《纺织复合材料》课程参考文献（3-D textile reinforcements in composite materials）03 Mechanical modelling of solid woven fabric composites

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3.1 Introduction Solid woven fabric composites represent a class of advanced composites which are reinforced by 2-D or 3-D woven preforms [1]. These materials offer new and exciting opportunities for tailoring the microstructure to specific thermomechanical applications in the fields of aerospace, marine, medicine and sports technology. The variables under control include fibre and matrix materials, yarn placement, yarn size and type. Together with this emerging ability to engineer composite materials comes the need to develop computationally efficient micromechanics models that can predict, with sufficient accuracy, the effect of the microstructural details on the internal and macroscopic behaviour of these new materials. Computational effi- ciency is indispensable because there are many parameters that must be varied in the course of engineering a composite material. This chapter addresses the issue of developing micromechanical models for solid woven fabric composites. In the future, it is probably inevitable that the optimization of the microstructure of a woven fabric composite will require the marriage of such micromechanical models and optimization algorithms. 3.2 Review on solid woven fabric composites 3.2.1 Introduction This section provides a survey of the literature. First, an overview of woven fabric composites is presented. Solid woven preforms vary considerably in terms of fibre orientation, entanglement and geometry. Second, in order to exploit the advantages of these composites fully, it is important to create a link between the microstructural geometry and the thermomechanical performance [2]. In the past decade, a variety of micromechanical models have been employed to study the overall thermo-elastic behaviour of orthogonal 2-D woven fabric composites based on the properties of the constituents 3 Mechanical modelling of solid woven fabric composites PHILIPPE VANDEURZEN, JAN IVENS AND IGNAAS VERPOEST 67 RIC3 7/10/99 7:37 PM Page 67 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:11 AM IP Address: 158.132.122.9

68 3-D textile reinforcements in composite materials 2-D Orthogonal:Plain,Twill,Satin Weaves Triaxial Layer-to-Layer Angle Interlock 3-D Through-Thickness Warp knit Orthogonal Interlock Sandwich Weave Weft knit 2-D Bias Triaxial Braids Tubular 3-D 2-Step Cartesian 4-Step Multi-Step 3.1 Classification of textile preforms for composite structures [31. and the fabric architecture.Some of these models also provide the oppor- WV:OS tunity to address strength properties.A review will assist in defining possi- ble modelling strategies for 3-D woven fabric composites. 2102 3.2.2 Classification Fibre reinforcement constitutes the structural backbone of a composite.The classification by Cox and Flanagan 3 of various textile preforms is repro- duced in Fig.3.1.The left column classifies textile preforms according to the machines and processes used to produce them.The major textile-forming techniques for composite reinforcements are weaving,knitting and braid- ing.Further,it is possible to make a distinction between the dimensional- ity of the textile preform.Following the definition of Cox [3],the division into 2-D and 3-D textile structures is determined by whether the fibre preform can transport an important load (higher than the load carried by the matrix alone)in two or three linearly independent directions. In general,an orthogonal 2-D woven fabric is made by weaving yarns together.A yarn is a continuous strand of textile fibres.The fabric is pro- duced on a loom that interlaces yarns at right angles to one another [2-8]. The lengthwise yarns are called warps,while the yarns that are shuttled across the loom are called fillings or wefts.The individual yarns in the warp and filling directions are also called an end and a pick,respectively.The interlacing of the yarns causes yarn undulation or yarn crimp.The weave type is determined by the method of interlacing both sets of yarns.Figure 3.2 shows three basic constructions:plain,twill and satin weave.Even in rather simple woven fabrics,there are important geometric differences between the warp and the weft direction.Those differences are the result

and the fabric architecture. Some of these models also provide the opportunity to address strength properties. A review will assist in defining possible modelling strategies for 3-D woven fabric composites. 3.2.2 Classification Fibre reinforcement constitutes the structural backbone of a composite.The classification by Cox and Flanagan [3] of various textile preforms is reproduced in Fig. 3.1.The left column classifies textile preforms according to the machines and processes used to produce them. The major textile-forming techniques for composite reinforcements are weaving, knitting and braiding. Further, it is possible to make a distinction between the dimensionality of the textile preform. Following the definition of Cox [3], the division into 2-D and 3-D textile structures is determined by whether the fibre preform can transport an important load (higher than the load carried by the matrix alone) in two or three linearly independent directions. In general, an orthogonal 2-D woven fabric is made by weaving yarns together. A yarn is a continuous strand of textile fibres. The fabric is produced on a loom that interlaces yarns at right angles to one another [2–8]. The lengthwise yarns are called warps, while the yarns that are shuttled across the loom are called fillings or wefts. The individual yarns in the warp and filling directions are also called an end and a pick, respectively. The interlacing of the yarns causes yarn undulation or yarn crimp. The weave type is determined by the method of interlacing both sets of yarns. Figure 3.2 shows three basic constructions: plain, twill and satin weave. Even in rather simple woven fabrics, there are important geometric differences between the warp and the weft direction. Those differences are the result 68 3-D textile reinforcements in composite materials 3.1 Classification of textile preforms for composite structures [3]. RIC3 7/10/99 7:37 PM Page 68 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:11 AM IP Address: 158.132.122.9

Mechanical modelling of solid woven fabric composites 69 (a) (b) (c) 3.2 Basic weave constructions:(a)plain,(b)twill and (c)5HS satin weave.The black box represents the fabric unit cell. of numerous constructional and process parameters such as weaving density,warp tension,weft tension and beating motion. The term 'hybrid'is used to describe fabrics containing more than one type of fibre material.Hybrid fabrics are attractive preforms for structural materials for two major reasons.First,these fabrics supply an even wider variety of material selection for designers.They offer the potential of improved composites'mechanical properties,weight saving or excellent impact resistance.Second,a more cost-effective use of expensive fibres can be obtained by replacing them partially with less expensive fibres.Hybrid fabrics are woven from fibrous materials such as glass,aramid,carbon, boron,ceramics and natural fibres. Advances in textile manufacturing technology are rapidly expanding the number and complexity of 3-D woven preforms.By changing the traditional weaving technique to produce 2-D fabrics,it is now possible to achieve a s much higher degree of integration in the thickness direction of the textile. The two major classes of solid 3-D weaving are through-thickness angle interlock weaving [10]and orthogonal interlock weaving [1-3].Angle inter- lock 3-D woven fabrics can be produced on a dobby loom or a jacquard loom.The warp yarns can now enter more than one layer of weft yarns. Other textile structures with laid-in straight yarns are also possible.By changing the number of layers,the pattern of repeat and the position of the laid-in yarns,an almost infinite number of geometric variations becomes possible.In an orthogonal interlock 3-D weave,the yarns are placed in three mutually orthogonal directions.These fabrics are produced principally by the multiple warp weaving method.Matrix-rich regions are created in com- posites reinforced with a 3-D woven orthogonal preform. In general,solid woven fabrics offer the advantages of handleability, dimensional stability,improved impact and damage resistance.However, these advantages are obtained at the cost of reduced stiffness and strength properties owing to the undulation of the yarns.There is thus a significant need to model the mechanical behaviour of these composites

of numerous constructional and process parameters such as weaving density, warp tension, weft tension and beating motion. The term ‘hybrid’ is used to describe fabrics containing more than one type of fibre material. Hybrid fabrics are attractive preforms for structural materials for two major reasons. First, these fabrics supply an even wider variety of material selection for designers. They offer the potential of improved composites’ mechanical properties, weight saving or excellent impact resistance. Second, a more cost-effective use of expensive fibres can be obtained by replacing them partially with less expensive fibres. Hybrid fabrics are woven from fibrous materials such as glass, aramid, carbon, boron, ceramics and natural fibres. Advances in textile manufacturing technology are rapidly expanding the number and complexity of 3-D woven preforms. By changing the traditional weaving technique to produce 2-D fabrics, it is now possible to achieve a much higher degree of integration in the thickness direction of the textile. The two major classes of solid 3-D weaving are through-thickness angle interlock weaving [10] and orthogonal interlock weaving [1–3].Angle interlock 3-D woven fabrics can be produced on a dobby loom or a jacquard loom. The warp yarns can now enter more than one layer of weft yarns. Other textile structures with laid-in straight yarns are also possible. By changing the number of layers, the pattern of repeat and the position of the laid-in yarns, an almost infinite number of geometric variations becomes possible. In an orthogonal interlock 3-D weave, the yarns are placed in three mutually orthogonal directions. These fabrics are produced principally by the multiple warp weaving method. Matrix-rich regions are created in composites reinforced with a 3-D woven orthogonal preform. In general, solid woven fabrics offer the advantages of handleability, dimensional stability, improved impact and damage resistance. However, these advantages are obtained at the cost of reduced stiffness and strength properties owing to the undulation of the yarns. There is thus a significant need to model the mechanical behaviour of these composites. Mechanical modelling of solid woven fabric composites 69 3.2 Basic weave constructions: (a) plain, (b) twill and (c) 5HS satin weave. The black box represents the fabric unit cell. RIC3 7/10/99 7:37 PM Page 69 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:11 AM IP Address: 158.132.122.9

70 3-D textile reinforcements in composite materials 3.2.3 Micromechanical models Considering the actual importance of 2-D woven fabric composites in the family of structural composites,the mechanical analyses of these compo- sites are now extensively reviewed and presented.Most of the published data are related to stiffness properties of plain weave laminae.There are few publications on the internal stress distribution and on the damage and strength analysis problem of general woven fabric composites.The possible extension of the different micromechanical models to analyse 3-D woven fabric composites will be discussed.It should also be stressed here that in this rapidly evolving field of study any review will soon be incomplete.New results are always being presented or printed. Models of Ishikawa and Chou In the 1980s,an extensive amount of work on the thermo-mechanical mod- elling of 2-D woven fabric composites was done by Ishikawa and Chou. They developed and presented three analytical 1-D elastic models [11-13]. These models are known as the mosaic model,the fibre crimp model and the bridging model.The classical lamination theory forms the basic analyt- ical tool for these developments [14. The models of Ishikawa and Chou are labelled 1-D models because they only consider the undulation of the yarns in the loading direction.Notice the total absence of any geometric analysis.That is,the actual yarn cross- sectional shape or the presence of a gap between adjacent yarns is not con- sidered.Therefore,no predictions are made for the out-of-plane yarn orientation and the fibre volume fraction.Moreover,these models consider balanced closed weaves only,whereas in practice the fabric can be unbal- anced and open.Since the classical laminated plate theory is the basis of each model only the in-plane elastic properties are predicted.The elastic models were extended to analyse the thermal properties,hybrid fabrics and the knee behaviour under uniaxial tensile loading along the filling direction only.However,an extension to treat 3-D woven preforms is not useful because of the geometric simplifications and the limitation to predicting only in-plane properties. Models of N.Naik,Shembekar and Ganesh N.Naik and Shembekar have developed 2-D elastic models for a 2-D non- hybrid plain weave fabric composite [15].These models are essentially an extension of the 1-D models of Ishikawa and Chou.However,these 2-D models take into account the undulation of both warp and weft yarns,the presence of a possible gap between adjacent yarns,the real cross-section of

3.2.3 Micromechanical models Considering the actual importance of 2-D woven fabric composites in the family of structural composites, the mechanical analyses of these composites are now extensively reviewed and presented. Most of the published data are related to stiffness properties of plain weave laminae. There are few publications on the internal stress distribution and on the damage and strength analysis problem of general woven fabric composites. The possible extension of the different micromechanical models to analyse 3-D woven fabric composites will be discussed. It should also be stressed here that in this rapidly evolving field of study any review will soon be incomplete. New results are always being presented or printed. Models of Ishikawa and Chou In the 1980s, an extensive amount of work on the thermo-mechanical modelling of 2-D woven fabric composites was done by Ishikawa and Chou. They developed and presented three analytical 1-D elastic models [11–13]. These models are known as the mosaic model, the fibre crimp model and the bridging model. The classical lamination theory forms the basic analytical tool for these developments [14]. The models of Ishikawa and Chou are labelled 1-D models because they only consider the undulation of the yarns in the loading direction. Notice the total absence of any geometric analysis. That is, the actual yarn crosssectional shape or the presence of a gap between adjacent yarns is not considered. Therefore, no predictions are made for the out-of-plane yarn orientation and the fibre volume fraction. Moreover, these models consider balanced closed weaves only, whereas in practice the fabric can be unbalanced and open. Since the classical laminated plate theory is the basis of each model only the in-plane elastic properties are predicted. The elastic models were extended to analyse the thermal properties, hybrid fabrics and the knee behaviour under uniaxial tensile loading along the filling direction only. However, an extension to treat 3-D woven preforms is not useful because of the geometric simplifications and the limitation to predicting only in-plane properties. Models of N. Naik, Shembekar and Ganesh N. Naik and Shembekar have developed 2-D elastic models for a 2-D nonhybrid plain weave fabric composite [15]. These models are essentially an extension of the 1-D models of Ishikawa and Chou. However, these 2-D models take into account the undulation of both warp and weft yarns, the presence of a possible gap between adjacent yarns, the real cross-section of 70 3-D textile reinforcements in composite materials RIC3 7/10/99 7:37 PM Page 70 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:11 AM IP Address: 158.132.122.9

Mechanical modelling of solid woven fabric composites 71 the yarn and the possible unbalanced nature of the plain fabric lamina.The representative unit cell is discretized into slices along or across the loading direction.These slices are further divided into different elements such as straight cross-ply or unidirectional regions,undulated cross-ply or uni- directional regions and pure matrix elements.In the analysis of Naik and Shembekar,two schemes for combining the in-plane stiffness matrices of the different elements are used:parallel-series and series-parallel.In the parallel-series (PS)model,the elements are first assembled in parallel across the loading direction with the isostrain assumption(adding the stiff- ness matrices,weighted by their volume fractions).Then,those multi- elements are assembled in series along the loading direction with the isostress assumption.In the second scheme,all the infinitesimal elements of a section along the loading direction are assembled with an iso- stress assumption (adding the compliance matrices,weighted by their volume fractions).Then,all the sections along the loading direction are assembled with an isostrain condition.Such a scheme is called a series- parallel(SP)model.Both schemes yield a full 2-D stiffness matrix for the plain woven fabric composite.A full mathematical treatment of the problem has been presented in reference [16].Based on experimental work, the PS model is recommended for the prediction of all in-plane elastic con- stants.Out-of-plane properties cannot be predicted.Hence,the extension of the model Recently,Naik and Ganesh have presented an extension of their thermo- elastic models to include the prediction of failure in plain weave compos- ites under on-axis static tensile loading [17,18.The load is assumed along the filling direction.Different stages of failure such as warp yarn transverse failure,filling yarn shear failure,filling yarn transverse failure,pure matrix 8 element failure and filling yarn longitudinal failure are considered.The newness of the model lies in the calculation procedure for the stresses in the matrix and yarn elements.However,this is exactly where the model is most confusing.A lot of effort has been spent on describing material non- linearities,geometric non-linearities and geometric effects of matrix element failures,while the available information on the stress prediction procedure is inadequate.The failure analysis is then carried out by com- paring the local element stresses or strains with the admissible values of stress or strain.The Tsai-Wu failure criterion [19]is used to predict the failure in the filling yarn elements.The maximum stress and strain criteria are used to predict the failure in the warp yarn and matrix elements.If an element fails,the stiffness of that element is reduced (degraded stiffness). The final failure of the unit cell laminate is assumed to have occurred if the fibres in the filling yarn are broken. In conclusion,some more practical drawbacks and disadvantages of the strength model of Naik are provided.First,the stress model lacks logic and

the yarn and the possible unbalanced nature of the plain fabric lamina. The representative unit cell is discretized into slices along or across the loading direction. These slices are further divided into different elements such as straight cross-ply or unidirectional regions, undulated cross-ply or unidirectional regions and pure matrix elements. In the analysis of Naik and Shembekar, two schemes for combining the in-plane stiffness matrices of the different elements are used: parallel–series and series–parallel. In the parallel–series (PS) model, the elements are first assembled in parallel across the loading direction with the isostrain assumption (adding the stiffness matrices, weighted by their volume fractions). Then, those multielements are assembled in series along the loading direction with the isostress assumption. In the second scheme, all the infinitesimal elements of a section along the loading direction are assembled with an isostress assumption (adding the compliance matrices, weighted by their volume fractions). Then, all the sections along the loading direction are assembled with an isostrain condition. Such a scheme is called a series– parallel (SP) model. Both schemes yield a full 2-D stiffness matrix for the plain woven fabric composite. A full mathematical treatment of the problem has been presented in reference [16]. Based on experimental work, the PS model is recommended for the prediction of all in-plane elastic constants. Out-of-plane properties cannot be predicted. Hence, the extension of the model to treat 3-D woven preforms is not useful. Recently, Naik and Ganesh have presented an extension of their thermoelastic models to include the prediction of failure in plain weave composites under on-axis static tensile loading [17,18]. The load is assumed along the filling direction. Different stages of failure such as warp yarn transverse failure, filling yarn shear failure, filling yarn transverse failure, pure matrix element failure and filling yarn longitudinal failure are considered. The newness of the model lies in the calculation procedure for the stresses in the matrix and yarn elements. However, this is exactly where the model is most confusing. A lot of effort has been spent on describing material nonlinearities, geometric non-linearities and geometric effects of matrix element failures, while the available information on the stress prediction procedure is inadequate. The failure analysis is then carried out by comparing the local element stresses or strains with the admissible values of stress or strain. The Tsai–Wu failure criterion [19] is used to predict the failure in the filling yarn elements. The maximum stress and strain criteria are used to predict the failure in the warp yarn and matrix elements. If an element fails, the stiffness of that element is reduced (degraded stiffness). The final failure of the unit cell laminate is assumed to have occurred if the fibres in the filling yarn are broken. In conclusion, some more practical drawbacks and disadvantages of the strength model of Naik are provided. First, the stress model lacks logic and Mechanical modelling of solid woven fabric composites 71 RIC3 7/10/99 7:37 PM Page 71 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:11 AM IP Address: 158.132.122.9

72 3-D textile reinforcements in composite materials simplicity (when and why is the PS model to be preferred over the SP model?).Second,only on-axis uniaxial tensile loads can be considered along the warp or weft direction.Third,the model does not account for thermal stresses which are known to be important in the stress and strength analy- sis of fibre composites.Finally,only a non-hybrid 2-D plain weave compos- ite can be considered in the present analysis. Model of Hahn and Pandy The 3-D thermo-elastic model of Hahn and Pandy [20]for non-hybrid plain fabric composites is simple in concept and mathematical implementation. This model is essentially an extension of the 2-D models of Naik.The geo- metric model accounts for the undulation of warp and weft yarns,the actual yarn cross-section and the presence of a gap between adjacent yarns.The yarn undulations are sinusoidal and described with shape functions.The gap 5 between two neighbouring yarns,however,is introduced by terminating the yarn at the start of the gap.Hence,for large gaps the yarn cross-section becomes quasi-rectangular,which is not realistic. In the thermo-elastic model,the strain is assumed to be uniform through- out the composite unit cell.Therefore the effective stiffness of the woven fabric composite is obtained as a volume average of the local stiffness properties of yarn and matrix elements.This is a so-called isostrain model.Closed-form expressions are provided for the 3-D effective elastic moduli and effective thermal expansion constants for a 2-D plain weave composite. The model has the advantage of being simple and easy to use.The iso- strain model can very easily be applied to analyse complex 3-D woven fabric composites.However,some disadvantages are here provided.First, the accuracy of the isostrain model still remains to be verified through more experimental verification of all 3-D elastic constants.It will be further shown in this chapter that the isostrain technique is not capable of accu- rately predicting all 3-D elastic constants [21].Second,the model can cer- tainly not be extended to solve the stress analysis problem accurately,and hence cannot be used for strength predictions. Model of R.Naik Recently,a micromechanics analysis tool labelled TexCad was developed by R.Naik to calculate the thermo-elastic properties along with damage and strength estimates for woven fabric composites [22].This tool can be used to analyse non-hybrid plain weave and satin weave composites.It dis- cretely models the yarn centreline paths within the repeating unit cell by assuming a sinusoidal undulation of the yarns.The 3-D effective stiffness matrix is computed by a yarn discretization scheme(which subdivides each

simplicity (when and why is the PS model to be preferred over the SP model?). Second, only on-axis uniaxial tensile loads can be considered along the warp or weft direction. Third, the model does not account for thermal stresses which are known to be important in the stress and strength analysis of fibre composites. Finally, only a non-hybrid 2-D plain weave composite can be considered in the present analysis. Model of Hahn and Pandy The 3-D thermo-elastic model of Hahn and Pandy [20] for non-hybrid plain fabric composites is simple in concept and mathematical implementation. This model is essentially an extension of the 2-D models of Naik. The geometric model accounts for the undulation of warp and weft yarns, the actual yarn cross-section and the presence of a gap between adjacent yarns. The yarn undulations are sinusoidal and described with shape functions.The gap between two neighbouring yarns, however, is introduced by terminating the yarn at the start of the gap. Hence, for large gaps the yarn cross-section becomes quasi-rectangular, which is not realistic. In the thermo-elastic model, the strain is assumed to be uniform throughout the composite unit cell. Therefore the effective stiffness of the woven fabric composite is obtained as a volume average of the local stiffness properties of yarn and matrix elements. This is a so-called isostrain model. Closed-form expressions are provided for the 3-D effective elastic moduli and effective thermal expansion constants for a 2-D plain weave composite. The model has the advantage of being simple and easy to use. The isostrain model can very easily be applied to analyse complex 3-D woven fabric composites. However, some disadvantages are here provided. First, the accuracy of the isostrain model still remains to be verified through more experimental verification of all 3-D elastic constants. It will be further shown in this chapter that the isostrain technique is not capable of accurately predicting all 3-D elastic constants [21]. Second, the model can certainly not be extended to solve the stress analysis problem accurately, and hence cannot be used for strength predictions. Model of R. Naik Recently, a micromechanics analysis tool labelled TexCad was developed by R. Naik to calculate the thermo-elastic properties along with damage and strength estimates for woven fabric composites [22]. This tool can be used to analyse non-hybrid plain weave and satin weave composites. It discretely models the yarn centreline paths within the repeating unit cell by assuming a sinusoidal undulation of the yarns. The 3-D effective stiffness matrix is computed by a yarn discretization scheme (which subdivides each 72 3-D textile reinforcements in composite materials RIC3 7/10/99 7:37 PM Page 72 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:11 AM IP Address: 158.132.122.9

Mechanical modelling of solid woven fabric composites 73 yarn into smaller,piecewise straight yarn slices)that assumed an isostrain state within the unit cell.Hence,as in the Hahn and Pandy model,the iso- strain model is applied.In the calculation for the strength,TexCad uses a curved beam-on-elastic-foundation model for yarn crimp regions together with an incremental approach in which stiffness properties for the failed yarn slices are reduced,based on the predicted yarn slice failure mode.Only on-axis tensile loadings and in-plane shear loadings were modelled and reported.Certainly,the most questionable assumption in this strength model is the calculation of the local stress fields in yarn and matrix slices based on the isostrain assumption.Basically,TexCad is well documented and easy to use.It is a thorough implementation of the isostrain approach which could be extended easily to analyse complex 3-D woven fabric com- posites.It will perform stiffness and failure analyses as correctly as can be expected for an isostrain model. Model of Paumelle,Hassim and Lene Paumelle et al.[23,24]developed a finite element method for analysing plain weave fabric composite structures.The periodic medium homoge- nization method is implemented.Basically,by applying periodic boundary conditions on the surface of the unit cell and by solving six elementary loading conditions on the unit cell,the complete set of elastic moduli of the homogenized structure can be computed.At the same time,the method pro- vides a good approximation of the local distribution of force and stress fields acting in the composite components and at their interface.These microscopic stress fields give a strong indication of the types of damage that will occur.To the best of our knowledge,Paumelle et al.have not yet 8 reported an extension of this finite element model to predict damage propa- gation or to analyse 3-D woven preforms.Moreover,outlined below are some problems encountered in a practical finite element analysis of solid woven fabric composites. First,this approach requires large computer calculation power and memory because of the 3-D nature and the complexity (size)of the yarn architecture.Second,a correct finite element model includes the generation of the fabric geometry and the finite element mesh of nodes and elements. Most of the time spent is related to the creation and the verification of a correct geometric model [25].Finally,there are major problems in analysing and interpreting the results in a 3-D domain of a rather complex geometry [26: Model of Blackketter Here,we will discuss in some detail the research work of Blackketter [27]. In our opinion,this work is certainly one of the first and most important

yarn into smaller, piecewise straight yarn slices) that assumed an isostrain state within the unit cell. Hence, as in the Hahn and Pandy model, the isostrain model is applied. In the calculation for the strength, TexCad uses a curved beam-on-elastic-foundation model for yarn crimp regions together with an incremental approach in which stiffness properties for the failed yarn slices are reduced, based on the predicted yarn slice failure mode. Only on-axis tensile loadings and in-plane shear loadings were modelled and reported. Certainly, the most questionable assumption in this strength model is the calculation of the local stress fields in yarn and matrix slices based on the isostrain assumption. Basically, TexCad is well documented and easy to use. It is a thorough implementation of the isostrain approach which could be extended easily to analyse complex 3-D woven fabric composites. It will perform stiffness and failure analyses as correctly as can be expected for an isostrain model. Model of Paumelle, Hassim and Léné Paumelle et al. [23,24] developed a finite element method for analysing plain weave fabric composite structures. The periodic medium homogenization method is implemented. Basically, by applying periodic boundary conditions on the surface of the unit cell and by solving six elementary loading conditions on the unit cell, the complete set of elastic moduli of the homogenized structure can be computed.At the same time, the method provides a good approximation of the local distribution of force and stress fields acting in the composite components and at their interface. These microscopic stress fields give a strong indication of the types of damage that will occur. To the best of our knowledge, Paumelle et al. have not yet reported an extension of this finite element model to predict damage propagation or to analyse 3-D woven preforms. Moreover, outlined below are some problems encountered in a practical finite element analysis of solid woven fabric composites. First, this approach requires large computer calculation power and memory because of the 3-D nature and the complexity (size) of the yarn architecture. Second, a correct finite element model includes the generation of the fabric geometry and the finite element mesh of nodes and elements. Most of the time spent is related to the creation and the verification of a correct geometric model [25]. Finally, there are major problems in analysing and interpreting the results in a 3-D domain of a rather complex geometry [26]. Model of Blackketter Here, we will discuss in some detail the research work of Blackketter [27]. In our opinion, this work is certainly one of the first and most important Mechanical modelling of solid woven fabric composites 73 RIC3 7/10/99 7:37 PM Page 73 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:11 AM IP Address: 158.132.122.9

74 3-D textile reinforcements in composite materials efforts to model damage propagation in 2-D woven composites.The approach could also be applied to 3-D woven fabric composites. Blackketter constructed a simplified 3-D unit cell of a single ply non- hybrid plain weave graphite/epoxy composite.From this description 3-D finite element models were generated.Twenty node isoparametric hexahe- dron elements were used in generating the finite element meshes.Limits on element refinement were imposed by the computational time required for solution.An incremental iterating finite element algorithm was developed to analyse loading response.Each iteration or load step required about 30 real-time minutes using a VAX8800 computer.Tension and shear loadings were modelled.The finite element model included capacities to model non- linear constitutive material behaviour and a scheme to estimate the effects of damage propagation by stiffness reduction.Results from this analysis compared extremely well with experimental stress-strain data.It was con- cluded that the non-linear stress-strain behaviour of the woven fabric com- posite is principally caused by damage propagation rather than by plastic deformation of the matrix. Let us describe now the damage propagation model as developed by Blackketter et al.At each Gaussian integration point (27 Gaussian quad- 银 rature integration points over each volume element),damage or failure was detected by comparing the actual stress state with a failure criterion.To 日 simulate damage at an integration point,the local stiffness properties were reduced.Therefore,each element in the model can contain both intact and failed Gaussian integration points.After the occurrence of damage,the volume considered was capable of sustaining only reduced loads and stresses had to be redistributed to surrounding volumes. PPV It is important to select an appropriate failure criterion for the matrix 8 and yarn materials.For the isotropic matrix material a maximum normal stress criterion was used to detect damage.If the principal stress exceeded the strength,the tensile modulus was reduced to 1%of its initial value.The shear modulus was reduced to 20%of its initial value.After failure,the matrix was no longer isotropic.For the transversely isotropic yarns,it is necessary to account both for the type of damage and the orientation of that damage.Blackketter compared the actual stresses in the local coor- dinate system (123)with the respective ultimate strengths.This is a maximum stress criterion.The 1-axis corresponds to the longitudinal yarn direction.Table 3.1 presents the different failure modes and the stiffness reduction factors used by Blackketter.Each Gaussian integration point was allowed to fail in one or all modes.Finally,catastrophic failure of the unit cell was characterized by large displacements compared with the previous values. The analysis by Blackketter of graphite/epoxy plain weave fabric com- posites has shown that by carefully modelling the fabric geometry,using

efforts to model damage propagation in 2-D woven composites. The approach could also be applied to 3-D woven fabric composites. Blackketter constructed a simplified 3-D unit cell of a single ply nonhybrid plain weave graphite/epoxy composite. From this description 3-D finite element models were generated. Twenty node isoparametric hexahedron elements were used in generating the finite element meshes. Limits on element refinement were imposed by the computational time required for solution. An incremental iterating finite element algorithm was developed to analyse loading response. Each iteration or load step required about 30 real-time minutes using a VAX8800 computer. Tension and shear loadings were modelled. The finite element model included capacities to model nonlinear constitutive material behaviour and a scheme to estimate the effects of damage propagation by stiffness reduction. Results from this analysis compared extremely well with experimental stress–strain data. It was concluded that the non-linear stress–strain behaviour of the woven fabric composite is principally caused by damage propagation rather than by plastic deformation of the matrix. Let us describe now the damage propagation model as developed by Blackketter et al. At each Gaussian integration point (27 Gaussian quadrature integration points over each volume element), damage or failure was detected by comparing the actual stress state with a failure criterion. To simulate damage at an integration point, the local stiffness properties were reduced. Therefore, each element in the model can contain both intact and failed Gaussian integration points. After the occurrence of damage, the volume considered was capable of sustaining only reduced loads and stresses had to be redistributed to surrounding volumes. It is important to select an appropriate failure criterion for the matrix and yarn materials. For the isotropic matrix material a maximum normal stress criterion was used to detect damage. If the principal stress exceeded the strength, the tensile modulus was reduced to 1% of its initial value. The shear modulus was reduced to 20% of its initial value. After failure, the matrix was no longer isotropic. For the transversely isotropic yarns, it is necessary to account both for the type of damage and the orientation of that damage. Blackketter compared the actual stresses in the local coordinate system (123) with the respective ultimate strengths. This is a maximum stress criterion. The 1-axis corresponds to the longitudinal yarn direction. Table 3.1 presents the different failure modes and the stiffness reduction factors used by Blackketter. Each Gaussian integration point was allowed to fail in one or all modes. Finally, catastrophic failure of the unit cell was characterized by large displacements compared with the previous values. The analysis by Blackketter of graphite/epoxy plain weave fabric composites has shown that by carefully modelling the fabric geometry, using 74 3-D textile reinforcements in composite materials RIC3 7/10/99 7:37 PM Page 74 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:11 AM IP Address: 158.132.122.9

Mechanical modelling of solid woven fabric composites 75 Table 3.1.Stiffness reduction scheme for the UD yarn elements,according to Blackketter [27] Failure mode Mechanical property and degradation factors En E22 E33 G23 G31 G12 Longitudinal tension on 0.01 0.01 0.01 0.01 0.01 0.01 2 Transverse tension 022 1.0 0.01 1.0 1.0 0.2 0.2 3 Transverse tension oaa 1.0 1.0 0.01 1.0 0.2 0.2 Transverse shear t2a 1.0 0.01 0.01 0.01 0.01 0.01 5 Longitudinal shear ti3 1.0 1.0 0.01 1.0 0.01 1.0 6 Longitudinal shear t12 1.0 0.01 1.0 1.0 1.0 0.01 correct constituent stiffness/strength data and by applying an appropriate stiffness reduction scheme,it is possible to predict the stress-strain behav- iour of woven fabric composites.The same ideas could certainly be applied poo to analyse 3-D woven fabric composites.However,Blackketter does not discuss in detail the limitations of the finite element modelling technique (meshing or calculation time). Models of Whitcomb oo/ Whitcomb and coworkers [28-30]have studied the effect of the yarn archi- tecture on the predicted elastic moduli and stresses in plain weave com- posites.The work is restricted to linear elastic analysis.Three-dimensional finite element models were used.Only simple plain weaves were studied because these offer sufficient complexity for the task.A refined model of 8 the complete unit cell would require immense amounts of computer memory and calculation time.However,by exploiting the geometric and material symmetries in the simple unit cell,it was sufficient to analyse 1/32 of the size of the complete plain weave unit cell.Twenty node isoparamet- ric hexahedral elements were used.Two different yarn architectures were investigated.The first was the 'translated architecture'where the complete yarn is created by keeping the cross-section vertical along the yarn path. The second was the 'extruded architecture'wherein the yarn cross-section is always placed perpendicular to the yarn path.The extruded yarn archi- tecture requires a more complex mesh generation. Whitcomb and coworkers also analysed progressive failure of plain weave fabric composites under in-plane tensile loading using a 3-D finite element model.The mechanical loading was parallel to one of the yarn directions.Thermal loading or thermal residual stresses were not consid- ered.The effects of various characteristics of the finite element model on predicted behaviour were examined.There is no 'right'way to model

correct constituent stiffness/strength data and by applying an appropriate stiffness reduction scheme, it is possible to predict the stress–strain behaviour of woven fabric composites. The same ideas could certainly be applied to analyse 3-D woven fabric composites. However, Blackketter does not discuss in detail the limitations of the finite element modelling technique (meshing or calculation time). Models of Whitcomb Whitcomb and coworkers [28–30] have studied the effect of the yarn architecture on the predicted elastic moduli and stresses in plain weave composites. The work is restricted to linear elastic analysis. Three-dimensional finite element models were used. Only simple plain weaves were studied because these offer sufficient complexity for the task. A refined model of the complete unit cell would require immense amounts of computer memory and calculation time. However, by exploiting the geometric and material symmetries in the simple unit cell, it was sufficient to analyse 1/32 of the size of the complete plain weave unit cell. Twenty node isoparametric hexahedral elements were used. Two different yarn architectures were investigated. The first was the ‘translated architecture’ where the complete yarn is created by keeping the cross-section vertical along the yarn path. The second was the ‘extruded architecture’ wherein the yarn cross-section is always placed perpendicular to the yarn path. The extruded yarn architecture requires a more complex mesh generation. Whitcomb and coworkers also analysed progressive failure of plain weave fabric composites under in-plane tensile loading using a 3-D finite element model. The mechanical loading was parallel to one of the yarn directions. Thermal loading or thermal residual stresses were not considered. The effects of various characteristics of the finite element model on predicted behaviour were examined. There is no ‘right’ way to model Mechanical modelling of solid woven fabric composites 75 Table 3.1. Stiffness reduction scheme for the UD yarn elements, according to Blackketter [27] Failure mode Mechanical property and degradation factors E11 E22 E33 G23 G31 G12 1 Longitudinal tension s11 0.01 0.01 0.01 0.01 0.01 0.01 2 Transverse tension s22 1.0 0.01 1.0 1.0 0.2 0.2 3 Transverse tension s33 1.0 1.0 0.01 1.0 0.2 0.2 4 Transverse shear t23 1.0 0.01 0.01 0.01 0.01 0.01 5 Longitudinal shear t13 1.0 1.0 0.01 1.0 0.01 1.0 6 Longitudinal shear t12 1.0 0.01 1.0 1.0 1.0 0.01 RIC3 7/10/99 7:37 PM Page 75 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:11 AM IP Address: 158.132.122.9

76 3-D textile reinforcements in composite materials damage evolution that is also practical [30].The most simple method to account for damage is to modify the constitutive matrix of the damaged finite element.Therefore,the failure analysis becomes a series of linear analyses.A maximum stress criterion was used to evaluate the damage of the matrix and yarn elements.Withcomb and coworkers have applied and compared three different techniques to modify the constitutive matrix after damage.First,the total constitutive matrix was reduced to essentially zero when any of the allowable strengths was exceeded.In the second technique, only specific rows and columns of the constitutive matrix were reduced according to the damage mode.Third,specific engineering moduli were reduced.This is the reduction scheme developed previously by Blackket- ter.Essentially,it was concluded that the predicted strength decreased con- siderably with increased waviness of the yarns.The modification technique of the constitutive matrix has a major effect on the predicted stress-strain curve.However,more numerical experiments are necessary to establish guidelines for an accurate failure analysis.No final conclusions have been given yet concerning the different reduction schemes.No extension is made to treat 3-D woven preforms. 3.2.4 Conclusions In the past 15 years,a variety of different micromechanical approaches has been developed to study the effective behaviour of 2-D woven fabric composites.Tables 3.2 and 3.3 summarize those micromechanical models. Basically,the literature review reveals that considerable work addressing the effects of fabric architecture on the effective elastic and thermal expan- sion properties was done.However,this work has not been systematic or exhaustive in general.Research has been too focused on material systems based on plain weave fabrics,limited ranges of fibre volume fractions and specific material thermo-elastic properties.Second,the stress and strength analyses are still in their infancy.Here,research has focused on specific loading directions,knee behaviour and damage mechanisms.There is certainly a need for reliable strength models.Finally,the extension of the models to consider 3-D preforms can only be achieved in a few cases (Tables 3.2 and 3.3). In the analytical methods we observe a predominant use of the isostrain assumption to predict the effective thermo-elastic and strength properties. No data are available to verify the accuracy of this approximation.More- over,most researchers have concentrated only on the primary determinant of mechanical and physical properties,namely the geometric orientation of the yarns.The idea that other geometric effects or boundary conditions could have an influence on the prediction of effective properties of woven fabric composites was ignored.The well-established finite element method

damage evolution that is also practical [30]. The most simple method to account for damage is to modify the constitutive matrix of the damaged finite element. Therefore, the failure analysis becomes a series of linear analyses. A maximum stress criterion was used to evaluate the damage of the matrix and yarn elements. Withcomb and coworkers have applied and compared three different techniques to modify the constitutive matrix after damage. First, the total constitutive matrix was reduced to essentially zero when any of the allowable strengths was exceeded. In the second technique, only specific rows and columns of the constitutive matrix were reduced according to the damage mode. Third, specific engineering moduli were reduced. This is the reduction scheme developed previously by Blackketter. Essentially, it was concluded that the predicted strength decreased considerably with increased waviness of the yarns. The modification technique of the constitutive matrix has a major effect on the predicted stress–strain curve. However, more numerical experiments are necessary to establish guidelines for an accurate failure analysis. No final conclusions have been given yet concerning the different reduction schemes. No extension is made to treat 3-D woven preforms. 3.2.4 Conclusions In the past 15 years, a variety of different micromechanical approaches has been developed to study the effective behaviour of 2-D woven fabric composites. Tables 3.2 and 3.3 summarize those micromechanical models. Basically, the literature review reveals that considerable work addressing the effects of fabric architecture on the effective elastic and thermal expansion properties was done. However, this work has not been systematic or exhaustive in general. Research has been too focused on material systems based on plain weave fabrics, limited ranges of fibre volume fractions and specific material thermo-elastic properties. Second, the stress and strength analyses are still in their infancy. Here, research has focused on specific loading directions, knee behaviour and damage mechanisms. There is certainly a need for reliable strength models. Finally, the extension of the models to consider 3-D preforms can only be achieved in a few cases (Tables 3.2 and 3.3). In the analytical methods we observe a predominant use of the isostrain assumption to predict the effective thermo-elastic and strength properties. No data are available to verify the accuracy of this approximation. Moreover, most researchers have concentrated only on the primary determinant of mechanical and physical properties, namely the geometric orientation of the yarns. The idea that other geometric effects or boundary conditions could have an influence on the prediction of effective properties of woven fabric composites was ignored. The well-established finite element method 76 3-D textile reinforcements in composite materials RIC3 7/10/99 7:37 PM Page 76 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:11 AM IP Address: 158.132.122.9

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