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