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 inplane 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 inaccurately 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 thermomechanical 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 development 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 attention on those tissues, and on the development of nanoscale biocomposite substitutes. Apparently, any successful use/critical design of synthetic composites or achievement in tissue engineering depends on a thorough understanding for the composite 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 simulated. However, without a good knowledge of the composite 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 structures, and each lamina involved is statically indeterminate (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 constitutive 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 material parameters involved in the model are minimal and can be measured or determined independently. Applications 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 simulation 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-mechanical 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 development and an accompanied one deals with the applications. 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 preform and woven, braided, and knitted fabric structures. One of the most critical factors that influence the composite response is matrix behaviour. A different matrix material may require a different theory for its constitutive 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 lamiFig. 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. However, 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 corresponding overall applied stress on the laminate is de- fined as a progressive failure strength. The remaining laminate is analyzed by discounting the stiffness contribution from the failed lamina. In this way, the ultimate strength of the laminate can be determined incrementally. Thus, a necessary and sufficient condition for understanding 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 discretized into a number of UD composites, see subsequent 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 incremental 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 following 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 Appendix A. In reality, the composite longitudinal property hardly depends on the fiber packing geometry. The corresponding 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 account for this sensitivity, two bridging parameters are incorporated into the corresponding independent bridging 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 inplane 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 parameters 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 between 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