《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_fibrous molithic-30

Computers structures PERGAMON Computers and Structures 80(2002)1159-1176 On a general constitutive description for the inelastic and failure behavior of fibrous laminates- Part I Lamina theory Zheng-Ming Huang Biomaterials Laboratory, Division of Bioengineering, Department of Mechanical Engineering, National Unirersity of singapore, Ria Received 10 April accepted 6 March 2002 abstract plane load, the ultimate failure of the laminate can correspond to its last-ply failure, and hence a stress failure criterion nay be sufficient to detect the maximum load that can be sustained by the laminate. Even in such case, the load shared by each lamina in the laminate cannot be correctly determined if the lamina instantaneous stiffness matrix is inaccu- rately provided, since the lamina is always statically indeterminate in the laminate. If, however, the laminate is subjected to a lateral load, its ultimate failure occurs before the last-ply failure and the only use of the stress failure criterion is no longer sufficient; an additional critical deflection or curvature condition must be employed as well. This necessitates development of an efficient constitutive relationship for laminated composites in order that the laminate strains/de flections until the ultimate failure can be accurately calculated. a general constitutive description for the thermo- mechanical response of a fibrous laminate up to the ultimate failure with applications to various fibrous laminates is presented in these two parts of papers. The constitutive relationship is obtained by combining the classical lamination theory with a recently developed bridging micromechanics model, through a layer -by-layer analysis. The present paper focuses on the lamina analysis. Attention has been given to the applicability of the constitutive theory to the fibrous laminates stacked with a wide variety of composite laminae, including multidirectional tape laminae, woven and braided fabric composites, and knitted fabric reinforced composites, which have different constituent behavior such as elasto-plasticity and elastic-visco-plasticity. The laminate analysis and the application examples will be presented in the subsequent paper. o 2002 Published by Elsevier Science Keywords: Laminated composite; Textile co te; Metal matrix composite; Composite structure; Mechanical property: Constitutive relationship: Lamina theory; Bridging micromechanics model cuon achievement poses on the available materials. In some ndustry, conventional monolithic materials are currently It has been recognised that technological develop- operating at or near their limits and do not offer the ment depends on advances in the field of materials. potential for meeting the demands of further technical Whatever the field may be, the final limitation on advancement [1]. In this regard, composites represent nothing but a giant step in the ever-lasting endeavour of optimisation in materials. Furthermore, most living E-mailaddresses:huangzm@mail.tongji.edu.cn,huangzm@tissuesofourbody,bothhardandsofttissuessuch as bones. skins. dentins. cartilages. and 02/- see front matter a 2002 Published by Elsevier Science Ltd. PI:S0045-7949(02)00074-3

On a general constitutive description for the inelastic and failure behavior of fibrous laminates––Part I: Lamina theory Zheng-Ming Huang Biomaterials Laboratory, Division of Bioengineering, Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Cresent, Singapore 119 260 Received 10 April 2001; accepted 6 March 2002 Abstract It is well known that a structural design with isotropic materials can be accomplished only based on a stress failure criterion. This is, however, generally not true with laminated composites. Only when the laminate is subjected to an in￾plane load, the ultimate failure of the laminate can correspond to its last-ply failure, and hence a stress failure criterion may be sufficient to detect the maximum load that can be sustained by the laminate. Even in such case, the load shared by each lamina in the laminate cannot be correctly determined if the lamina instantaneous stiffness matrix is inaccu￾rately provided, since the lamina is always statically indeterminate in the laminate. If, however, the laminate is subjected to a lateral load, its ultimate failure occurs before the last-ply failure and the only use of the stress failure criterion is no longer sufficient; an additional critical deflection or curvature condition must be employed as well. This necessitates development of an efficient constitutive relationship for laminated composites in order that the laminate strains/de- flections until the ultimate failure can be accurately calculated. A general constitutive description for the thermo￾mechanical response of a fibrous laminate up to the ultimate failure with applications to various fibrous laminates is presented in these two parts of papers. The constitutive relationship is obtained by combining the classical lamination theory with a recently developed bridging micromechanics model, through a layer-by-layer analysis. The present paper focuses on the lamina analysis. Attention has been given to the applicability of the constitutive theory to the fibrous laminates stacked with a wide variety of composite laminae, including multidirectional tape laminae, woven and braided fabric composites, and knitted fabric reinforced composites, which have different constituent behavior such as elasto-plasticity and elastic-visco-plasticity. The laminate analysis and the application examples will be presented in the subsequent paper.
2002 Published by Elsevier Science Ltd. Keywords: Laminated composite; Textile composite; Metal matrix composite; Composite structure; Mechanical property; Constitutive relationship; Lamina theory; Bridging micromechanics model 1. Introduction It has been recognised that technological develop￾ment depends on advances in the field of materials. Whatever the field may be, the final limitation on achievement poses on the available materials. In some industry, conventional monolithic materials are currently operating at or near their limits and do not offer the potential for meeting the demands of further technical advancement [1]. In this regard, composites represent nothing but a giant step in the ever-lasting endeavour of optimisation in materials. Furthermore, most living tissues of our body, both hard and soft tissues such as bones, skins, dentins, cartilages, and even cells, are Computers and Structures 80 (2002) 1159–1176 www.elsevier.com/locate/compstruc E-mail addresses: huangzm@mail.tongji.edu.cn, huangzm@ email.com (Z.-M. Huang). 0045-7949/02/$ - see front matter
2002 Published by Elsevier Science Ltd. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 0 7 4 - 3

Z-M. Huang/Computers and Structures 80(2002)1159-1176 essentially fibrous composites. An expanding interest in fiber reinforced composite [3-7 until rupture. The ma- biological engineering attracts people's increasing atten terial parameters involved in the model are minimal and tion on those tissues, and on the development of nano- can be measured or determined independently. Appli- scale biocomposite substitutes cations of this model to a number of fibrous composites Apparently, any successful use/critical design of including various textile(woven, braided, and knitted synthetic composites or achievement in tissue engineer fabric reinforced composites have been successfully ing depends on a thorough understanding for the com- achieved [7-1l] However, the composites considered posite whole properties. However, according to the were mainly subjected to in-plane load conditions and critical survey organised by UK Science Engineering not enough attentions have been given to the responses Research council and uk institution of mechanica of laminated composites. For the convenience or an Engineers, the composite theories in current use are sti nite element method based structural analysis, it is useful less successful [2]. The main reason for this is that only to bring all the information, i.e. the systematic simula- the linear elastic constitutive equations of the composite tion procedure as well as its applications to a broad have been well established, and are essentially used for range of instructive examples, into a summary article. analyses in the current literature. Little is known about The purpose of these two parts of papers is to present a the composite inelastic behaviors. There is hardly any unified constitutive description for the thermo-mechan- commercial finite element analysis software package that ical response of a lamin has incorporated an efficient material constitutive model variety of application examples of which several are with which the response of a composite structure out of new. The present paper focuses on the theoretical de- a linear elastic deformation range can be directly simu velopment and an accompanied one deals with the ap- lated. However, without a good knowledge of the com- plications posite inelastic behavior, the composite load carrying The constitutive relationship is established through capacity cannot be well assessed, and hence, a critical combining the bridging model, for lamina analysis, with design of a composite material/structure cannot be made. the classical lamination theory, for laminate analysis As This is because most composites are in laminated struc- mentioned earlier, one purpose for the lamina analysis tures, and each lamina involved is statically indetermi is to obtain instantaneous stiffness/compliance matrix. nate(Fig. 1). The lamina load share depends on Considered in the paper are various laminae reinforced constitutive equations. Just before the laminate attains with different fiber preforms, including UD fiber pre- its failure status. some laminae in the laminate must form and woven braided and knitted fabric structures have undergone more or less inelastic deformations. The One of the most critical factors that influence the com- lamina load share cannot be correctly determined if only posite response is matrix behaviour. a different matrix its linear elastic constitutive relationship is used material may require a different theory for its constitu- throughout. In the case of a living tissue especially a soft tive description. In the present paper, an additional ssue. the inelastic deformation is even more distinct constitutive theory, the Bodner-Partom unified theory All these necessitate the development of a rational con- was employed to describe the response of titanium stitutive model for describing the composite inelastic matrix material at high temperature. However, the behavior up to failure bridging model is developed based on an incremental Very recently, the present author proposed one sucl odel, called the bridging model, which best applicable. As the Bodner-Partom theory uses a simulate the inelastic behaviour of a unidirectional (ud) total stress-total strain description in the differential form,a transformation between the Bodner-Partom description and the Prandth-Reuss description has been illustrated in the paper. The laminate analysis and the numerical examples will be presented in the subsequent 2. Simulation procedure Based on an incremental solution strategy which is best applicable to nonlinear problems, a detailed flow chart to show the simulation procedure for a gener laminated composite is indicated in Fig. 2. Essentially, two steps are involved in the simulation. Fig.i.compaRisonofanisolatedlaminawiththattakenfromInthefirststepalaminateanalysisisperformedThisis a laminate accomplished in this work by using the classical lami-

essentially fibrous composites. An expanding interest in biological engineering attracts people’s increasing atten￾tion on those tissues, and on the development of nano￾scale biocomposite substitutes. Apparently, any successful use/critical design of synthetic composites or achievement in tissue engineer￾ing depends on a thorough understanding for the com￾posite whole properties. However, according to the critical survey organised by UK Science & Engineering Research Council and UK Institution of Mechanical Engineers, the composite theories in current use are still less successful [2]. The main reason for this is that only the linear elastic constitutive equations of the composite have been well established, and are essentially used for analyses in the current literature. Little is known about the composite inelastic behaviors. There is hardly any commercial finite element analysis software package that has incorporated an efficient material constitutive model with which the response of a composite structure out of a linear elastic deformation range can be directly simu￾lated. However, without a good knowledge of the com￾posite inelastic behavior, the composite load carrying capacity cannot be well assessed, and hence, a critical design of a composite material/structure cannot be made. This is because most composites are in laminated struc￾tures, and each lamina involved is statically indetermi￾nate (Fig. 1). The lamina load share depends on its constitutive equations. Just before the laminate attains its failure status, some laminae in the laminate must have undergone more or less inelastic deformations. The lamina load share cannot be correctly determined if only its linear elastic constitutive relationship is used throughout. In the case of a living tissue especially a soft tissue, the inelastic deformation is even more distinct. All these necessitate the development of a rational con￾stitutive model for describing the composite inelastic behavior up to failure. Very recently, the present author proposed one such model, called the Bridging Model, which can fairly well simulate the inelastic behaviour of a unidirectional (UD) fiber reinforced composite [3–7] until rupture. The ma￾terial parameters involved in the model are minimal and can be measured or determined independently. Appli￾cations of this model to a number of fibrous composites including various textile (woven, braided, and knitted) fabric reinforced composites have been successfully achieved [7–11]. However, the composites considered were mainly subjected to in-plane load conditions and not enough attentions have been given to the responses of laminated composites. For the convenience of an fi- nite element method based structural analysis, it is useful to bring all the information, i.e. the systematic simula￾tion procedure as well as its applications to a broad range of instructive examples, into a summary article. The purpose of these two parts of papers is to present a unified constitutive description for the thermo-mechan￾ical response of a laminated composite and to show a variety of application examples of which several are new. The present paper focuses on the theoretical de￾velopment and an accompanied one deals with the ap￾plications. The constitutive relationship is established through combining the bridging model, for lamina analysis, with the classical lamination theory, for laminate analysis. As mentioned earlier, one purpose for the lamina analysis is to obtain instantaneous stiffness/compliance matrix. Considered in the paper are various laminae reinforced with different fiber preforms, including UD fiber pre￾form and woven, braided, and knitted fabric structures. One of the most critical factors that influence the com￾posite response is matrix behaviour. A different matrix material may require a different theory for its constitu￾tive description. In the present paper, an additional constitutive theory, the Bodner–Partom unified theory, was employed to describe the response of a titanium matrix material at high temperature. However, the bridging model is developed based on an incremental solution strategy with which the Prandtl–Reuss theory is best applicable. As the Bodner–Partom theory uses a total stress–total strain description in the differential form, a transformation between the Bodner–Partom description and the Prandtl–Reuss description has been illustrated in the paper. The laminate analysis and the numerical examples will be presented in the subsequent paper. 2. Simulation procedure Based on an incremental solution strategy which is best applicable to nonlinear problems, a detailed flow chart to show the simulation procedure for a general laminated composite is indicated in Fig. 2. Essentially, two steps are involved in the simulation. In the first step, a laminate analysis is performed. This is accomplished in this work by using the classical lami￾Fig. 1. Comparison of an isolated lamina with that taken from a laminate. 1160 Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176

Z-M. Huang / Computers and Structures 80(2002)1159-1176 1161 materials. The latter are dependent on the fiber and L Overal stiffness matrix matrix current stress states. On the other hand, having explicitly known the internal stresses in the fiber and strain a analysis匚今ilne matrix materials, the effective properties as well as stress-strain response of the lamina can be completely Stress shared by eachlamina identified. The lamina failure status can be detected by m checking whether the fiber or the matrix has attained its ultimate stress state or not. if the lamina fails the cor- Check lamina tailure les responding overall applied stress on the laminate is de- ilure strengt fined as a progressive failure strength. The remaining Check laminate ultimate failureS aminate is analyzed by discounting the stiffness contri- bution from the failed lamina. In this way, the ultimate strength of the laminate can be determined incremen- tally Thus, a necessary and sufficient condition for un Fig. 2. A flow chart to show analysis procedure for a fibrous derstanding the inelastic and strength behavior of a fi- laminate brous composite is to explicitly determine the internal stresses in its constituent fiber and matrix materials. all nation theory. The purpose of the laminate analysis is to ness matrix. the internal stresses in the fiber. and the obtain the in-plane strain and curvature increments of internal stresses in the matrix, pertaining to the lamina the laminate, and further to determine the stresses analysis at any load level can be obtained by using the sheared by each lamina in the laminate, as indicated Fig. 3 for a multidirectional tape laminate. The most bridging micromechanics model, which is summarized in the next section important quantity involved in this step analysis is the instantaneous stiffness matrix of the lamina. which will be used to construct the laminate overall stiffness matrix Initially, all the laminae in the laminate are in linear and 3. Summary of the bridging model elastic deformation (provided that the laminate is not Similarly as other composite theories, the bridgir subjected to high thermal residual stresses), and their model is developed with respect to a UD composite iffness matrices can be defined in a usual way. How- This is because other fibrous composites can be dis- ever, with the increase of load level, some laminae may cretized into a number of UD composites, see subse- ergo inelastic deformation. Their initial (elastic) stifness matrices are no longer applicable. Thus, the element (RVE) of a ud composite is indicated in Fig 4 second step analysis, i.e lamina analysis, is concerned with determination of the lamina instantaneous stiffness There are two material elements in the rve, i.e., the matrix even if an inelastic deformation has occurred fiber and the matrix. No other element such as fiber- This instantaneous stiffness matrix cannot be determined matrix interface region has been distinctly considered in ithout an explicit knowledge of the internal stress the bridging model Let us deal with a plane problem first. The incre- states generated in the fiber and matrix materials of the mental stresses in the fiber and matrix can be correlated lamina. This is because the lamina instantaneous stiff- through a bridging matrix, [a. ness matrix is closely related with the instantaneous tiffness matrices of the constituent fiber and matrix {da}=4]{do where da)= don, doz, do1) with f and m referring to the fiber and matrix, respectively. Based on(1), the three basic quantities of the composite are found to be {dd}=(F+4]){d}={]{d} {d}=团l(+h4)-{da}=]l{d},(3) ]=(]+S"4)(h+V团4 Fig. 3. A schematic show for the analysis of a multidirectional (S]+VmSlADIB (4)

nation theory. The purpose of the laminate analysis is to obtain the in-plane strain and curvature increments of the laminate, and further to determine the stresses sheared by each lamina in the laminate, as indicated in Fig. 3 for a multidirectional tape laminate. The most important quantity involved in this step analysis is the instantaneous stiffness matrix of the lamina, which will be used to construct the laminate overall stiffness matrix. Initially, all the laminae in the laminate are in linear and elastic deformation (provided that the laminate is not subjected to high thermal residual stresses), and their stiffness matrices can be defined in a usual way. How￾ever, with the increase of load level, some laminae may undergo inelastic deformation. Their initial (elastic) stiffness matrices are no longer applicable. Thus, the second step analysis, i.e., lamina analysis, is concerned with determination of the lamina instantaneous stiffness matrix even if an inelastic deformation has occurred. This instantaneous stiffness matrix cannot be determined without an explicit knowledge of the internal stress states generated in the fiber and matrix materials of the lamina. This is because the lamina instantaneous stiff- ness matrix is closely related with the instantaneous stiffness matrices of the constituent fiber and matrix materials. The latter are dependent on the fiber and matrix current stress states. On the other hand, having explicitly known the internal stresses in the fiber and matrix materials, the effective properties as well as stress–strain response of the lamina can be completely identified. The lamina failure status can be detected by checking whether the fiber or the matrix has attained its ultimate stress state or not. If the lamina fails, the cor￾responding overall applied stress on the laminate is de- fined as a progressive failure strength. The remaining laminate is analyzed by discounting the stiffness contri￾bution from the failed lamina. In this way, the ultimate strength of the laminate can be determined incremen￾tally. Thus, a necessary and sufficient condition for un￾derstanding the inelastic and strength behavior of a fi- brous composite is to explicitly determine the internal stresses in its constituent fiber and matrix materials. All the three quantities, i.e., the lamina instantaneous stiff- ness matrix, the internal stresses in the fiber, and the internal stresses in the matrix, pertaining to the lamina analysis at any load level can be obtained by using the bridging micromechanics model, which is summarized in the next section. 3. Summary of the bridging model Similarly as other composite theories, the bridging model is developed with respect to a UD composite. This is because other fibrous composites can be dis￾cretized into a number of UD composites, see subse￾quent sections for detail. A representative volume element (RVE) of a UD composite is indicated in Fig. 4. There are two material elements in the RVE, i.e., the fiber and the matrix. No other element such as fiber– matrix interface region has been distinctly considered in the bridging model. Let us deal with a plane problem first. The incre￾mental stresses in the fiber and matrix can be correlated through a bridging matrix, ½A, as fdrmg¼½Afdrf g; ð1Þ where fdrg¼fdr11; dr22; dr12gT with f and m referring to the fiber and matrix, respectively. Based on (1), the three basic quantities of the composite are found to be [3,7] fdrf g¼ðVf ½I þ Vm½AÞ 1 fdrg¼½Bfdrg; ð2Þ fdrmg¼½AðVf½I þ Vm½AÞ 1 fdrg¼½A½Bfdrg; ð3Þ and ½S¼ðVf½Sf þ Vm½Sm½AÞðVf½I þ Vm½AÞ 1 ¼ ðVf½Sf þ Vm½Sm½AÞ½B: ð4Þ Fig. 2. A flow chart to show analysis procedure for a fibrous laminate. Fig. 3. A schematic show for the analysis of a multidirectional tape laminate. Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176 1161

Z.M. Huang / Computers and Structures 80(2002)1159-1176 Fiber Matrix Fig. 4. A RVE of a UD composite In the above, [n is a unit matrix, V and Vm are the Note that the matrix [B in(2H4) has the same fiber and matrix volume fractions. [S]and [sm] are the. structure as [4], with its elements given by planar instantaneous compliance matrices of the fiber and matrix materials, respectively, whose definiti bu=(r+Vmay)(r+Vma33)/c, will be highlighted in Section 6. The (da) in(2)and (3) b12=-(ma1)(r+Vma33)/c, (7.1) are the overall applied stress increments on the UD composite. Therefore, it is crucial to determine the b1=[(ma1)(ma23)-(r+Vmaz)(ma13)/c, bridging matrix, which is expressed in the following form[34,7] b22=(V+na)(H+Va3)/e, (7.2) 0 00a33 (5) b3=(+Vman)(+Mann)/c, (7.3) The bridging elements on the diagonal, all, azz and c=(r+Vman)(+Ima2)(+Vma33) a33 are independent, whereas the others are dependent The dependent elements can be determined by requiring Thus, the most important task is to determine the that the overall compliance matrix of the composite, independent bridging elements. They are, however, given by Eq(4), be symmetric. This results in the fol functions of two kinds of variables:(a) constituent fibe and matrix properties, and(b) fiber packing geometry a12=(S2-S)(a1-a2)/S1-Sm) ( including fiber volume fraction, fiber arrangement (6. 1) pattern in the matrix, fiber cross-sectional shape, fiber- matrix interface bonding, etc. ) When both the fiber and d2Bur-d1B 413= the matrix are in elastic deformation, we can always B1B22-B12B2 (6.2) express the independent elements as -d2B1 1(1-P/E1)+ (8.1) (6.3) (8.2) d=S1(a1-a3) (64) a3=1+3(1-G/G12)+ d2= Sm(Vr+Vman)(a22-a33)+Su(r +Vma33)a12 where E and g are the Youngs and shear moduli of (6.5) the matrix; Ef, Ef2,and Gn are the longitud B1=S12 B12=S-S1, transverse, and in-plane shear moduli of the fiber. The expansion coefficient ii can only depend on the fiber B2=(+la2)(S出-S2) (6.6) acking geometry, but are independent of the materi properties. By virtue of some well-established composite B2l=m(Si2-p)a1(r+man)(S22-sm).(6.7) expansion coefficients can be derived, being 3. 71

In the above, ½I is a unit matrix, Vf and Vm are the fiber and matrix volume fractions. ½Sf and ½Sm are the planar instantaneous compliance matrices of the fiber and matrix materials, respectively, whose definitions will be highlighted in Section 6. The fdrg in (2) and (3) are the overall applied stress increments on the UD composite. Therefore, it is crucial to determine the bridging matrix, which is expressed in the following form [3,4,7] ½A ¼ a11 a12 a13 0 a22 a23 0 0 a33 2 4 3 5: ð5Þ The bridging elements on the diagonal, a11, a22 and a33 are independent, whereas the others are dependent. The dependent elements can be determined by requiring that the overall compliance matrix of the composite, given by Eq. (4), be symmetric. This results in the fol￾lowing expressions a12 ¼ ðSf 12 Sm 12Þða11 a22Þ=ðSf 11 Sm 11Þ; ð6:1Þ a13 ¼ d2b11 d1b21 b11b22 b12b21 ; ð6:2Þ a23 ¼ d1b22 d2b12 b11b22 b12b21 ; ð6:3Þ d1 ¼ Sm 13ða11 a33Þ; ð6:4Þ d2 ¼ Sm 23ðVf þ Vma11Þða22 a33Þ þ Sm 13ðVf þ Vma33Þa12; ð6:5Þ b11 ¼ Sm 12 Sf 12; b12 ¼ Sm 11 Sf 11; b22 ¼ ðVf þ Vma22ÞðSm 12 Sf 12Þ; ð6:6Þ b21 ¼ VmðSf 12 Sm 12Þa12 ðVf þ Vma11ÞðSf 22 Sm 22Þ: ð6:7Þ Note that the matrix ½B in (2)–(4) has the same structure as [A], with its elements given by b11 ¼ ðVf þ Vma22ÞðVf þ Vma33Þ=c; b12 ¼ ðVma12ÞðVf þ Vma33Þ=c; ð7:1Þ b13 ¼ ½ðVma12ÞðVma23Þ ðVf þ Vma22ÞðVma13Þ=c; b22 ¼ ðVf þ Vma11ÞðVf þ Vma33Þ=c; ð7:2Þ b23 ¼ ðVma23ÞðVf þ Vma11Þ=c; b33 ¼ ðVf þ Vma22ÞðVf þ Vma11Þ=c; ð7:3Þ c ¼ ðVf þ Vma11ÞðVf þ Vma22ÞðVf þ Vma33Þ: ð7:4Þ Thus, the most important task is to determine the independent bridging elements. They are, however, functions of two kinds of variables: (a) constituent fiber and matrix properties, and (b) fiber packing geometry (including fiber volume fraction, fiber arrangement pattern in the matrix, fiber cross-sectional shape, fiber– matrix interface bonding, etc.). When both the fiber and the matrix are in elastic deformation, we can always express the independent elements as a11 ¼ 1 þ k11ð1 Em=Ef 11Þþ ; ð8:1Þ a22 ¼ 1 þ k21ð1 Em=Ef 22Þþ ; ð8:2Þ a33 ¼ 1 þ k31ð1 Gm=Gf 12Þþ ; ð8:3Þ where Em and Gm are the Young’s and shear moduli of the matrix; Ef 11, Ef 22, and Gf 12 are the longitudinal, transverse, and in-plane shear moduli of the fiber. The expansion coefficient kij can only depend on the fiber packing geometry, but are independent of the material properties. By virtue of some well-established composite elasticity theories, a set of explicit expressions for the expansion coefficients can be derived, being [3,7] Fig. 4. A RVE of a UD composite. 1162 Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176

Z-M. Huang / Compute Structures80(2002)l591170 (10.1), is valid in most cases. On the other hand, the 0.5, sensitive to the specific fiber packing geometry. To ac- The most important feature is that Eq(9)should be count for this sensitivity, two bridging parameters are valid until rupture of the composite. This is because the incorporated into the corresponding independent bridg fiber packing geometry does not change or only varies ing elements, i.e.[71 very little when the composite deforms from an elastic region to an inelastic one. Thus. if the fiber material is a2=B+(1-) El, 0≤B≤1, inearly elastic until rupture and the matrix is elastic- plastic, we should have a3=x+(1-2)x,0≤x≤1 an=Em/E (10.1) The bridging parameters B and a can be adjusted by a2=0.5(1+Ean/E2 comparing the predicted effective transvers a3=0.5(1+Gm/G12) (10.3) plane shear moduli, Ezz and G12 of the composite, 1.e (+ mau(+man) where(refer to Fig. 5) (Vr+man)(,2+a22/m S2)+VVm(S2-Sm2)a12 E ∫E,when≤ 理, when a (104) G vr/Gn2+Ima33/Gm (12.2) Gn={05/1+四,when≤哩 > with measured ones. The so calibrated bridging meters can be used in(ll. 1)and (11. 2)for later inelastic ym is matrix Poissons ratio and analysis. If no other information is available. B=a=0.5 can be employe 啁=V(G)2+(璺)2-())+3(唱)2(106 is the matrix von misses effective stress. when the 4. Thermal load effect problem under consideration is fully three-dimensional the corresponding bridging matrix, A, is given in Ap- Suppose that the working temperature of the UD pendⅸxA composite, Ti is different from a reference temperature In reality, the composite longitudinal property hardly To at which the internal stresses of the fiber and the depends on the fiber packing geometry. The matrix are already known. Because of mismatch be- sponding independent bridging element formula, Eq. tween the coefficients of thermal expansion of the fiber Oy=yield strength E=tan(aYoung s modulus ng modulus E Fig. 5. An elastic-plastic stress-strain curve with definition of material parameters

k11 ¼ 1; k21 ¼ k31 ¼ 0:5; and all the other kij ¼ 0: ð9Þ The most important feature is that Eq. (9) should be valid until rupture of the composite. This is because the fiber packing geometry does not change or only varies very little when the composite deforms from an elastic region to an inelastic one. Thus, if the fiber material is linearly elastic until rupture and the matrix is elastic– plastic, we should have a11 ¼ Em=Ef 11; ð10:1Þ a22 ¼ 0:5ð1 þ Em=Ef 22Þ; ð10:2Þ a33 ¼ 0:5ð1 þ Gm=Gf 12Þ; ð10:3Þ where (refer to Fig. 5) Em ¼ Em; when rm e 6 rm Y Em T ; when rm e > rm Y;  ð10:4Þ Gm ¼ 0:5Em=ð1 þ mmÞ; when rm e 6 rm Y Em T =3; when rm e > rm Y:  ð10:5Þ mm is matrix Poisson’s ratio and rm e ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrm 11Þ 2 þ ðrm 22Þ 2 ðrm 11Þðrm 22Þ þ 3ðrm 12Þ 2 q ð10:6Þ is the matrix von Misses effective stress. When the problem under consideration is fully three-dimensional, the corresponding bridging matrix, ½A, is given in Ap￾pendix A. In reality, the composite longitudinal property hardly depends on the fiber packing geometry. The corre￾sponding independent bridging element formula, Eq. (10.1), is valid in most cases. On the other hand, the composite transverse and in-plane shear responses are sensitive to the specific fiber packing geometry. To ac￾count for this sensitivity, two bridging parameters are incorporated into the corresponding independent bridg￾ing elements, i.e. [7] a22 ¼ b þ ð1 bÞ Em Ef 22 ; 0 6 b 6 1; ð11:1Þ a33 ¼ a þ ð1 aÞ Gm Gf 12 ; 0 6 a 6 1; ð11:2Þ The bridging parameters b and a can be adjusted by comparing the predicted effective transverse and in￾plane shear moduli, E22 and G12 of the composite, i.e. E22 ¼ ðVf þ Vma11ÞðVf þ Vma22Þ ðVf þ Vma11ÞðVfSf 22 þ a22VmSm 22Þ þ VfVmðSm 12 Sf 12Þa12 ; ð12:1Þ G12 ¼ Vf þ Vma33 Vf=Gf 12 þ Vma33=Gm ; ð12:2Þ with measured ones. The so calibrated bridging para￾meters can be used in (11.1) and (11.2) for later inelastic analysis. If no other information is available, b ¼ a ¼ 0:5 can be employed. 4. Thermal load effect Suppose that the working temperature of the UD composite, T1 is different from a reference temperature, T0 at which the internal stresses of the fiber and the matrix are already known. Because of mismatch be￾tween the coefficients of thermal expansion of the fiber Fig. 5. An elastic–plastic stress–strain curve with definition of material parameters. Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176 1163

Z.M. Huang /Computers and Structures 80(2002)1159-1176 and the matrix, thermal stresses will be generated in the The overall thermal expansion coefficients of the constituent materials during the temperature variation, composite are thus determined from dT=T-To. The general constitutive equations of the fiber, matrix, and the composite are thus modified to {x}=H{x}+{}+(S]-S]){b}.(17 {d}={s]{dd}+{x}dr, (131) If there is no mechanical load applied to the lamina namely, Ida)=10f, the pure thermal stress increments de"=s"do"1+(a")dT. (13.2) in the constituents are simply expressed as idom]=bm)dT and do])=-bmdT Ide)=[sfda)+iadr where a, a, and ai, respectively, are the thermal ex pansion coefficients of the fiber, matrix, and the com- From Eq (18), the thermal residual stresses in the con- posite at the initial temperature To. In the case of a plane stituents of the lamina can be estimated.Furthermore,if problem, we have of=of=0. It should be noted that the initial interval of the temperature variation, [To, Til. the constituent compliance matrices in Egs (13.1) and is large, a sub-division is required and the final thermal (13.2),S] and Im], are also defined at the initial tem- stresses, o o and om, are obtained by the sum- perature To. For example, sm may not be merely the mation of those from each sub-interval contribution On the other hand the internal stress increments can be related to the overall applied stress incre idah 5. Failure criteria and the temperature variation, dT, through The total stresses in the constituent materials are Ido)=([n+Iml)do)+b)dT obtained by summing up the thermal-mechanical stress =[]{da}+{b}d7 (14.1) wIth residual stress components. Namely, the current total stresses in the en by {d}=团(+m4)-{da}+{b=}dr {d}+{o =[B]{da}+{b"}dT (14.2) where [B]([B) and [B] are called the stress concen tration matrices of the fiber and matrix materials. and where (b) and (bm) are thermal stress concentration factors of the fibers and the matrix, respectively. It is evident o)+=fo0M-)+ do)m-n that the latter two factors satisfy K=0,1,…,.with{o}={0} (19.3) H{b}+n{b}={(0} Therefore, only one concentration factor needs to be om) M-n, K+=fom) M-n,K+ dom)(M-n, determined. This factor is uniquely expressible in terms of the concentration matrices. Choosing bmI as inde 1,…,with{o"n0={0} (194) pendent, Benveniste and Dvorak [12] derived a rigorous The superscript"R refers to residual stresses. If no relation as follows other but thermal residual stresses are involved, they are {b}=(-{=)(]-S)-({2}-{x} calculated using Eq (18). The superscript"M-T"indi- cates that the quantity involved is resulted from the By means of the bridging matrix, the last equation coupled thermo-mechanical effect. The stress incre- calculated from }=(-4(+=)-)s Eqs. (14. 1)and(14.2), rather than from(2)and (3), with constituent properties specified at the current temper S])-({x2}-{x}). (16.1) ture. T. When the stress state in either the fiber or the matrix has attained its ultimate value, the corresponding The fiber thermal stress concentration factor is then composite is considered to have failed. In this way, the given by constituent materials (16.2) In detecting the constituent failure, the maximum

and the matrix, thermal stresses will be generated in the constituent materials during the temperature variation, dT ¼ T1 T0. The general constitutive equations of the fiber, matrix, and the composite are thus modified to fde f g¼½Sf fdrf gþfaf gdT ; ð13:1Þ fde mg¼½SmfdrmgþfamgdT ; ð13:2Þ and fdeg¼½SfdrgþfagdT ð13:3Þ where af i , am i , and ai, respectively, are the thermal ex￾pansion coefficients of the fiber, matrix, and the com￾posite at the initial temperature T0. In the case of a plane problem, we have af 3 ¼ am 3 ¼ 0. It should be noted that the constituent compliance matrices in Eqs. (13.1) and (13.2), ½Sf and ½Sm, are also defined at the initial tem￾perature T0. For example, ½Sm may not be merely the elastic component. On the other hand, the internal stress increments can be related to the overall applied stress increments, fdrg, and the temperature variation, dT , through fdrf g¼ðVf ½I þ Vm½AÞ 1 fdrgþfbf gdT ¼ ½Bf fdrgþfbf gdT ; ð14:1Þ fdrmg¼½AðVf ½I þ Vm½AÞ 1 fdrgþfbmgdT ¼ ½BmfdrgþfbmgdT ; ð14:2Þ where ½Bf (½B) and ½Bm are called the stress concen￾tration matrices of the fiber and matrix materials, and fbf g and fbmg are thermal stress concentration factors of the fibers and the matrix, respectively. It is evident that the latter two factors satisfy Vffbf g þ Vmfbmg¼f0g: ð15Þ Therefore, only one concentration factor needs to be determined. This factor is uniquely expressible in terms of the concentration matrices. Choosing fbmg as inde￾pendent, Benveniste and Dvorak [12] derived a rigorous relation as follows fbmg ¼ ð½I ½BmÞð½Sf ½SmÞ 1 ðfamg faf gÞ: By means of the bridging matrix, the last equation becomes fbmg¼ ½I  ½AðVf ½I þ Vm½AÞ 1 ð½Sf ½SmÞ 1 ðfamg faf gÞ: ð16:1Þ The fiber thermal stress concentration factor is then given by fbf g¼ Vm Vf fbmg: ð16:2Þ The overall thermal expansion coefficients of the composite are thus determined from fag ¼ Vffaf g þ Vmfamg þ Vmð½Sm ½Sf Þfbmg: ð17Þ If there is no mechanical load applied to the lamina, namely, fdrg¼f0g, the pure thermal stress increments in the constituents are simply expressed as fdrmg ðTÞ ¼ fbmgdT and fdrf g ðTÞ ¼ Vm Vf fbmgdT ð18Þ From Eq. (18), the thermal residual stresses in the con￾stituents of the lamina can be estimated. Furthermore, if the initial interval of the temperature variation, [T0; T1], is large, a sub-division is required and the final thermal stresses, frf gðTÞ and frmgðTÞ , are obtained by the sum￾mation of those from each sub-interval contribution. 5. Failure criteria The total stresses in the constituent materials are obtained by summing up the thermal–mechanical stress with residual stress components. Namely, the current total stresses in the constituents are given by rf Kþ1 ¼ rf ðRÞ þ rf ðM–TÞ;Kþ1 ; ð19:1Þ rm f gKþ1 ¼ rm f gðRÞ þ rm f gðM–TÞ;Kþ1 ; ð19:2Þ where rf ðM–TÞ;Kþ1 ¼ rf ðM–TÞ;K þ drf ðM–TÞ ; K ¼ 0; 1; ... ; with rf ðM–TÞ;0 ¼ f0g; ð19:3Þ rm f gðM–TÞ;Kþ1 ¼ rm f gðM–TÞ;K þ drm f gðM–TÞ ; K ¼ 0; 1; ... ; with rm f gðM–TÞ;0 ¼ f0g: ð19:4Þ The superscript ‘‘R’’ refers to residual stresses. If no other but thermal residual stresses are involved, they are calculated using Eq. (18). The superscript ‘‘M–T’’ indi￾cates that the quantity involved is resulted from the coupled thermo-mechanical effect. The stress incre￾ments, fdrf gðM–TÞ and fdrmg ðM–TÞ are calculated from Eqs. (14.1) and (14.2), rather than from (2) and (3), with constituent properties specified at the current tempera￾ture, T. When the stress state in either the fiber or the matrix has attained its ultimate value, the corresponding composite is considered to have failed. In this way, the composite strength is defined in terms of those of its constituent materials. In detecting the constituent failure, the maximum normal stress criterion is among the best applicable. This 1164 Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176

Z-M. Huang / Computers and Structures 80(2002)1159-1176 criterion is certainly applicable to isotropic matrix ma instantaneous compliance matrix of the matrix material terial. For transversely isotropic fiber, however, the. is given below application is still reasonable. Due to its small cross- ectional dimension. the failure of the fiber material in Sm] when≤哩 he composite is most probably resulted from an exces- [sm+[sm, when om >oy, ive stress in its longitudinal direction, which corre- sponds to the maximum normal stress. Hence, the where composite failure is assumed if either of the following 0 conditions is satisfied 0 d≤d,≤(-d),山≤,≤(-m)(20 where af and af are the first and the third principal stresses in the fiber, of and o are the fiber dndn dadu 20nd tensile and compressive strengths, am and 4M(o) and the third principal stresses in the matrix, and om and symmetry 4012d1 ous are the matrix ultimate tensile and compressive strengths, respectively It has been recognised that the maximum normal EEm stress criterion may not be very accurate if two or three (22.3) principal stresses are close to each other. In such a case, a generalised maximum normal stress criterion [13] may be pertinent. Details are referred to Ref. [13] d 0,ifi≠j i,j=1, 6. Constitutive description for constituent materials lI.if i=i' The three-dimensional Prandth-Reuss theory formu- It is seen from Eq(4)that the overall compliance lae are summarized in Appendix B matrix of the composite relies upon the instantaneous compliance matrices of the fiber and matrix materials. In practice, the fiber material used is generally at most 6.2. Bodner and Partom theory transversely isotropic. Furthermore, most fibers such Some materials such as titanium alloys display sig- glass, carbon/graphite,Kevlar, boron, alumina, etc can nificant temperature dependent thermoelastic/viscoplas be regarded as linearly elastic until rupture. The com- tic behavior. Their constitutive relationships can be best pliance matrix of such a fiber is simply defined using represented using the unified Bodner and Partom model Hooke's law, and keeps unchanged up to failure On the other hand, most matrix materials especially with directional hardening [I5. This unified theory as- metal and polymer materials possess ability of undergo sumes that the total strain of the material is the sum of ing significant inelastic deformation before failure. These the elastic, thermal, and viscoplastic strains. The in taneous compliance matrices of them can be described 6. strains are controlled by the following flow rule g a number of well-developed constitutive theories 1(z+ For an explanation purpose, only two such theories D summarized in this section. One is the prandtReuss theory and another is the Bodner-Partom theory. The where former is applicable to an elasto-plastic material, whereas the latter to an elastic-visco-plastic material. If the ma- J2=l0 d, d=di-doHdjit trix used is a rubber or elastomer material. its instanta- cous compliance matrix can be defined using a model given in Ref. [6 z=mW(z1-2)-A1Z1 6.1. Prandtl-Reuss theory Z\aZ he Prandtl-Reuss plastic flow theory is well known in the literature [4, 14], and only related formulae are Wp=dj 4p z(0)=z0 summarized herein. According to this theory, the planar

criterion is certainly applicable to isotropic matrix ma￾terial. For transversely isotropic fiber, however, the application is still reasonable. Due to its small cross￾sectional dimension, the failure of the fiber material in the composite is most probably resulted from an exces￾sive stress in its longitudinal direction, which corre￾sponds to the maximum normal stress. Hence, the composite failure is assumed if either of the following conditions is satisfied: r1 f 6 rf u; r3 f 6 ð rf u;cÞ; r1 m 6 rm u ; r3 m 6 ð rm u;cÞ ð20Þ where r1 f and r3 f are the first and the third principal stresses in the fiber, rf u and rf u;c are the fiber longitudinal tensile and compressive strengths, r1 m and r3 m are the first and the third principal stresses in the matrix, and rm u and rm u;c are the matrix ultimate tensile and compressive strengths, respectively. It has been recognised that the maximum normal stress criterion may not be very accurate if two or three principal stresses are close to each other. In such a case, a generalised maximum normal stress criterion [13] may be pertinent. Details are referred to Ref. [13]. 6. Constitutive description for constituent materials It is seen from Eq. (4) that the overall compliance matrix of the composite relies upon the instantaneous compliance matrices of the fiber and matrix materials. In practice, the fiber material used is generally at most transversely isotropic. Furthermore, most fibers such as glass, carbon/graphite, Kevlar, boron, alumina, etc. can be regarded as linearly elastic until rupture. The com￾pliance matrix of such a fiber is simply defined using Hooke’s law, and keeps unchanged up to failure. On the other hand, most matrix materials especially metal and polymer materials possess ability of undergo￾ing significant inelastic deformation before failure. These materials, however, are generally isotropic. The instan￾taneous compliance matrices of them can be described using a number of well-developed constitutive theories. For an explanation purpose, only two such theories are summarized in this section. One is the Prandtl–Reuss theory and another is the Bodner–Partom theory. The former is applicable to an elasto-plastic material, whereas the latter to an elastic–visco-plastic material. If the ma￾trix used is a rubber or elastomer material, its instanta￾neous compliance matrix can be defined using a model given in Ref. [6]. 6.1. Prandtl–Reuss theory The Prandtl–Reuss plastic flow theory is well known in the literature [4,14], and only related formulae are summarized herein. According to this theory, the planar instantaneous compliance matrix of the matrix material is given below. Sm ½ ¼ Sm ½ e ; when rm e 6 rm Y Sm ½ e þ Sm ½ p ; when rm e > rm Y;  ð21Þ where ½Sm e ¼ 1 Em mm Em 0 1 Em 0 symmetric 1 Gm 2 6 4 3 7 5; ð22:1Þ Sm ½ p ¼ 9 4Mm T ðrm e Þ 2 r0 11r0 11 r0 22r0 11 2r0 12r0 11 r0 22r0 22 2r0 12r0 22 symmetry 4r0 12r0 12 2 6 4 3 7 5 rij¼rm ij ; ð22:2Þ Mm T ¼ EmEm T Em Em T ; ð22:3Þ r0 ij ¼ rij 1 3 ðr11 þ r22Þdij; dij ¼ 0; if i 6¼ j 1; if i ¼ j  ; i;j ¼ 1; 2: ð22:4Þ The three-dimensional Prandtl–Reuss theory formu￾lae are summarized in Appendix B. 6.2. Bodner and Partom theory Some materials such as titanium alloys display sig￾nificant temperature dependent thermoelastic/viscoplas￾tic behavior. Their constitutive relationships can be best represented using the unified Bodner and Partom model with directional hardening [15]. This unified theory as￾sumes that the total strain of the material is the sum of the elastic, thermal, and viscoplastic strains. The in￾elastic strains are controlled by the following flow rule [16,17] e_ I ij ¼ D0 exp " 1 2 ðZI þ ZDÞ 2 3J2 !n# r0 ij ffiffiffiffi J2 p ; ð23Þ where J2 ¼ 1 2 r0 ijr0 ij; r0 ij ¼ rij 1 3 rkkdij; Z_ I ¼ m1W_ pðZ1 ZI Þ A1Z1 ZI Z2 Z1  r1 þ T_ ZI Z2 Z1 Z2   oZ1 oT  þ Z1 ZI Z1 Z2   oZ2 oT  ; W_ p ¼ rije_ I ij; ZD ¼ bijuij; uij ¼ rij ffiffiffiffiffiffiffiffiffiffiffi rklrkl p ; ZI ð0Þ ¼ Z0; Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176 1165

Z.M. Huang / Computers and Structures 80(2002)1159-1176 =m取(2-1=)-42(z In the above, Ar is a time increment. The subscript k refers to the current time and k-l the previous time. It is noted that when a quantity, such as Z, depends ex- a7,1(0)=0 plicitly on temperature, the subscript k indicates that this quantity(Zu) takes value at the current temperature In the above, Do, n,mi,Zi,Ai,r1,Z,Zo,m2,Z3,a T which is the temperature at the current time and r? are material-dependent parameters, which control constant, we must have(21+B)=a0,or hardening characteristics. The overhead dot of a quan-(Z+B2)=ok (27) tity designates a differentiation with respect to time t. It is seen that after integration the Bodner-Partor where x=[-2In(v3E! /2Do)=. Substituting Eq.(27) model is expressed in total stress and total strain forms, into(25)and (26), we can solve for Zi and Be, and fur- which are inconvenient to be incorporated in the ther, obtain ok from Eq(27). The total strain, Ek, is then Bridging Model. To resolve this problem, the matrix calculated from based on the bodner partom model. from this curve. E-6+a /E=f-1+ Ae +oR/E the material parameters required by the Prandtl-Reuss In this way, the uniaxial stress-strain curve can b constitutive description, Eqs. (21),(22.1 22. 4)can be obtained incrementally defined. Thus, we only need to consider the one- Sometime, a total strain rate, i.e. 8. rather than a dimensional Bodner-Partom equation Under a uniaxial plastic strain rate is provided. In such case, we have tension condition, the general Bodner-Partom Eq(23) e≈E-△a/(△E) =Doexp-2 02 Furthermore, the three temperature derivatives in- (24) volved in Eqs. ( 25)and(26), i.e., (0Z,/aT).(aZ2/0T),and (aZ /ar), can be constructed using the corresponding where a and a' are, respectively, the stress and inelastic experimental data, as illustrated in an example shown in strain of the material under a uniaxial load. with the accom Euler trapezoidal scheme, a set of discretized evolution equations are obtained as follows 7. Analysis of a woven or braided fabric lamina 2=2k-1+=,-1(21k-1-24-1)+01(21k-2 In this work, we refer to a lamina as a composite that △L1 ZI-z made by reinforcing a single layer fibrous preform with a matrix material. When this preform is constructed by simply arranging straight fibers in the same direction the composite is called a UD lamina. If, however, this preform is fabricated using a textile technique such as 2 weaving or braiding, the resulting composite is called a aT/k-l woven or braided fabric lamina Z1-2k Compared with traditional tape laminates, textile k-1 fabric reinforced composites have several advantages such as ease of fabrication, feasibility to complicated structural contour, balanced in-plane property, high ZI-z ability in energy absorption, etc. They are receiving in- (25) creasing attentions in various aspects of engineering [18- 20. The analysis of textile composites is important for efficient design and application. For somewhat simplic △m,gl ity, only isothermal analysis with dr=0 will be carried B=B4-1+-,[ak-1(Z3k-1-B4-1)+0(Zxk-B out in this paper for a textile composite. △t4 any factors which can influence the me- +Zik chanical performance of a textile composite. In fact, an +h及/2 of textile pre using existing technology. However, the analysis proce- (26) same, as depicted in Fig. 6, consisting of a sub-division

_ bij ¼ m2W_ pðZ3uij bijÞ A2Z1 bij ffiffiffiffiffiffiffiffiffiffiffi bklbkl p ffiffiffiffiffiffiffiffiffiffiffi bklbkl p Z1 !r2 þ T_ bij Z3 oZ3 oT ; bijð0Þ ¼ 0: In the above, D0, n, m1, Z1, A1, r1, Z2, Z0, m2, Z3, A2, and r2 are material-dependent parameters, which control the time and temperature-dependent effects and the hardening characteristics. The overhead dot of a quan￾tity designates a differentiation with respect to time t. It is seen that after integration the Bodner–Partom model is expressed in total stress and total strain forms, which are inconvenient to be incorporated in the Bridging Model. To resolve this problem, the matrix uniaxial stress–strain curve is plotted at every moment based on the Bodner–Partom model. From this curve, the material parameters required by the Prandtl–Reuss constitutive description, Eqs. (21),(22.1)–(22.4) can be defined. Thus, we only need to consider the one￾dimensional Bodner–Partom equation. Under a uniaxial tension condition, the general Bodner–Partom Eq. (23) reads e_ I ¼ D0 exp " 1 2 ðZI þ bÞ 2 r2 !n# 2 ffiffiffi 3 p ; ð24Þ where r and eI are, respectively, the stress and inelastic strain of the material under a uniaxial load. With the Euler trapezoidal scheme, a set of discretized evolution equations are obtained as follows: ZI k ¼ ZI k 1 þ Dtm1e_ I 2 ½rk 1ðZ1;k 1 ZI k 1Þ þ rk ðZ1;k ZI k Þ DtA1 2 Z1;k ZI k Z2;k Z1;k  r1  þ Z1;k 1 ZI k 1 Z2;k 1 Z1;k 1  r1 þ Dt 2 T_ k 1 ZI k 1 Z2 Z1 Z2   k 1 oZ1 oT   k 1   þ Z1 ZI k 1 Z1 Z2   k 1 oZ2 oT   k 1  þ T_ k ZI k Z2 Z1 Z2   k oZ1 oT   k  þ Z1 ZI k Z1 Z2   k oZ2 oT   k ; ð25Þ bk ¼ bk 1 þ Dtm2e_ I 2 ½rk 1ðZ3;k 1 bk 1Þ þ rk ðZ3;k bk Þ DtA2 2 Z1;k 1 bk 1 Z1;k 1  r2  þ Z1;k bk Z1;k  r2 þ Dt 2 T_ k 1 bk 1 Z3;k 1 oZ3 oT   k 1  þ T_ k bk Z3;k oZ3 oT   k  : ð26Þ In the above, Dt is a time increment. The subscript k refers to the current time and k 1the previous time. It is noted that when a quantity, such as Z1 depends ex￾plicitly on temperature, the subscript k indicates that this quantity (Z1;k ) takes value at the current temperature Tk which is the temperature at the current time. As the plastic strain rate e_ I at the specific moment is constant, we must have ðZI þ bÞ ¼ ar, or, ðZI k þ bk Þ ¼ ark ; ð27Þ where a ¼ 2 ln ffiffiffi 3 p e_ I =2D0     1 2n . Substituting Eq. (27) into (25) and (26), we can solve for ZI k and bk , and fur￾ther, obtain rk from Eq. (27). The total strain, ek , is then calculated from ek ¼ e I k þ rk=E ¼ e I k 1 þ Dte_ I þ rk=E: ð28Þ In this way, the uniaxial stress–strain curve can be obtained incrementally. Sometime, a total strain rate, i.e., e_, rather than a plastic strain rate is provided. In such case, we have e_ I  e_ Dr=ðDtEÞ: ð29Þ Furthermore, the three temperature derivatives in￾volved in Eqs. (25) and (26), i.e., (oZ1=oT ), (oZ2=oT ), and (oZ3=oT ), can be constructed using the corresponding experimental data, as illustrated in an example shown in the accompanied paper. 7. Analysis of a woven or braided fabric lamina In this work, we refer to a lamina as a composite that is made by reinforcing a single layer fibrous preform with a matrix material. When this preform is constructed by simply arranging straight fibers in the same direction, the composite is called a UD lamina. If, however, this preform is fabricated using a textile technique such as weaving or braiding, the resulting composite is called a woven or braided fabric lamina. Compared with traditional tape laminates, textile fabric reinforced composites have several advantages such as ease of fabrication, feasibility to complicated structural contour, balanced in-plane property, high ability in energy absorption, etc. They are receiving in￾creasing attentions in various aspects of engineering [18– 20]. The analysis of textile composites is important for efficient design and application. For somewhat simplic￾ity, only isothermal analysis with dT ¼ 0 will be carried out in this paper for a textile composite. There are many factors which can influence the me￾chanical performance of a textile composite. In fact, an unlimited number of textile preforms can be developed using existing technology. However, the analysis proce￾dure for various textile composites is essentially the same, as depicted in Fig. 6, consisting of a sub-division 1166 Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176

Z-M. Huang/Computers and Structures 80(2002)1159-1176 Textile lamina 45 through a rotation by 45.(see, Ref. [8]), only the analysis for a diamond braid lamina is elaborated in this section. Furthermore. no thermal residual stress is as- sumed to occur in the lamina (this is the case when th Unit cel amina is fabricated at room temperature as in a sub sequent example). RVE a schematic diagram to show the sub-division of a braided fabric lamina is indicated in Fig. 7.a diamond structure is shown in Fig. 8, which is fabricated from UD composites two yarns(called fill and warp yarns respectively) in terracing one after another (Fig. 7(a)). The braiding Fig. 6. Analysis procedure for a textile fabric reinforced lam- angle, 0, is defined as an inclined angle between the yarn axis and the fabric longitudinal direction(Figs. 8 and 7(b)). As the fabric structure can be constructed by re- peating some unit cell(Fig. 7(b)), the analysis for the and an assemblage. The textile lamina is first sub- braid lamina can be achieved by that for a unit cell as divided into a number of UD composites, to which the long as all the unit cells in the composite are under the bridging model can be applied. Then all the UD cor same load/deformation condition. However, the unit cell posites are assembled together to obtain the three basic shown in Fig. 7(b) can be further divided into four sub quantities of the original textile lamina, i.e., the internal cells that are identical or symmetrical. Thus, we only stress increments in the fiber and matrix materials and need to analyze one sub-cell, which is called a rvE for the lamina instantaneous compliance matrix. the braid lamina The most customised work in the analysis of a textile The Rve is first sub-divided in the fabric plane into composite is to identify the unit cell or the rve geo- sub-elements [8], as shown in Fig. 7(c). Suppose that the metry of the textile structure under consideration. This total number of the sub-elements is M. Each sub-ele- identification is necessary for the sub-division as well as ment, Fig. 7(d), can have at most four material layers, for the assemblage. In this paper, only the simplest i.e., the braider yarn l, the braider yarn 2, and the top extile structures are taken into account, for easy illus- and bottom pure matrix layers. These material layers are ration purpose. For woven and braided fabrics, the. considered as UD composites in their respective local simplest structures are a plain weave and a diamond coordinate systems(both the pure matrix layers can be braid [21]. As the plain weave can be obtained geomet regarded as a UD composite with a zero fiber volume Single layer lat L, y, local system Subdivision of a suh-ele Yarn 2 Fig. 7. A schematic diagram to show analysis procedure for a braided fabric lamina

and an assemblage. The textile lamina is first sub￾divided into a number of UD composites, to which the bridging model can be applied. Then all the UD com￾posites are assembled together to obtain the three basic quantities of the original textile lamina, i.e., the internal stress increments in the fiber and matrix materials and the lamina instantaneous compliance matrix. The most customised work in the analysis of a textile composite is to identify the unit cell or the RVE geo￾metry of the textile structure under consideration. This identification is necessary for the sub-division as well as for the assemblage. In this paper, only the simplest textile structures are taken into account, for easy illus￾tration purpose. For woven and braided fabrics, the simplest structures are a plain weave and a diamond braid [21]. As the plain weave can be obtained geomet￾rically from a diamond braid with a braiding angle of 45 through a rotation by 45 (see, Ref. [8]), only the analysis for a diamond braid lamina is elaborated in this section. Furthermore, no thermal residual stress is as￾sumed to occur in the lamina (this is the case when the lamina is fabricated at room temperature as in a sub￾sequent example). 7.1. Sub-division A schematic diagram to show the sub-division of a braided fabric lamina is indicated in Fig. 7. A diamond structure is shown in Fig. 8, which is fabricated from two yarns (called fill and warp yarns respectively) in￾terlacing one after another (Fig. 7(a)). The braiding angle, h, is defined as an inclined angle between the yarn axis and the fabric longitudinal direction (Figs. 8 and 7(b)). As the fabric structure can be constructed by re￾peating some unit cell (Fig. 7(b)), the analysis for the braid lamina can be achieved by that for a unit cell as long as all the unit cells in the composite are under the same load/deformation condition. However, the unit cell shown in Fig. 7(b) can be further divided into four sub￾cells that are identical or symmetrical. Thus, we only need to analyze one sub-cell, which is called a RVE for the braid lamina. The RVE is first sub-divided in the fabric plane into sub-elements [8], as shown in Fig. 7(c). Suppose that the total number of the sub-elements is M. Each sub-ele￾ment, Fig. 7(d), can have at most four material layers, i.e., the braider yarn 1, the braider yarn 2, and the top and bottom pure matrix layers. These material layers are considered as UD composites in their respective local coordinate systems (both the pure matrix layers can be regarded as a UD composite with a zero fiber volume Fig. 6. Analysis procedure for a textile fabric reinforced lam￾ina. Fig. 7. A schematic diagram to show analysis procedure for a braided fabric lamina. Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176 1167

Z.M. Huang /Computers and Structures 80(2002)1159-1176 geometric coordinate systems as [8](refer to Fig. 10) Longitudinal Y-XI cot(20) Y,, cot(20) X COS Fig. 8. Photograph of a diamond braided fabric structure Y2+ x2 cot(20) on), as indicated in Fig. 7(eHg). To define inclined X COS (31.1) of the yarn I and yarn 2 in Fig. 7(e)and(f, their positions in the Rve must be identified 7. 2. RVE geometry For the braider I and the braider 2, we assign two (x2 geometric coordinate systems, (X,. Y1, Z, and (X,, Y,, Z2) respectively, with origins on the respective yarn central a1+g1 planes, as indicated in Fig 9(b). These two systems are u= sin(20) and u2 sin( 20) (32) introduced for the convenience of geometric descrip- tions. It is noted that YI and y2 are along the yarn axia In the above, t is the yarn thickness, a is the yarn directions. It also deserves mentioning that the geo- width, and g is an inter-yarn gap(Fig 9(a)). The sub- metric coordinate system for the yarn is different from a scripts,Iand2 refer to the braider I and the braider 2 local (material) coordinate system. (x1, x2, x3), pertaining respectively. In most cases, the two yarns are the same in to a UD composite, where xi is always along the fiber geometry. Moreover, the inter-yarn gaps are generally axial direction (hence, xi coincides with Y or Y2). Ori made zero(see e.g., Fig. 8)in order to achieve a high entations of these two yarns in the Rve can be com- performance. Thus, only two geometric parameters, i.e., pletely specified by their respective top and bottom the yarn thickness(o)and width(a), are required, which A-A Fig. 9. Schematic diagram of (a)a repeating unit cell of a diamond braid (b)a RvE

fraction), as indicated in Fig. 7(e)–(g). To define inclined angles of the yarn 1and yarn 2 in Fig. 7(e) and (f), their relative positions in the RVE must be identified. 7.2. RVE geometry For the braider 1and the braider 2, we assign two geometric coordinate systems, (X1; Y1; Z1) and (X2; Y2; Z2) respectively, with origins on the respective yarn central planes, as indicated in Fig. 9(b). These two systems are introduced for the convenience of geometric descrip￾tions. It is noted that Y1 and Y2 are along the yarn axial directions. It also deserves mentioning that the geo￾metric coordinate system for the yarn is different from a local (material) coordinate system, (x1; x2; x3), pertaining to a UD composite, where x1 is always along the fiber axial direction (hence, x1 coincides with Y1 or Y2). Ori￾entations of these two yarns in the RVE can be com￾pletely specified by their respective top and bottom surface shape functions, which are expressed in the geometric coordinate systems as [8] (refer to Fig. 10) Zlower 1 ¼ t1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2X1 a1  2 s þ t2 2  cos p Y1 X1 cotð2hÞ 2u1  ; ð30:1Þ Zupper 1 ¼ t1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2X1 a1  2 s þ t2 2  cos p Y1 X1 cotð2hÞ 2u1  ; ð30:2Þ Zlower 2 ¼ t2 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2X2 a2  2 s t1 2  cos p Y2 þ X2 cotð2hÞ 2u2  ; ð31:1Þ Zupper 2 ¼ t2 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2X2 a2  2 s t1 2  cos p Y2 þ X2 cotð2hÞ 2u2  ; ð31:2Þ u1 ¼ a2 þ g2 sinð2hÞ ; and u2 ¼ a1 þ g1 sinð2hÞ : ð32Þ In the above, t is the yarn thickness, a is the yarn width, and g is an inter-yarn gap (Fig. 9(a)). The sub￾scripts, 1and 2 refer to the braider 1and the braider 2 respectively. In most cases, the two yarns are the same in geometry. Moreover, the inter-yarn gaps are generally made zero (see e.g., Fig. 8) in order to achieve a high performance. Thus, only two geometric parameters, i.e., the yarn thickness ðtÞ and width ðaÞ, are required, which Fig. 8. Photograph of a diamond braided fabric structure. Fig. 9. Schematic diagram of (a) a repeating unit cell of a diamond braid (b) a RVE. 1168 Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176