shape of a yarn is the generator. This generator can be translated along the 
centerline curve of the yarn path in order to generate a 3D solid model.  
? Yarn Path Representation  
Adanur and Liao proposed a geometric modeling technique by generative model which 
can actually generate all sorts of fabric structures such as woven, knitted and braided [8]. 
However, the yarn path function they used for the description of undulation of yarn is limited 
to cosine and sine. Cosine and sine functions are probably appropriate to describe undulation 
of the yarn in simple woven fabric; however, they are no longer precise for the undulation of 
38 
warp yarn in the two-layer woven fabric. Therefore, a more accurate yarn path function has to 
be used to represent the centerline path of the yarn. 
The path of a yarn can be considered as one dimensional line representing the yarn?s 
centerline in three dimensional space. Thus, the yarn path can be defined in 3D space as a 
function of distance along the yarn. The most generic and accurate way to describe such a 
yarn path is to specify a number of discrete positions along the yarn length, known as master 
nodes, and have the rest part interpolated [57] between these points rather than to define a 
certain cosine or sine function which is not precise enough to represent yarn path in this case. 
Generally speaking, the more master nodes are defined, the more accurate model of the yarn 
path. In order to get an accurate yarn path model, it is sufficient to specify one or two master 
nodes per crossover as long as the interpolation function is suitable. The continuity C must be 
required in the interpolation function so that there is no gap in the yarn path and the tangent 
of yarn path varies smoothly. A common solution to make sure the continuity and smoothness 
of the yarn path is to use spline functions. A spline is a function defined piecewise by 
polynomials. In its most general form a polynomial spline S : [a, b] consists of polynomial 
pieces Si : [ti, ti+1], where 
a = t0 < t1 < ? ? ? < tk?2 < tk?1 = b                       (14)                               
That is: 
                          
 
 
 
 
                      
                   
 
 
 
                          
                       (15)                           
39 
Since it is impractical to measure so many master nodes along the yarn path, another 
requirement of this interpolation function is that it can generate the curve of yarn path with 
the master nodes as little as possible. In this case no extra points need to be defined in order 
to specify the yarn path. A natural cubic spline function is used for the model of yarn path 
[57]. A detailed definition of natural cubic spline function is introduced below: 
The continuity conditions C0, C1 and C2 are applied by the equation: 
                                                                  (16)                   
                                                                        (17) 
                                                                    (18) 
The boundary conditions at t = a and t = b are given in equation: 
                                                                       (19) 
The derivatives of the spline at the knots can be calculated by solving the matrix equations: 
          
 
 
 
 
 
 
       
   
   
   
   
 
       
 
   
   
   
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
  
     
      
      
 
        
        
         
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                    
         
              
 
                  
                  
                   
 
 
 
 
 
 
 
           (20) 
Once the derivatives are known, the coefficients of the spline pieces can easily be determined. 
For each cubic spline piece, there are 4 equations with 4 unknown coefficients ai, bi, ci and di: 
                                                                         (21) 
                                                                          (22) 
                                                                         (23) 
                                                                        (24) 
The natural spline interpolation function also has a very important characteristic of 
40 
minimization of the total curvature. Equation 25 is the expression for approximation of the 
total curvature through the entire domain. The natural spline interpolation method can 
minimize the value of the total curvature. Since the total energy of elastic strip is proportional 
to the curvature, it turns out that the natural spline interpolation is actually the approximation 
of geometry of elastic strip with minimal energy constraint in k knots [57].  
                                                  (25) 
? Yarn Cross Section 
As discussed previously, the 3D solid model of a yarn is generated through the translation 
of cross section of yarn along its centerline yarn path. Thus, the shape of cross section is 
critical to determine the 3D solid model of a yarn. The cross section of yarn is defined as the 
2D surface cut by plane perpendicular to the yarn path tangent. When it comes to 
multifilament which is assembled by fibers, the cross section shape would no longer be 
constant shape and rather difficult to determine. Therefore, the cross section of the 
multifilament yarn can be assumed to be the smallest region which encompasses all of the 
fibers within the yarn. The outline of cross sections can be defined using parametric 
equations in 2 dimensions. Various shapes have been explored including the ellipse proposed 
by Peirce [13], power ellipse [58] and a modified lenticular shape [59]. All of these cross 
sections are summarized as follows. 
? Ellipse 
The elliptical cross-section is one of the simplest approximations. With given width A and 
height B the equation is defined as follows: 
                                                                    (26) 
41 
                                                                      (27) 
                                                                       (28) 
? Lenticular 
The lenticular cross-section is defined by the intersection part of two circles of radius r1 
and r2 each of which offsets vertically by distances o1 and o2 respectively. The parameters r1, 
r2, o1 and o2 can be calculated from w, h and distortion d in the following equations (Figure 
17):  
 
                                                                      (29) 
                                                                         (30) 
                                                             (31) 
                                                              (32) 
 
Figure 17 Schematic of lenticular cross sections 
 
The outline of cross-section can be defined as: 
                                        
                
                                (33) 
                                           
                  
                          (34) 
where: 
42 
                       
               
 
         
                
 
         
                    (35) 
? Power ellipse 
The power ellipse is a cross-sectional shape between ellipse and lenticular. The power 
ellipse is a slight modification to the elliptical cross-section where the y coordinate is 
assigned a power n to make the section resemble a rectangle with rounded edges when n < 1 
or a shape similar to a lenticular cross-section when n > 1 [57]. It is defined as follows: 
                                                                (36) 
                        
 
          
         
                        
                   (37) 
 
? Specification of Centroids. 
The computer software associated with the microscope device (CellSells Standard 1.8.1 
Olympus Corporation) is used in this research to measure the geometry including major and 
minor radius of each yarn, spacing between top and bottom layer, spacing between warp 
yarns and spacing between weft yarns as well as configuration. In this case, the spacing is 
defined as the distance between centroids of cross-sections. Another critical factor for 
accuracy of 3D solid model is the crimp height at crossover points. Theoretical speaking, 
crimp of weft yarns for two-layer fabric structure is not developed until heat-setting 
procedure. Therefore, 3D model is of little practical value without taking into account the 
crimp of weft yarns. This is another reason to use the generative modeling technique and 
interpolation method. Two assumptions are made in order to simplify the model without 
losing much accuracy. First, the cross-section of each yarn is elliptical. Second, there is no 
43 
penetration at contact region and yarns contact perfectly at tangent plane to the line crossing 
two centroids at crossover point. 
The number of centerline nodes is determined from the number of weft yarns and warp 
yarns in one repeating unit. For instance, Figure 9 exhibits the configuration of two-layer 
fabric sample 2. This two-layer fabric structure has 16 weft yarns in one unit cell. Horizontal 
number from 1~8 indicates the horizontal position of each weft yarn and vertical number 1, 2 
represent for vertical position of each weft yarn. There are at least 10 nodes to be defined for 
warp yarn in order to smoothly draw the curve of centerline of warp yarn. Likewise, 
minimum 10 nodes need to be defined along the weft yarn to smoothly draw the weft yarn. 
The related variables are defined as follows: 
The number of columns of weft yarn in one repeating unit: M 
The number of warp yarns in one repeating unit: N 
One single warp yarn could be described by a vector:  
                                                                         (38)                         
                                   represents the coordinate of centerline node. Due to 
the periodic relationship between adjacent warp yarns, the relationship between adjacent 
warp yarns could be described by the following equation: 
                                            (39) 
where   stands for the periodic vector.  
This could be applied to weft yarns: 
                                                     (2M means 2 weft layers) 
                                                                           (40) 
44 
This method could be applied to 3D modeling software such as Solidworks. All 
configurations of single-layer woven fabrics and two-layer woven fabrics are given in Figure 
18. 
 
Single-layer fabric sample 1 
 
Single-layer fabric sample 2 
 
Single-layer fabric sample 3 
 
Single-layer fabric sample 4 
45 
 
Single-layer fabric sample 5 
 
Single-layer fabric sample 6 
 
Two-layer fabric sample 1 
 
Two-layer fabric sample 2 
 
Two-layer fabric sample 3 
 
46 
Two-layer fabric sample 4 
 
Two-layer fabric sample 5 
Figure 18 3D solid model of unit cell of all fabric samples 
 
2.3.2.2 Material Property Definition 
The data of modulus and Poisson?s ratio are used as inputs retrieved from the 
micro-mechanical model.  
2.3.2.3 Meshing 
In order to perform the finite element analyses, the geometry of the fabric must be 
meshed. The only specific requirement about the mesh is that the cross section mesh at one 
end must be identical to that at the other end in a unit cell. This is due to the periodicity of the 
unit cell. The overall fabric is periodically made of repeating units. Hence, the nodes at both 
cross-sections at the end must be periodically related. The meshes at both cross-section ends 
have to be identical to ensure that the elements generated from meshing are the same. Figure 
19 shows a typical mesh of cross-sections at both ends. This requirement can be achieved by 
the mapped face mesh with triangle and rectangle elements in Ansysworkbench. The rest of 
the meshing steps could follow the standard procedure of meshing in Ansysworkbench. 
Figures 20, 21 and 22 present the coarse, medium and fine mesh of single-layer fabric sample 
1 respectively. Table 8 presents the mesh data of coarse, medium and fine mesh.  
47 
 
Figure 19 Schematic of the meshing of cross section by triangles and rectangles 
 
 
Figure 20 Coarse mesh of unit cell of single-layer fabric sample 1 
 
 
48 
 
Figure 21 Medium mesh of unit cell of single-layer fabric sample 1 
 
 
Figure 22 Fine mesh of unit cell of single-layer fabric sample 1 
 
Table 8 Mesh statistics 
 Number of Nodes Number of elements 
Coarse mesh 8576 4152 
Medium mesh 13476 6876 
Fine mesh 29943 16175 
 
49 
2.3.2.4 Contact Definition 
? Contact Pair 
In this case, a surface to surface contact model is used for contact behavior at crossover 
points. The contact pairs are defined according to the contact algorithm in ANSYS to handle 
the interpenetration. 3D eight-node surface-to-surface contact elements 
CONTA174/TARGE170 are used. CONTA174 is an 8-node element that is intended for 
general rigid-flexible and flexible-flexible contact analysis [60].  
In studying the contact between two bodies, the surface of one body is conventionally 
taken as a contact surface and the surface of the other body as a target surface. For 
rigid-flexible contact, the contact surface is associated with the deformable body; and the 
target surface must be the rigid surface. For flexible-flexible contact, both contact and target 
surfaces are associated with deformable bodies. The contact and target surfaces constitute a 
"Contact Pair" (Figure 23). The yarn is considered to be flexible body, so flexible-flexible 
contact is applied. 
 
Figure 23 A contact pair 
 
 
50 
 
? Frictional Model 
In this research, the yarn is considered to be contacted with another yarn at the crossover 
point and the friction effect between them sustains the stability of interlaced structure. 
Therefore, a Coulomb friction model is used according to Ansys contact options [60]. In the 
basic Coulomb friction model, two contacting surfaces can carry shear stresses. When the 
equivalent shear stress is less than a limit frictional stress (?lim), no motion occurs between the 
two surfaces. This state is known as sticking. The Coulomb friction model is defined as: 
                                                            (41) 
                                                             (42) 
where: 
    = limit frictional stress 
  = equivalent stress for 3D contact 
? = coefficient of friction for isotropic material 
P = contact normal pressure  
b = contact cohesion  
Once the equivalent frictional stress exceeds ?lim, the contact and target surfaces will 
slide relative to each other. This state is known as sliding. The sticking/sliding calculations 
determine when a point transitions from sticking to sliding or vice versa. The contact 
cohesion provides sliding resistance even with zero normal pressure. The contact cohesion is 
assumed to be zero for the yarn to yarn contact. The friction coefficients in each fabric 
samples are experimentally measured. 
51 
2.3.2.5 Periodic Boundary Conditions 
Because the overall fabric is considered to be made of repeatable repeating units or 
representative volumes, a periodic boundary condition is imposed on the model to ensure 
symmetry and repeatability of the repeating unit. By applying such boundary conditions, the 
model is equivalent to an infinitely large fabric undergoing uniform deformation. If a large 
fabric needs to be modelled with non-uniform deformation as is usually the case, the unit cell 
model is still useful. A macroscopic model is required which takes effective properties from 
the unit cell model in a multi-scale modelling approach. Since this research only concentrates 
on the unidirectional tensile behaviour of fabrics, there is no need to use a well established 
macroscopic model to calculate the macroscopic behaviour. That is to say, in this particular 
case, periodic boundary condition serves as the equivalent function of homogenization which 
calculates the macroscopic response of fabrics.  
 
 
Figure 24 Periodicity of boundary of a yarn [57] 
Figure 24 illustrates a section of a periodic yarn, where the yarn is repeatable along the 
repeat vector P. Nodes Ai and Bi lie on opposite sides of the yarn section. Their positions are 
initially related by the following equation: 
                                                                       (43) 
52 
 
Where Bi is the coordinates of points on one surface, Ai is the coordinates of points on 
another surface and     is the periodicity vector. 
This is a requirement for applying periodic boundary conditions and it is a consideration 
to be taken into account when the mesh is generated for the yarn. 
2.3.2.6 Derivation of Periodic Boundary Conditions 
The global constitutive law describes the relation between macroscopic stresses ( ) and 
strains (E) which are defined as volumetric averages of the relevant microscopic stresses (?) 
and strains (?) respectively:  
                                                             (44) 
                                                             (45)       
The microscopic stress and strain fields must fulfill the periodicity of the heterogeneous 
material [61-62]. This principle can be expressed by the following equation: 
                                                          (46) 
where: 
   = a rigid displacement of the unit cell 
  = anti-symmetric tensor related to the small rigid rotation of the unit cell 
  = position vector in the RV 
  = periodic part of microscopic displacement field 
For the problem of elastic behavior of unit cell, Equation 46 could be further simplified 
to: 
                                                                      (47) 
53 
                                                                      (48) 
Then we get: 
                                                                      (49) 
The equation could be further rearranged into [62]: 
                                                                             (50)                                       
                                      
?   
 ?  
   
                         (51) 
where             and         are displacement vector of corresponding nodes on the surfaces 
of one side and the other, respectively.     is average strain tensor applied on the repeating 
unit,     is the periodicity vector. The periodic boundary conditions can be set up by 
constraint equations in Ansys [60]. Detailed information is given in the following. Equation 
(50) is a more general form of Equation (49) in mechanical model. 
2.3.2.7 Implementation of Boundary Conditions 
There two types of boundary conditions defined for the unit cell. The first one is actually 
the displacement constraints on the edge in order to make the convergence of solution easier. 
In particular, nodes on the edge have no displacement along the y axis while fabric is 
subjected to tension along the x axis. This can be achieved by specifying the displacements 
along the y axis to be zero (Figure 25).  
 
Figure 25 Displacement constraints on the edge 
54 
The second one is periodic boundary condition which could be implemented as follows: 
U1-U2=E(X1-X2)                         (52)                                 
U3-U4=E(X3-X4)                         (53) 
where U1, U2, U3, and U4 are displacement vectors of points on the edges of 1, 2, 3, and 4 
respectively (Figure 26). E stands for macroscopic strain tensor of the fabric  
 
Figure 26 Periodic boundary conditions on the single-layer fabric sample 5 
2.3.3 Macro-Mechanical Modeling 
As discussed previously, there is no need to apply a specific macro-mechanical model for 
the calculation of macro-mechanical behavior. The macro-mechanical behavior can be 
obtained from the meso-mechanical model as long as the boundary conditions are 
appropriately established. On the macro-scale level, the fabric is considered as homogeneous 
material which can be described by the constitutive equation between macro strains E and 
macro-stresses  :  
                                                                     (54) 
The aim of macro-mechanical model is to find macro-stiffness matrix C. The solution to 
this question is actually the homogenization process. The homogenization process is 
55 
approached by considering the unit cell of a textile as a repetitive part of an infinite array of 
identical cells [1, 35, 38 124]. Then the meso-mechanical modeling of unit cell can calculate 
the stiffness matrix Cc of infinite array of identical cells through the appropriate periodic 
boundary conditions with given macro strain E. The macro stress can be calculated by 
volumetric average of microscopic stresses throughout the unit cell.  
2.4 Results and Analysis 
Figures 27-37 exhibit the contour of stress distribution throughout the unit cell under 
undeformed state and 10% deformed state in unidirectional tension for single-layer and 
two-layer fabrics. It is clearly indicated that stress concentration occurred at crossover points. 
The middle part between the edge and crossover point has less stress concentration than the 
crossover point. The positions at the crossover points in the fabric structure have the highest 
possibility to break or deform. Moreover, the translation of stress from weft yarns to warp 
yarns of single-layer fabrics and warp yarns to weft yarns of two-layer fabrics at the contact 
region can be observed. The detailed stress distribution indicates that stress from tensile force 
is translated to the yarns in perpendicular direction through the crossover points. This 
information could be further used for fracture analysis and ballistic resistance. 
Figure 38-48 show the comparison between simulation data and experimental data. 
Generally speaking, the simulation data agreed well with the experimental data at initial stage, 
i.e. below 40%~50% of total deformation. However, the variation between simulation and 
experimental data is continuously increasing with growing deformation. This is probably due 
to the fact that the solid yarn is going to have plastic deformation at certain point which no 
longer indicates linear tensile behavior. Another reason is probably the simplification by 
56 
using isotropic properties instead of anisotropic properties. However, since this work mainly 
focuses on the mechanical properties under unidirectional tensile tests, the assumption of 
isotropic properties does not affect the simulation results much. Isotropic property 
assumption is a reasonable choice to simplify the model without losing the accuracy in the 
simulation. There are variations between contact simulations and actual contact conditions, 
which could also result in the inaccuracy of the model. 
The warp direction usually has higher force to extension ratio because the warp yarn has 
higher modulus than filling yarn. Even though the number of weft yarns is more than the 
number of warp yarns, the slope of force to extension in warp direction is still higher than the 
slope in weft direction. The initial modulus dominates unidirectional tensile property.   
 
 
 
 
 
 
 
 
57 
 
Figure 27 The contour of stress distribution throughout the unit cell of single-layer fabric 
sample 1 in undeformed state (top) and 10% deformed state (bottom) in the warp direction 
 
 
 
 
 
 
 
 
 
 
58 
 
Figure 28 The contour of stress distribution throughout the unit cell of single-layer fabric 
2 in undeformed state (top) and 10% deformed state (bottom) in the warp direction 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
59 
 
Figure 29 The contour of stress distribution throughout the unit cell of single-layer fabric 
3 in undeformed state (top) and 10% deformed state (bottom) in the warp direction 
 
 
 
 
 
 
 
 
 
 
 
60 
 
Figure 30 The contour of stress distribution throughout the unit cell of single-layer fabric 
4 in undeformed state (top) and 10% deformed state (bottom) in the weft direction 
 
 
 
 
 
 
 
 
 
 
 
 
 
61 
 
Figure 31 The contour of stress distribution throughout the unit cell of single-layer fabric 
5 in undeformed state (top) and 10% deformed state (bottom) in the warp direction 
 
 
 
 
 
 
 
 
 
 
 
 
62 
 
Figure 32 The contour of stress distribution throughout the unit cell of single-layer fabric 
6 in undeformed state (top) and 10% deformed state (bottom) in the weft direction 
 
 
 
 
 
 
 
 
 
 
 
63 
 
Figure 33 The contour of stress distribution throughout the unit cell of two-layer fabric 1 
in undeformed state (top) and 10% deformed state (bottom) warp direction 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
64 
 
Figure 34 The contour of stress distribution throughout the unit cell of two-layer fabric 2 
in undeformed state (top) and 10% deformed state (bottom) warp direction 
 
 
 
 
 
 
 
 
65 
 
Figure 35 The contour of stress distribution throughout the unit cell of two-layer fabric 3 
in undeformed state (top) and 10% deformed state (bottom) warp direction 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
66 
 
Figure 36 The contour of stress distribution throughout the unit cell of two-layer fabric 4 
in undeformed state (top) and 10% deformed state (bottom) warp direction 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
67 
 
Figure 37 The contour of stress distribution throughout the unit cell of two-layer fabric 5 
in undeformed state (top) and 10% deformed state (bottom) warp direction 
 
 
 
 
 
 
68 
 
Figure 38 Comparison between experimental data and simulation data of single-layer 
fabric sample 1 (1: simulation data in warp direction, 2: experimental data for warp direction, 
3: simulation data in weft direction, 4: experimental data in weft direction) 
 
 
Figure 39 Comparison between experimental data and simulation data of single-layer 
fabric sample 2 (1: simulation data in warp direction, 2: experimental data for warp direction, 
3: simulation data in weft direction, 4: experimental data in weft direction) 
              
4 
3 2 
1 
0 
500 
1000 
1500 
2000 
2500 
3000 
0 10 20 30 40 
Fo
rce
(N
) 
Extension(mm) 
2 
4 
1 
3 
0 
500 
1000 
1500 
2000 
2500 
3000 
0 5 10 15 20 25 30 35 40 
Fo
rce
(N
) 
Extension(mm) 
69 
 
Figure 40 Comparison between experimental data and simulation data of single-layer 
fabric sample 3 (1: simulation data in warp direction, 2: experimental data for warp direction, 
3: simulation data in weft direction, 4: experimental data in weft direction) 
 
Figure 41 Comparison between experimental data and simulation data of single-layer 
fabric sample 4 (1: simulation data in warp direction, 2: experimental data for warp direction, 
3: simulation data in weft direction, 4: experimental data in weft direction) 
2 
1 
3 
4 
0 
1000 
2000 
3000 
4000 
5000 
6000 
0 5 10 15 20 25 30 35 40 
Fo
rce
(N
) 
Extension(N) 
2 
1 
4 
3 
0 
500 
1000 
1500 
2000 
2500 
3000 
3500 
4000 
4500 
0 5 10 15 20 25 30 35 
Fo
rce
(N
) 
Extension(mm) 
70 
 
Figure 42 Comparison between experimental data and simulation data of single-layer 
fabric sample 5 (1: simulation data in warp direction, 2: experimental data for warp direction, 
3: simulation data in weft direction, 4: experimental data in weft direction) 
 
Figure 43 Comparison between experimental data and simulation data of single-layer 
fabric sample 6 (1: simulation data in warp direction, 2: experimental data for warp direction, 
3: simulation data in weft direction, 4: experimental data in weft direction) 
 
4 
3 
2 
1 
0 
500 
1000 
1500 
2000 
2500 
0 5 10 15 20 25 30 35 40 45 50 
Fo
rce
(N
) 
Extension(mm) 
4 
3 
2 
1 
0 
500 
1000 
1500 
2000 
2500 
0 5 10 15 20 25 30 35 40 45 
Fo
rce
(N
) 
Extension(mm) 
71 
 
Figure 44 Comparison between experimental data and simulation data of two-layer fabric 
sample 1 (1: simulation data in warp direction, 2: experimental data for warp direction, 3: 
simulation data in weft direction, 4: experimental data in weft direction) 
 
Figure 45 Comparison between experimental data and simulation data of two-layer fabric 
sample 2 (1: simulation data in warp direction, 2: experimental data for warp direction, 3: 
simulation data in weft direction, 4: experimental data in weft direction) 
 
0 
200 
400 
600 
800 
1000 
1200 
0 5 10 15 20 25 
Fo
rce
(N
) 
Extension(mm) 
1 
2 
3 
4 
0 
500 
1000 
1500 
2000 
2500 
3000 
3500 
0 5 10 15 20 25 30 35 
Fo
rce
(N
) 
Extension(mm) 
1 
2 
3 
4 
72 
 
Figure 46 Comparison between experimental data and simulation data of two-layer fabric 
sample 3 (1: simulation data in warp direction, 2: experimental data for warp direction, 3: 
simulation data in weft direction, 4: experimental data in weft direction) 
 
Figure 47 Comparison between experimental data and simulation data of two-layer fabric 
sample 4 (1: simulation data in warp direction, 2: experimental data for warp direction, 3: 
simulation data in weft direction, 4: experimental data in weft direction) 
0 
500 
1000 
1500 
2000 
2500 
3000 
3500 
4000 
4500 
0 5 10 15 20 25 30 35 
Fo
rce
(N
) 
Extension(mm) 
1 
2 
3 
4 
0 
500 
1000 
1500 
2000 
2500 
0 5 10 15 20 25 30 
Fo
rce
(N
) 
Extension(mm) 
1 
2 
3 
4 
73 
 
Figure 48 Comparison between experimental data and simulation data of two-layer fabric 
sample 5 (1: simulation data in warp direction, 2: experimental data for warp direction, 3: 
simulation data in weft direction, 4: experimental data in weft direction) 
 
2.5 Conclusions 
A mechanical model is built to simulate tensile behavior of monofilament woven fabrics. 
The model is based on the multi-level mechanical theory in which the properties of the yarn 
and geometry of the repeating unit are taken into account. 
A generative model is used to efficiently generate the 3D solid model of woven fabrics. 
This technique can be further used to all sorts of complex fabric structures. 
The simulation data is in reasonably good agreement with the experimental data for all. 
The variation between experimental data and simulation data probably result from the 
difference between isotropic material property and an-isotropic material property. 
This model is based on FEA and shows the stress-strain field at local points. Stress 
concentration occurs at crossover points and stress translation takes place at crossover points, 
which can be further integrated into fatigue and damage analysis of fabrics and fabric 
reinforced composites. 
2 
1 
4 
3 
0 
500 
1000 
1500 
2000 
2500 
0 5 10 15 20 25 
Fo
rce
(N
) 
Extension(mm) 
74 
Chapter 3 Mechanical Analysis of Auxetic Behavior of a Novel Braiding Tubular 
Structure by FEM 
3.1 Literature Review 
Auxetic materials [63, 64] which are materials with negative Poisson?s ratio, have 
received considerable attention over the last three decades because of their counter intuitive 
mechanical behavior: the auxetic material elongates in the force direction and expands 
perpendicularly to the force direction simultaneously. This interesting behavior has been 
studied and found that auxetic property could improve many fundamental bulk properties 
such as the indentation resistance, the shear modulus, fracture toughness and synclastic 
curvature [65]. The analysis of auxetic behaviour and exploration of auxetic materials 
demonstrated that auxetic materials exist in a wide range of fields from natural single crystals 
[66] and cubic metals [67] to biological materials such as skin [68] and bones [69] and 
synthetic materials such as foams [70], honeycomb structures [71] and composites [72]. 
Among all other auxetic materials discovered, synthetic auxetic materials are demonstrated to 
be very promising since they can be tailored for specific purpose by controlling the structure 
and properties. 
Numerous efforts has been done to scope the fundamental mechanism behind auxetic 
behavior. The research of crystal arsenic [73] and zeolites [74] drew a conclusion that auxetic 
behavior attributes to molecular bond rotation. Meanwhile, auxetic fibers demonstrated that 
they can be woven into technical textiles endowing the auxetic property to the overall 
structure. Shanahan and Piccirelli [75] fabricated woven fabric which exhibited auxetic 
behavior as a result of geometric effect by woven structure. Ugbolue et al. [76] invented a 
75 
knitted textile structure by carefully knitting the structure resulting in a Poisson?s ratio of -0.6 
over the fabric width. Hall et al. [77] successfully produced carbon nanotube sheets with an 
in-plane Poisson?s ratio as low as a -0.2 by combination of single and multi-walled carbon 
nanotubes. Auxetic yarn structure was fabricated by Hook et al. [1] by wrapping a 
monofilament yarn with higher modulus and smaller diameter onto another yarn with lower 
modulus and larger diameter. This helical structure of combination of two separate yarns 
indicated a net increase in the width of the structure when subjected to a tensile load. Miller 
et al. [78] used helical auxetic yarns in weaving to manufacture a woven fabric made of 
helical auxetic yarns and tested the respective mechanical properties.  
In this research, a tubular fabric structure with auxetic behavior was made by braiding. 
Moreover, a finite element model of this tubular fabric structure was built for the analysis of 
auxetic behavior. 
3.2 Material Preparation 
Two polyester yarns were prepared for the manufacture of helical auxetic yarn. One 
polyester yarn has much higher modulus and smaller diameter than the other. The auxetic 
yarn was manufactured on the braiding machine and then helical auxetic yarns were used as 
yarn component to fabricate the tubular braided structure by braiding. 
The modulus of warp yarn and core yarn are tested according to the ASTM D3822 ? 07 
Monofilament. The data are given in Table 9.  
 
 
 
76 
Table 9 Mechanical properties of wrap and core yarns as component of the helical auxetic 
yarn 
 
 Core Wrap 
Young?s modulus (MPa) 13 1360 
Ultimate strength (MPa) 31 650 
Strain at break (%) 88 6.2 
 
3.3 Approach  
The auxetic yarn is manufactured on braiding machine. The core yarn is inserted in the 
system taking place of the position of mandrel in the middle of the braiding machine. The 
core yarn is stationary. The wrap is wrapped onto the core yarn with spindle. An illustration 
of the manufacture process is given in Figure 49. Figure 50 presents one spindle rotating in 
the trajectory. Figure 51 shows the braiding machine used. 
 
Figure 49 Schematic of manufacture of helical auxetic yarn 
 
Figure 50 The cross sectional view of rotating surface 
77 
 
Figure 51 The braiding machine used 
The geometry of helical auxetic yarn is analyzed by microscope. According to the 
manufacture process of helical auxetic yarn, the diameter of core and wrap yarns (Dc and 
Dw), the starting angle (?) and pitch (L) are three important parameters determining the 
geometry (Figure 52 and 53). The data are given in Table 10. 
 
 
Figure 52 Geometry of a helical auxetic yarn 
 
Figure 53 Geometrical parameters 
Each geometric parameter is correlated by the following equations: 
                                                                    (55)                                                              
78 
                                                                                                    (56) 
                              
 
                                (57) 
                                                          (58) 
Ac and Aw are the cross sectional area of core yarn and wrap yarn in helical auxetic structure. 
Table 10 Geometric parameters for the auxetic yarn 
Core diameter (Dc) 1.32 mm 
Warp diameter (Dw) 0.27 mm 
Starting angle (?) 17.4?  
Pitch (L) 13.2mm 
Then, the biaxial tubular structure was fabricated on the braiding where 8 spindles of 
helical auxetic yarns were used. A mandrel was used in the core, around which 8 spindles of 
auxetic yarns are rotating. Figure 54 illustrates the geometric structure of the biaxial tubular 
structure. Figure 55 presents the picture of biaxial tubular structure. 
 
 
 
 
 
79 
 
Figure 54 Schematic of the biaxial tubular structure 
The geometry information of the biaxial tubular structure is given in Table 11. 
 
Table 11 Geometry of the biaxial tubular structure 
Starting angle 47?  
Pitch (mm) 31.45 
Mandrel diameter (mm) 10.75 
 
Figure 55 The biaxial tubular structure 
 
80 
3.4 Analysis of the Properties by FEM 
The original purpose of the biaxial tubular fabric was to fabricate a novel 3D structure 
with variable permeability on the surface. This characteristic is achieved through the auxetic 
behavior of auxetic yarns in the biaxial tubular structure. Theoretically speaking, the increase 
of width of auxetic yarns in the structure would change the open area on the surface leading 
to variable permeability. The permeability can be quantified by measurement of open area on 
the surface under different longitudinal strain. The area of open surface can be measured by 
the universal mechanical test machine associated with a camera device while the biaxial 
structure is subjected to tensile loading. Then the curve of the open surface area versus 
longitudinal strain can be drawn according to the experimental data. The FEM model of the 
biaxial tubular structure is built to calculate the open surface area under different longitudinal 
strain. The results were validated by comparing the experimental data and simulation data. 
The objective of FEM model of biaxial tubular structure is to obtain the open surface 
area as a function of longitudinal strain. The change of the open surface area is due to three 
parameters: the Poisson?s ratio of helical auxetic yarn, the edge length of unit cell and the 
angle between the two cross yarns. 
3.4.1 Micro-Mechanical Model 
The micro-mechanical model of biaxial tubular structure is to simulate the mechanical 
properties of the helical auxetic yarn. An FEM model of the auxetic yarn is established. The 
3D solid geometry model is made according to the geometric data experimentally measured. 
The same generative modeling technique in Chapter 2 is employed here for the 3D 
construction of the helical auxetic yarn. Figure 56 shows the 3D solid geometry of the helical 
81 
auxetic yarn.  
 
Figure 56 3D solid model of the helical auxetic yarn 
The material is defined as isotropic material. The boundary conditions to simulate the 
tensile test are derived from the compatibility of equations on the two ends. The boundary 
conditions are given in the following equations. 
                          Uac=Uaw=U                       (59) 
                           Ubc=Ubw=-U                       (60) 
where U is the prescribed displacement which can be defined in an incremental form. 
The FEM model is established such that the stress ? is determined with strain U 
prescribed. Then the stress-strain curve of auxetic yarn can be obtained and validated by 
comparison with experimental data. 
The implementation of boundary conditions is illustrated in Figure 57. Figure 58 shows 
the contour of stress distribution of helical auxetic yarn before and after the deformation. The 
simulation and experimental data are exhibited in Figure 59. The simulation data is fit quite 
well with the experimental data below the strain point at around 0.05. The inaccuracy of 
simulation data increased probably due to the plastic deformation and the rough contact 
algorithm. 
 
82 
 
Figure 57 Application of boundary conditions 
 
Figure 58 The contour of stress distribution before (up) and after (down) deformation 
 
 
Figure 59 Comparison between simulation data and experimental data for tensile 
property 
0 
10 
20 
30 
40 
50 
60 
70 
0 0.05 0.1 0.15 0.2 0.25 
Str
es
s(M
Pa
) 
Strain 
Experimental data 
Simulation data 
83 
 
It is indicated in Figure 59 that the mechanical property of auxetic yarn is highly 
non-linear. Therefore, a bilinear isotropic property is defined for the material property of 
auxetic yarn. The stress-strain curve is divided into two parts by the onset point where the 
slope of the curve is changed abruptly. The turning of the curves can also be considered as the 
yield point. The first part is fitted with a linear function which represents the linear isotropic 
property; the second part is also fitted with another linear function which represents the 
plastic behaviour (Figure 60). These two linear functions are used to define the material 
property of the helical auxetic yarn in the meso-mechanical model.  
 
Figure 60 Curve fitting with bilinear function 
After the stress-strain property of the helical auxetic yarn is obtained from the simulation, 
a 3D solid cylinder model with modified diameter is generated to represent the helical auxetic 
yarn (Figure 61). The modified diameter is given in the following equation: 
                                                                  (61)                                   
 
y = 5.67E+08x 
y = 2E+08x + 2E+07 
0 
10 
20 
30 
40 
50 
60 
70 
0 0.05 0.1 0.15 0.2 0.25 
Str
ess
 (M
pa
) 
First part 
Second part 
Strain 
84 
 
Figure 61 Representation of helical auxetic yarn 
Then, the only mechanical property left for the helical auxetic yarn is the Poisson?s ratio. 
A Poisson?s ratio has to be no more than 0.5 for the linear isotropic part. According to the 
geometry of the modified yarn, the overall lateral strain is the sum of wrap yarn?s lateral 
strain and core yarn?s lateral strain. The Poisson?s ratio for the linear isotropic property of the 
helical auxetic yarn can be calculated by considering the laminate theory of parallel structures 
in the following equation:  
                           
 
   
 
   
 
                    (62) 
where    and    stand for the Poisson?s ratio of wrap yarn and core yarn. 
Furthermore, the probe can be defined in the FEM to measure the lowest position and 
highest position in transverse direction of the helical auxetic yarn, which gives the strain in 
transverse (  ) direction. Then the Poisson?s ratio can be calculated by Equation (63). The 
curves of the Poisson?s ratio versus longitudinal strain of the warp yarn, the core yarn, the 
experimental data of HAY and the simulation data of HAY are given in Figure 62.  
                                    
 
                         (63) 
85 
 
Figure 62 Poisson?s ratio versus longitudinal strain 
 
3.4.2 Meso-Mechanical Model 
? Geometry of unit cell of the biaxial tubular structure  
The biaxial tubular structure can be considered as a 2D interlaced fabric sheet by 
unwinding the structure. The periodic representative volume is a 2?2 plain woven structure 
with a certain angle between the warp and weft yarn at crossing point. The 3D solid 
geometric is generated by the same generative modelling technique used for woven fabrics. 
The geometry information of the unit cell of biaxial tubular structure is measured by 
microscope. It must be clarified that the crossing angle between the two interlaced yarns is 
2?, which is continuously changing while the shear force is increasing. Figure 63 shows the 
3D solid model of the unit cell. 
-3 
-2 
-1 
0 
1 
2 
3 
4 
5 
6 
0 0.1 0.2 0.3 0.4 P
oisson
's r
ati
o 
Longitudinal strain 
Experimental data of HAY 
Warp 
core 
Simulation data of HAY 
86 
 
Figure 63 3D solid model of the unit cell 
? Simulation 
The tensile test of the biaxial tubular structure along the longitudinal direction is 
considered to be equivalent to pure shear test. The relationship between the longitudinal force 
and shear force is illustrated in Figure 64 and given in the following equations. 
                                                                    (64) 
                                                                    (65) 
 
Figure 64 Illustration of tension and shear 
 
The periodic boundary conditions are still valid for the unit cell of biaxial tubular 
structure due to the periodicity of the unit cell while the overall structure is subjected to 
deformation. In particular, periodic boundary conditions can be substituted by the boundary 
conditions of prescribed forces on the edges of unit cell. The nodes on both edges of one yarn 
have exactly the opposite forces to have periodicity. The displacement in x axis must be zero 
87 
to avoid rigid body displacement. Figure 65 shows the application of boundary conditions. 
The boundary conditions in terms of displacements are given in the following equation: 
                                                                              (66) 
where     is the macro strain tensor which can be defined as an incremental form. This 
could be further transformed into the boundary condition in the form of forces: 
                                                          (67) 
where                        are four pairs of opposite forces which can be defined in 
Ansysworkbench. 
In addition, the condition           is applied at the cross over point (Figure 65) to 
avoid the rigid displacement. A bonded tie contact is defined at each cross over point. 
 
 
Figure 65 Application of boundary conditions 
3.5 Results and Discussion 
The shear deformation can be calculated by angle between the crossing yarns. 
                                                                     (68) 
where   is the change of angle,    is the angle after deformation and    is the angle 
before deformation  
It is assumed that the shape of the unit cell is always a parallelogram regardless of 
different crossing angles. According to the geometry of the unit cell (Figure 66), the open 
88 
area (A) is: 
                                                           (69) 
 
 
Figure 66 Geometry of the unit cell 
Suppose the edge length at a given tensile force F is   ; then the final edge length l is 
given by the following equation: 
                                                                       (70) 
where    is the lateral strain of the helical auxetic yarn, 
           is the displacement of the width of the helical auxetic yarn, 
          
      is the displacement projected on hte axis direction of the helical auxetic yarn, 
resulted from the increase of width of the helical auxetic yarn under tension. 
The value     and   can be calculated from the FEM model of the unit cell as a function of 
incremental tensile force F. The lateral strain    is a function of the longitudinal strain    
which equals to: 
                                          
 
                         (71) 
The initial length was l0 = 9.83 mm 
 
89 
 
Figure 67 The contour of stress distribution before deformation (left) and after deformation 
(right) 
 
Figure 68 Experimental data and simulation results of each parameter  
Figure 67 presents the contour of stress distribution before deformation and after 
deformation. Figure 68 shows the final results of the model. It is clearly observed that the 
open area is decreased while the structure is subjected to incremental tension. The 
experimental data of open area versus axial force indicates that the open area is going to 
decrease at the beginning stage and then gradually increase after the lowest point at around 
250 N. This phenomenon can be explained by the combination effect of auxetic behaviour of 
the helical auxetic yarn and the decrease of the crossing angle under axial tension. The 
0 
20 
40 
60 
80 
100 
120 
140 
160 
180 
0 100 200 300 400 500 
Force (N) 
Crossing angle 
(Degree) 
Unmodified edge 
length (mm) 
Modified edge length 
(mm) 
Experimental data of 
open area  
Model data of open 
area 
(mm2) 
(mm2) 
90 
increase in width of the auxetic yarn contributes to the decrease of the open area and the 
decrease of the crossing angle has the opposite effect. The FEM model based on auxetic 
behaviour and crossing angle is validated through the comparison between simulation data 
and experimental data. The major inaccuracy of the model is probably due to the relative 
sliding between helical auxetic yarns and measurement errors. The yarn?s relative sliding, 
which occurs in reality, is not taken into account in the model. The relative sliding, if it is big 
enough, could severely damage the periodicity and stability of the unit cell. If the unit cells 
do not periodically constitute the overall structure, the model is no longer valid.  
3.6 Conclusions 
A novel biaxial tubular fabric made of helical auxetic yarn is fabricated by braiding 
method. The tubular braiding fabric has variable permeability property while it is subjected to 
incremental axial forces. An FEM based mechanical model including the micro-mechanical 
and meso-mechanical model is established to investigate the mechanical properties. The 
model is validated by comparison between simulation data and experimental data. The 
auxetic behaviour of the yarn and decreasing crossing angle under axial force are the two 
major parameters to influence the permeability.    
 
 
 
 
 
 
91 
 
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