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 methoddamage 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 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