http://jcm.sagepub.com/ Journal of Composite Materials http://jcm.sagepub.com/content/48/6/735 The online version of this article can be found at: DOI: 10.1177/0021998313477171 2014 48: 735 originally published online 1 March 2013 Journal of Composite Materials Jingjing Li, Meiying Zhao, Xiaosheng Gao, Xiaopeng Wan and Jun Zhou composites Modeling the stiffness, strength, and progressive failure behavior of woven fabric-reinforced Published by: http://www.sagepublications.com On behalf of: American Society for Composites can be found at: Journal of Composite Materials Additional services and information for http://jcm.sagepub.com/cgi/alerts Email Alerts: http://jcm.sagepub.com/subscriptions Subscriptions: http://www.sagepub.com/journalsReprints.nav Reprints: http://www.sagepub.com/journalsPermissions.nav Permissions: http://jcm.sagepub.com/content/48/6/735.refs.html Citations: What is This? - Mar 1, 2013 OnlineFirst Version of Record - Mar 5, 2014 Version of Record >> at OhioLink on August 26, 2014 jcm.sagepub.com Downloaded from at OhioLink on August 26, 2014 jcm.sagepub.com Downloaded from
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http://jcm.sagepub.com/Journal of Composite Materials
http://jcm.sagepub.com/content/48/6/735The online version of this article can be found at:
DOI: 10.1177/0021998313477171
2014 48: 735 originally published online 1 March 2013Journal of Composite MaterialsJingjing Li, Meiying Zhao, Xiaosheng Gao, Xiaopeng Wan and Jun Zhou
compositesModeling the stiffness, strength, and progressive failure behavior of woven fabric-reinforced
Published by:
http://www.sagepublications.com
On behalf of:
American Society for Composites
can be found at:Journal of Composite MaterialsAdditional services and information for
Fiber-reinforced woven composites have been widelyused in several industries, such as aerospace, automo-bile and marine, due to their good performances onstrength, corrosion resistance, thermal expansion andin-plane properties compared with traditional mater-ials. In particular, plain weave composites are oftenused to overcome low damage tolerance, to enhanceimpact resistance and to construct curvature surfaces.They own better balanced ply properties comparedwith unidirectional laminated composites. However,their material nonlinearity and geometry nonlinearitycaused by the complex architecture make them difficultto be simulated and analyzed. There are two mainmethods to analyze woven composite materials. Thefirst method is to perform detailed finite element ana-lysis, where the geometries of matrix, warp and fillyarns are explicitly modeled using three-dimensional(3D) solid elements or shell elements, respectively.It is a directive way to obtain the properties of wovencomposites in the meso-scale level. This method hasbeen employed by many researchers to investigatestress concentration and progressive failure in thematerial. Blackketter et al.1 analyzed the damagebehavior of a plain weave graphite/epoxy fabric-
reinforced composite using a 3D unit cell (UC)model. Potluri and Thammandra2 studied the influenceof fabric tensions applied during processing/consolida-tion on the mechanical properties of a finished compos-ite. Tang and Whitcomb3 investigated the damageinitiation and evolution and studied the progressivefailure behavior of woven composites.
Compared with the method described above, thesecond method does not require to establish thedetailed model in the meso-scale level. The mostfamous analytical model for investigating the stiffnessand strength of woven composites was posted byIshikawa and Chou.4,5 In their work, Ishikawa andChou used three models, the mosaic model, the fiberundulation model and the bridging model, to describethe mechanics of the woven composites. These modelsare called laminate theory models since they are based
1School of Aeronautics, Northwestern Ploytechnical University, Xi’an,
China2Department of Mechanical Engineering, University of Akron, Akron,
OH, USA
Corresponding author:
Xiaosheng Gao, Department of Mechanical Engineering, University of
on the classical lamination theory. However, both themosaic and the fiber undulation models considered a1D strip of a fabric. Accuracy was sacrificed because ofsuch simplification. Naik and Shembekar6–8 extendedthe fiber undulation model to 2D. This model was fur-ther modified by Naik and Ganesh9–11 and applied toanalyze the failure behavior of plain weave composites.These models take into account the actual fabric struc-ture by considering the fiber undulation and continuityalong both the warp and fill directions and were used tostudy the effect of fabric and laminate architecture.Tanov and Tabiei12 introduced two micro-mechanicalmodels, the four-cell model and the single-cell model,utilizing the representative volume cell (RVC) approachto analyze the plain weave composites. Tabiei and Yicompared several methods used to predict the elasticproperties of woven fabric composite materials andfound that the four-cell method and their simplifiedmethod of cells are the most computationally efficientmethod as compared to other methods.13 The four-cellmodel has since been accepted by many researchers.Tabiei et al. used it to predict the failure strength ofwoven composites.14 Wen and Aliabadi15 infused thismethod to their mesh-free model for modeling damagein plain woven fabric composites.
In this study, a progressive damage model (the PDmodel) is developed based on the four-cell model. Twotypes of plain weave composites, the balanced and theunbalanced plain weave composites, are considered.The objective of this study is three-fold. The first is toestablish a homogenized nonlinear woven compositemodel that accounts for the geometric nonlinearity.The second is to include the nonlinear behavior dueto progressive material failure in the model. And thethird is to predict the failure strength of these plainweave fabric composites. This model is not only ableto predict the nonlinear material properties and failurestrength but also trace the PD sequence in the material.It is much more efficient than the full 3D finite elementmethod in the meso-scale and can be used toobtain material properties for large-scale structuralanalysis.
Formulation of the micromechanicalmodels
Micro-mechanical models are very important in obtain-ing the homogenized properties and tracing the pro-gressive failure behavior of woven compositematerials. Using the micromechanical model, strainsand stresses related to each constituent of the fill/warp yarn and matrix can be obtained and monitoredto judge the failure statement of each constituent.
The parallel-series assumption is adopted here inorder to obtain the mechanical properties of the
constituents at each homogenization level. In a parallelmodel, the displacements due to the longitudinal loadare considered to be identical. The load is shared inver-sely proportional to the stiffness of the constituent.In contrast, in a series model the load is identical andthe displacement is inversely proportional to the stiff-ness of the constituent. The interface between each con-stituent is assumed to be perfectly bounded. Thisassumption is validated and used in many micromecha-nical models for anisotropic materials. More detailsand general expressions of these models can be foundin Appendices 1–3.
Homogenization procedures
Figure 1 presents an RVC of a woven composite, whereFigure 1(a) shows a schematic of the geometryand Figure 1(b) indicates the geometrical parameters.This geometry will be used for deriving the microme-chanical relations. Because of the symmetry of thisRVC, a quarter of it is selected as the UC to be ana-lyzed. For example, the UC of the balanced plain weavecomposite, T1, can be further divided into four subcells, Sec-1 to Sec-4, representing the fill-warp crosssection, the undulated fill yarn section, the undulatedwarp yarn section and matrix section, respectively.Figure 2 shows this UC model and the schematic ofeach sub cell.
In order to obtain the final homogenized propertiesof the whole woven composite, two levels of homogen-ization process are performed. The first-level homogen-ization is to obtain the properties of the sub cells,
Figure 1. The schematic of the representative volume cell
(RVC): (a) geometry and (b) geometrical parameters.
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for which the sub cell is further divided into differentconstituents. Sec-1 contains fill yarn, warp yarn, matrixbetween the fill and warp yarn and upper/down matrix,Sec-2 contains fill yarn and matrix, Sec-3 contains warpyarn and matrix and Sec-4 is the pure matrix section.After the properties of each sub cell are obtained, thesecond-level homogenization process will be conductedto get the homogenized properties of the wovencomposite.
There are two parameters to describe the direction ofthe yarn in each sub cell, the braid angle � and theundulation angle �. Under the uniaxial tension condi-tion, the braid angle � will not change. The undulationangles in both fill and warp yarns will be the same in thebalanced plain weave composites. However, �f and �ware different in unbalanced woven composites. Figure 3shows a typical photomicrograph of a plain weavefabric composite cross section. The undulation anglevaries slightly along the fill yarn. For simplicity, thissmall variation is ignored and an average value isused. The undulation angles of the fill and warp yarnscan be calculated by equation (1), where parametersHw, Aw, Gw, Hf, Af and Gf are geometry variables asshown in Figure 1(b). The values of these geometricalparameters will change because of the deformation ofthe sub cells and need to be updated during the loadinghistory.
�f ¼ tan�1Hw
Aw þ Gw
� �
�w ¼ tan�1Hf
Af þ Gf
� � ð1Þ
The first-level homogenization in each section
The first-level homogenization procedure is based onthe Z-model described in Appendix 1. The geometryof Sec-1 shown in Figure 2(d) contains five parts: fillyarn, warp yarn, mid-matrix, up-matrix and down-matrix. The mid-matrix part is introduced to analyzethe interaction behavior between the fill and warp yarnsand the up- and down-matrices are introduced to studythe delamination behavior between the lamina.The thicknesses of these five constituents are denotedby Hf, Hw, Hmu, Hmm and Hmd as indicated inFigure 1(b). In order to get the effective thicknessratios required by the Z-model, the effective thickness ofeach constituent is needed. The effective thickness canbe derived from the shape functions given by Ganeshand Naik.9 By integrating the shape functions, thecross-section area of the fill yarn can be obtained as
Sfill ¼Hf
2�ðAf þ Gf Þ sin
�
2�
Af
Af þ Gf
� �
þAf ðAf þ Gf Þ
�ðAf þ 2Gf Þsin
�
2�
Af þ 2Gf
Af þ Gf
� �ð2Þ
Figure 2. The unit cell (UC) model. (a) Top view of the UC containing four sub cells: Sec-1 (cross section), Sec-4 (matrix section),
Sec-2 and Sec-3 (undulation section); (b) the schematic of Sec-3; (c) the schematic of Sec-2 and (d) the schematic of Sec-1.
Figure 3. The two-dimensional (2D) photomicrograph of the
plain weave fabric16.
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A similar expression can be obtained for the cross-section area of the warp yarn
Swarp ¼Hw
2�ðAw þ GwÞ sin
�
2�
Aw
Aw þ Gw
� �
þAwðAw þ GwÞ
�ðAw þ 2GwÞsin
�
2�
Aw þ 2Gw
Aw þ Gw
� �ð3Þ
Therefore the effective thickness of the fill yarn andthe warp yarn can be expressed as
Hef ¼
Vfill
Af
2 �Aw
2
¼SfillAw
Af
2 �Aw
2
� �cos�f
ð4Þ
Hew ¼
Vwarp
Af
2 �Aw
2
¼SwarpAf
Af
2 �Aw
2
� �cos �w
ð5Þ
The effective thickness of other parts can beexpressed as
Hemm ¼ �1 �H ð6Þ
Hemu ¼ He
md ¼H�He
f �Hew �He
mm
2ð7Þ
Here, the thickness of the mid-matrix is set to be asmall number, which occupies 0.1% of the whole lam-ina’s thickness. With the effective thickness of everypart being determined, the parameters kA and kB inthe Z-model can be figured out accordingly.
There are some assumptions in this model. Thematrix components are assumed to be isotropic andthe properties of the fill and warp yarns areassumed to be similar to those of the unidirectionalcomposites. Stellbrink compared several methods toobtain the basic properties of unidirectional laminatesin Reference [17]. Here, we adopt the methodsuggested by Stellbrink to evaluate elastic propertiesof the yarns.
Ey1 ¼ �
yf � Ef
1 þ 1� �yf
� �� Em ð8Þ
Ey2 ¼
Em
1� 1� Em
Ef2
� ��
ffiffiffiffiffi�yf
q ð9Þ
Gy12 ¼
Gm
1� 1� Gm
Gf12
� ��
ffiffiffiffiffi�yf
q ð10Þ
Gy23 ¼
�yf þ 0:62 1� �yf
� ��yf
Gf23
þ0:62 1��y
f
� �Gm
ð11Þ
In the above equations, the fiber volume fraction �yfin the yarns can be calculated from the overall fibervolume fraction �of as
�yf ¼ �of �
Vo
Vo � Vmð12Þ
where Vo and Vm are the overall and matrix volumes ofthe sub cell, respectively.
The initial stiffness matrix of fill yarn written in thematerial coordinates is
C½ �0fill¼
C11 C12 C13
C12 C11 C13
C13 C13 C33
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
C44 0 0
0 C55 0
0 0 C66
2666666666666664
3777777777777775
ð13Þ
The stiffness matrix of warp yarn and fill yarn in theglobal system can be obtained by transforming this fillstiffness matrix into the global coordinates. The trans-formation matrix in 2D is
R ¼
l21 m21 n21
l22 m22 n22
l23 m23 n23
l1m1 m1n1 n1l1
l2m2 m2n2 n2l2
l3m3 m3n3 n3l3
2l1l2 2m1m2 2n1n2
2l2l3 2m2m3 2n2n3
2l3l1 2m3m1 2n3n1
l1m2 þ l2m1 m1n2 þm2n1 n1l2 þ n2l1
l2m3 þ l3m2 m2n3 þm3n2 n2l3 þ n3l2
l3m1 þ l1m3 m3n1 þm1n3 n3l1 þ n1l3
266666666664
377777777775
ð14Þ
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l1 ¼ cos � cos � l2 ¼ � sin � l3 ¼ � sin � cos �m1 ¼ cos � sin � m2 ¼ cos � m3 ¼ � sin � sin �
n1 ¼ sin � n2 ¼ 0 n3 ¼ cos �
ð15Þ
Hence, the stiffness matrix in the global system canbe written as
C½ �fill =warp¼ R
C½ �0fill R T
ð16Þ
where � and � are the yarn orientation and undulationangles, respectively.
By using the Z-model, the homogenized propertiesof sub cell Sec-1 can be obtained. Homogenized proper-ties of Sec-2 and Sec-3 can be obtained similarly.
The second-level homogenization in UC
With the homogenized properties of the four sub cellsbeen determined, the X-model and the Y-model,described in Appendices 2 and 3, respectively, will beutilized to obtain the properties of the whole UC. Forthe UC shown in Figure 2(a), the Y-model is first uti-lized in part A (Sec-3 and Sec-4) and part B (Sec-2 andSec-1), respectively. Then the X-model is used betweenpart A and part B to obtain the overall properties ofthe RVC.
This homogenization method can be used in othertypes of plain weave composites, e.g., T2. For differenttypes of composite, the sub cells are different. However,the homogenization process remains the same.
A PD model
Damage modes of unidirectional composites have beenextensively investigated and identified using both opti-cal and scanning election microscopes. Generallyspeaking, the failure mechanisms in fiber-reinforcedcomposites include delamination, intralaminar matrixcracking, longitudinal matrix splitting, fiber/matrixdebonding, fiber pull-out and fiber fracture.18
Numerous damage criteria have been developed at themacroscopic level for unidirectional composites.19
However, due to the complicated architecture of thewoven composite, it is impossible to find a singlemacroscopic criterion that works for this materialunder all circumstances. The micro-mechanical modeldeveloped above leads a way to model the PD process.
In order to trace the damage behavior of wovencomposites, the material has been divided into manyconstituents as described above. The strains and stres-ses of each constituent have already been provided
by the X-model, Y-model and Z-model during thehomogenization process. They are monitored to checkif and when damage takes place. Failure status vari-ables are introduced to represent the damage effect onthe material property. Each failure status variable con-tains six components
FD½ � ¼ fd1 fd2 fd3 fd4 fd5 fd6 T
ð17Þ
where fd1, fd2 and fd3 are the failure statement variablesof the longitudinal, transverse and thickness directions,respectively, and fd4, fd5 and fd6 are the failure state-ment variables of the in-plane and out-plane shear dir-ections 12, 13 and 23, respectively.
To avoid the updated stiffness matrix becoming sin-gular, the failure status variable is restricted to a rangeof 1.0 (no damage) to a very small number �2 (the com-ponent is completely damaged). Here �2 is set tobe 10�8. The stiffness will be degraded and the stiffnessmatrix will be updated based on the failure status vari-ables. The element will be deleted when it is completelydamaged.
Rowlands summarized the commonly used compos-ite strength theories in Reference [19]. For simplicity,the maximum strain criteria and maximum stress cri-teria are used here. The Maximum stress failure criter-ion will be used in the situation of iso-strain while themaximum strain failure criterion will be used in thesituation of iso-stress. This arrangement is reasonablewith the parallel-series modeling approach.
Matrix damage is considered as one of the maindamage modes in composite failure. Many researchershave investigated the relationship between the stiffnessdegradation and matrix cracking in unidirectionallamina20–22 and found transverse cracking in matrixaffects the fiber direction stiffness greatly, especiallyfor E-glass/epoxy materials.23 A bilinear relationshipbetween stress and strain is proposed when matrixdamage happens.20 The fibers in the loading directionare assumed to carry a reduced load by a coefficient �.In order to determinate �, the critical stress �cri must bedefined first. The critical stress is defined as the value ofthe longitudinal stress when the failure status variablefd2 in fill yarns is activated. Therefore
� ¼�1 � �cri�0s � �cri
ð18Þ
where �1 is the longitudinal stress and �0s is the initiallongitudinal failure strength of the fill yarn. As given byIannucci in Reference [24], the longitudinal failurestrength of the fill yarn, �s, also decreases with degrad-ation of the Young’s modulus in the fill yarn.
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When the failure status variables are activated, thestiffness matrix will be updated by the followingequation
C½ �updated¼ ½Bij�½C�
½Bij� ¼fdi i ¼ j0 i 6¼ j
�ð19Þ
where subscripts i and j take values from 1 to 6.Here, a user material subroutine of this PD model is
developed for the finite element software ABAQUS.Figure 4 shows the flow chart of this user subroutine.
Results and discussions
The basic mechanical properties of the E-glass/epoxyplain weave composite materials considered in this
study can be found in Reference [10] and are listed inTable 1. The geometry of RVC is shown in Figure 1and the geometrical parameters are given in Table 2.10
Two kinds of woven composites are illustrated:the balanced (sample code ‘T1’) and the unbalanced(sample code ‘T2’) woven composites. The balancedwoven composites (T1-A) have the same architecturein both longitudinal and transverse directions whilethe unbalanced ones (T2-B and T2-C) represent the dif-ferences in these directions.
Elastic property prediction
The PD model is first to be verified by its predictabilityof the initial elastic properties of the samples. By usingthe effective thickness of the fabric and the homogen-ization method described above, Table 3 shows thehomogenized elastic properties at the first time step
Figure 4. Flowchart of the progressive damage model.
Table 1. Properties of fiber and matrix.
Material E (GPa) G (GPa) Poisson ratio, g
Longitudinal tensile
strength, XT (MPa) Shear strength, XS (MPa)
E-glass 72 27.7 0.30 1995.0a —
Epoxy resin 3.5 3.5 0.35 36.6b 43.0c 60.0b 100.0c
aPractical value calculated from UD lamina properties.bEpoxy as a part of e-glass/epoxy UD composite.cPure epoxy resin.
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when damage does not occur. Comparing with theexperimental data presented in Reference [10], theerror of our model prediction is below 10%. For com-parison, the results of Naik and Ganesh’s analyticalmodel (average of three analyses) are also shownin Table 3.
Progressive failure behavior
Figures 5–7 compare the model-predicted stress–straincurves with experimental data. The dashed lines repre-sent the experimental data given in Reference [10] underthe uniaxial tension loading while the solid lines are theresults predicted by the current model. The model pre-dictions match the experimental data very well.
The balanced plain weave composite T1-A1 is con-sidered in Figure 5. The nonlinear behavior is capturedby the PD model. The damage sequence can be tracedby monitoring the failure status variables (FD). Table 4shows the evolution of these failure status variablesduring the loading history. In notation FDN_C(d),‘‘FD’’ refers to failure damage variable, ‘‘N’’ refers tothe section ID (sub cell), ‘‘C’’ refers to the componentin the sub cell and ‘‘d’’ denotes the direction in localcoordinates. For example, FD1_MU(1) represents thefailure damage variable of the upper matrix in Sec-1 inthe longitudinal direction. Table 4 suggests that mater-ial failure occurs first by transverse damage in Sec-1and Sec-3 denoted by FD1_W(2) and FD3_W(2),which corresponds to point A in Figure 5. Materialfailure caused by the matrix damage in Sec-1 and Sec-3 occurs second and this corresponds to point B inFigure 5. Material failure caused by matrix damage inSec-2 and Sec-4 happens third, which corresponds topoint C and point D is the critical point whereFD1_F(2) and FD2_F(2) become 1. It indicates thetransverse matrix damage of the fill yarn, which willactivate the fill yarn’s stiffness degradation. All thesefailure status variables affect the properties of the
plain weave composites during loading history andeach drop in the stress on the stress–strain curve cor-responds to component failure in the composite mater-ial. Finally, the element will be deleted when the fillyarn is completely damaged in the longitudinaldirection.
Figure 6 shows the progressive failure behavior ofthe unbalanced plain weave composite T2-B1. The UCof this material is also assembled by four sub cells.However, because of the differences of strand tex, theproperties in the longitudinal and transverse directionsare different. For this material, the sequence of failuremode is similar to T1-A1. The first damage mode is thewarp transverse damage, followed by the matrix failurein Sec-1 and Sec-3, and then the matrix failure in Sec-2and Sec-4. At point C in Figure 6, the stiffness degrad-ation of the fill yarn is activated.
Figure 7 shows the strain–stress curve of T2-C1. Thiskind of the unbalanced plain weave composite is differ-ent from the one described above. The UC is composedof only two sub cells, Sec-1 and Sec-2, and three kindsof failure modes occur in the following sequence: thewarp transverse damage mode, the matrix damagemode and fill transverse failure mode.
From Figures 5 to 7, it can be found that with theincrease of the overall fiber volume fraction, the effectof matrix failure on the overall stress–strain behavior
Table 3. Experimental and predicted elastic properties.
No.
Young’s modulus (GPa)
Ea1,exp Eb
1,pre Errorb (%) Ec1,pre Errorc (%)
A1 17.6 15.9 9.6 13.9 21.0
B1 23.2 21.2 8.6 18.3 21.1
C4 23.2 23.8 –2.5 20.2 12.9
aExperimental data presented in Reference [10].bPredictions of the current model.cPredicted results by Naik and Ganesh.10
reduces. This is reflected by the relatively smootherstress–strain curve shown in Figure 7 comparing tothat shown in Figure 5. Furthermore, the relativelystrong nonlinear behavior predicted by the model athigh strain levels (beyond 2–2.5%) due to the activationof the fill yarn longitudinal stiffness degradation sup-ports the previous claims that transverse crackingin matrix greatly affects the fiber direction stiffness ofE-glass/epoxy materials.
From the above examples, it can be seen the PDmodel developed in this study is applicable to differenttypes of plain weave composites. The model predictionsof the stress–strain behavior match the experiment
results very well. Furthermore, the failure status vari-able provides a means to track the damage sequence inthe composite material.
Failure strength prediction
Naik and Ganesh provided experimental data of thefailure strength of the above three plain weave compos-ites under uniaxial tension.10 Using the developed PDmodel, the failure strength of these composites can bepredicted and compared with experimental data. Asindicated in Table 5, the PD model-predicted failurestrength agrees with experimental data very well. The
Figure 5. Comparison of the predicted stress–strain curve of the T1-A1 composite with experimental results.
Figure 6. Comparison of the predicted stress–strain curve of the T2-B1 composite with experimental results.
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errors in all but one case are less than 10%. For com-parison, the results of Naik and Ganesh’s analyticalmodel are also shown in Table 5. It is clear that the PDmodel shows a better prediction of the failure strength.
Conclusions
A PD model for analyzing plain weave composites isdeveloped, in which the RVC is divided into sub cells,the overall mechanical properties of the composite areobtained by two levels of homogenization process, thecriteria for failure of the sub cell constituents areincluded and the resulting material property degrad-ation is considered. Model predictions show good com-parisons with experimental data. The main conclusionsof this study can be summarized as follows
1. The PD model provides a general approach toobtain homogenized properties of woven compos-ites. It can be easily implemented into finite elementcodes for structural analysis. As an example, themodel is implemented into ABAQUS via a user-defined subroutine, UMAT.
2. The progressive failure behavior can be traced by thefailure status variable of each constituent of theRVC. The results suggest that the warp longitudinalfailure occurs first, followed by the fill transversefailure. All these failure modes are accompanied bythe failure of matrix and contribute to the nonlinear-ity of the overall stress–strain curve. The longitu-dinal failure of the fill is the critical failure mode toindicate the final failure of the RVC.
3. The predicted overall elastic properties and ultimatefailure strength agree well with experimental data.The error is within 10%, which is very good, con-sidering the variability of the geometrical parametersused in the analysis and the assumptions made inmodel development.
Funding
This research received no specific grant from any funding
agency in the public, commercial, or not-for-profit sectors.
Conflict of interest
None declared.
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Appendix 1
The Z-model
The Z-model is used to obtain the homogenized proper-ties of two different materials stacked in Z direction,Figure 8. The strain and stress components based onthe parallel-series assumption are expressed as
"A11 ¼ "B11 ¼ "11 "A22 ¼ "
B22 ¼ "22 "A12 ¼ "
B12 ¼ "12
�A33 ¼ �B33 ¼ �33 �A13 ¼ �
B13 ¼ �13 �A23 ¼ �
B23 ¼ �23
ð20Þ
The rest of stresses and strains can be obtained by
kA�A11 þ kB�
B11 ¼ �11
kA�A22 þ kB�
B22 ¼ �22
kA�A12 þ kB�
B12 ¼ �12
kA"A33 þ kB"
B33 ¼ "33
kA"A13 þ kB"
B13 ¼ "13
kA"A23 þ kB"
B23 ¼ "23
ð21Þ
where kA and kB be the effective thickness ratios of partA and B, respectively.
Let CijA and CijB be the components of the stiffnessmatrices of A and B, respectively. By combining equa-tions (20) and (21), we can get the general expressionsof the homogenized stiffness matrix for the Z-model,Cij, as given in equation (22). The stiffness matrix will
be symmetric if the two materials are transversely iso-tropic, i.e. Cij¼Cij.
Figure 9. The strain and stress components based onthe parallel-series assumption are expressed as
"A22 ¼ "B22 ¼ "22 "A33 ¼ "
B33 ¼ "33 "A23 ¼ "
B23 ¼ "23
�A11 ¼ �B11 ¼ �11 �A12 ¼ �
B12 ¼ �12 �A13 ¼ �
B13 ¼ �13
ð25Þ
The rest of stresses and strains can be written as
kA�A22 þ kB�
B22 ¼ �22
kA�A33 þ kB�
B33 ¼ �33
kA�A23 þ kB�
B23 ¼ �23
kA"A11 þ kB"
B11 ¼ "11
kA"A12 þ kB"
B12 ¼ "12
kA"A13 þ kB"
B13 ¼ "13
ð26Þ
By combining equations (25) and (26), we can getgeneral expressions of homogenized stiffness matrixfor this X-model, as given in equation (27).
C11 ¼ p1C11AC11B
C12 ¼ p1ðkBC11AC12B þ kAC11BC12AÞ
C13 ¼ p1ðkBC11AC13B þ kAC11BC13AÞ
C21 ¼ p1ðkAC21AC11B þ kBC21BC11AÞ
C22 ¼ kAC22A þ kBC22B
þ kAkBp1ðC21A � C21BÞðC12B � C12AÞ
C23 ¼ kAC23A þ kBC23B
þ kAkBp1ðC21A � C21BÞðC13B � C13AÞ
C31 ¼ p1ðkAC31AC11B þ kBC31BC11AÞ
C32 ¼ kAC32A þ kBC32B
þ kAkBp1ðC31A � C31BÞðC12B � C12AÞ
C33 ¼ kAC33A þ kBC33B
þ kAkBp1ðC31A � C31BÞðC13B � C13AÞ
C44 ¼ p2C44AC44B
C55 ¼ p3C55AC55B
C66 ¼ kAC66A þ kBC66B ð27Þ
where
p1 ¼1
kAC11B þ kBC11Ap2 ¼
1
kAC44B þ kBC44A
p3 ¼1
kAC55B þ kBC55A
ð28Þ
The strain components in each constituent can beexpressed as
"11A ¼ p1C11B"11 þ kBp1ðC12B � C12AÞ"22
þ kBp1ðC13B � C13AÞ"33
"11B ¼ p1C11A"11 � kAp1ðC12B � C12AÞ"22
� kAp1ðC13B � C13AÞ"33
"22A ¼ "22B ¼ "22
"33A ¼ "33B ¼ "33
"12A ¼ p2C44B"12
"12B ¼ p2C44A"12
"13A ¼ p3C55B"13
"13B ¼ p3C55A"13
"23A ¼ "23B ¼ "23 ð29Þ
Appendix 3
The Y-model
The Y-model is used to obtain the homogenized prop-erties of two different materials stacked in Y direction,Figure 10. The strain and stress components based onthe parallel-series assumption are expressed as
"A11 ¼ "B11 ¼ "11 "A33 ¼ "
B33 ¼ "33 "A13 ¼ "
B13 ¼ "13
�A22 ¼ �B22 ¼ �22 �A12 ¼ �
B12 ¼ �12 �A23 ¼ �
B23 ¼ �23
ð30Þ
Figure 10. Y-model.Figure 9. X-model.
746 Journal of Composite Materials 48(6)
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The rest of stresses and strains can be obtained by
kA�A11 þ kB�
B11 ¼ �11
kA�A33 þ kB�
B33 ¼ �33
kA�A13 þ kB�
B13 ¼ �13
kA"A22 þ kB"
B22 ¼ "22
kA"A12 þ kB"
B12 ¼ "12
kA"A23 þ kB"
B23 ¼ "23
ð31Þ
Combining equations (30) and (31), we can get gen-eral expressions of homogenized stiffness matrix for theY-model. Based on the parallel-series assumption, thecomponents of the homogenized stiffness matrix aregiven in equation (32).
C11 ¼ kAC11A þ kBC11B
þ kAkBp1ðC12A � C12BÞðC21B � C21AÞ
C12 ¼ p1ðkAC12AC22B þ kBC12BC22AÞ
C13 ¼ kAC13A þ kBC13B
þ kAkBp1ðC12A � C12BÞðC23B � C23AÞ
C21 ¼ p1ðkBC22AC21B þ kAC22BC21AÞ
C22 ¼ p1C22AC22B
C23 ¼ p1ðkBC22AC23B þ kAC22BC23AÞ
C31 ¼ kAC31A þ kBC31B
þ kAkBp1ðC32A � C32BÞðC21B � C21AÞ
C32 ¼ p1ðkAC32AC22B þ kBC32BC22AÞ
C33 ¼ kAC33A þ kBC33B
þ kAkBp1ðC32A � C32BÞðC23B � C23AÞ
C44 ¼ p2C44AC44B
C55 ¼ kAC55A þ kBC55B
C66 ¼ p3C66AC66B ð32Þ
where
p1 ¼1
kAC22B þ kBC22Ap2 ¼
1
kAC44B þ kBC44A
p3 ¼1
kAC66B þ kBC66A
ð33Þ
The strains in each constituent can be expressed as
"11A ¼ "11B ¼ "11
"22A ¼ p1C22B"22 þ kBp1ðC21B � C21AÞ"11
þ kBp1ðC23B � C23AÞ"33
"22B ¼ p1C22A"22 � kAp1ðC21B � C21AÞ"11
� kAp1ðC23B � C23AÞ"33
"33A ¼ "33B ¼ "33
"12A ¼ p2C44B"12
"12B ¼ p2C44A"12
"23A ¼ p3C66B"23
"23B ¼ p3C66a"23
"13A ¼ "13B ¼ "13 ð34Þ
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