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Multiscale modeling of textile

composites

Thiam Wai Chua

MT 10.19

Master of Science graduation project report

January, 2011

Supervisors:dr.ir. Varvara Kouznetsovadr.ir. Wouter Wilsonprof.dr.ir. Marc Geers

Mechanics of MaterialsComputational and Experimental Mechanics

Department of Mechanical EngineeringTechnische Universiteit Eindhoven

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Abstract

In this project, a coupled multiscale framework is developed for analysis of fiberreinforced composites. The developed framework is a versatile tool to study the mechan-ical behavior at the macroscopic level and simultaneously the impact of the mechanicalbehavior from the microscopic level to macroscopic level. In the framework, the macro-scopic material constitutive behavior at each integration point is obtained through theunderlying microscopic composite structure. Abaqus subroutines are used to link themicroscopic level to the macroscopic level structural analyses and vice versa. The fo-cus of this project is to document and present the development and added value of theframework to the analysis of the textile composites. Thus, the quantitative comparisonbetween the analysis results and experimental data has not been attempted. However,the qualitative comparison has been made successfully to verify the correctness of theframework functionality. Multiscale analyses on the tensile test of a crossply laminateand compression test on a notched quasiisotropic laminate are performed in order toillustrate and demonstrate the power and utility of the framework. This framework canbe seen as a powerful and useful tool for future engineering analysis and design of com-posite structure due to its flexibility to apply on many textile composites which hardly

to be achieved using other analytical approaches. Thus, engineers enable to utilize thisframework at the beginning stage of the composite structures design and developmentfor the savings of cost and time.

Keywords: coupled multiscale analysis, composites, Abaqus.

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Contents

Abstract i

1 Introduction 1

1.1 Composite aerospace landing gear structures . . . . . . . . . . . . . . . . 1

1.2 Textile composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Modeling approaches for textile composites . . . . . . . . . . . . . . . . . 2

1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Scope and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Multiscale modeling framework description 6

2.1 Basic hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Microscopic level boundary value problem . . . . . . . . . . . . . . . . . 7

2.3 Macromicro levels coupling . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.3 Internal work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Microstructure boundary value problem . . . . . . . . . . . . . . 11

2.4.2 Macroscopic stress . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.3 Macroscopic tangent stiffness . . . . . . . . . . . . . . . . . . . . 12

2.5 Modeling framework implementation using Abaqus . . . . . . . . . . . . 13

2.6 Modeling framework verification . . . . . . . . . . . . . . . . . . . . . . . 13

3 Microstructural modeling 19

3.1 Textile and noncrimp composite models . . . . . . . . . . . . . . . . . . 193.2 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Microscopic level analyses . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.1 Tension test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.1.1 Uniaxial tension . . . . . . . . . . . . . . . . . . . . . . 25

3.4.1.2 Biaxial tension . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.2 Compression test . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4.3 Shear test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Application of the multiscale modeling framework 314.1 Tensile test on a crossply laminate . . . . . . . . . . . . . . . . . . . . . 31

4.2 Compression test on a notched quasiisotropic laminate . . . . . . . . . . 36

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Contents iii

5 Conclusions and recommendations 475.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . 48

5.2.1 Multiscale modeling framework . . . . . . . . . . . . . . . . . . . 485.2.2 Microstructural modeling . . . . . . . . . . . . . . . . . . . . . . 495.2.3 Application of the multiscale modeling framework . . . . . . . . 50

Bibliography 51

Appendices 58

A Literature survey 59

B Macromicro levels coupling 64

C Microscopic level boundary value problem 67

D Macroscopic stress calculation 70

E Condensation of the microscopic stiffness 74

F Consistent tangent stiffness for the Kirchhoff stress tensor 75

G Total matrix of the reduced stiffness matrix 78

H Incrementaliterative multiscale modeling approach 80

I Multiscale analyses of unidirectional fiber reinforced composite 82

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Chapter 1

Introduction

1.1 Composite aerospace landing gear structures

Composite materials are very attractive to engineers because of their advantages com-pared with many conventional engineering materials (e.g. steel, aluminum, metallic al-loys, etc.), such as high strengthtoweight ratio, dimensional stability, superior corrosionresistance and relatively high impact resistance [62]. Besides that, the multiphase andmultilayer composites can be tailored through different fiber directions, lamina thick-nesses and stacking sequences for specific design requirements.

In the near future, the aerospace landing gear or undercarriage (Figure 1.1(a)) compo-nents currently made of conventional engineering materials such as steel will be replacedby the composite materials due to aforementioned advantages. Figure 1.1(b) shows atrailing arm from a landing gear structure (at macrolevel) made of textile composite

material (at microlevel). The macroscopic trailing arm structure can be assumed as ho-mogeneous anisotropic composite materials, however, at microlevel, the structure has tobe treated as heterogeneous anisotropic due to the distinguishable constituents of fibersand resin.

(a)

macroscale

microscale

(b)

Figure 1.1: (a) A landing gear (or undercarriage) layout for an aircraft, and (b) a homogeneousanisotropic composite trailing arm structure after assembly of the steel wheel axleand the wheel at the macrolevel [76] with its heterogeneous anisotropic plain

weave pattern composite at the microlevel [4].

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2 Chapter 1

1.2 Textile composites

Textile composites (also know as fabric composites) are a relatively new class of advancedcomposites [89]. They offer advantages and opportunities for designing and tailoringthe textile structures for aerospace structures in order to meet the strict design andfunctional requirements. For instance, they provide an excellent impact resistance dueto the bidirectional fiber configuration compared with the conventional unidirectionalfiber configuration in a lamina [83]. In addition, the production of geometrically complextextile composites becomes relatively efficient and costeffective due to the significantdevelopment of the weaving manufacturing technology based on the loom technique [89].Kamiya et al. [32] reviewed some of the recent advanced techniques in the fabrication andthe design of textile preforms. Some of the applications of the advanced textile compositesin aircraft, marine, automobiles, civil infrastructures as well as medical prosthesis werereviewed by Mouritz et al. [53].

An enormous number of weave topologies for the textile composites. They can be

obtained by changing the interlacing style, repetitive pattern and the proportion of yarns(also know as tows, consisting of fibers bundle aligned in warp and weft (or fill) directions)embedded within matrix. Two of the textile composite types are shown in Figure 1.2.

In order to reduce the relatively expensive experimental time and costs on these newlydeveloped composites, the development of analytical and numerical models for predictingthe mechanical performance of these composites is indispensable.

x (warp)

y

z (weft)

(a)

x (warp)

y

z (weft)

(b)

Figure 1.2: Schematics of the common weaves used in the aerospace landing gear structures.(a) Plain weave and (b) twill weave (The yarns, indicated in green color, are embeddedin the rectangular-cuboid-shape matrix; generated by TexGen [75]).

1.3 Modeling approaches for textile composites

Various analytical and numerical models has been developed for predicting overall prop-erties of the plain weave textile composites in the past two decades. The simplest methodto determine the overall properties for a textile composite is based on rule of mixture andclassical laminate theory (CLT). King et al. [36] modeled the woven fabrics in macroscopiccontinuum constitutive model to determine the macroscopic stresses based on both rela-tively simple modeling approaches. A macroscopic landing gear structure finite element(FE) model, documented in National Aerospace Laboratory (NLR) report written by

Thuis [76], the composite layered shell element type was used in the model for simplifica-tion. The applicability of these conventional analytical methods to a macroscopic textilecomposite structure becomes questionable due to the complexity of yarn geometry in

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Introduction 3

threedimensional (3D) space at microscopic level and unable to describe the mechanicalbehavior of a composite microscopically.

A relatively sophisticated onedimensional (1D) mosaic model developed by Ishikawaand Chou [27] for the plain weave textile composite. The major drawback of this model

is that the yarns continuity is not taken into account. To improve on this, Ishikawaand Chou [29] proposed the crimp model as an extension of their mosaic model. Thismodel has taken into account the continuity and the undulation of yarns, however, onlyalong the loading direction were considered. The transverse yarn undulation and theactual crosssectional geometry should also be taken into account because they are theimportant factors of the homogenized properties of textile composite.

Next, two twodimensional (2D) slice array model (SAM) and element array model(EAM) models proposed by Naik and Ganesh [54], for the plain weave textile composites.In these models, the continuity and the undulation of yarns in warp and weft directionsas well as the presence of the gap between adjacent yarns were considered. The differentmaterials and geometrical properties of warp and weft yarns can be modeled.

In order to extend to the 3D analysis and to include the detailed geometrical descrip-tions of the yarns, the method of cell partition has been developed by many researchers( [31, 70, 71,79]). Through this method, the representative cell is discritized into subcells,where the effective properties of each subcell are obtained.

The limitation of the aforementioned analytical methods were developed specificallyapplied on one type of the textile composite. Thus, they do not offer flexibility to beapplied on other textile composite types, i.e. the analytical methods need to be remodeled for each textile composite. Due to this consequence, a relatively flexible analysisapproach is needed to analyze on any textile composite without the necessity of remodeling on the analysis approach.

In recent years, due to the advancement of structural and material modeling tech-nology, a relatively accurate geometrical textile composites models have been developedthrough computeraided engineering (CAE) and textile geometric modeling software,such as WiseTex [82], TechText CAD [24] and TexGen [63]. As a result, the analyticaltextile model can be circumvented. Enormous number of research papers are availableregarding the geometric modeling using this approach and numerical analysis thereof( [3, 34,40, 59, 64, 65, 84, 87, 88, 93]).

By taking the advantage of FE method and the advancement of CAE and textilegeometric modeling software, an analysis approach for performing the analysis which linkthe macroscopic and microscopic structures can be developed. The FE analysis approach

is known as global/local method, it was developed by Whitcomb [85]. This approach hasbeen further implemented by Srirengan et al. [66] and Xu et al. [90] for the 3D stress anal-ysis of the textile composite structures and further improved by Whitcomb et al. [86], whoused homogenized engineering material properties to accelerate the global stress analysisfor the textile composites. Generally, a global FE analysis was performed on a globalregion with the global coarse finite elements, followed by a detailed local FE analysisperformed in a local region with the independent 3D local fine finite elements. However,the main drawback of this approach is that the local level mechanical mechanisms canonly be studied on the region of interest (ROI) on global level.

To address this problem, multiscale computational homogenization for heterogenous

materials is proposed by Kouznetsova et al. [38,39], Miehe et al. [50], Miehe [48], Moulinecand Suquet [52], Miehe and Koch [49], Michel et al. [46], Feyel and Chaboche [13], Ter-ada et al. [74]. This approach can be used to study the microscopic mechanical phe-

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4 Chapter 1

nomenon within a macroscopic structure collectively and it provides a promising multiscale solution to many engineering materials with heterogeneous microscopic structures,such as metal alloy systems, porous media, polycrystalline materials [16, 37]. Hence, itis reasonable to expect that the multiscale computational homogenization approach hasthe capability to solve the relatively complex textile composites problems by bridging themacroscopic and microscopic scales.

The multiscale computational homogenization method as well as the global/localmethod in the textile composites analysis have their own pros and cons. The selectionbetween both methods should be based on the descriptions of a specific engineering prob-lem. The global/local method is essentially a FE analysis through the mesh refinementtechnique on a region of interest (ROI), e.g. crack tips or specific regions with high stressesor strains. Whereas, the multiscale computational homogenization approach is used op-timally for predicting the macroscopic structure mechanical response with the underlyingmicroscopic structure phenomenon collectively, i.e. the microstructural phenomenon ateach macroscopic region can be studied in detail. The comprehensive literature survey of

various modeling approaches for textile composites can be found in Appendix A.

1.4 Motivation

A numerical analysis framework for predicting the mechanical and physical responses ofcomposite landing gear materials macroscopically and microscopically is indispensablefor improving the structural design and integrity.

Therefore, based on the literature survey on various textile composite analysis ap-proaches as discussed in Section 1.3, the multiscale computational homogenization ap-

proach is utilized in this project, because the microstructural phenomenon in variousmacroscopic regions can be studied simultaneously for predicting the failure modes at themicroscopic level.

Besides that, another advantage of this approach is its flexibility for adaptation. Forinstance, the overall elastic properties for variation of geometrical parameters (volumefraction, shape, orientations, etc.) of the constituents can be easily performed withoutthe necessity of reimplementing the analysis framework.

Furthermore, this framework has an additional advantage that the material propertiesare defined on the constituent scale, thereby the macroscopic constitutive equation isunnecessary. Thus, the framework can be incorporated into the design and analysisprocess at early stage.

1.5 Scope and outline

The aim of this project is to investigate the applicability of the coupled macromicrocomputational homogenization approach to the analysis of textile composite structures.Hence, the approach is implemented within the FE framework and its power, utility andlimitation are demonstrated.

In chapter 2, the multiscale computational homogenization approach is introducedin brief. The concise descriptions of the microscopic boundary value problem and micro

macro coupling framework are given. Then, the implementation of the computationalhomogenization within FE framework is briefly discussed. The quantitative compari-son between numerical results with the analytical solutions and experimental data for

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Introduction 5

unidirectional fiberreinforced composite is performed to verify the correctness of theframework.

Next, in chapter 3, the geometric modeling and material properties for the textile andnoncrimp composite representative volume elements (RVE) are presented. The textileand noncrimp RVEs are modeled for the comparison purposes. Then, the analyses onthe RVEs are performed to exemplify the utility of the computational homogenizationapproach at the microscopic level.

In order to illustrate the power and utility of the framework on textile composite,chapter 4 presents the application of the framework on two types of macroscopic structurescoupled with textile RVEs. Firstly, the multiscale framework is used to analyze themacroscopic laminate [45o/45o/45o/45o]S with two different material and geometricalproperties within RVEs to illustrates the flexibility of the framework. Secondly, theanalysis on notched quasiisotropic laminate [45/0/ 45/90]S to demonstrate thecapabilities of the framework to analyze a relatively complex macroscopic structure.

Finally, in chapter 5, the limitation of the framework is discussed. Thus, several

solutions are recommended in order to reduce the computational time and cost of theframework. Then, the future development of the framework for the benefits of engineerswill be discussed.

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Chapter 2

Multiscale modeling frameworkdescription

In the context of solving macroscopic composite structure problems, computational ho-mogenization is used to obtain the overall macroscopic constitutive response from the un-derlying microscopic heterogenous representative volume element (RVE). The advantageof this approach is that the macroscopic constitutive description can be circumvented,where the constitutive behavior at macroscopic level is determined through the micro-scopic constitutive behavior. In this chapter, the crucial components of the multiscalecomputational homogenization scheme and its FE implementation within Abaqus aretreated in brief.

2.1 Basic hypothesesA composite material is assumed to be macroscopically sufficiently homogeneous butmicroscopically inhomogeneous (consist of matrix and bundles of fibers), as schematicallyshown in Figure 2.1. Based on the principle of scale separation, the characteristic wavelength of RVE, m should be considerably smaller than the wave length of the macroscopicloading or the characteristic size of the macroscopic structure counterpart, M, as follows

m M, (2.1)

where the subscript M and m refer to macroscopic and microscopic quantities, re-

spectively. Due to the morphology of composite materials in a landing gear structure, theglobal periodicity can be assumed, where the fibers bundle (yarns) pattern is repeated inevery macroscopic point, as shown in Figure 2.2 1.

In the computational homogenization scheme [38], the macroscopic deformation gra-dient tensor, FM is calculated at every macroscopic point, i.e. integration points of afinite element (FE) meshed macrostructure. The FM of a particular macroscopic pointis imposed upon RVE (which is assigned to that macroscopic point) to prescribe its totaldeformation. Upon the solution of the RVE boundary value problem, the macroscopicstress tensor (e.g. macroscopic Cauchy stress tensor, M ) is obtained through the micro-scopic stress tensor (e.g. microscopic Cauchy stress tensor, m ) using the RVE volume

1Note that the selected periodic plain weave textile composite as shown in Figure 2.2 is only forillustration and schematic purposes. Several possibilities of the periodic RVE for plain weave textilecomposite exist.

6

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Multiscale modeling framework description 7

M

Figure 2.1: Continuum macrostructure of a drag brace component and heterogeneous RVE ofa composite at a macroscopic point, M.

weft yarn warp yarn matrix (resin)

Figure 2.2: Schematic representative of a global periodic plain weave textile compositemacrostructure in two dimensional view.

averaging technique. Additionally, the local macroscopic consistent tangent (e.g. consis-tent tangent defined from Kirchhoff stress tensor, 4CM) can be derived from the RVEtotal stiffness. The explained computational homogenization framework is schematicallyillustrated in Figure 2.3.

boundary value problem

MICRO

MACRO

FMM

4C

M

Figure 2.3: Computational homogenization scheme.

2.2 Microscopic level boundary value problem

A RVE contains the geometrical and material properties of a composite microstructure.Figure 2.4 shows the corner nodes numbering of a RVE in three dimensional (3D) view

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8 Chapter 2

and the Cartesian coordinate system with the position vector of node 1, x1 from theorigin. The two dimensional (2D) views of a plain weave textile composite RVE fromyz-plane and xz-plane are depicted in Figure 2.5.

12

34

5 6

78

x

yz

x1

Figure 2.4: Schematic picture of the corner nodes numbering for a RVE in 3D view.

+

+

4 1

58 top

bottom

frontback

n n +

(a) yz-plane view

+

+

1 2

5 6top

bottom

left right

n n +

(b) xz-plane view

Figure 2.5: The schematic picture of a plain weave textile composite RVE in (a) yz-plane viewand (b) xz-plane view.

In the RVE deformation field, a point with the initial position vector, x0 in the initialconfiguration, V0 and the current position vector, x in the current configuration, V can

be described by microscopic deformation gradient tensor, Fm =

0mxT

, where the

gradient operator2, 0m is taken with respect to the V0 and the superscript T indicatestransposition.

In the state of equilibrium, the equilibrium equation (in absence of the body forces)

for the RVE in terms of the microscopic Cauchy stress tensor m can be formulated asm m = 0 in V, (2.2)

Within Abaqus FE package, the constitutive law of linear elastic material behaviorfor yarn and resin is defined as

= Del : el, (2.3)

where Del is the fourthorder elasticity tensor, and el is the logarithm of strain tensorin finitestrain problems.

Based on the assumption of global periodicity of a composite structure (Figure 2.2),

the BCs of its RVE should reflect periodicity characteristic. Terada et al. [74] and van

2The gradient operator in Cartesian coordinate system is defined as = e1

x1+ e2

x2

+ e3

x3

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Multiscale modeling framework description 9

der Sluis et al. [77] verified that for a given microstructural cell size, the periodic BCsprovide the better estimation for the overall properties, than the prescribed displacementand prescribed traction BCs. An arbitrary periodic deformation of a textile composite isillustrated in Figure 2.6.

Figure 2.6: Typical periodic deformation of the RVEs for a textile composite.

The general form of the periodicity conditions for a RVE are written as

x + x = FM

x +0 x

0

, (2.4a)

p + = p , (2.4b)

to represent the periodic deformation (2.4a) and antiperiodic tractions (2.4b) on theRVE surfaces. The x + and x are the current position vectors, x +0 and x

0 are theinitial position vectors at the corresponding nodes on the right, right and left, left (top,top and bottom, bottom; front, front and back, back) surfaces, respectively. The unit

normal, n+

and n

(n

= n+

) are defined at the corresponding nodes on the oppositesurfaces, +0 and

0 , respectively (see Figure 2.5). Next, based on explained n+ and n

definition, p + and p are the tractions defined at the corresponding opposite nodes.The periodicity condition (2.4a) is prescribed on an initial periodic RVE surfaces in orderto preserve its periodicity at deformed state.

2.3 Macromicro levels coupling

The macromicro coupling is based on averaging theorems. In this section, the macromicro coupling of the deformation tensor and stress tensor based on volume averagingtheorem are discussed briefly. The importance of the energy averaging theorem, knownas HillMandel condition or macrohomogeneity condition is discussed in brief as well.

2.3.1 Deformation

The volume averaging theorem regarding to the macromicro coupling of the kinematicquantities state that the FM is the volume average of the Fm as follow

FM =1

V0 V0 Fm dV0. (2.5)where V0 is the initial RVE volume.

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10 Chapter 2

According to the divergence (Gauss) theoremV

a(x) dV =

n(x) a(x) d, (2.6)

where a(x) is a vector function on domain V and n(x) is the outward pointing unit normalto the surface of the domain V. The application of the divergence theorem (2.6) and

Fm =

0mxT

lead to the transformation of (2.5) from volume integral into surface

integral

FM =1

V0

V0

Fm dV0 =1

V0

0

xn0 d0. (2.7)

The utilization of the periodic BCs on a RVE leads to the satisfaction of ( 2.7), as shownin Appendix B.

2.3.2 Stress

The averaging theorem for the first PiolaKirchhoff stress tensor, P is given as

PM =1

V0

V0

Pm dV0. (2.8)

Next, through applying the relation (with account for microscopic equilibrium equation)

of, 0m PTmx0 = Pm, the divergence theorem (2.6), and the definition of the micro-scopic first PiolaKirchhoff stress vector, p = n0 P

Tm. The PM can be defined on the

RVE surface as follow

PM =1

V0

0

px0 d0. (2.9)

Similarly, the application of the averaging theorem on the m over current RVE vol-ume, V and the transformation from volume integral to surface integral can be elaboratedas follow

M=

1

V

tx d. (2.10)

where t is the definition of the Cauchy stress vector (t = n m).

In the case for kinematic quantities, the nonlinearity between stress measures shouldbe taken with cautiousness, i.e. not all macroscopic stress quantities obtained from thevolume averaging theorem is valid. Therefore, the M should be defined as

M =1

det(FM)PM F

TM. (2.11)

However, in the case of periodic BCs, it can be shown that

M and M are equivalent, asshown in Appendix D. The complete derivation of (2.9), (2.10) are shown in Appendix B,and the derivation of (2.11) can be found in [45].

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Multiscale modeling framework description 11

2.3.3 Internal work

The averaging theorem for the micro-macro energy transition, known as HillMandel ormacrohomogeneity condition [25,26,68] has to be satisfied for the conservation of specificenergy through the scale transition. This condition states that the microscopic volume

average for the variation of work performed upon RVE is equal to the local variation ofwork at macrolevel.The condition is given as

1

V0

V0

Pm : FTm dV0 = PM : F

TM, x, (2.12)

which is formulated in terms of the work conjugated pair of (P and F). The lefthandside of (2.12) can be defined in terms of RVE quantities as follow

W0M =1

V0

V0

Pm : FTm dV0 =

1

V0

0

p x d0, (2.13)

where the divergence theorem (2.6), microscopic equilibrium (2.2), the definition of themicroscopic first PiolaKirchhoff stress vector, p = n0 P

Tm have been used.

Next, the surface integral of the macroscopic first PiolaKirchhoff stress tensor (2.9),the periodic BCs (2.4a) and antiperiodic tractions (2.4b) have been used to prove thesatisfaction of HillMandel condition (2.12), as shown in Appendix B.

2.4 Numerical implementation

2.4.1 Microstructure boundary value problem

The RVE boundary value problem imposed with the periodic BCs can be solved nu-merically using FE method. It is assumed that the FE discritization is performed uponRVE, such that the corresponding opposite surfaces of the RVE are discretized into anequivalent distribution of nodes due to the initial periodicity characteristic of compos-ite (Figure 2.2). Thus, in order to perserve the periodicity in the deformed state, therespective pair of nodes on the corresponding opposite surfaces can be written as

xtop = xbottom + x5 x1, (2.14a)

xright = xleft + x2 x1, (2.14b)

xfront = xback + x1 x4, (2.14c)

where xp is the current position vector of the four prescribed corner nodes (also knownas control nodes, master nodes [11] or control vertices [78]), p (p = 1, 2, 4, 5), as shown inFigure 2.4. The xright and xleft, xfront and xback, xtop and xbottom are the current positionvectors for the respective pair of nodes on the right and left, front and back, top andbottom surfaces of the RVE, respectively. The applied constraint equations (2.14) impliesthat the total deformation of the RVE can be imposed by the displacement of the fourcorner nodes.

Besides prescribing the position vectors for every respective pair of nodes, they canbe formulated in terms of displacements, u, as

utop = ubottom + u5 u1, (2.15a)

uright = uleft + u2 u1, (2.15b)

ufront = uback + u1 u4. (2.15c)

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12 Chapter 2

The relation ofup with x0p can be expressed as

up = (FM I) x0p , p = 1, 2, 4, 5, (2.16)

where I is the identity tensor. Further details can be found in Appendix C.

2.4.2 Macroscopic stress

Next, The RVE volume averaged stress has to be extracted after analyzing the RVE. Ofcourse, the PM can be computed numerically through the volume integral (2.8). However,it is computationally inefficient because the RVE volume for a textile composite can betypically several orders of magnitude larger than the unidirectional (UD) RVE volume,i.e. the number of finite elements for the textile RVE can be considerably higher thanunidirectional RVE [33]. Therefore, it is computationally more efficient to compute thePM through the surface integral [37], from (2.9), the PM can be further simplified for

the case of the periodic BCs as

PM =1

V0

fe1 x01 +

fe2 x02 +fe4 x04 +

fe5 x05

=

1

V0

p=1,2,4,5

fepx0p , (2.17)

where fep is the reaction (resulting) external forces acted on four corner nodes, p (p =1, 2, 4, 5), the derivation of (2.17) is shown in Appendix D. The (2.17) shows that onlythe external forces at the four corner nodes (p = 1, 2, 4, 5) contribute to PM. As stated insection 2.2, the stress measures for Abaqus is defined in Cauchy stress tensor, therefore,the computed PM (2.17) has to be transformed into M according to (2.11). In addition,

following the derivation steps for PM (as shown in Appendix D), the M can be computedfor the case of periodic BCs as

M =1

V

fe1 x1 +

fe2 x2 +fe4 x4 +

fe5 x5

=

1

V

p=1,2,4,5

fepxp, (2.18)

where V is the current RVE volume.

2.4.3 Macroscopic tangent stiffness

For the nonlinear FE analysis framework, beside macroscopic stress, the stiffness matrixat every macroscopic integration point is required. The stiffness matrix is determinednumerically from the relation between variation of the macroscopic stress and variationof the macroscopic deformation at every macroscopic integration point. The method ofcondense the microscopic stiffness to the local macroscopic stiffness [37,38,49] is dicussedin Appendix E.

Within Abaqus FE scheme, the consistent tangent stiffness is defined as

M =4C

M : DM, (2.19)

which related to the macroscopic Kirchhoff stress tensor, M = det(FM)M and variation

in virtual rate of deformation, DM = 12LM + LTM, where LM is the symmetric rate

of deformation tensor. Then, the departure point of obtaining the 4CM is by varying the

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Multiscale modeling framework description 13

definition of the macroscopic Kirchhoff stress tensor, M = det (FM)M = PM FTM from

(2.11), as shown in Appendix F. Next, the (2.19) is written as

M = 1

V0 i j x(i)K

(ij)M x(j)

LT

: 4IS + 4I M : DM. (2.20)where K

(ij)M is the component of RVE reduced stiffness matrix, KM (Appendix G), with

x(i) and x(j) are the current position vectors for the corner nodes i and j (i, j = 1, 2, 4, 5).The 4IS and 4I are the symmetric fourthorder unit tensor and fourthorder unit tensor,respectively. The superscript LT denotes left transposition3.

After that, by comparing the (2.20) with (2.19), the tangent stiffness tensor, 4CM isidentified as

4C

M =1

V0 i j x(i)K

(ij)M x(j)

LT

: 4IS + 4I M (2.21)

The detailed derivation of 4CM is shown in Appendix F.

2.5 Modeling framework implementation using Abaqus

The numerical framework of the multiscale modeling approach as explained in Section 2.4has been implemented within Abaqus. The analyses of the macroscopic structure andRVEs are executed in the Abaqus simultaneously in the nested manner. At the macro-scopic level, the subroutine UMAT is used to define the material mechanical constitutivebehavior at integration points through coupling to their underlying RVEs.

Firstly, the deformation gradient tensor, FM will be computed by Abaqus at everymacroscopic integration point (IP) at each macroscopic incremental load and at eachmacroscopic iteration. After that, the computed FM at a particular macroscopic IP willbe transferred to the RVE which is assigned to that macroscopic IP for prescribing theRVE total deformation using subroutine DISP. Next, after the completion of the RVEanalysis at microscopic level, the M is computed within subroutine URDFIL and thereduced stiffness matrix, KM is obtained through substructure generation [1]. Finally,the computed M and KM have to be returned to UMAT where the macroscopic tangentstiffness, 4CM will be calculated.

Figure 2.7 and Figure 2.8 show the utilization of subroutines UMAT, DISP and URD-

FIL working with the macroscopic structure and RVE models within Abaqus, as explainedbriefly above. The incrementaliterative approach for the macromicro computational ho-mogenization method is schematically illustrated in Appendix H. The complete codingof the subroutines is documented in the technical report [7].

2.6 Modeling framework verification

In this section, the verification of the correctness of the developed coupled multiscaleframework implementation is performed using unidirectional (UD) fiberreinforced metalmatrix composite. This composite is selected for verification purposes due to the avail-

ability of the experimental data and analytical solutions.

3The left transposition for a fourthorder tensor, 4T is defined as TLTijkl = Tjikl.

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14 Chapter 2

MACROSTEP 1: At the beginning of each increment and each iteration, Abaqus calculates the deforma-

tion gradient tensor FM at every macroscopic IP which is passed into UMAT. Then,UMAT is programmed to write the received FM in FM.txt.

STEP 2: A new Abaqus job is initialized for the RVE analysis job.

MICROSTEP 3: The written FM is read by the DISP for prescribing the RVE total deformation through

the displacements of the four corner nodes with the RVE imposed by the periodic bound-ary conditions.

STEP 4: Solve the RVE boundary value problem.

STEP 5: At the RVE analysis level, the reduced stiffness matrix, KM of the RVE is generated(through substructure generation) and written into KM.mtx.

STEP 6: ThePM is calculated using (2.17), then, the M is calculated through the computed PMusing (2.11) within subroutine URDFIL. The computed M is written into stressM.txt.

STEP 7: Besides KM, additional information needed for the computation of4C

M(2.21) within

UMAT has to be written in various text files, such as RVE initial volume, V0 (written toRVEvolume.txt), current coordinates of the four corner nodes, xp (written to Coords.txt)and the Kirchhoff stress tensor, M (written to KirchhoffstressM.txt).

MACROSTEP 8: The M in stressM.txt and KM in KM.mtx (together with RVEvolume.txt, Coords.txt

and KirchhoffstressM.txt) are read by UMAT for updating the Cauchy stress tensor andcomputing the macroscopic tangent stiffness, respectively.

Figure 2.7: Work flow steps of the multiscale modeling implementation within Abaqus usingsubroutine UMAT, DISP and URDFIL.

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Multiscale modeling framework description 15

Abaqus

Macro-structureAbaqus

Micro-structuremacro_structure.inp

UMAT.for

micro_structure.inp

user_subroutine.for

FM.txt

*Heading

...

*Boundary, user

*Substructure

Generate

*Substructure matrix

output, stiffness=YES

KM.mtx

subroutine DISP

subroutine URDFIL

*Heading

...

*user material

subroutine UMAT

execute Abaqus micro-

structure analysis job

STRESS(NTENS)

DFGRD1(I,J)

DDSDDE(NTENS,

NTENS)

CauchystressM.txt

KirchhoffstressM.txt

RVEvolume.txt

Coords.txt

Figure 2.8: Schematic view of the multiscale numerical framework implemented withinAbaqus.

The UD RVE is shown in Figure 2.9 with the geometrical parameters given in Ta-ble 2.2. The matrix and the fiber are aluminum and boron, respectively, with bothmaterials considered isotropic. The material properties of both constituents are givenin Table 2.3, which is taken from [35]. The fiber material with volume fraction of 0.47is embedded centrally in the matrix material. Next, the matrix and fiber volumes aremeshed with the element type of C3D8 (8node 3D linear brick element) with the equiv-alent distribution of the nodes on the corresponding opposite surfaces for the applicationof the periodic boundary conditions (Figure 2.10). The damage modeling with the co-hesive behavior at the contacting interface is not included in the UD RVE because thedetailed study of this damage mode is out of scope for the framework verification pur-

poses. Therefore, the contacting interface between matrix and fiber is tied together usingmesh tied constraints.

Table 2.2: Geometrical parameters for unidirectional RVE displayed in Figure 2.9 (all param-eter values are shown in unit of millimeter).

Parameter Value Description

wmatrix 0.2 matrix widthtmatrix 0.1 matrix thicknesshmatrix 0.2 matrix height

dfiber 0.1547 fiber diameter

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16 Chapter 2

y

xz

hmatrix

tmatrix

wmatrix

(a) matrix

y

xz

dfiber

(b) fiber

Figure 2.9: The unidirectional RVE composed of (a) matrix and (b) fiber materials modeled

using Abaqus [1], with the geometrical parameters are given in Table 2.2.

Table 2.3: Material properties of constituent materials for the UD RVE model [35](the elasticmodulus, E is shown in unit of GPa).

Property E v

Boron 379.3 0.1Aluminum 68.3 0.3

(a) matrix (b) fiber

Figure 2.10: The unidirectional RVE composed of (a) matrix and (b) fiber materials modelmeshed with element type of C3D8 (8node 3D linear brick element).

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Multiscale modeling framework description 17

The macroscopic structure (meshed with 8 elements) used to couple with the UDRVE is shown in Figure 2.11, with the dimensions shown in Table 2.4. The x-, y- andz-axis on the global coordinate system of the UD RVE and macroscopic structure areoriented in parallel. The macroscopic longitudinal modulus, Ez and Poissons ratio, vxzare obtained through tensile loading in z-direction. The macroscopic transverse modulus,Ex and Poissons ratio, vxy are obtained through tensile loading in x-direction. Thelongitudinal shear modulus, Gxz and the transverse shear modulus, Gxy are obtainedthrough the longitudinal shear loading and transverse shear loading, respectively. Thedetailed description of the boundary conditions to these loading cases are given in [67].

Table 2.4: Geometrical parameters for the macroscopic structure displayed in Figure 2.11 (allparameter values are shown in unit of millimeter).

Parameter Value Description

W 20.0 widthL 40.0 lengthH 0.4 height

X

Y

Z

y

xz

W

L

H

Figure 2.11: Macroscopic structure for the coupled multiscale analyses meshed with 8 ele-ments of type C3D8 with the dimensions shown in Table 2.4.

Sun et al. [67] used the UD RVE to predict its mechanical properties based on principleof strain energy with applying the periodic BCs. Chamis [5] presented solutions basedon unified set of composite micromechanics equations. The analytical solution of Hashin

and Rosen [23] is based on energy variational principles which provides the bounds forthe elastic moduli. The analytical solutions and experimental data shown in Table 2.5are obtained from [67].

The elastic moduli at microlevel and macrolevel shown in Table 2.5, are computedwith the developed framework through prescribing the aforementioned BCs on UD RVEindividually and the macroscopic structure, respectively. Regarding to the analysis resultsunder longitudinal and transverse shear loadings at macrolevel, the Gxz and Gxy arecomputed from the maximum longitudinal and transverse shear stresses, respectively. Ingeneral, good agreement is observed between the micro and macroscale computationalhomogenization solutions, FE prediction [67], analytical solutions [5,23] and experimental

data [35].In the case of elastic moduli solutions at the microscopic and macroscopic levels, a

small deviation is observed. This is due to a small numerical difference of the user-defined

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18 Chapter 2

FM for prescribing the BCs on UD RVE at microlevel and the FM computed by Abaqusinternally to impose its underlying UD RVE BCs in the multiscale analyses. The stresscontour of the analyzed UD RVE and macroscopic structure are shown in Appendix I.

Lastly, through the comparison between analysis results, experimental data and theanalytical solutions, good agreement has been achieved. Thus, it can be concluded thatthe correctness for the implementation of the framework has been verified.

Table 2.5: Elastic constants comparison for unidirectional RVE (the elastic modulus, E andshear modulus, G are shown in unit of GPa)

Elastic UD RVE UD FE model strain energy Energy variation Experimentconstants (micro) (macro) [67] method [5] principle [23] [35]

Ez 214.5 214.9 215.0 214 215 216Ex 142.9 143.4 144.0 156 131.4 - 139.1 140

Gxz 54.2 54.15 57.2 62.6 53.9 52Gxy 45.6 45.8 45.9 43.6 50.0 - 54.6 -vxz 0.19 0.19 0.19 0.20 0.195 0.29vxy 0.26 0.25 0.29 0.31 0.28 - 0.31 -

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Chapter 3

Microstructural modeling

3.1 Textile and noncrimp composite models

The geometrical plain weave textile composite and a 0o/90o noncrimp RVEs are mod-eled using TexGen [75], as shown in Figure 3.1. The textile composite is characterizedby the interlacement of the warp and weft yarns orthogonal to each other, resulting incrimp (undulation or waviness) in both directions. Both models are an openpackingand unbalanced RVEs, i.e. spacings exist between two adjacent yarns in the same di-rection and the geometrical dimensions are different in both warp and weft yarns butthe material properties of the yarns are identical (Figure 3.1(a)). The 0o/90o noncrimpRVE (Figure 3.1(b)) is modeled for the comparison purposes with the textile RVE in themechanical analyses.

x (warp)

y

z (weft)

(a)

x (warp)

y

z (weft)

(b)

Figure 3.1: (a) Plain weave textile composite RVE model and (b) 0o/90o noncrimp compositeRVE created by using TexGen [75].

In practice, the volume of a yarn is not entirely occupied by fibers due to the flowof epoxy through the fibrous preform (porous medium) during fabrication process suchas Resin Transfer Moulding (RTM). However, for the RVEs modeled in this project,the fibers within a yarn are not modeled individually, instead the yarns are representedas a solid volume consisting of fiber bundle and matrix, because this representation isconsiderably more practical for the computational effort (in terms of the CPU processorspeed and memory) [63]. Therefore, the number of fibers is represented by the volumeof a yarn, i.e. a higher yarn volume represents higher number of fibers and vice versa.In general, the cross-sections of yarns are not circular or perfect elliptical due to the

interlacing of the yarns and the processing of the composite, which can lead to yarnflattening [79]. Therefore, a lenticular shape is selected to describe the cross-sectionof the yarns on the basis of microscopic observations by Vandeurzen et al. [79]. An

19

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20 Chapter 3

accurate geometrical modeling of the textile RVE is beyond the scope of this project andis not discussed in detail. The geometrical variables for the textile RVE are depicted inFigure 3.2 and their values are tabulated in Table 3.1. The noncrimp composite RVE hasthe identical geometrical parameters as textile one, except undulation. The constructedRVEs have the yarn volume fraction of 0.2478 and 0.0348 in warp and weft directions,respectively.

Both RVEs consisting of yarns and solid domain (matrix) volumes are imported intoAbaqus/Standard version 6.9 [1]. The matrix pocket is generated through cutting outthe solid matrix volume occupied by the yarns. After that, the individual parts of yarnsand matrix are assembled to produce a RVE model.

WRVE

LRVEHRVE

hgap warpWwarp

Hwarp

Vgap

Wweft

hgap weft

Hweft

Figure 3.2: The textile RVE with the geometrical parameters as given in Table 3.1.

Table 3.1: Geometrical parameters for the plain weave textile composite RVE displayed inFigure 3.2 (all parameter values are shown in unit of millimeter).

Parameter Value Description Parameter Value Description

Wwarp 0.3 warp yarn width Wweft 0.3 weft yarn widthHwarp 0.1514 warp yarn thickness Hweft 0.0757 weft yarn thicknesshgap warp 0.09 warp yarn gap hgap weft 1.2 weft yarn gapLRVE 3.0 RVE length Vgap 0.012 gap between yarnsWRVE 0.8 RVE widthHRVE 0.3 RVE thickness

3.2 Material properties

The matrix is considered as an isotropic elastic material and the yarns are considered astransversely isotropic elastic material. The matrix material is described by two elasticconstants (elastic modulus E and Poissons ratio v). Whereas, the yarn material model

requires six elastic constants, i.e. E1, E3, G12, G13, v12, v13. These six elastic constants aredefined with respect to the assigned material (or local) coordinate system in the yarns,as shown in Figure 3.3. The longitudinal direction is defined by 3-axis, which follows the

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Microstructural modeling 21

yarn path tangentially, the transverse plane is described by the 1-axis and 2-axis, whichare represent by the plane of isotropy (Note that the x-, y- and z-axis are used for theglobal coordinate system, whereas, the local coordinate system is denoted by 1-, 2- and3-axis.). Since the yarn is considered as transversely isotropic, hence,

E1

= E2, G

13= G

23, v

13= v

23.

(a) weft yarns (b) warp yarns

Figure 3.3: The assigned material (or local) coordinate system for (a) weft yarns and (b)warp yarns (12 plane is the plane of isotropy and 3axis follows the yarn pathtangentially).

The material properties of the yarns (combination of carbon fibers and epoxy resin)and the matrix (epoxy resin) are tabulated in Table 3.2. These values are taken from [73],where they are measured experimentally on a unidirectional carbon fiberreinforced poly-

mer (CFRP) and pure epoxy resin, respectively.

Table 3.2: Material properties of constituent materials for the textile and noncrimp RVEmodels with the carbon fiberreinforced plastic (CFRP) on warp and weft yarnsand the epoxy on matrix [73] (the elastic modulus, E and shear modulus, G areshown in unit of GPa).

Property E1 E2 E3 G12 G13 G23 v12 v13 v23Yarn (CFRP) 40.0 40.0 230.0 14.3 24.0 24.0 0.26 0.26 0.26

Property E v

Matrix (epoxy) 3.5 0.35

3.3 Finite element model

The warp and weft yarns are meshed using the swept meshing technique with the elementof type C3D8 (8-node 3D linear brick element). Whereas, the element of type C3D4 (4-node linear tetrahedron element) is used to mesh the matrix volume due to its geometrical

complexity. According to the periodicity characteristic of the composite (Section 2.4.1),the RVEs are meshed such that the corresponding opposite surfaces (left and right; frontand back; top and bottom) are discretized into equivalent distribution of nodes. Due to

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22 Chapter 3

the incompatible element types of both volumes, the contacting interfaces between theyarns and matrix are tied together with the mesh tie constraint.

In practice, the interfacial debonding between the fiber and matrix can occur duringthe fabrication process (due to the mismatch of their thermal expansion coefficients) [81].

However, the interfacial debonding between the fiber and matrix is not considered in thisproject because the microstructure constituted of the fibers and the matrix within theyarn is not modeled, as explained earlier. Therefore, the mesh tie constraint between theyarns and matrix interfaces is used to simulate the perfect bonding between both volumes.Moreover, the perfect bonding is justified to reduce the computational cost and time, sincethe interface formulation can increase the cost and time dramatically [14, 92, 95].

Four levels of meshes are created for the textile RVE, from level 1 (coarse) to level 4(fine), for obtaining the optimum meshed RVE in terms of the accuracy of analysis resultsand CPU time. Figure 3.4 shows the different mesh densities of the matrix volume andyarns volume and Table 3.3 shows the number of nodes, number of elements, total numberof degrees of freedom (DOF), and the relative CPU time for the analysis (tensile in x-axis)in each mesh density level. The deformation gradient tensor, F of

F =

1 + xx xy xzyx 1 + yy yzzx zy 1 + zz

= 1.02 0 00 1.0 0

0 0 1.0

is applied on the textile RVE for prescribing the RVE total deformation through thedisplacement of the four corner nodes and imposed by the periodic boundary conditions(BCs), where and are the normal strain and shear strain, respectively. The subscriptsx, y and z are denoted for the directions of the axes on the global coordinatesystem (see Figure 3.1). The material properties of yarns and matrix are consideredas transversely isotropic elastic (CFRP) defined on the local coordinate systems andisotropic elastic (epoxy), respectively, as described in Section 3.2.

Figure 3.5 shows the analysis results (RVE volume averaged engineering stress in x-direction) for varying mesh density levels and Table 3.3 tabulates the relative CPU timefor each mesh level. There is a clear difference between the coarse mesh (level 1) andother three meshes, however the mesh density level 3 and level 4 (fine) show very similarresults. The mesh density level 3 gives a relatively low relative CPU time comparedwith the fine meshed RVE. Thus, it can be concluded from the Figure 3.4 and Table 3.3that the mesh density level 3 is sufficient to provide the accurate results. Moreover, themaximum mises equivalent stress of each mesh density level is shown in Figure 3.6, itshows that the maximum stress becomes constant through mesh refinement and mesh

density level 3 is sufficient to provide the maximum mises equivalent stress proximity tomesh density level 4. Hence, further simulations are performed using mesh density level3. The noncrimp composite RVE is meshed with the identical mesh density level 3 dueto the same conclusion.

3.4 Microscopic level analyses

The mechanical behavior of the RVEs under uniaxial tensile strain in warp and weft

directions, different prescribed biaxial tensile strains, compression strain and inplaneshear strain are analyzed in this section. The material of yarns and matrix are consideredas CFRP and epoxy, respectively (Section 3.2).

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Microstructural modeling 23

(a) level 1 (coarse mesh) (b) level 2

(c) level 3 (d) level 4 (fine mesh)

Figure 3.4: Textile composite RVE meshes with different mesh density levels (matrix on theleft and yarns on the right).

Table 3.3: Textile RVE mesh statistics

Mesh density level Number of nodes Number of elements DOF Relative CPU time

1 (coarse) 2155 5429 6465 1.002 4662 13009 13986 3.933 18191 32905 54573 13.174 (fine) 28901 61380 86703 41.67

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24 Chapter 3

0 0.5 1 1.5 20

200

400

600

800

1000

1200

x

(%)

PMx

(MPa)

Engineering stress in xdirection averaged over RVE volume for textile RVE

1 (coarse)

23

4 (fine)

Figure 3.5: Engineering stress in x-direction averaged over RVE volume versus strain in thesame direction for different mesh densities of the textile RVE.

0 1 2 3 4 5 6 7 8 9

x 104

0

1000

2000

3000

4000

5000

6000

Degrees of freedom of the microscopic model

Maximumm

isesequivalentstress(MPa)

The maximum mises equivalent stress versus degrees of freedom for the mesh refinement

level 1 (coarse)

level 2level 3

level 4 (fine)

Figure 3.6: The maximum mises equivalent stress for different mesh densities of the textileRVE.

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Microstructural modeling 25

3.4.1 Tension test

The characteristic of crimp of the yarns has a significant influence on the mechanicalbehavior of a composite structure. In the case of uniaxial tensile strain on a textile RVE,yarns under tensile strain tend to straighten, whereas, the crimp of yarns in transverse

direction tend to increase. This phenomenon is called crimp interchange and it occurs inthe case of biaxial tensile strain as well, i.e. the yarns in one direction has an influence onthe behavior of the yarns in transverse direction. The nature of crimp interchange can beinfluenced by the geometrical properties of the yarns and it has a significant influence onthe strength of a composite structure [59]. The mechanical response of the RVEs underuniaxial tension tests is studied first, followed by the biaxial tension tests.

3.4.1.1 Uniaxial tension

The prescribed deformation applied on the RVEs for the uniaxial tension in warp and

weft directions are

Funiaxial warp =

1 + xx xy xzyx 1 + yy yzzx zy 1 + zz

= 1 + xx 0 00 unprescribed 00 0 unprescribed

,and

Funiaxial weft =

1 + xx xy xzyx 1 + yy yzzx zy 1 + zz

= unprescribed 0 00 unprescribed 0

0 0 1 + zz

,respectively, where the unprescribed in the deformation gradient implied that the RVEis free to contract or expand in the corresponding direction. The RVE is imposed by theperiodic boundary conditions.

Through the comparison between Figure 3.7(a) and Figure 3.7(b), the stress value inwarp direction is higher than the weft direction in the textile RVE because of the highervolume fraction in warp yarns compared to weft yarns which characterizes the number offibers. Moreover, the effective longitudinal modulus for warp yarns is higher comparedwith weft yarns due to the lower maximum undulation angle of warp yarns at the initialstate, i.e. the waviness of the weft yarns are higher than warp yarns [54]. Note that thestress value for the noncrimp RVE is slightly higher than the textile RVE under uniaxialtensile strain in warp and weft directions because of the aforementioned explanation.Although the non-crimp RVE enables to sustain slightly higher stress compared withtextile RVE, the development and application of textile composites in aerospace structures

is tremendous successfully due to the cost efficient and ease to produce a low-cost andreliable composite structural components with complex shapes [2, 89].

3.4.1.2 Biaxial tension

The biaxial tensile strain can be achieved by applying the strain in the longitudinaldirection of warp and weft yarns in various biaxial strain ratios, k

k =21

, (3.1)

where 1 is the primary textile strain direction and 2 is the secondary textile straindirection. The primary and secondary textile strain directions correspond either to thewarp or weft yarn strain depending on whether warp or weft strain is plotted. The RVE

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26 Chapter 3

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

1200

x

(%)

xx

(M

Pa)

Cauchy stress of xxcomponent averaged over RVE volume for uniaxial tensile in xdirectio

textile RVE

noncrimp RVE

(a) uniaxial strain: warp

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

1200

z

(%)

zz

(M

Pa)

Cauchy stress of zzcomponent averaged over RVE volume for uniaxial tensile in zdirectio

textile RVE

noncrimp RVE

(b) uniaxial strain: weft

Figure 3.7: The Cauchy stress averaged over RVE volume for uniaxial tensile strain of (a)warp or (b) weft direction.

is free to contract in y-axis (thickness direction). The deformation gradient tensor forthe biaxial tension is

F =

1 + xx xy xzyx 1 + yy yzzx zy 1 + zz

= 1 + xx 0 00 unprescribed 0

0 0 1 + zz

,where either xx or zz is the primary strain for the particular k value. The RVE isimposed by the periodic boundary conditions.

The mechanical behavior under biaxial strain tension with the primary strain direction(1) either warp or weft is studied separately due to the unbalanced yarn nature in thetextile RVE. Figure 3.8 and Figure 3.9 show the computed volume averaged Cauchy stressof xx- and zz-component for different k values of weft and warp as primary direction,respectively. By comparing both figures, the strain ratio has smaller influence on theresponse of the warp yarns because the weft yarns are not able to impede the straiteningof warp yarns due to the higher crimp (i.e. higher waviness) of the weft yarns (Figure 3.8).On the contrary, the behavior in the weft yarns is strongly influenced by the strain ratio,i.e. the mechanical behavior of weft yarns is strongly depended on the mechanical behaviorof the warp yarns (Figure 3.9). Note that the stress value in warp direction (Figure 3.8(a)and Figure 3.9(a)) is higher than the stress value in weft direction (Figure 3.8(b) and

Figure 3.9(b)) due to the aforementioned explanation (yarns waviness and yarn volumefraction in warp and weft directions) in Section 3.4.1.1.

3.4.2 Compression test

The mechanical response of the RVEs under inplane uniaxial compressive strain appliedin warp or weft direction is studied in this section. The applied uniaxial compressiondeformation in warp and weft directions are

Fcompression warp = 1 + xx xy xz

yx 1 + yy yzzx zy 1 + zz =

0.99 0 0

0 unprescribed 00 0 unprescribed,

and

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Microstructural modeling 27

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

1200

x

(%)

xx

(MPa)

Cauchy stress of xxcomponent averaged over RVE volume for k=warp

/weft

k = 0k = 0.5

k = 1k = 2

(a)

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

1200

z

(%)

zz

(MPa)

Cauchy stress of zzcomponent averaged over RVE volume for k=warp

/weft

k = 0k = 0.5

k = 1k = 2

(b)

Figure 3.8: The Cauchy stress of (a) xx- and (b) zz-component averaged over RVE volumefor different biaxial strain ratio, k = warp/weft (primary strain in weft direction).

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

1200

x

(%)

xx

(MPa)

Cauchy stress of xxcomponent averaged over RVE volume for k=weft

/warp

k = 0k = 0.5

k = 1k = 2

(a)

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

1200

z

(%)

zz

(MPa)

Cauchy stress of zzcomponent averaged over RVE volume for k=weft

/warp

k = 0k = 0.5

k = 1k = 2

(b)

Figure 3.9: The Cauchy stress of (a) xx- and (b) zz-component averaged over RVE volumefor different biaxial strain ratio, k = weft/warp (primary strain in warp direction).

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28 Chapter 3

Fcompression weft =

1 + xx xy xzyx 1 + yy yzzx zy 1 + zz

= unprescribed 0 00 unprescribed 0

0 0 0.99

,respectively. The RVE is imposed by the periodic boundary conditions.

Figure 3.10 shows the Cauchy stress-strain response in compression. The compressive

stress value in the warp direction is higher than that in the weft direction. This isattributed by the warp yarns are relatively straighter compared with the weft yarns atthe deformed geometrical state, as shown in Figure 3.11, and the difference of the yarnvolume fraction in the warp and weft directions in the RVE for sustaining the compressionloading in the corresponding direction.

Besides that, the results in Figure 3.10 indicate that the noncrimp RVE has highercompression stress as compared to the textile one, as a result of the yarns waviness in thetextile RVE. This study only shows qualitatively that the compressive stress decreaseswith increasing the yarn waviness. However, the study of the influence of waviness pa-rameters, such as wavelength and amplitude, on the compressive mechanical response is

out of scope of this project.

1 0.8 0.6 0.4 0.2 01200

1000

800

600

400

200

0

x

(%)

xx

(MPa)

Cauchy stress of xxcomponent averaged over RVE volume for uniaxial compression in war

textile RVEnoncrimp RVE

(a)

1 0.8 0.6 0.4 0.2 01200

1000

800

600

400

200

0

z

(%)

zz

(MPa)

Cauchy stress of zzcomponent averaged over RVE volume for uniaxial compression in we

textile RVEnoncrimp RVE

(b)

Figure 3.10: The Cauchy stress averaged over textile and noncrimp RVEs under compressivein (a) warp or (b) weft direction.

3.4.3 Shear testIn this section, the RVEs are analyzed to the inplane shear strain (xz = 0.01 andzx = 0.01) with the following applied deformation,

F =

1 + xx xy xzyx 1 + yy yzzx zy 1 + zz

= 0 0 0.010 unprescribed 0

0.01 0 0

.The RVE is imposed by the periodic boundary conditions.

The mechanical behavior of the RVE under tensile strain is yarn dominated (i.e. fiberdominated). However, the mechanical behavior of the RVE is matrix dominated undershear loading, which has less influenced by the yarns architecture [10]. It can be ob-

served that the stress-strain curves for textile and non-crimp RVEs are nearly coincide(with the difference of 2.56% compared with 7.52% and 7.44% for unixial tension (Fig-ure 3.7(a)) and compression (Figure 3.10(a)) in warp direction, respectively), as shown

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Microstructural modeling 29

(a) (b)

Figure 3.11: The von Mises equivalent stress of the yarns under uniaxial compressive strainin (a) warp and (b) weft direction. (Stress contour plot values from 40 to 2950 MPa,deformation scaled up 15)

in Figure 3.12. Besides that, the stress contour of the textile (Figure 3.13(a)) and noncrimp (Figure 3.13(b)) RVEs are comparable. This may indicate that the crimp effectplays a minor role in the mechanical response of the textile RVE under inplane shearloading. From Figure 3.14, it can be observed that the mechanical response of the RVEis dominated by matrix where the maximum principal logarithmic strain in the matrix ishigher compared with yarn.

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

xz (%)

xz

(MPa)

Cauchy stress of xzcomponent averaged over RVE volume for inplane shear ( xz

& zx

)

textile RVE

noncrimp RVE

Figure 3.12: The Cauchy stress averaged over RVE volume for the inplane shear strain (xzand zx)

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30 Chapter 3

(a) (b)

Figure 3.13: The von Mises equivalent stress of the (a) textile and (b) noncrimp yarns in theRVE under inplane shear strain (xz = 0.01 and zx = 0.01) (Deformation scaledup 15).

(a) (b)

Figure 3.14: The maximum principal logarithmic strain field of the (a) matrix and (b) yarnsin the RVE under inplane shear strain (xz = 0.01 and zx = 0.01) (Deformationscaled up 15).

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Chapter 4

Application of the multiscalemodeling framework

In this chapter, the applicability of the multiscale framework to the analysis on themacroscopic textile structures will be illustrated. It will be shown that this is a valuableand powerful tool to redesign and improve the structural integrity of the textile structureswithout costly prototyping and testing.

4.1 Tensile test on a crossply laminate

The applicability of the multiscale framework to predict the initiation of failure modesin the [45o/45o/45o/45o]s laminated composite under tensile load is evaluated. The45o tensile test is typically performed for the mechanical analysis of textile compositesto shear [60].

Figure 4.1 shows the twill weave laminated composite sample with stacking sequence of[45o/45o/45o/45o]s under tensile test performed at the National Aerospace Laboratory(NLR). It has been noted that the failure of the sample occurs near to the edge of gageregion. For the sake of simplification, only the gage region of the composite sample ismodeled for the multiscale analysis, as shown in Figure 4.2. Due to the symmetricalply layup sequence, only half of the thickness is modeled. The dimensions of the modelare shown in Table 4.1 which correspond to the dimensions of a standard tension testingsample of textile composite [43]. The model is meshed with (Width)3 (Length)5 (Height)4 elements. The model is subjected to longitudinal extension loading (in x-

direction), where the nodes on surface at x=0 and x=L are restricted to the movementin y- and z-directions to represent the clamping effect, i.e. to restrict the compositesample from rotation as in the actual composite testing sample. Symmetrical BCs areapplied at the bottom of xz-plane to represent the symmetrical ply layup sequenceof the macroscopic model. The BCs applied on the macroscopic model are shown inFigure 4.3. Every integration point is assign to the textile representative volume element(RVE) (Figure 3.4(c)). The textile RVEs are rotated to +45o or 45o with respect toyaxis at the global coordinate system, corresponding to the +45o or 45o layer of themacroscopic model, respectively. The material of yarns and matrix are considered ascarbon fiberreinforced plastic (CFRP) and epoxy, respectively (see Table 3.2).

The Figure 4.4 shows the result of the simulation, where the high strain concentrationregions in the macroscopic model. By qualitatively comparing this high strain concen-tration region with the region of failure of the tested composite sample (Figure 4.1), the

31

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32 Chapter 4

tensile loading direction

Figure 4.1: The tested composite tension test sample with shear failure mode. (The markedregion represents shear failure mode dominated by matrix material (insert) of thesample.) (Photo courtesy of Fokker Landing Gear B.V.)

X

Y

Z

y

xL

W

H

z

layer 45o

layer -45o

layer 45o

layer -45o

Figure 4.2: The macroscopic structure model (with the dimensions shown in Table 4.1) forthe tensile test is meshed with (Width)3 (Length)5 (Height)4 elements. Thezoom in view shows the model is meshed with four elements through thicknessand the stacking sequence of the model.

Table 4.1: The dimensions of the tension test on a crossply composite laminate[45o/45o/45o/45o]s (Figure 4.2).

width (W) length (L) height (H)2.5mm 130.0mm 1.2mm

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Application of the multiscale modeling framework 33

y

xz

boundary conditions of the side clampand tension

Dx= 0.13

D

Dy

z

= 0

= 0

boundary conditions of the side c lamp

Dx= 0

D

Dy

z

= 0

= 0

boundary conditions ofthe symmetrical plane

Rx= 0

D

Ry

z

= 0

= 0

Figure 4.3: The applied boundary conditions of the [45o/45o/45o/45o]s composite model.(The D and R indicate the displacement (in millimeter) and rotation (in radian), respec-tively. The subscript x, y and z are the directions in x, y and z-axis at the globalcoordinate system, respectively.)

high strain concentration region in the macroscopic structure correctly indicate the failureinitiation region, this illustrate the multiscale framework in principle enables to predictthe regions of failure quantitatively, provided that the RVEs are adequate to representthe actual textile microscopic structure. From Figure 4.4, the maximum principal loga-rithmic strain of the RVE corresponding to macroscopic high strain region is shown. It isshown that the local strain in the matrix is approximately 6 times higher than the macro-scopic strain. Thus, the prediction of the mechanical response at the microscopic levelwhich may contribute to the macroscopic failure is considerably important and should be

investigated in order to improve the structural integrity.Since the deformation of the sample is shear dominated within all layers for the 45

composite tensile test and mainly dominated by matrix [10,60]. This can also be observedfrom Figure 4.1, where the sample failed mainly due to the shearing of matrix material.Therefore, the initiation of this principal failure mode in the matrix by shearing shouldbe predicted first, followed by the failure modes of yarns.

Figure 4.5(a) and 4.5(b) show the maximum principal logarithmic strain of the matrixand yarns, respectively. It can be observed that the strain in the matrix is higher than inthe yarns, as matrix material has lower stiffness compared with yarns. The high strainstend to concentrate in the matrix between the warp and weft yarns region (interlacing

region), as shown in Figure 4.6, because that region has relatively few matrix material tobear the stress. This finding is consistent with Ref. [33] where it has been concluded thehigh strains tend to occur between the yarns. Therefore, this may indicate the earliestfracture initiation of the matrix in the interlacing region.

Recall that the yarn, in reality, is composed of matrix and fibers, although bothconstituents are not modeled distinguishably in the yarns (see Section 3.1). However,the failure of matrix within yarn should be expected in actual textile composite sample.Figure 4.5(b) shows the local high strain regions in the warp yarns. Note that the strainvalue in the warp yarns is higher compared to the weft yarns, this is due to the differenceof the effective shear stiffness, G of the warp and weft yarns. The effective shear stiffness

is higher for the yarns with lower maximum undulation angle [54]. Hence, the effectiveshear stiffness is higher for the warp yarns due to the lower maximum undulation angle(i.e. lower waviness) of the yarns. This is consistent with the studies of failure behavior

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34 Chapter 4

macroscopic high strain region

macroscopicstructure

textile RVE matrix

textile RVE yarns

Figure 4.4: The maximum principal logarithmic strain contour for the macroscopic compositemodel under tension test with the orientation of [45o/45o/45o/45o]s, coupledwith 45o textile RVEs; inserts: the deformed +45 textile RVE (matrix andyarns), corresponding to the high strain region in the macroscopic structure. Themaximum principal logarithmic strain contour bar for the macroscopic structureand RVE are shown at the position of lower left and upper right, respectively.(The deformation scaled up 400 and 300 for the macroscopic structure and the RVE,respectively.)

of the plain weave textile laminates under inplane shear loading by Ganesh et al. [15].

As a result, in reality, microcracking of matrix within the yarns will exist in a 45ocomposite tensile testing sample, the warp yarn may fail prior to weft yarn due to thehigher strain borne by the warp yarn [57, 58] (also known as intrayarn microcracking).The intrayarn microcracking from the warp yarns may spread within the yarns resultsin interfacial delamination between the matrix and the fibers and spread into the matrixrich part results in fracture of matrix.

Finally, with the prediction of the combination failures of the matrix (in the interlacingregion) and the yarns, the final macroscopic catastrophic shear failure mode of the sample(Figure 4.1) can be predicted through the multiscale analysis.

In order to improve the structural integrity, the matrix with high strain in the in-

terlacing region (Figure 4.6) has to be reduced. According to the architecture of themodeled plain weave textile RVE (Figure 3.1(a)), the warp yarns are considerably moreimportant than the weft yarns, because the warp yarns are able to sustain higher tensileand compression stresses compared to the weft yarns. To achieve this, the weft yarnvolume fraction can be reduced, i.e. increase the matrix volume fraction, particularlyat the interlacing region. Thus, more matrix material in the interlacing region will beavailable to bear the stress. Thus, the modified RVE has warp and weft yarn volumefractions of 0.2478 and 0.0175, respectively (compared with the yarn volume fraction of0.2478 and 0.0348 in warp and weft directions, respectively, before the reduction of theweft yarn volume fraction). The modified RVE is coupled to the macroscopic composite

model (Figure 4.2) and the same boundary conditions (Figure 4.3) are applied on themacroscopic structure model.

Figure 4.7 shows the maximum principal logarithmic strain in the matrix of which

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Application of the multiscale modeling framework 35

(a)

local high strain regionsin the warp yarns

(b)

Figure 4.5: The maximum principal logarithmic strain contour in (a) the matrix and (b) yarnmaterials for the +45 textile RVE in the macroscopic high strain region of the45 composite model. (The region marked in (b) indicates the local high strain regionof warp yarns. The deformation scaled up 300)

local high strainregion in the matrix

cross section planeview side

Figure 4.6: The maximum principal logarithmic strain contour in a cross section view of the+45 textile RVE matrix material in the macroscopic high strain region of the45 composite model. (The region marked in the figure indicate the local high strainregion; the deformation is scaled up 300.)

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36 Chapter 4

the volume fraction has been increased, the strain color bar has been reset equivalentto the color bar in Figure 4.6 for comparison purposes. As can be seen in Figure 4.7, thehigh strain concentration in the interlacing region has been reduced due to the increaseof the matrix material in that region.

Next, as explained previously, since the warp yarns are considerably more importantthan the weft yarns, the high strain concentration region within the yarns has to be shiftedfrom the warp to weft directions, such that the weft yarns preferably fail prior to warpyarns. Therefore, glass fibersreinforced plastic (GFRP) is selected for the weft yarns tosubstitute the carbon fibersreinforced plastic (CFRP) due to the lower shear stiffnessof the GFRP compared with CDFP, i.e. the strain in weft yarns is higher comparedwith warp yarns. This may result the failure initiation in the weft yarns, as shown inFigure 4.8. The material properties of the GFRP, CFRP and the matrix are shown inTable 4.2.

By comparing the Figure 4.9 and Figure 4.4, as a result of improving the microscopicstructural integrity through the modification of the RVEs, the structural integrity of the

macroscopic composite model has been improved. It can be seen that the maximum strainof the composite model has been reduced with approximately 1.5%. This indicates thatthe macroscopic composite model with the microscopic structural integrity improvementwill be able to fail later than the macroscopic composite model without the microscopicstructural integrity improvement.

Table 4.2: Material properties of constituent materials for the textile RVE models of carbonfibersreinforced plastic (CFRP) on warp yarns [73] and glass fiberreinforced plas-tic (GFRP) on weft yarns [20](the elastic modulus, E and shear modulus, G areshown in unit of GPa).

Property E1 E2 E3 G12 G13 G23 v12 v13 v23Warp yarn (CFRP) 40.0 40.0 230.0 14.3 24.0 24.0 0.26 0.26 0.26Weft yarn (GFRP) 9.9 9.9 36.6 3.8 4.1 4.1 0.30 0.27 0.27

Property E vMatrix (epoxy) 3.5 0.35

As a concluding remark to this section, it has been demonstrated that the modificationof the RVE properties, e.g. material and geometrical properties in order to improve the

structural integrity can easily be done in the developed multiscale framework. Thisdemonstrates the power, added value and the flexibility of the framework. Next, themultiscale analysis will be performed on the notched laminate composite, to illustratethe capabilities of the framework on a relatively complex composite structure.

4.2 Compression test on a notched quasiisotropic

laminate

In this section, the multi-scale analysis is performed on a notched textile composite. In

many engineering structures, holes are required to be drilled or mouldedin into laminatesto facilitate bolting or riveting to the main loadbearing structures [18]. As a valuablereference, Figure 4.10 and Figure 4.11 show the failure modes of the fourharness satin

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Application of the multiscale modeling framework 37

cross section planeview side

Figure 4.7: The maximum principal logarithmic strain contour in a cross section view of the+45 textile RVE matrix material (after the increase of the matrix volume frac-tion) in the macroscopic high strain region of the 45 composite model. (Thelogarithmic strain contour bar is reset equivalent to Figure 4.6 for comparison purposes;the deformation is scaled up 300.)

local high strain regionsin the weft yarns

Figure 4.8: The maximum principal logarithmic strain contour of the +45 textile RVE yarnmaterial where warp yarns are the carbon fibers-reinforced and weft yarns arethe glass fibers-reinforced (from the macroscopic high strain region) for the 45

composite. (The region marked in the figure indicats the local high strain region of theweft yarns; the deformation is scaled up 300.)

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38 Chapter 4

macroscopic high strain region

macroscopicstructure

Figure 4.9: The maximum principal logarithmic strain contour for the macroscopic compos-

ite model under tension test with the orientation of [45o

/45o

/45o

/45o

]s, cou-pled with 45o textile RVEs which the matrix volume fraction has been increaseand the weft yarns has been substituted from the carbon fiberreinforced plastic(CFRP) to glass fiberreinforced plastic (GFRP). The properties of these materi-als are shown in Table 4.2.

textile composite [41] and the quasiisotropic eightshaft satin cloth woven composite [18]under compressive load. Notice that the macroscopic failure modes for a notched textilecomposite are relatively complex. Therefore, it is interesting to understand the com-

pressive behavior of the textile composite materials microscopically and investigate theapplicability of the developed multiscale framework on this structure under compressionloading.

Figure 4.12 shows the finite element (FE) model of a notched macroscopic laminatedstructure with the layup sequence of [45o/0o/45o/90o]S. Due to the symmetrical plylayup sequence, only half of the thickness is modeled. The dimensions of the modelare shown in Table 4.3 which correspond to standard dimensions for a notched textilecomposite testing sample [43]. A circular hole of 6.0mm in diameter is centrally positionedin the model. A symmetrical quarter of the model cannot be modeled due to the existenceof 45o layups. The model is meshed with 64 elements of type C3D8 (8-node 3D

linear brick element). The model is subjected to longitudinal compression loading (inx-direction), where the nodes on surface at x=0 and x=L are restricted in the movementto y- and z-direction to represent the clamping effect preventing the rotation of thesample, as is the case in a standard testing of the composite. Due to the symmetricalply layup sequence, the symmetrical BCs are applied on the bottom of xz-plane (seeFigure 4.13). Every integration point of the notched quasiisotropic composite is assignedwith the textile RVE (Figure 3.4(c)). the textile RVE is rotated to +45o, 0, 45o or 90o

with respect to y-axis on the global coordinate system, to represent the +45o, 0, 45o

or 90oply on the macroscopic model, respectively. The material of yarns and matrixare considered as carbon fiberreinforced plastic (CFRP) and epoxy, respectively (see

Table 3.2).Figure 4.14 shows the stress contour of the model at each layer. At each layer, high

stress region near to the hole edge on the model may represent the initiation of the local

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Application of the multiscale modeling framework 39

localizedfailure region

com

pressionloading

direction

(a)

delamination

(b)

compression loadingdirection

(c)

compression loadingdirection

(d)

Figure 4.10: The failure modes of the notched fourharness satin weave textile compositesample under compression load, which are (a) localized damage in the vicinity to

the hole edge, (b) delamination inside the edge of hole, and the final catastrophicfailure of the sample from (c) inplane view and (d) thickness view [41].

compression loadingdirection

(a)

compr

essionloading

direction

(b)

Figure 4.11: The failure modes near to the drilled hole for a quasiisotropic eightshaft satintextile composite under compression loading, (a) thickness view and (b) inplaneview [18].

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40 Chapter 4

y W

L

H

layer 45o

layer -45o

layer 0o

layer 90o

x

z

Figure 4.12: The meshed notched macroscopic structure for compression testing with dimen-sions of (Width)36.0mm (Length)36.0mm (Height)1.2mm and the circularhole of 6.0mm in diameter is located centrally in the model through the thickness.

Table 4.3: The dimensions of the compression test on a notched composite laminate[45o/0o/45o/90o]S (Figure 4.12).

width (W) length (L) height (H)

36.0mm 36.0mm 1.2mm

y

xz

boundary conditions of the side clampand compression

Dx= -0.036

D

Dy

z

= 0

= 0

boundary conditions of the side clamp

Dx= 0

D

Dy

z

= 0

= 0

boundary conditions ofthe symmetrical plane

Rx= 0

D

Ry

z

= 0

= 0

Figure 4.13: The applied boundary conditions of the notched composite model with stackingsequence of [45o/0o/45o/90o]S. (The D and R indicate the displacement (in mil-limeter) and rotation (in radian), respectively. The subscript x, y and z are the

directions in x, y and z-axis at the global coordinate system, respectively.)

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Application of the multiscale modeling framework 41

failure, as shown in Figure 4.10(a). Note, that the high stress regions on layer 45 and45 differ due to different ply direction, i.e. the high stress region contour for layer45o and 45o have opposite directions (see Figure 4.14(a) and 4.14(c)). However, thehigh stress region contour for layer 0 and 90 are comparable, except the stress value ishigher for layer 0 compared to layer 90 because the compressive stress in warp directionis higher than weft direction due to the higher yarn volume fraction of the warp yarn. Ithas been observed experimentally [41] that the localized failure region at the hole edgecontributed by the microscopic level failure and may lead to catastrophic failure, as shownin Figure 4.10(c). Therefore, the analysis and prediction of failure modes through theunderlying textile RVE at the hole edge from each layer should be made. Figure 4.15shows the stress contour for the cross sectional view (half of the model width through yz-plane view) of the notched composite model with the underlying deformed textile RVEsfrom the high stress region in every layer.

x

z

(a)

x

z

(b)

x

z

(c)

x

z

(d)

Figure 4.14: The von Mises stress contours of the n