《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-35

Computers structures PERGAMON Computers and Structures 80(2002)1177-1199 On a general constitutive description for the inelastic and failure behavior of fibrous laminates Part ll Laminate theory and applications Zheng-Ming Huang Biomaterials Laboratory, Division of bioengineering, Department of Mechanical Engineering, National University of Singapore, Received 10 April 2001; accepted 6 March 2002 These two parts of papers report systematically a constitutive description for the inelastic and strength behavior of laminated composites reinforced with various fiber preforms. The constitutive relationship is established microme- chanically, through layer-by-layer analysis. Namely, only the properties of the constituent fiber and matrix materials of the composites are required as input data. In the previous part( Comput. Struct(submitted)), the lamina theory was presented. Three fundamental quantities of the laminae, i.e. the internal stresses generated in the constituent fiber and matrix materials and the instantaneous compliance matrix, with different fiber preform (including woven, braided, and knitted fabric) reinforcements were explicitly obtained by virtue of the bridging micromechanics model. In the present paper, the laminate stress analysis is shown. The purpose of this analysis is to determine the load shared by each lamina in the laminate, so that the lamina theory can be applied. Incorporation of the constitutive equations into an FEM tware package is illustrated. A number of application examples are given in the paper to demonstrate the efficiency of the constitutive theory established. The predictions thus made include: failure envelopes of multidirectional laminates subjected to biaxial in-plane loads, thermo-mechanical cycling stress-strain curves of a titanium metal matrix composite laminate, S-n curves of multilayer knitted fabric reinforced laminates under tensile fatigue, and bending load- deflection plots and ultimate bending strengths of laminated braided fabric reinforced beams subjected to lateral loads. All these predictions are based on the constituent properties which were measured or available independently, and are compared with experimental results. Reasonably good correlations have been found in all the cases. It is expected that the present constitutive relationship can benefit the critical design and strength analysis of a primarily loaded structure made of composite materials. o 2002 Published by Elsevier Science Ltd. words Laminated Textile com Metal matrix composite; Composite structure; Mechanical property; Co relationship: In-plane failure: Thermo-mechanical fatigue; Flexural failure; Load-deflection curve, Stiffness discount; Strength prediction; Bridging micromechanics model; FEM structural analysis cuon nized as attractive candidates in most modern industries the usages of them are still relatively limited compared Although fiber reinforced composites have been in with their counterparts, i.e. homogeneous and isotropic practice for nearly half a century, and have been recog- materials such as metals, ceramics, and polymers. A major limitation is in the lack of an efficient and versatile constitutive description for the composite materials [2] Specifically, the composite load carrying capacity has not mailaddresses:huangzm@mail.tongji.edu.cn,huangzm@beenwellunderstood[3,4].Assuch,asignificantad- om亿ZM. Huang) vancement in composite failure theory is necessary. 0045-7949/02/- see front matter o 2002 Published by Elsevier Science Ltd. PI:S0045-7949(02)00075-5

On a general constitutive description for the inelastic and failure behavior of fibrous laminates––Part II: Laminate theory and applications Zheng-Ming Huang Biomaterials Laboratory, Division of bioengineering, Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 Received 10 April 2001; accepted 6 March 2002 Abstract These two parts of papers report systematically a constitutive description for the inelastic and strength behavior of laminated composites reinforced with various fiber preforms. The constitutive relationship is established microme￾chanically, through layer-by-layer analysis. Namely, only the properties of the constituent fiber and matrix materials of the composites are required as input data. In the previous part (Comput. Struct. (submitted)), the lamina theory was presented. Three fundamental quantities of the laminae, i.e. the internal stresses generated in the constituent fiber and matrix materials and the instantaneous compliance matrix, with different fiber preform (including woven, braided, and knitted fabric) reinforcements were explicitly obtained by virtue of the bridging micromechanics model. In the present paper, the laminate stress analysis is shown. The purpose of this analysis is to determine the load shared by each lamina in the laminate, so that the lamina theory can be applied. Incorporation of the constitutive equations into an FEM software package is illustrated. A number of application examples are given in the paper to demonstrate the efficiency of the constitutive theory established. The predictions thus made include: failure envelopes of multidirectional laminates subjected to biaxial in-plane loads, thermo-mechanical cycling stress–strain curves of a titanium metal matrix composite laminate, S–N curves of multilayer knitted fabric reinforced laminates under tensile fatigue, and bending load– deflection plots and ultimate bending strengths of laminated braided fabric reinforced beams subjected to lateral loads. All these predictions are based on the constituent properties which were measured or available independently, and are compared with experimental results. Reasonably good correlations have been found in all the cases. It is expected that the present constitutive relationship can benefit the critical design and strength analysis of a primarily loaded structure made of composite materials.
2002 Published by Elsevier Science Ltd. Keywords: Laminated composite; Textile composite; Metal matrix composite; Composite structure; Mechanical property; Constitutive relationship; In-plane failure; Thermo-mechanical fatigue; Flexural failure; Load–deflection curve; Stiffness discount; Strength prediction; Bridging micromechanics model; FEM structural analysis 1. Introduction Although fiber reinforced composites have been in practice for nearly half a century, and have been recog￾nized as attractive candidates in most modern industries, the usages of them are still relatively limited compared with their counterparts, i.e. homogeneous and isotropic materials such as metals, ceramics, and polymers. A major limitation is in the lack of an efficient and versatile constitutive description for the composite materials [2]. Specifically, the composite load carrying capacity has not been well understood [3,4]. As such, a significant ad￾vancement in composite failure theory is necessary. Computers and Structures 80 (2002) 1177–1199 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 5 - 5

Z-M. Huang/ Computers and Structures 80(2002)1177-1199 The present author has recently proposed a useful shown in Fig. 2 of Ref [ID. This will be done based on the composite theory, the bridging model [5]. By combining classical lamination theory. Suppose that the laminate the bridging model with the classical lamination theory, consists of a sequence of fibrous laminae, stacking in a general constitutive relationship can be established for different ply-angles. Each lamina can have different con- fibrous composite laminate. In this two-parts of stituent materials, different fiber volume fraction, and papers, a systematic description for the general constit even different fiber preform reinforcement. We take a utive theory is presented. In the first part [I], the lamina global coordinate system(X, y, z) with X and y in the analysis based on the bridging model has been described. laminate plane and z along the thickness direction. Let Any single layer continuous fiber reinforced composite the fiber direction of the kth lamina have an inclined ply- lamina was eventually subdivided into a number of uni- angle Ak with the global X direction, as shown in Fig 3 directional (UD) composites to which the bridging model of Ref [l] for a UD lamina. According to the classical can be applied. The response of the original lamina was lamination theory, only in-plane stress and strain in- obtained by means of an assemblage. The purpose of the crements, i. e idol= doxx, do yy, doxy)and deh lamina analysis under an arbitrary load condition was IdExx, dey, 2dexr, are retained, where G refers to the obtain three fundamental quantities, i.e. the instanta global coordinate system. The out-of plane strain com- neous compliance matrix of the lamina, which will be ponents, dExz, dayz, and dez, are assumed to be zero, immediately seen in this paper to be critical for laminate whereas the out-of-plane stress components can be de- analysis, the internal stress state in the constituent fibers, termined using the lamina three-dimensional constitutive and the internal stresses in the constituent matrix mat relationship if necessary. The averaged stress increments rial. Based on the latter two quantities, the lamina load on the kth lamina can be determined from(refer to eq carrying capacity can be understood from the knowledge(13.3)of Ref [ D of the constituent load carrying abilities. In the second part, i.e. the present paper, the lami nate analysis is elaborated. As the laminate consists of doe laminae, the external load applied on the laminate must [(CO)defT -) dT (1) be shared by each lamina involved. This load share can be determined through the classical lamination theor where However, the instantaneous compliance matrix of the lamina is required, since the lamina is a statically inde- terminate structure in the laminate. For the same reason d-{4++2,4 the laminate response can be determined once the lam- ina behavior has been fully understood. Towards this Zk+zk-I dx yr, idexx +(zk+zk-idK purpose, one additional incorporation is of stiffness discount: whenever a lamina fails. its stiffness contribu tion to the remaining laminate must be reduced. a total reduction is adopted in the present paper, which is ver- fied to be pertinent by some experimental evidence Extensive comparisons between simulations and e periments have been demonstrated in the paper using {B={(B1,(B2),(B3)}2=([T1)(S)-2{(xh xample laminates. These examples cover a large variety of laminates reinforced with different fiber preforms and having different constituent materials, under static or depx, dapy, and depr and dxpx, dnr, and dir are the fatigue load conditions, and with or without thermal laminate in- plane strain and curvature increments. Zk load effect. For all these composites, only the material and Zk-I are the Z coordinates of the top and the bottom parameters of their constituent fibers and matrixes are surfaces of the kth lamina. I is the lamina instanta- sed as input data, which are measured independently. neous compliance matrix given in the ply coordinate The predicted ultimate strengths or the stress-strain system, which can be obtained using the formulae pre curves up to failure of all the composite laminates agree sented in Ref. [] for various fiber preform reinforced reasonably well with their experimental counterparts laminae. Specifically, for a UD lamina, its ply coordi- nate system coincides with its local one. Hence, Sk is simply given by Eq (4)of Ref [1]. If the lamina under 2. Laminate analysis ion is a single layer woven/braided fabric composite, [ SI is defined by equation(41)of Ref. Having obtained the instantaneous compliance (see Section 6 of this paper for additional discussion) tries of various fibrous laminae [l], we can now per- Further, Sk is given by equation(43. 1)of Ref [I] if the form the laminate analysis(referring to the flow chart lamina is a single layer knitted fabric composite. The

The present author has recently proposed a useful composite theory, the bridging model [5]. By combining the bridging model with the classical lamination theory, a general constitutive relationship can be established for any fibrous composite laminate. In this two-parts of papers, a systematic description for the general constit￾utive theory is presented. In the first part [1], the lamina analysis based on the bridging model has been described. Any single layer continuous fiber reinforced composite lamina was eventually subdivided into a number of uni￾directional (UD) composites to which the bridging model can be applied. The response of the original lamina was obtained by means of an assemblage. The purpose of the lamina analysis under an arbitrary load condition was to obtain three fundamental quantities, i.e. the instanta￾neous compliance matrix of the lamina, which will be immediately seen in this paper to be critical for laminate analysis, the internal stress state in the constituent fibers, and the internal stresses in the constituent matrix mate￾rial. Based on the latter two quantities, the lamina load carrying capacity can be understood from the knowledge of the constituent load carrying abilities. In the second part, i.e. the present paper, the lami￾nate analysis is elaborated. As the laminate consists of laminae, the external load applied on the laminate must be shared by each lamina involved. This load share can be determined through the classical lamination theory. However, the instantaneous compliance matrix of the lamina is required, since the lamina is a statically inde￾terminate structure in the laminate. For the same reason, the laminate response can be determined once the lam￾ina behavior has been fully understood. Towards this purpose, one additional incorporation is of stiffness discount: whenever a lamina fails, its stiffness contribu￾tion to the remaining laminate must be reduced. A total reduction is adopted in the present paper, which is ver￾ified to be pertinent by some experimental evidence. Extensive comparisons between simulations and ex￾periments have been demonstrated in the paper using example laminates. These examples cover a large variety of laminates reinforced with different fiber preforms and having different constituent materials, under static or fatigue load conditions, and with or without thermal load effect. For all these composites, only the material parameters of their constituent fibers and matrixes are used as input data, which are measured independently. The predicted ultimate strengths or the stress–strain curves up to failure of all the composite laminates agree reasonably well with their experimental counterparts. 2. Laminate analysis Having obtained the instantaneous compliance ma￾trixes of various fibrous laminae [1], we can now per￾form the laminate analysis (referring to the flow chart shown in Fig. 2 of Ref. [1]). This will be done based on the classical lamination theory. Suppose that the laminate consists of a sequence of fibrous laminae, stacking in different ply-angles. Each lamina can have different con￾stituent materials, different fiber volume fraction, and even different fiber preform reinforcement. We take a global coordinate system (X, Y, Z) with X and Y in the laminate plane and Z along the thickness direction. Let the fiber direction of the kth lamina have an inclined ply￾angle hk with the global X direction, as shown in Fig. 3 of Ref. [1] for a UD lamina. According to the classical lamination theory, only in-plane stress and strain in￾crements, i.e. fdrg G ¼ fdrXX ; drYY ; drXY gT and fdegG ¼ fdeXX ; deYY ; 2deXY gT , are retained, where G refers to the global coordinate system. The out-of plane strain com￾ponents, deXZ ; deYZ , and deZZ , are assumed to be zero, whereas the out-of-plane stress components can be de￾termined using the lamina three-dimensional constitutive relationship if necessary. The averaged stress increments on the kth lamina can be determined from (refer to Eq. (13.3) of Ref. [1]) fdrgG k ¼ ð½T  cÞk ð½S P k Þ 1 ð½T  T c ÞfdegG k  fbgG k dT ¼ ½ðCG ij Þk fdegG k  fbgG k dT ; ð1Þ where fdegG k ¼ de 0 XX
þ Zk þ Zk1 2 dj0 XX ; de 0 YY þ Zk þ Zk1 2 dj0 YY ; 2de 0 XY þ ðZk þ Zk1Þdj0 XY T ; ð2:1Þ and fbgG k ¼ fðb1Þ G k ;ðb2Þ G k ;ðb3Þ G k g T ¼ ð½T  cÞk ð½S P k Þ 1 fagk : ð2:2Þ de0 XX ; de0 YY , and de0 XY and dj0 XX , dj0 YY , and dj0 XY are the laminate in-plane strain and curvature increments. Zk and Zk1 are the Z coordinates of the top and the bottom surfaces of the kth lamina. ½S P k is the lamina instanta￾neous compliance matrix given in the ply coordinate system, which can be obtained using the formulae pre￾sented in Ref. [1] for various fiber preform reinforced laminae. Specifically, for a UD lamina, its ply coordi￾nate system coincides with its local one. Hence, ½S P k is simply given by Eq. (4) of Ref. [1]. If the lamina under consideration is a single layer woven/braided fabric composite, ½S P k is defined by equation (41) of Ref. [1] (see Section 6 of this paper for additional discussion). Further, ½S P k is given by equation (43.1) of Ref. [1] if the lamina is a single layer knitted fabric composite. The 1178 Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199

g/ Computers and Structures 80(2002)1177-1199 79 global stresses can be transformed into the ply coordi- dg =2()(Zk-Zk-1)dT, nate system through d}=(){d}, where(do= doar, do,x, do The coordinate trans- d=∑(B}(4-2=)dT formation matrixes in(1)and (3), [Te] and [T), are the same as equations(44)and(45)of Ref [l] but with volved s being replaced by the ply angle 0. Substituting N is the total number of lamina plies in the lami Eq.(3) into the right hand sides of equations(14. 1) and nate. (Cy) are the stiffness elements of the kth lam 14.2)of Ref. [u if this ply is a UD lamina, equations ina in the global coordinate system, see Eq.(D). In (42. 1)and (42. 2)of Ref [I] if the ply is a braided fabric Eq.(5), dNxx, dRy, and dNr and dMxx,dMr,and lamina(see Section 6 for additional discussion), or drr are, respectively, the externally applied in-plane equations(43. 2)and(43.) of Ref [I] if the ply is a force and moment increments per unit length on the knitted fabric lamina, the averaged stress increments in laminate the fiber and matrix phases of this lamina can be eval- Apparently, different lamina ply in the laminate uated. It is thus only necessary to determine the in-plane carries different load share. With the increase of the strains and curvature increments. Note that the internal external load, some ply must fail first before the other stress resultants of Eq. (1)must be balanced with the Once some koth lamina fails, the corresponding overall externally applied forces and moments. For instance, we applied load on the laminate is defined as a progressive must have failure strength(e.g. the first-ply or the second- ply fail ure strength, etc. ) As the ply failure has already been dNax= do dz-2 (do d) z, (4.1) d he ne respon dihe failure or de s automaic ml tea. tified. It is either the matrix fail the fiber frac- ture. or the failure of the both constituents that cause doxxzdz (doxx)Zdz,(4.2) the ply failure. Meanwhile, the failed lamina cannot sustain any additional load share, see the experimen- tal evidence shown in Section 6. The additional exter where h is the thickness of the laminate. Substituting(1) nal load must be shared by the remaining un-failed into ( 4. I)and (4.2), and the like and performing the laminae. Namely, Egs. (4.1)and(4.2)should be modi ing integra fied dNxx +dQ Nry +dQ dOxy+dQ dNa s dora dz dMy+dg dMr +d2 Zdz= (doxx),ZdZ. (8.2) el ol2 @is gll oll o dex Oir o2 gb3 g1 o2) oli de Thus, the post-failure analysis is still based on eq oll ol? @l3 @ll 012 gl3 dxe but with reduced overall stiffness elements and equiva- 望望盟②盟|dkn lent thermal loads which are given by el=>(C c=∑(CG)(-4- G (CG)(42-2=1) g=3∑(cA(n- (CG)(2-2-1)

global stresses can be transformed into the ply coordi￾nate system through fdrgP k ¼ ð½T  T s Þk fdrgG k ; ð3Þ where fdrgP ¼ fdrxx; dryy ; drxyg T . The coordinate trans￾formation matrixes in (1) and (3), ½Tc and ½Ts, are the same as equations (44) and (45) of Ref. [1] but with in￾volved n being replaced by the ply angle h. Substituting Eq. (3) into the right hand sides of equations (14.1) and (14.2) of Ref. [1] if this ply is a UD lamina, equations (42.1) and (42.2) of Ref. [1] if the ply is a braided fabric lamina (see Section 6 for additional discussion), or equations (43.2) and (43.3) of Ref. [1] if the ply is a knitted fabric lamina, the averaged stress increments in the fiber and matrix phases of this lamina can be eval￾uated. It is thus only necessary to determine the in-plane strains and curvature increments. Note that the internal stress resultants of Eq. (1) must be balanced with the externally applied forces and moments. For instance, we must have dNXX ¼ Z h=2 h=2 drXX dZ ¼ XN k¼1 Z Zk Zk1 ðdrXX Þk dZ; ð4:1Þ dMXX ¼ Z h=2 h=2 drXX Z dZ ¼ XN k¼1 Z Zk Zk1 ðdrXX ÞkZ dZ; ð4:2Þ where h is the thickness of the laminate. Substituting (1) into (4.1) and (4.2), and the like and performing the resulting integrations, we obtain the following equation dNXX þ dXI 1 dNYY þ dXI 2 dNXY þ dXI 3 dMXX þ dXII 1 dMYY þ dXII 2 dMXY þ dXII 3 8 >>>>>>>>>>>>>>>>>: 9 >>>>>>>>>= >>>>>>>>>; ¼ QI 11 QI 12 QI 13 QII 11 QII 12 QII 13 QI 12 QI 22 QI 23 QII 12 QII 22 QII 23 QI 13 QI 23 QI 33 QII 13 QII 23 QII 33 QII 11 QII 12 QII 13 QIII 11 QIII 12 QIII 13 QII 12 QII 22 QII 23 QIII 12 QIII 22 QIII 23 QII 13 QII 23 QII 33 QIII 13 QIII 23 QIII 33 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 de0 XX de0 YY 2de0 XY dj0 XX dj0 YY 2dj0 XY 8 >>>>>>>>>>>>>>>>>>>>>: 9 >>>>>>>>>>>= >>>>>>>>>>>; ; ð5Þ QI ij ¼ XN k¼1 ðCG ij Þk ðZk  Zk1Þ; QII ij ¼ 1 2 XN k¼1 ðCG ij Þk ðZ2 k  Z2 k1Þ; QIII ij ¼ 1 3 XN k¼1 ðCG ij Þk ðZ3 k  Z3 k1Þ; ð6Þ dXI i ¼ XN k¼1 ðbiÞ G k ðZk  Zk1ÞdT ; dXII i ¼ 1 2 XN k¼1 ðbi Þ G k ðZ2 k  Z2 k1ÞdT : ð7Þ N is the total number of lamina plies in the lami￾nate. ðCG ij Þk are the stiffness elements of the kth lam￾ina in the global coordinate system, see Eq. (1). In Eq. (5), dNXX ; dNYY , and dNXY and dMXX , dMYY , and dMXY are, respectively, the externally applied in-plane force and moment increments per unit length on the laminate. Apparently, different lamina ply in the laminate carries different load share. With the increase of the external load, some ply must fail first before the others. Once some k0th lamina fails, the corresponding overall applied load on the laminate is defined as a progressive failure strength (e.g. the first-ply or the second-ply fail￾ure strength, etc.). As the ply failure has already been defined upon the failure of one constituent material, the corresponding failure mode is automatically iden￾tified. It is either the matrix failure, or the fiber frac￾ture, or the failure of the both constituents that causes the ply failure. Meanwhile, the failed lamina cannot sustain any additional load share, see the experimen￾tal evidence shown in Section 6. The additional exter￾nal load must be shared by the remaining un-failed laminae. Namely, Eqs. (4.1) and (4.2) should be modi- fied to dNXX ¼ Z h=2 h=2 drXX dZ ¼ XN k¼1 k6¼k0 Z Zk Zk1 ðdrXX Þk dZ; ð8:1Þ dMXX ¼ Z h=2 h=2 drXX Z dZ ¼ XN k¼1 k6¼k0 Z Zk Zk1 ðdrXX ÞkZ dZ: ð8:2Þ Thus, the post-failure analysis is still based on Eq. (5), but with reduced overall stiffness elements and equiva￾lent thermal loads which are given by QI ij ¼ XN k¼1 k62fk0g ðCG ij Þk ðZk  Zk1Þ; QII ij ¼ 1 2 XN k¼1 k62fk0g ðCG ij Þk ðZ2 k  Z2 k1Þ; QIII ij ¼ 1 3 XN k¼1 k62fk0g ðCG ij Þk ðZ3 k  Z3 k1Þ; ð9Þ Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199 1179

Z-M. Huang / Comput and Structures 80(2002)1177-1199 the failure behavior and ultimate strength of laminated dg2=∑(MB)(4-2-)dT, composites reinforced with various fiber preforms and (10) having dierent constituent materials. In the present dg=∑(B)(z-2-)dT are analyzed. a number of multidirectional laminates were subjected to in-plane biaxial loads. Those problems were designed to verify and compare the efficiency of the In the above, kol represents all those laminae which outmost composite strength theories in the current lit have already failed and the symbol"stands for"not erature. No experimental data for any of the exercised belonging to. Continued in this way, the ultimate fail laminates were given in Ref [6]. Predictions from the ure strength of the laminate is determined using the most famous theories in current use were presented in overall applied load at which all the plies have failed. Ref [4]. Although no comparison with experiments was It must be pointed out that the stiffness reduction, as able to make, one can still recognize that quite large per Eqs.(9)and (10), up to the last ply failure is appli- discrepancy existed among the predictions of the differ- load. If, however, the laminate is involved with a lateral noted that almost all of the theories taken part in the (out-of-plane) load, the stiffness reduction must be per exercises were phenom gical [3, 4]. Namely, a com- formed more carefully. For instance, let the laminate be plete set of critical strength parameters as well as elastic subjected to only a bending load. Using the stiffness properties measured from the composite (lamina or lam- reduction procedure given in Eqs. (9)and (10), the inate)had been employed in the predictions. In the con- predicted failure process (as well as load-deflection trast, the present constitutive relationship makes use of curve)may no longer be correct after the failure of an the constituent materials intermediate ply, which corresponds to the ultimate Four material systems, i.e. the E-Glass 21 x K43 failure. The reason is that in reality the ultimate bending Gevetex and LY556/HT907/DY063 epoxy, the Sile- load (which is defined as the maximum load that the nka E-glass 1200tex and Mr750/HY917/DY063 epoxy laminate can sustain) is generally attained by the lami- the graphite AS4 and 3501-6 epoxy, and the graph nate before its last ply failure. After the ultimate bending ite T300 and BSL914C epoxy, were used in the exercise ad, the load-deflection curve of the laminated beam is [6]. The objective is to predict the failure envelop loa downward, a phen named material softening. or stress-strain response of the laminates up to failure However, using the incremental solution strategy de- scribed above the predicted load-deflection curve is always upward till the last-ply failure. Under the pure UD laminae together with their fiber volume fractions bending condition, the middle plane strain increm used for the exercise problems have been specified in deox etc, are negligibly small. The remaining bending Ref. [6]. They are summarized in Tables 1-4. In the curvature will have very little, if any, stress contribution present calculations, all the fibers are assumed to be to the last ply failure or the last two-ply failures, ac linearly elastic until rupture. The matrix of each UD cording to Eqs. (1),(2.1), and(2.2). Thus, the last ply lamina, however, must be considered as elastic-plastic will not fail at all, but the deflection(curvature)can be This is because the in-plane shear stress-shear strain increased unlimitedly. In a subsequent example, we will curve of every lamina provided in Ref. [6]is nonlinear see experimental evidence that some plies of the beam From Eq.(4)of Ref. [I], we can clearly see that the could not be forced to failure due to the limitation of the composite can display a nonlinear deformation if and testing apparatus to excessive beam deflection. Namely, only if at least one of its constituent materials undergoes only partial layers of the laminated beam failed after the such a deformatio bending test. This is consistent with our theory. This All the thermo-elastic properties of the constituent also indicates that if a fiexural load is involved, the only materials of the four laminae as well as their fiber vol- use of a stress failure criterion will not be enough to ume fractions are remained the same as those provided determine the ultimate failure; an additional critical in Ref. [6]. However, the bridging parameters a and B order that the ultimate strength of the laminate can be used in defining the independent bridging elements a33 deflection/curvature condition has to be adopted and a2?(see Eq (I1) of Ref [I] have to be calibrated against the transverse and in-plane shear moduli of the laminae in such a way that the predicted moduli agree airly well with the measured values. The parameter B has 3. Biaxial failure of multidirectional laminates been chosen as 0.45 for all the four laminae. whereas the parameters a has been taken to be 0.3 for one lamina and In this and the subsequent sections, we will apply the 0.35 for the others, as indicated in Tables 1-4. Both the general constitutive relationship developed to identify ridging parameters of each lamina remain unchanged

dXI i ¼ XN k¼1 k62fk0g ðbiÞ G k ðZk  Zk1ÞdT ; dXII i ¼ 1 2 XN k¼1 k62fk0g ðbiÞ G k ðZ2 k  Z2 k1ÞdT : ð10Þ In the above, fk0g represents all those laminae which have already failed and the symbol ‘‘62’’ stands for ‘‘not belonging to’’. Continued in this way, the ultimate fail￾ure strength of the laminate is determined using the overall applied load at which all the plies have failed. It must be pointed out that the stiffness reduction, as per Eqs. (9) and (10), up to the last ply failure is appli￾cable only when the laminate is subjected to an in-plane load. If, however, the laminate is involved with a lateral (out-of-plane) load, the stiffness reduction must be per￾formed more carefully. For instance, let the laminate be subjected to only a bending load. Using the stiffness reduction procedure given in Eqs. (9) and (10), the predicted failure process (as well as load–deflection curve) may no longer be correct after the failure of an intermediate ply, which corresponds to the ultimate failure. The reason is that in reality the ultimate bending load (which is defined as the maximum load that the laminate can sustain) is generally attained by the lami￾nate before its last ply failure. After the ultimate bending load, the load–deflection curve of the laminated beam is downward, a phenomenon named material softening. However, using the incremental solution strategy de￾scribed above, the predicted load–deflection curve is always upward till the last-ply failure. Under the pure bending condition, the middle plane strain increments, de0 XX etc., are negligibly small. The remaining bending curvature will have very little, if any, stress contribution to the last ply failure or the last two-ply failures, ac￾cording to Eqs. (1), (2.1), and (2.2). Thus, the last ply will not fail at all, but the deflection (curvature) can be increased unlimitedly. In a subsequent example, we will see experimental evidence that some plies of the beam could not be forced to failure due to the limitation of the testing apparatus to excessive beam deflection. Namely, only partial layers of the laminated beam failed after the bending test. This is consistent with our theory. This also indicates that if a flexural load is involved, the only use of a stress failure criterion will not be enough to determine the ultimate failure; an additional critical deflection/curvature condition has to be adopted in order that the ultimate strength of the laminate can be assessed. 3. Biaxial failure of multidirectional laminates In this and the subsequent sections, we will apply the general constitutive relationship developed to identify the failure behavior and ultimate strength of laminated composites reinforced with various fiber preforms and having different constituent materials. In the present section, the failure exercise problems as posed in Ref. [6] are analyzed. A number of multidirectional laminates were subjected to in-plane biaxial loads. Those problems were designed to verify and compare the efficiency of the outmost composite strength theories in the current lit￾erature. No experimental data for any of the exercised laminates were given in Ref. [6]. Predictions from the most famous theories in current use were presented in Ref. [4]. Although no comparison with experiments was able to make, one can still recognize that quite large discrepancy existed among the predictions of the differ￾ent theories for any of the exercise problems [4]. It is noted that almost all of the theories taken part in the exercises were phenomenological [3,4]. Namely, a com￾plete set of critical strength parameters as well as elastic properties measured from the composite (lamina or lam￾inate) had been employed in the predictions. In the con￾trast, the present constitutive relationship makes use of the constituent materials. Four material systems, i.e. the E-Glass 21  K43 Gevetex and LY556/HT907/DY063 epoxy, the Sile￾nka E-glass 1200tex and MY750/HY917/DY063 epoxy, the graphite AS4 and 3501-6 epoxy, and the graph￾ite T300 and BSL914C epoxy, were used in the exercise [6]. The objective is to predict the failure envelope or stress–strain response of the laminates up to failure [6]. Measured properties of the fibers, matrixes, and four UD laminae together with their fiber volume fractions used for the exercise problems have been specified in Ref. [6]. They are summarized in Tables 1–4. In the present calculations, all the fibers are assumed to be linearly elastic until rupture. The matrix of each UD lamina, however, must be considered as elastic–plastic. This is because the in-plane shear stress–shear strain curve of every lamina provided in Ref. [6] is nonlinear. From Eq. (4) of Ref. [1], we can clearly see that the composite can display a nonlinear deformation if and only if at least one of its constituent materials undergoes such a deformation. All the thermo-elastic properties of the constituent materials of the four laminae as well as their fiber vol￾ume fractions are remained the same as those provided in Ref. [6]. However, the bridging parameters a and b used in defining the independent bridging elements a33 and a22 (see Eq. (11) of Ref. [1]) have to be calibrated against the transverse and in-plane shear moduli of the laminae in such a way that the predicted moduli agree fairly well with the measured values. The parameter b has been chosen as 0.45 for all the four laminae, whereas the parameters a has been taken to be 0.3 for one lamina and 0.35 for the others, as indicated in Tables 1–4. Both the bridging parameters of each lamina remain unchanged 1180 Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199

Z-M. Huang /Computers and Structures 80(2002)1177-1199 Table I Properties of E-glass 21 x K43 Gevetex and LY556/HT907/DY063 epoxy UD lamina Fiber Provided Predicted Provided Used Provided Used Eu(gPa) Ex(GPa 177 4.38 3.35 3.35 Gn(Gpa) 5.72 33.33 33 l.24 0.278 0.35 GY),(MPa)y (ar)2(MP (aY)3(MPa) 44.7 (OY)(MPa) (aY)s(MPa) (ar),(MPa)a GY)(mPa) (ET),(mPa)a Er),(MPa)a ET)(MPa) TTT 1140 2150 9089 55.7 (MPa) MPar 114 GS( (MPay 2.132 2.227 0.692 0.644 1087 2a12(% 7.638 x2(×10-O)26.4 9 58 20.478 (MPa) 62000 -6.863 l28 Fiber volume fraction: Vr =0.62; stress-free temperature: 120C; working temperature: 25C: bridging parameters used: B=0.45 and Assumed to be the same in both tension and compression Longitudinal compression d transverse tension Transverse compression. subsequent predictions Predicted thermal residual stress, from stress-free temperature to working temperature in the prediction for the whole response of the corre- hand, the in-plane shear stress-shear strain curves of the sponding laminate tems were provided [6]. These curves are used to retry UD laminae fabricated from all the four material sy strength and hardening modulus) measured using the plastic parameters of the respective matrix material a It is expected that the matrix plastic parameters(yield monolithic material specimens can be directly employed Compared with a transverse stress-strain response, the just as in the employment of the constituent thermo- in-plane shear stress curve generally displays more elastic properties. However, no detailed information distinct nonlinear behavior. It is noted that whenever about the matrix plasticity was given [6]. On the other possible the matrix not

in the prediction for the whole response of the corre￾sponding laminate. It is expected that the matrix plastic parameters (yield strength and hardening modulus) measured using monolithic material specimens can be directly employed, just as in the employment of the constituent thermo￾elastic properties. However, no detailed information about the matrix plasticity was given [6]. On the other hand, the in-plane shear stress–shear strain curves of the UD laminae fabricated from all the four material sys￾tems were provided [6]. These curves are used to retrieve the plastic parameters of the respective matrix materials. Compared with a transverse stress–strain response, the in-plane shear stress–strain curve generally displays more distinct nonlinear behavior. It is noted that whenever possible the matrix plastic parameters should not be Table 1 Properties of E-glass 21  K43 Gevetex and LY556/HT907/DY063 epoxy UD lamina Properties Lamina Fiber Resin Provided Predicted Provided Used Provided Used E11 (GPa) 53.48 50.87 80 80 3.35 3.35 E22 (GPa) 17.7 14.38 80 80 3.35 3.35 G12 (Gpa) 5.83 5.72 33.33 33.33 1.24 1.24 m12 0.278 0.257 0.2 0.2 0.35 0.35 ðrYÞ1 (MPa)a – – – – – 31 .9 ðrYÞ2 (MPa)a – – – – – 38.4 ðrYÞ3 (MPa)a – – – – – 44.7 ðrYÞ4 (MPa)a – – – – – 49.9 ðrYÞ5 (MPa)a – – – – – 53.6 ðrYÞ6 (MPa)a – – – – – 56.1 ðrYÞ7 (MPa)a – – – – – 58.1 ðrYÞ8 (MPa)a – – – – – 60.0 ðETÞ1 (MPa)a – – – – – 1 566 ðETÞ2 (MPa)a – – – – – 1 337 ðETÞ3 (MPa)a – – – – – 944 ðETÞ4 (MPa)a – – – – – 584 ðETÞ5 (MPa)a – – – – – 338 ðETÞ6 (MPa)a – – – – – 245 ðETÞ7 (MPa)a – – – – – 1 97 rL u (MPa)b 1140 1140 2150 1804.1 80 56.5 rL u;c (MPa)c 570 570 1450 908.9 120 55.7 rT u (MPa)d 35 72 – – – – rT u;c (MPa)e 1 1 41 1 4– – – – rS u (MPa)f 72 84.8 –––– e11 (%)b 2.132 2.227 – 5 5.5434g e11;c (%)c 1.065 1.123 –––– e22 (%)d 0.197 0.692 –––– e22;c (%)e 0.644 1.087 –––– 2e12 (%)f 3.8 7.638 –––– a1ð106/C) 8.6 6.23 4.9 4.9 58 58 a2ð106/C) 26.4 20.62 4.9 4.9 58 58 r11 (MPa)h –0– 12.55g – 20.47g r22 (MPa)h –0– 6.86g – 11.2g r12 (MPa)h –0–0–0 Fiber volume fraction: Vf ¼ 0:62; stress-free temperature: 120 C; working temperature: 25 C; bridging parameters used: b ¼ 0:45 and a ¼ 0:35. a Assumed to be the same in both tension and compression. bLongitudinal tension. c Longitudinal compression. dTransverse tension. e Transverse compression. f In-plane shearing. g Not for use in subsequent predictions. h Predicted thermal residual stress, from stress-free temperature to working temperature. Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199 1181

Z.M. Huang /Computers and Structures 80(2002)1177-1199 Table 2 Properties of Silenka E-glass 1200tex and MY750/HY917/DY063 epoxy UD lamina Properties Lamina Provided Predicted Provided Used Provided 45.6 Ex(gPa 16.2 13.45 3.35 3.35 30.8 30.8 124 0.278 0.2 0.35 0.35 GY),(MPa) (ar)2(MPa (aY)3(MPa 6.8 (OY)(MPa (aY)s(MP 56 (ay)(MP (ar),(MPa)a EEEEEE PPPPPPP (ET),(mPa)a .0882746 ET),(MPa) 80 2150 2092.8 9 74.8 8.8 aT(MPa GS(MP 77048 ElL.c (%)c 0.246 212(%y x1(×10-6°C) 9 x2(×10-O)264 2962000 4.9 13.558 880 02(MPa产 7.32 10.988 Fiber volume fraction: Vr=0.60: stress-free temperature: 120C; working temperature: 25C: bridging parameters used: B=0.45 and a assumed to be the same in both tension and compression b Longitudinal tension d transverse tension Transverse compression. In-plane shearing. Not for use in subsequent predictions. Predicted thermal residual stress, from stress-free temperature to working temperature back calculated based on the overall longitudinal re- ulus of the matrix at a specific loading range was spec- sponse of the composite, since in some cases the matrix ified as material may not sustain a full load share (i.e. to its maximum load carry ning capacit) when the composite is=(=),when()≤四≤(四) longitudinally loaded up to failure. The retrieved uni- axial stress-strain curve of each matrix was assumed to consist of eight linear segments. Hence, hardening mod- (ET)o=Em,oY)=0

back calculated based on the overall longitudinal re￾sponse of the composite, since in some cases the matrix material may not sustain a full load share (i.e. to its maximum load carrying capacity) when the composite is longitudinally loaded up to failure. The retrieved uni￾axial stress–strain curve of each matrix was assumed to consist of eight linear segments. Hence, hardening mod￾ulus of the matrix at a specific loading range was spec￾ified as Em T ¼ Em T   i ; when rm Y   i 6 rm e 6 rm Y   iþ1; i ¼ 0; 1; ... ; 7; Em T   0 ¼ Em; rm Y   0 ¼ 0; ð11:1Þ Table 2 Properties of Silenka E-glass 1200tex and MY750/HY917/DY063 epoxy UD lamina Properties Lamina Fiber Resin Provided Predicted Provided Used Provided Used E11 (GPa) 45.6 45.74 74 74 3.35 3.35 E22 (GPa) 16.2 13.45 74 74 3.35 3.35 G12 (Gpa) 5.83 5.31 30.8 30.8 1.24 1.24 m12 0.278 0.26 0.2 0.2 0.35 0.35 ðrYÞ1 (MPa)a ––––– 32.6 ðrYÞ2 (MPa)a ––––– 39.9 ðrYÞ3 (MPa)a ––––– 46.8 ðrYÞ4 (MPa)a – – – – – 52 ðrYÞ5 (MPa)a ––––– 55.6 ðrYÞ6 (MPa)a – – – – – 58 ðrYÞ7 (MPa)a ––––– 60.1 ðrYÞ8 (MPa)a ––––– 62.0 ðETÞ1 (MPa)a – – – – – 1 698 ðETÞ2 (MPa)a – – – – – 1 387 ðETÞ3 (MPa)a – – – – – 91 8 ðETÞ4 (MPa)a – – – – – 542 ðETÞ5 (MPa)a – – – – – 31 7 ðETÞ6 (MPa)a – – – – – 244 ðETÞ7 (MPa)a – – – – – 1 86 rL u (MPa)b 1280 1280 2150 2092.8 80 60.9 rL u;c (MPa)c 800 800 1450 1311.8 120 74.8 rT u (MPa)d 40 78.8 – – – – rT u;c (MPa)e 145 145 – – – – rS u (MPa)f 73 90 – – – – e11 (%)b 2.807 2.820 – – 5 5.7704g e11;c (%)c 1.754 1.749 – – – – e22 (%)d 0.246 0.835 – – – – e22;c (%)e 1.2 2.918 – – – – 2e12 (%)f 4 9.875 – – – – a1ð106/C) 8.6 6.46 4.9 4.9 58 58 a2ð106/C) 26.4 21.67 4.9 4.9 58 58 r11 (MPa)h –0– 13.55g – 20.33g r22 (MPa)h –0– 7.32g – 10.98g r12 (MPa)h –0–0–0 Fiber volume fraction: Vf ¼ 0:60; stress-free temperature: 120 C; working temperature: 25 C; bridging parameters used: b ¼ 0:45 and a ¼ 0:35. a Assumed to be the same in both tension and compression. bLongitudinal tension. c Longitudinal compression. dTransverse tension. e Transverse compression. f In-plane shearing. g Not for use in subsequent predictions. h Predicted thermal residual stress, from stress-free temperature to working temperature. 1182 Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199

Z-M. Huang Computers and Structures 80(2002)1177-1199 Table 3 Properties of graphite AS4 and 3501-6 UD lamina Properties Provided Predicted Provided Used Provided Used Eu(gPa) Ex(GPa 9.23 Gn(Gpa) 6.6 1567 0.256 GY),(MPa)y (ar)2(MP 4l.8 (aY)3(MPa) (OY)(MPa) (aY)s(MPa) 4.0 (ar),(MPa)a 12 GY)(mPa) (ET),(mPa)a 507 Er),(MPa)a 2530 2072 ET)(MPa) TTT 1950 3206.4 1480 2458.6 1164 (MPa) 659 MPar GS( (MPay 98.8 1420 2.5424 1.175 0.436 3.404 2a12(% -0.5 45 45 -22.238 (MPa) 02000 5533 9.063 Fiber volume fraction: Vr =0.60: stress-free temperature: 177C; working temperature: 25C: bridging parameters used: B=0.45 and Assumed to be the same in both tension and compression Longitudinal compression d transverse tension Transverse compression. subsequent predictions Predicted thermal residual stress, from stress-free temperature to working temperature and理=()y, when a≥(q) (11.2) that the predicted plastic strain at this stress level was equal to 0.02 CY/Em. The retrieved matrix plastic pa- The recovering was performed as follows. Starting from rameters of the four laminae are summarized in Tables the given elastic modulus, Em the remaining hardening 1-4, respectively moduli of the matrix were adjusted in such a way that It is evident that the most important parameters the predicted in-plane shear stress-shear strain curve the present modeling approach to the composite strength was as close to the provided one as possible. The yield are the ultimate strength data of the constituent ma- strength, a=(o), was determined using a condition terials. The tensile and compressive strengths of the

and Em T ¼ Em T   7; when rm e P rm Y   8: ð11:2Þ The recovering was performed as follows. Starting from the given elastic modulus, Em the remaining hardening moduli of the matrix were adjusted in such a way that the predicted in-plane shear stress–shear strain curve was as close to the provided one as possible. The yield strength, rm Y ðrm YÞ1, was determined using a condition that the predicted plastic strain at this stress level was equal to 0.02 rm Y=Em. The retrieved matrix plastic pa￾rameters of the four laminae are summarized in Tables 1–4, respectively. It is evident that the most important parameters in the present modeling approach to the composite strength are the ultimate strength data of the constituent ma￾terials. The tensile and compressive strengths of the Table 3 Properties of graphite AS4 and 3501-6 epoxy UD lamina Properties Lamina Fiber Resin Provided Predicted Provided Used Provided Used E11 (GPa) 126 136.7 225 225 4.2 4.2 E22 (GPa) 11 9.23 15 15 4.2 4.2 G12 (Gpa) 6.6 5.54 15 15 1.567 1.567 m12 0.28 0.256 0.2 0.2 0.34 0.34 ðrYÞ1 (MPa)a – – – –– 38.1 ðrYÞ2 (MPa)a – – – – – 41 .8 ðrYÞ3 (MPa)a – – – –– 46.1 ðrYÞ4 (MPa)a – – – –– 50.1 ðrYÞ5 (MPa)a – – – –– 54.0 ðrYÞ6 (MPa)a – – – –– 57.6 ðrYÞ7 (MPa)a – – – – – 61 .2 ðrYÞ8 (MPa)a – – – –– 64.6 ðETÞ1 (MPa)a – – – –– 2507 ðETÞ2 (MPa)a – – – –– 2530 ðETÞ3 (MPa)a – – – –– 2072 ðETÞ4 (MPa)a – – – – – 1 721 ðETÞ5 (MPa)a – – – – – 1 409 ðETÞ6 (MPa)a – – – – – 1 202 ðETÞ7 (MPa)a – – – – – 991 rL u (MPa)b 1950 1950 3350 3206.4 69 65.6 rL u;c (MPa)c 1480 1480 2500 2458.6 250 116.4 rT u (MPa)d 48 65.9 – – – – rT u;c (MPa)e 200 200 – – – – rS u (MPa)f 79 98.8 – – 50 – e11 (%)b 1.38 1.420 – – 1.7 2.5424g e11;c (%)c 1.175 1.084 – – – – e22 (%)d 0.436 0.792 – – – – e22;c (%)e 2.0 3.404 – – – – 2e12 (%)f 2 3.187 – – – – a1ð106/C) 10.06 0.5 0.5 45 45 a2ð106/C) 26 27.9 15 15 45 45 r11 (MPa)h –0– 22.23g – 33.35g r22 (MPa)h –0– 9.06g – 13.6g r12 (MPa)h – 0 – 0– 0 Fiber volume fraction: Vf ¼ 0:60; stress-free temperature: 177 C; working temperature: 25 C; bridging parameters used: b ¼ 0:45 and a ¼ 0:3. a Assumed to be the same in both tension and compression. bLongitudinal tension. c Longitudinal compression. dTransverse tension. e Transverse compression. f In-plane shearing. g Not for use in subsequent predictions. h Predicted thermal residual stress, from stress-free temperature to working temperature. Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199 1183

Z-M. Huang / Computers and Structures 80(2002)1177-1199 Table 4 Properties of graphite T300 and bSL914C epoxy UD lamina Provided Predicted Provided Used Provided Used Ex(gPa 0.35 (ar)2(MPa (aY)3(MPa (OY)(MPa (aY)s(MP EEEEEE TTTTTT PPPPPPPP 1500 00 14994 l16.8 MPa)d aT(MPa GS(MP 81.2 1087 ElL.c (%)c 0.652 0.245 0.641 2c(%)° 1.818 212(%y 4.578 x1(×10-6/C) 0.06 -0.7 -0.7 554 02(MPa产 000 -7499 11243 Fiber volume fraction: Vr=0.60: stress-free temperature: 120C; working temperature: 25C: bridging parameters used: B=0.45 and a assumed to be the same in both tension and compression b Longitudinal tension ransverse tension Transverse compression. s Not for use in subsequent predictions. Predicted thermal residual stress, from stress-free temperature to working temperature constituents are retrieved from the ultimate strengths of(-570)MPa, a transverse tension of 35 MPa, a trans- the Ud laminae. The retrieval is accomplished by ap- plying the ultimate uniaxial loads to the respective Ud shearing of 72 MPa to the E-Glass 21 x K43 Gevetex lamina. The resulting largest and smallest maximum and and ly556/Ht907/DY063 epoxy Ud lamina with a fiber minimum stresses in the fiber and matrix materials are volume fraction of 0.62(see table 1), the calculated defined as their respective tensile and compressive maximum or minimum stresses in the fiber and matrix trengths. For example, when applying a longitudinal materials are: omax= 1804. 1 MPa and om.=56.5 MPa tension of 1140 MPa, a longitudinal compression of(corresponding to the longitudina ension). 0

constituents are retrieved from the ultimate strengths of the UD laminae. The retrieval is accomplished by ap￾plying the ultimate uniaxial loads to the respective UD lamina. The resulting largest and smallest maximum and minimum stresses in the fiber and matrix materials are defined as their respective tensile and compressive strengths. For example, when applying a longitudinal tension of 1140 MPa, a longitudinal compression of (570) MPa, a transverse tension of 35 MPa, a trans￾verse compression of (114) MPa, and an in-plane shearing of 72 MPa to the E-Glass 21  K43 Gevetex and LY556/HT907/DY063 epoxy UD lamina with a fiber volume fraction of 0.62 (see Table 1), the calculated maximum or minimum stresses in the fiber and matrix materials are: rf max ¼ 1804:1MPa and rm max ¼ 56:5 MPa (corresponding to the longitudinal tension), rf min ¼ Table 4 Properties of graphite T300 and BSL914C epoxy UD lamina Properties Lamina Fiber Resin Provided Predicted Provided Used Provided Used E11 (GPa) 138 139.6 230 230 4.0 4.0 E22 (GPa) 11 9.09 15 15 4.0 4.0 G12 (Gpa) 5.5 5.04 15 15 1.481 1.481 m12 0.28 0.26 0.2 0.2 0.35 0.35 ðrYÞ1 (MPa)a – – – – – 41 .6 ðrYÞ2 (MPa)a – – – –– 49.6 ðrYÞ3 (MPa)a – – – –– 55.8 ðrYÞ4 (MPa)a – – – –– 59.9 ðrYÞ5 (MPa)a – – – –– 63.1 ðrYÞ6 (MPa)a – – – –– 66.3 ðrYÞ7 (MPa)a – – – –– 68.9 ðrYÞ8 (MPa)a – – – – – 71 .4 ðETÞ1 (MPa)a – – – – – 201 5 ðETÞ2 (MPa)a – – – – – 1 384 ðETÞ3 (MPa)a – – – – – 769 ðETÞ4 (MPa)a – – – – – 548 ðETÞ5 (MPa)a – – – – – 457 ðETÞ6 (MPa)a – – – – – 324 ðETÞ7 (MPa)a – – – – – 275 rL u (MPa)b 1500 1500 2500 2462.5 75 56.4 rL u;c (MPa)c 900 900 2000 1499.4 150 116.8 rT u (MPa)d 27 55.8 – – – – rT u;c (MPa)e 200 200 – – – – rS u (MPa)f 80 81.2 – – 70 – e11 (%)b 1.087 1.073 – – 4 5.4139g e11;c (%)c 0.652 0.645 – – – – e22 (%)d 0.245 0.641– – – – e22;c (%)e 1.818 6.891 – – – – 2e12 (%)f 4 4.578 – – – – a1ð106/C) 1 0.06 0.7 0.7 55 55 a2ð106/C) 26 29.6 12 12 55 55 r11 (MPa)h –0– 16.59g – 24.89g r22 (MPa)h –0– 7.49g – 11.24g r12 (MPa)h – 0 – 0– 0 Fiber volume fraction: Vf ¼ 0:60; stress-free temperature: 120 C; working temperature: 25 C; bridging parameters used: b ¼ 0:45 and a ¼ 0:35. a Assumed to be the same in both tension and compression. bLongitudinal tension. c Longitudinal compression. dTransverse tension. e Transverse compression. f In-plane shearing. g Not for use in subsequent predictions. h Predicted thermal residual stress, from stress-free temperature to working temperature. 1184 Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199

Z.M. Huang/ Computers and Structures 80(2002 )1177-119 l185 908.9 MPa and omn =-37.6 MPa( to the longitudinal compression), ofmax=43. 8 MPa and om,=20.7 MPa (to the transverse tension ofin=-90. 1 MPa and heory (fnal faure) -55.7 MPa(to th pression), ane =94.9 MPa and amax= 34.7 MPa(to the in-plane 9 earing). Choosing the largest and the smallest values of them, the fiber and the matrix tensile and compressive strengths are determined as: of= 1804. 1 MPa, ol 55.7 MPa. The fiber and matrix strengths of the other three material ems are determined similarly. Results are given in tables 1-4 -to Using the material parameters of Tables 1-4, the strength envelopes or stress-strain responses up to fail- x-directional stress(MPa) ure of the exercised laminates as posed in Ref [6] can be Fig. 2. Measured and predicted ultimate biaxial failure stresses easily predicted. For simplicity, only the results of four for [900/+ 30] laminate subjected to combined ou and txy laminates, each made of a different material system, are Material system is taken from Table I presented here, as shown in Fig. I for the exercise roblem 2, in Fig. 2 for the problem 5, in Fig. 3 for the problem 7, and in Fig. 4 for the problem 9. Here, the problem number corresponds to that specified in Ref. 匚.m [6]. see also the illustration to each of Figs. 1-4. More details can refer to Ref. 7 Fortunately, the experimental data for the exer- cised problems are now available, and are plotted in Figs. 1-4 for comparison. These data were provided by he exercise organizers to the author once the predic tions, Ref. [7, had been finished. It is seen that the agreement between the theory and the experiments is 4. Cyclic response of MMC laminate under thermome- chanical load Fig. 3. Measured and predicted ow VS Ey and o,vS. err curves for90°/±45°/0°, laminate under uniaxial tension(ay/(a 1/0). Material system en from Table 3 In this section, we consider a cross-ply metal matrix omposite(MMC) tape laminate, 0/90, subjected to Theory gina failure ctional stress (MPal stress(MPa Fig. 1. Predicted and measured ultimate biaxial failure stresse Fig. 4. Measured and predicted ultimate biaxial failure stresses for 0 lamina subjected to combined Ou and tay. The material for[+55 laminate subjected to combined oy and our. Material system is taken from Table 4 system is taken from Table 2

908:9 MPa and rm min ¼ 37:6 MPa (to the longitudinal compression), rf max ¼ 43:8 MPa and rm max ¼ 20:7 MPa (to the transverse tension), rf min ¼ 90:1MPa and rm min ¼ 55:7 MPa (to the transverse compression), and rf max ¼ 94:9 MPa and rm max ¼ 34:7 MPa (to the in-plane shearing). Choosing the largest and the smallest values of them, the fiber and the matrix tensile and compressive strengths are determined as: rf u ¼ 1804:1MPa, rf u;c ¼ 908:9 MPa, rm u ¼ 56:5 MPa, and rm u;c ¼ 55:7 MPa. The fiber and matrix strengths of the other three material systems are determined similarly. Results are given in Tables 1–4. Using the material parameters of Tables 1–4, the strength envelopes or stress–strain responses up to fail￾ure of the exercised laminates as posed in Ref. [6] can be easily predicted. For simplicity, only the results of four laminates, each made of a different material system, are presented here, as shown in Fig. 1for the exercise problem 2, in Fig. 2 for the problem 5, in Fig. 3 for the problem 7, and in Fig. 4 for the problem 9. Here, the problem number corresponds to that specified in Ref. [6], see also the illustration to each of Figs. 1–4. More details can refer to Ref. [7]. Fortunately, the experimental data for the exer￾cised problems are now available, and are plotted in Figs. 1–4 for comparison. These data were provided by the exercise organizers to the author once the predic￾tions, Ref. [7], had been finished. It is seen that the agreement between the theory and the experiments is reasonable. 4. Cyclic response of MMC laminate under thermome￾chanical load In this section, we consider a cross-ply metal matrix composite (MMC) tape laminate, ½0=90 2s, subjected to Fig. 1. Predicted and measured ultimate biaxial failure stresses for 0 lamina subjected to combined rxx and sxy . The material system is taken from Table 4. Fig. 2. Measured and predicted ultimate biaxial failure stresses for ½90=  30 s laminate subjected to combined rxx and sxy . Material system is taken from Table 1. Fig. 3. Measured and predicted ryy vs. eyy and ryy vs. xx curves for ½90=  45=0 s laminate under uniaxial tension ðryy=rxx ¼ 1=0Þ. Material system is taken from Table 3. Fig. 4. Measured and predicted ultimate biaxial failure stresses for ½55 s laminate subjected to combined ryy and rxx. Material system is taken from Table 2. Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199 1185

Z.M. Huang / Computers and Structures 80(2002)1177-1199 both thermal and in-plane mechanical fatigue loads. The ( composite was made from ceramic SCS-6 fibers and TIMETAL 21S matrix, with a fiber volume fraction of 0.35[ 8]. Both the constituent materials are isotropic where (delg is obtained from Eq.(2. 1). Calculate in- Whereas the SCs-6 fibers are linearly elastic until rup- equations. (14. 1) and(14.2) of Ref. [i. and update ture. the titanium TIMetAL 2IS matrix exhibits ar elastic-visco-plastic behavior [8,9], for which the Bod Io=oh+ido]k and omh=omh+ido Check if there is any ply ko such that one of the updated er-Partom constitutive model as summarized in Sec tion 6.2 of Ref [1] should be employed stresses,o h, and omIk, has attained its ultimate value Suppose that the composite laminate cools down If yes, ko=ko+1. If all the plies in the laminate have failed, stop calculation. from a stress-free temperature, Tzero to a reference tem- (7) Calculate the plastic strain rate in the matrix of perature, Trer Let the cooling rate be denoted by Tool. If the kth lamina according to Tiero# Tref thermal residual stresses will be generated in the composite. For a general purpose, let the composite 1=(-s=)(ah ycling variation in temperature from Tini to Ifin. If any 2oM(S( je (12) pair of them coincides such as Tini Tin the thermal load and the mechanical load are independently applied. We where [se mlk is the elastic component of the instanta- Tin simultaneously. According to the theories presented equivalent plastic strain rate of us trix material.The can thus suppose that the composite reaches alfin and neous compliance matrix of the in Ref. [I] and in Section 2, a simulation procedure for determined from the formula the in-plane thermal-mechanical fatigue response of the titanium mMc laminate is summarized below (2))=V2(em)(em)/3 (13) (1)LetE,v,r,Pm,四,m,喟,聞,o,,,, f=n, and 0) be given. Calculate all coordinate Note that Eaml should also be included in Eq(13),be- transformation matrixes, ((Te) cause it is not always zero according to a three-dimen- according to equations(44)and(45)of Ref [l] with O sional constitutive description. Suppose that from Eq instead of s. Determine the thermal residual stresses in (12), we hav Sudu+Sdn+s23 d21. Then from the fiber and matrix phases, of) and omIR, of each k- dim)=0 it follows that Es(m)=s2dI+s23d21.Hence, ply of the laminate. Apply to the composite the initial load increment from (0) to (o) increment from Tref to Tini and calculate the initial in ternal stress increments in the fiber and matrix, olk (8)Update T=T+AT and t=t+Af. Define fom. Let oh=o +oh and oml. uniaxial stress-strain curve of the matrix material in the kth lamina using the Bodner-Partom model and the (2)Take 1=0 and M>0. Define dT=(Tin-Tni ) current plastic strain rate. (e)gm). Define(Em)g,(vm L, dt= 1/2oM, and do)(@=(o] -io)ini)/ where o is the cycling frequency(Hz). Define the stress (如)(),(碑)k,(),and(m) Go to step(3) The above process should be repeated until all the and temperature rates as o=2o(alfin-ialini) laminae in the laminate have failed, or, until the maxi- and T= 2o(Tfin-Tini) Calculate the laminate stiffness mum incremental step, M, is attained. In the latter case, elements, @l, Oll, and oll, from Eq (6). Set ko= null. a reversed process(with a changed sign of the tempe 3)According to the current matrix sti Home, ature and load increments) begins. It must be realised the given material parameters, calculate sm], (Ak, that a complete cycling consists of a forward and a Bebe,(aJk, and IBIg(see Ref [ID respectively. reversed processes. If the composite is subjected to a Set (b)=-Vm(bm)/V. Evaluate [(CH)=([Tle) tensile fatigue, the resulting matrix will be generally ()-([r2)4 subjected to a loading condition only in one directional (4)Calculate the load incremental vectors, IdN process, either in the forward or in the reversed process dNrY, dNxy, dMxx, dMn, dMxr. Update the laminate depending on the thermal residual stress level and the stifness and equivalent thermal load elements using Eqs. laminate lay-up arrangement. While in the other direc- (9)and(10), if necessary tional process, the matrix will be subjected to an un- ()Calculate the laminate in-plane strain and cur- loading. In the case of the unloading, only the elastic ature increments, daxx, dery, and dexy and dxxx, dry, component of the material instantaneous compliance and dx Dy, using Eq. (5) matrix will retain. In other words the matrix constitu- (6) For each un-failed kth lamina, calculate its load tive equations must be described using Hooke's law share in the local coordinate system according to during the unloading

both thermal and in-plane mechanical fatigue loads. The composite was made from ceramic SCS-6 fibers and TIMETAL 21S matrix, with a fiber volume fraction of 0.35 [8]. Both the constituent materials are isotropic. Whereas the SCS-6 fibers are linearly elastic until rup￾ture, the titanium TIMETAL 21S matrix exhibits an elastic-visco-plastic behavior [8,9], for which the Bod￾ner–Partom constitutive model as summarized in Sec￾tion 6.2 of Ref. [1] should be employed. Suppose that the composite laminate cools down from a stress-free temperature, Tzero to a reference tem￾perature, Tref Let the cooling rate be denoted by T_ cool. If Tzero 6¼ Tref thermal residual stresses will be generated in the composite. For a general purpose, let the composite be subjected to a cycling load from frgini to frgfin and a cycling variation in temperature from Tini to Tfin. If any pair of them coincides such as Tini ¼ Tfin the thermal load and the mechanical load are independently applied. We can thus suppose that the composite reaches frgfin and Tfin simultaneously. According to the theories presented in Ref. [1] and in Section 2, a simulation procedure for the in-plane thermal–mechanical fatigue response of the titanium MMC laminate is summarized below. (1) Let Ef , mf , af ; Em, mm, am, rm Y, Em T , rf u, rf u;c; rm u , rm u;c, fzkgðNÞ k¼0, and fhkgðNÞ k¼1 be given. Calculate all coordinate transformation matrixes, ð½T  cÞk  ðNÞ k¼1 and ð½T  sÞk  ðNÞ k¼1, according to equations (44) and (45) of Ref. [1] with hk instead of n. Determine the thermal residual stresses in the fiber and matrix phases, frf gR k and frmgR k , of each k￾ply of the laminate. Apply to the composite the initial load increment from {0} to frgini and the temperature increment from Tref to Tini and calculate the initial in￾ternal stress increments in the fiber and matrix, frf g 0 k and frmg0 k . Let frf gk ¼ frf g R k þ frf gk and frmgk ¼ frmgR k þ frmg0 k . (2) Take t ¼ 0 and M > 0. Define dT ¼ ðTfin  TiniÞ/ M, dt ¼ 1/2xM, and fdrgðGÞ ¼ ðfrgfin  frginiÞ=M, where x is the cycling frequency (Hz). Define the stress and temperature rates as fr_gðGÞ ¼ 2x frgfin  frgini ð Þ and T_ ¼ 2xð Þ Tfin  Tini . Calculate the laminate stiffness elements, QI ij; QII ij , and QIII ij , from Eq. (6). Set k0 ¼ null. (3) According to the current matrix stresses, frmgk , and the given material parameters, calculate ½Sm k ; ½A k , ½B k , fbmgk , fagk , and fbgG k (see Ref. [1]) respectively. Set fbf gk ¼ Vmfbmgk=Vf . Evaluate ½ðCG ij Þk  ¼ ð½T  cÞk  ð½S k Þ 1 ð½T  T c Þk . (4) Calculate the load incremental vectors, fdNXX ; dNYY ; dNXY ; dMXX ; dMYY ; dMXY gT. Update the laminate stiffness and equivalent thermal load elements using Eqs. (9) and (10), if necessary. (5) Calculate the laminate in-plane strain and cur￾vature increments, de0 XX , de0 YY , and de0 XY and dj0 XX , dj0 YY , and dj0 XY , using Eq. (5). (6) For each un-failed kth lamina, calculate its load share in the local coordinate system according to fdrgk ¼ ð½T  T s Þk ½ðCG ij Þk fdeg G k   fbgG k dT  ; where fdegG k is obtained from Eq. (2.1). Calculate in￾ternal stress increments, fdrf gk and fdrmgk , from equations. (14.1) and (14.2) of Ref. [1], and update frf gk ¼ frf gk þ fdrf gk and frmgk ¼ frmgk þ fdrmgk . Check if there is any ply k0 such that one of the updated stresses, frf gk and frmgk , has attained its ultimate value. If yes, k0 ¼ k0 þ 1. If all the plies in the laminate have failed, stop calculation. (7) Calculate the plastic strain rate in the matrix of the kth lamina according to e_ I n oðmÞ k ¼ ð½Sm k  ½Se;m k Þfr_gk ¼ 2xMð½Sm k  ½Se;m k Þfdrgk ; ð12Þ where ½Se;m k is the elastic component of the instanta￾neous compliance matrix of the matrix material. The equivalent plastic strain rate of the matrix, ðe_ I Þ ðmÞ k , is determined from the formula, ðe_ I Þ ðmÞ k ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðe_ I;ðmÞ ij Þk ðe_ I;ðmÞ ij Þk=3 q : ð13Þ Note that e_ I;ðmÞ 33 should also be included in Eq. (13), be￾cause it is not always zero according to a three-dimen￾sional constitutive description. Suppose that from Eq. (12), we have e_ I;ðmÞ 22 ¼ 121r_ 11 þ 122r_ 22 þ 123r_ 21. Then from r_ I;ðmÞ 33 ¼ 0 it follows that e_ I;ðmÞ 33 ¼ 121r_ 11 þ 123r_ 21. Hence, ðe_ I Þ ðmÞ k ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2½ðe_ I;ðmÞ 11 Þ 2 k þ ðe_ I;ðmÞ 22 Þ 2 k þ ðe_ I;ðmÞ 33 Þ 2 k þ 2ðe_ I;ðmÞ 12 Þ 2 k =3 q : (8) Update T ¼ T þ DT and t ¼ t þ Dt. Define the uniaxial stress–strain curve of the matrix material in the kth lamina using the Bodner–Partom model and the current plastic strain rate, ðe_ I Þ ðmÞ k . Define ðEmÞk , ðmmÞk , ðamÞk , ðrm YÞk , ðEm T Þk , ðrm u Þk , and ðrm u;cÞk . Go to step (3). The above process should be repeated until all the laminae in the laminate have failed, or, until the maxi￾mum incremental step, M, is attained. In the latter case, a reversed process (with a changed sign of the temper￾ature and load increments) begins. It must be realised that a complete cycling consists of a forward and a reversed processes. If the composite is subjected to a tensile fatigue, the resulting matrix will be generally subjected to a loading condition only in one directional process, either in the forward or in the reversed process depending on the thermal residual stress level and the laminate lay-up arrangement. While in the other direc￾tional process, the matrix will be subjected to an un￾loading. In the case of the unloading, only the elastic component of the material instantaneous compliance matrix will retain. In other words, the matrix constitu￾tive equations must be described using Hooke’s law during the unloading. 1186 Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199