《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_Prediction of effective elastic modulus of plain.pdf_大学文库

Prediction of effective elastic modulus of plain weave multiphase and multilayer silicon carbide ceramic matrix composite Y.. Xu*, W. H. Zhang and h. B Wang The bottom up multiscale finite element modelling that is based on the sequential consideration of the fibre/interface/matrix scale and the tow/matrix/coat scale is employed to predict the effective elastic modulus of the plain wave multiphase and multilayer silicon carbide ceramic matrix composite. Instead of using the homogenisation method, the strain energy based method that is advantageous in computing efficiency and numerical implementation is adopted for the multiscale analysis and evaluation of effective properties. First, the effective properties of tows are computed on the fibre/interface/matrix scale. They are then incorporated into the tow/matrix/coat scale to evaluate the effective properties of the composite. Numerical results obtained by the proposed method and model show a good agreement with the results measured experimentally Keywords: Multiphase and multilayer, Silicon carbide ceramic matrix composite, Plain weave, Multiscale analysis, Strain energy based method Introduction (ii alternate the infiltration of silicon carbide and pyrolytic carbon to form the matrix. Note that in Silicon carbide ceramic matrix composites(CMC-SiC e second step, the multiphase and multilayer are widely used in aerospace engineering because of their matrix consisting of silicon carbide and pyrolytic strength and high temperature resis- tance. As is known. the traditional CMC- SiC has a carbon is formed around the fibre first and the multiphase and multilayer matrix is formed single phase and single layer architecture in which each around the tow after the space between the of the coat, matrix and interface consists of only one fibres is filled by the matrix kind of material. In contrast, for the multiphase and (iii) the coat is formed by infiltrating silicon carbide. multilayer CMC-SiC, each of the coat, matrix and Considering such a forming process, the mult interface consists of multiphase materials. In fact, the scale analysis methods2 are often needed for the latter can effectively resist the diffusion of oxidising d medium in composite to prevent the oxidation of tow The exact prediction of the effective properties is essential and improve high temperature mechanical behaviours, to design the microstructure of composite materials e.g. the toughness of the composite. Therefore, the and understand the mechanical behaviours. To do this, advantages of the multiphase and multilayer CMC-Sic the homogenisation method is widely used. -It is based make it attractive in aerospace applications on the mathematical theory of small parameter asympto- The present paper will focus on the prediction of the tic expansion. However, owing to the complexity of effective elastic modulus of plain weave multiphase and multiphase and multilayer architecture, the homogenisa- multilayer CMC-SiC. The modelling of this kind of tion method becomes time consuming and is greatly composite is as follows. First, hexagonal arrangements limited in the prediction of multiphase and multilayer of fibres whose diameters are several micrometres CMC-SiC. constitute the tow. Second, the tows are interlaced to In the present paper, the relationship between the form the 2x2 braided mat which are then stacked to st energy of the microstructure and the he form the preformed laminate. Finally, the composite neous equivalent model is studied for the orthotror ructure is obtained by means of the following chemical material and a strain energy based method is proposed vapour infiltration process: to predict the effective elastic properties using the strain infiltration of the pyrolytic carbon into the energy of the microstructure under specific boundary preformed laminate to form the interface. conditions. Compared with the homogenisation method the strain energy based method demonstrates its simplicity in numerical implementation and its efficiency nd Aerospace Computing, High In computing time. ey Laboratory of Contemporary The present paper is organised as follows. First, the Northwestem Polytechnic University, strain energy based method is introduced followed by XIan 710072. Chin descriptions of the multiscale analysis approach and the corrEspondingauthoremailxyj1014@yahoo.com.cn multiscale finite element modelling. Then numerical a 2008 Institute of Materials, Min Received 3 pted 14 January 2008 Materials Science and Technology 2008 VOL 24 No 4 435

Prediction of effective elastic modulus of plain weave multiphase and multilayer silicon carbide ceramic matrix composite Y. J. Xu*, W. H. Zhang and H. B. Wang The bottom up multiscale finite element modelling that is based on the sequential consideration of the fibre/interface/matrix scale and the tow/matrix/coat scale is employed to predict the effective elastic modulus of the plain wave multiphase and multilayer silicon carbide ceramic matrix composite. Instead of using the homogenisation method, the strain energy based method that is advantageous in computing efficiency and numerical implementation is adopted for the multiscale analysis and evaluation of effective properties. First, the effective properties of tows are computed on the fibre/interface/matrix scale. They are then incorporated into the tow/matrix/coat scale to evaluate the effective properties of the composite. Numerical results obtained by the proposed method and model show a good agreement with the results measured experimentally. Keywords: Multiphase and multilayer, Silicon carbide ceramic matrix composite, Plain weave, Multiscale analysis, Strain energy based method Introduction Silicon carbide ceramic matrix composites (CMC–SiC) are widely used in aerospace engineering because of their low weight, high strength and high temperature resistance. As is known, the traditional CMC–SiC has a single phase and single layer architecture in which each of the coat, matrix and interface consists of only one kind of material. In contrast, for the multiphase and multilayer CMC–SiC, each of the coat, matrix and interface consists of multiphase materials. In fact, the latter can effectively resist the diffusion of oxidising medium in composite to prevent the oxidation of tow and improve high temperature mechanical behaviours, e.g. the toughness of the composite.1 Therefore, the advantages of the multiphase and multilayer CMC–SiC make it attractive in aerospace applications. The present paper will focus on the prediction of the effective elastic modulus of plain weave multiphase and multilayer CMC–SiC. The modelling of this kind of composite is as follows. First, hexagonal arrangements of fibres whose diameters are several micrometres constitute the tow. Second, the tows are interlaced to form the 262 braided mat which are then stacked to form the preformed laminate. Finally, the composite structure is obtained by means of the following chemical vapour infiltration process: (i) infiltration of the pyrolytic carbon into the preformed laminate to form the interface. (ii) alternate the infiltration of silicon carbide and pyrolytic carbon to form the matrix. Note that in the second step, the multiphase and multilayer matrix consisting of silicon carbide and pyrolytic carbon is formed around the fibre first and the multiphase and multilayer matrix is formed around the tow after the space between the fibres is filled by the matrix. (iii) the coat is formed by infiltrating silicon carbide. Considering such a forming process, the multiscale analysis methods2,3 are often needed for the study. The exact prediction of the effective properties is essential to design the microstructure of composite materials and understand the mechanical behaviours. To do this, the homogenisation method is widely used.4–7 It is based on the mathematical theory of small parameter asymptotic expansion. However, owing to the complexity of multiphase and multilayer architecture, the homogenisation method becomes time consuming and is greatly limited in the prediction of multiphase and multilayer CMC–SiC. In the present paper, the relationship between the strain energy of the microstructure and the homogeneous equivalent model is studied for the orthotropic material and a strain energy based method is proposed to predict the effective elastic properties using the strain energy of the microstructure under specific boundary conditions. Compared with the homogenisation method, the strain energy based method demonstrates its simplicity in numerical implementation and its efficiency in computing time. The present paper is organised as follows. First, the strain energy based method is introduced followed by descriptions of the multiscale analysis approach and the multiscale finite element modelling. Then numerical Laboratory of Engineering Simulation and Aerospace Computing, High Performance Computing Centre, The Key Laboratory of Contemporary Design and Integrated Manufacturing, Northwestern Polytechnic University, Xi’an 710072, China *Corresponding author, email xyj1014@yahoo.com.cn 2008 Institute of Materials, Minerals and Mining Published by Maney on behalf of the Institute Received 3 November 2007; accepted 14 January 2008 DOI 10.1179/174328408X282056 Materials Science and Technology 2008 VOL 24 NO 4 435

Xu et al. Prediction of effective elastic modulus of cmc-sic Equivalent process E=;(01i+a2+033+0112+0223 +31a31)V(4) 1 llustration of a heterogeneous microstructure and b As discussed below, with the help of nine different load homogeneous equivalent model cases, the combination of equations (3)and (4)can be used to deduce the effective elastic constants CikI results are then compared with the experimental results Load case 1 for the validation. Finally, concluding remarks and Suppose a unit initial strain is imposed in the yr advanced researches are provided direction,i.e.ED=(100000.Note that the super- script(1)represents the first load case. The correspond Strain energy based method ing strain energy can be expressed as Prediction of effective elastic tensors of material microstructure ElD=IoDEDY Without loss of generality, the authors consider a representative volume cell(RvC) with a heterogeneous a)=(CH1CH2C出13000 kinds of materials (different colours represent different By replacing a and a into equation (5), the following homogeneous equivalent model depicted in Fig. Ib by the equivalent method. In the equivalent model, the E)=C出1y average or effective stress and strain of the microstr ture,o and E correspond to Load case 2 (1) Suppose a unit initial tensile strain is imposed in the y2 direction, i.e. 24)=(0 10000). The corresponding and both are related by the effective elastic tensors CH, stress Is Note that v is the volume of the microstructure The strain energy can be expressed as Consider the case of three-dimensional orthotropic materials, equation (2)can be written as Load case 3 The loading to be imposed in this case is a unit initial tensile strain in the y3 direction, i.e. 25)=(001000) The corresponding stress is C3000 (10) The strain energy in this case can be expressed as CH2C出3000 (11) The loading to be imposed is a unit initial shear strain, i.e24)=(000100). The corresponding stress 00000c It can be demonstrated that the strain energy related to the microstructure is equal to that of the homogeneou The strain energy in this case can be expressed as equivalent model under uniformly distributed loading E4=04d4)v=5CHn2v boundary conditions such that 436 Materials Science and Technol 2008 VOL 24 NO 4

results are then compared with the experimental results for the validation. Finally, concluding remarks and advanced researches are provided. Strain energy based method Prediction of effective elastic tensors of material microstructure Without loss of generality, the authors consider a representative volume cell (RVC) with a heterogeneous microstructure depicted in Fig. 1a. It consists of two kinds of materials (different colours represent different materials) and can be macroscopically regarded as the homogeneous equivalent model depicted in Fig. 1b by the equivalent method. In the equivalent model, the average or effective stress and strain of the microstructure, s and e correspond to s~ 1 V ð V sdV, e~ 1 V ð V edV (1) and both are related by the effective elastic tensors CH ijkl sij~CH ijklekl (2) Note that V is the volume of the microstructure. Consider the case of three-dimensional orthotropic materials, equation (2) can be written as s11 s22 s33 s12 s23 s31 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 ~ CH 1111 CH 1122 CH 1133 000 CH 1122 CH 2222 CH 2233 000 CH 1133 CH 2233 CH 3333 000 000 CH 1212 0 0 0000 CH 2323 0 00000 CH 3131 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 e11 e22 e33 e12 e23 e31 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 (3) It can be demonstrated that the strain energy related to the microstructure is equal to that of the homogeneous equivalent model under uniformly distributed loading boundary conditions such that E~ ð V 1 2 ðs11e11zs22e22zs33e33zs12e12zs23e23 zs31e31ÞdV ~ 1 2 ðs11e11zs22e22zs33e33zs12e12zs23e23 zs31e31ÞV (4) As discussed below, with the help of nine different load cases, the combination of equations (3) and (4) can be used to deduce the effective elastic constants CH ijkl involved in equation (3) for the RVC. Load case 1 Suppose a unit initial strain is imposed in the y1 direction, i.e. e (1)~(1 0 0 0 0 0)T. Note that the superscript (1) represents the first load case. The corresponding strain energy can be expressed as E(1)~1 2 s(1)e (1)V (5) s(1)~(CH 1111 CH 1122 CH 1133 0 0 0)T (6) By replacing s(1) and e (1) into equation (5), the following expression of strain energy can be obtained E(1)~1 2 CH 1111V (7) Load case 2 Suppose a unit initial tensile strain is imposed in the y2 direction, i.e. e (2)~(0 1 0 0 0 0)T. The corresponding stress is s(2)~(CH 1122 CH 2222 CH 2233 0 0 0)T (8) The strain energy can be expressed as E(2)~1 2 s(2)e (2)V~ 1 2 CH 2222V (9) Load case 3 The loading to be imposed in this case is a unit initial tensile strain in the y3 direction, i.e. e (3)~(0 0 1 0 0 0)T. The corresponding stress is s(3)~(CH 1133 CH 2233 CH 3333 0 0 0)T (10) The strain energy in this case can be expressed as E(3)~1 2 s(3)e (3)V~ 1 2 CH 3333V (11) Load case 4 The loading to be imposed is a unit initial shear strain, i.e. e (4)~(0 0 0 1 0 0)T. The corresponding stress is s(4)~(0 0 0 CH 1212 0 0)T (12) The strain energy in this case can be expressed as E(4)~1 2 s(4)e (4)V~ 1 2 CH 1212V (13) 1 Illustration of a heterogeneous microstructure and b homogeneous equivalent model Xu et al. Prediction of effective elastic modulus of CMC–SiC 436 Materials Science and Technology 2008 VOL 24 NO 4

Xu et aL. Prediction of effective elastic modulus of CMc-Sic Load case 5 The loading to be imposed in this case is a unit initial [2E-巴理=e=E000 shear strain, i.e.2s)=(000010). The corresponding 0 stress 000 a)=(0000C230) (14) 2E(4) 0(24 The strain energy in this case can essed as y=3C出23 (15 The elastic constants can be derived by inversing the above matrix. In practice, the considered rvC will Load case 6 be discretised as a finite element model on which the initial strain will The loading to be imposed in this case is a unit initial osed to evaluate the shear strain, i.e. 2o)=(00000 1). The corresponding stress Numerical comparison with homogenisation a6)=(00000C31) method The strain energy in this case can be expressed as The considered homogenisation method is a rigorous approach that is based on the two scale asymptotic pansion, macroscale coordinate system x and micro- (7) scale coordinate system y describing the fast variation of the material properties of the material microstructure in the macroscale coordinate system. Hence, the scale ratio Load case 7 between x and y is E==<<l. Therefore, the displace Smppsese unin inthe tensileantdains ar irectitaseouse. ment e an arbitray m a terial powt is ale elastin body (1 10000). The corresponding stresses are f(x)=u(x, y)+Eu'(x,y)+2u(x, y) CHu+CHn CH2+C222 CH33+C2233000)(18) The elastic equilibrium equation syste en can expressed as The strain energy can be expressed as E =5(CH1+2C2+C2)(19) E b; SuidO+ t; Suidr SUi Load case 8 ose unit initial tensile strains are simultaneously where Eijk denotes the elastic tensor of constitutive osed in the y2 and y3 directions, i.e materials, and bi and ti refer to the body force and a8)=(011000). The corresponding stresses are traction respectively After the substitution of equation(25) into equa- tion(26) and the mathematical development, the 00) (20) following expression of effective elastic tensor can be The strain energy in this case can be expressed as Eijkl- eijmn )d ES)=-08E8)V=-(CH2,+2CH V(21) where r denotes the volume of the unit cell, i is the y periodic admissible displacement field associated with Load case 9 load case kl. Obviously, Eijk denotes the average elastic Suppose unit initial tensile strains are simultaneously tensor depending only upon the material volume directions. i.e actions of constituents as evaluated with the classical (101000). The corresponding stresses are mixture rule whereas Ejm ayn reflects the infiuence of the material microstructure of the RvC. In practice, the considered rvc will be discretised as a finite element CHn+CHi33 CH22+C233 C1133+C33300 0)(22) model to evaluate EH, in which x d by energy in this case can be expressed as running the finite element analysis procedure. As an example, two kinds of 3D microstructures are tested E9)=0989v=s(CHu +2CHn +cHu)v (23) below by means of the homogenisation method and the train energy method respectively. Effective elastic In conclusion, the effective elastic matrix can be written tensors are compared to show the validity of the strain energy based method Materials Science and Technology 2008 VOL 24 No 4 437

Load case 5 The loading to be imposed in this case is a unit initial shear strain, i.e. e (5)~(0 0 0 0 1 0)T. The corresponding stress is s(5)~(0 0 0 0 CH 2323 0)T (14) The strain energy in this case can be expressed as E(5)~1 2 s(5)e (5)V~ 1 2 CH 2323V (15) Load case 6 The loading to be imposed in this case is a unit initial shear strain, i.e. e (6)~(0 0 0 0 0 1)T. The corresponding stress is s(6)~(0 0 0 0 0 CH 3131) T (16) The strain energy in this case can be expressed as E(6)~1 2 s(6)e (6)V~ 1 2 CH 3131V (17) Load case 7 Suppose unit initial tensile strains are simultaneously imposed in the y1 and y2 directions, i.e. e (7)~(1 1 0 0 0 0)T. The corresponding stresses are s(7)~ (CH 1111zCH 1122 CH 1122zCH 2222 CH 1133zCH 2233 0 0 0)T (18) The strain energy can be expressed as E(7)~1 2 s(7)e (7)V~ 1 2 (CH 1111z2CH 1122zCH 2222)V (19) Load case 8 Suppose unit initial tensile strains are simultaneously imposed in the y2 and y3 directions, i.e. e (8)~(0 1 1 0 0 0)T. The corresponding stresses are s(8)~ (CH 1122zCH 1133 CH 2222zCH 2233 CH 2233zCH 3333 0 0 0)T (20) The strain energy in this case can be expressed as E(8)~1 2 s(8)e (8)V~ 1 2 (CH 2222z2CH 2233zCH 3333)V (21) Load case 9 Suppose unit initial tensile strains are simultaneously imposed in the y1 and y3 directions, i.e. e (9)~(1 0 1 0 0 0)T. The corresponding stresses are s(9)~ (CH 1111zCH 1133 CH 1122zCH 2233 CH 1133zCH 3333 0 0 0)T (22) The strain energy in this case can be expressed as E(9)~1 2 s(9)e (9)V~ 1 2 (CH 1111z2CH 1133zCH 3333)V (23) In conclusion, the effective elastic matrix can be written as 2Eð Þ1 V Eð Þ7 {Eð Þ2 {Eð Þ1 V Eð Þ9 {Eð Þ3 {Eð Þ1 V 000 2Eð Þ2 V Eð Þ8 {Eð Þ3 {Eð Þ2 V 000 2Eð Þ3 V 000 2Eð Þ4 V 0 0 2Eð Þ5 V 0 sym 2Eð Þ6 V 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 (24) The elastic constants can be derived by inversing the above matrix. In practice, the considered RVC will be discretised as a finite element model on which the initial strain will be imposed to evaluate the strain energy. Numerical comparison with homogenisation method The considered homogenisation method is a rigorous approach that is based on the two scale asymptotic expansion, macroscale coordinate system x and microscale coordinate system y describing the fast variation of the material properties of the material microstructure in the macroscale coordinate system. Hence, the scale ratio between x and y is e~ y x %1. Therefore, the displacement of an arbitrary material point in an elastic body can be approximated by a two scale asymptotic expansion7 ue (x)~u0 (x,y)zeu1 (x,y)ze 2 u2 (x,y)z::: (25) The elastic equilibrium equation system can be expressed as ð V Eijkl Luk Lxl Ldui Lxj dV~ ð V biduidVz ð V tiduidC Vdui (26) where Eijkl denotes the elastic tensor of constitutive materials, and bi and ti refer to the body force and traction respectively. After the substitution of equation (25) into equation (26) and the mathematical development, the following expression of effective elastic tensor can be obtained EH ijkl~ 1 j j Y ð Y Eijkl{Eijmn Lxkl m Lyn dY (27) where j j Y denotes the volume of the unit cell, xkl is the Y periodic admissible displacement field associated with load case kl. Obviously, Eijkl denotes the average elastic tensor depending only upon the material volume fractions of constituents as evaluated with the classical mixture rule whereas Eijmn Lxkl m Lyn reflects the influence of the material microstructure of the RVC. In practice, the considered RVC will be discretised as a finite element model to evaluate EH ijkl in which xkl is computed by running the finite element analysis procedure. As an example, two kinds of 3D microstructures are tested below by means of the homogenisation method and the strain energy method respectively. Effective elastic tensors are compared to show the validity of the strain energy based method. Xu et al. Prediction of effective elastic modulus of CMC–SiC Materials Science and Technology 2008 VOL 24 NO 4 437

Xu et aL. Prediction of effective elastic modulus of CMc-Sic □Tow a g model of Rvc coat is removed to reveal itecture); b geometry model of RVC shown from tative volume cell model o multiphase and er woven The fibre/interface/matrix finite element models of the RVC with two and four layers of matrixes are depicted in Fig. 6. Both models use hexahedron and wedge elements for the discretisation of different material phases ensure a smooth connection along the interface. ow/matrix/ coat scale and finite element model with two layers of matrixes: b Rvc with four Similarly to the above scale, the problem is here element models of rvc on fibre//matrix concerned with the modelling of multiphase and mult ayer plain weave composite. Over the last decades, extensive investigations of plain weave composites have Fibre/interface/ matrix scale and finite element for been conducted. a variety of models were presented-I5 or the determination of the mechanical properties of model plain weave composites, but almost all of them were In fact, the fibre/interface/matrix scale concerns the tows concerned with the single layer plain weave composite which are considered as multiphase and multilayer instead of the multiphase and multilayer plain weave unidirectional fibre reinforced composites. As seen in composites. Tabiei et al. 3-5 presented a computation- Fig 5a, a hexagonal arrangement of fibres is assumed. a ally efficient model to predict the elastic modules of single layer interface and a multiphase and multilayer woven fabric composite. Although cross-sections of the matrix are distributed around the fibres. In Fig. 5b, fill and warp tows are assumed to be rectangular and characteristic geometric parameters of the RVC model their undulating form is approximated by a single with three layers of matrixes are given: r, fibre diameter; average undulation angle with the horizontal plane, Lf, centre distance between two adjacent fibres; d, the predicted properties are fairly accurate. In the interface layer thickness; d2, thickness of the first layer of present paper, this model is extended to be adapted to the matrix; d3, thickness of the second layer of the the more complex case of the new multiphase and matrIx multilayer plain weave RVC. As shown in Fig. 5b, the arrangements of material The geometric model of the RvC consisting of tw phases from inside to outside are in the following order: layers of matrixes and single layer coat is depicted in fibre-single layer pyrolytic carbon interface- Fig. 7. Related parameters are: w, width of fill tow; af, multiphase and multilayer matrix consisting of silicon width of warp tow, hw, thickness of fill tow; hf, thickness carbide and pyrolytic carbon. Elastic constants of the of warp tow, trand t2, thickness of two layers of matrixes fibres, interface and matrix are the basic properties 6, average undulation angle with the horizontal plane required to obtain the effective modulus by means of the The length L, width w and height H of the RVC are finite element method expressed as follows Table 2 Prediction of effective elastic tensor in case 2 Strain energy based method nisation metho 3255406 325.5406 81.9406325.5406 2.16902.16906.6654 2.2219 0121.80 121.80 022218 2.2222 2.2218 Materials Science and Technology 2008 VOL 24 No 4 439

Fibre/interface/matrix scale and finite element model In fact, the fibre/interface/matrix scale concerns the tows which are considered as multiphase and multilayer unidirectional fibre reinforced composites. As seen in Fig. 5a, a hexagonal arrangement of fibres is assumed. A single layer interface and a multiphase and multilayer matrix are distributed around the fibres. In Fig. 5b, characteristic geometric parameters of the RVC model with three layers of matrixes are given: wf, fibre diameter; Lf, centre distance between two adjacent fibres; d1, interface layer thickness; d2, thickness of the first layer of the matrix; d3, thickness of the second layer of the matrix. As shown in Fig. 5b, the arrangements of material phases from inside to outside are in the following order: fibreRsingle layer pyrolytic carbon interfaceR multiphase and multilayer matrix consisting of silicon carbide and pyrolytic carbon. Elastic constants of the fibres, interface and matrix are the basic properties required to obtain the effective modulus by means of the finite element method. The fibre/interface/matrix finite element models of the RVC with two and four layers of matrixes are depicted in Fig. 6. Both models use hexahedron and wedge elements for the discretisation of different material phases to ensure a smooth connection along the interface. Tow/matrix/coat scale and finite element model Similarly to the above scale, the problem is here concerned with the modelling of multiphase and multilayer plain weave composite. Over the last decades, extensive investigations of plain weave composites have been conducted. A variety of models were presented8–15 for the determination of the mechanical properties of plain weave composites, but almost all of them were concerned with the single layer plain weave composite instead of the multiphase and multilayer plain weave composites. Tabiei et al.13–15 presented a computationally efficient model to predict the elastic modules of woven fabric composite. Although cross-sections of the fill and warp tows are assumed to be rectangular and their undulating form is approximated by a single average undulation angle with the horizontal plane, the predicted properties are fairly accurate. In the present paper, this model is extended to be adapted to the more complex case of the new multiphase and multilayer plain weave RVC. The geometric model of the RVC consisting of two layers of matrixes and single layer coat is depicted in Fig. 7. Related parameters are: aw , width of fill tow; af, width of warp tow; hw, thickness of fill tow; hf,, thickness of warp tow; t1and t2, thickness of two layers of matrixes; h, average undulation angle with the horizontal plane. The length L, width W and height H of the RVC are expressed as follows Table 2 Prediction of effective elastic tensor in case 2 Strain energy based method Homogenisation method 325:5406 81:7747 325:5406 2:1690 2:1690 6:6654 0 0 0 121:80 0 0 0 02:2218 0 0 0 0 02:2218 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 325:5406 81:9406 325:5406 2:2219 2:2219 6:6657 0 0 0 121:80 0 0 0 02:2222 0 0 0 0 02:2222 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 (a) (b) a RVC with two layers of matrixes; b RVC with four layers of matrixes 6 Finite element models of RVC on fibre/interface/matrix scale (a) (b) a geometry model of RVC (coat is removed to reveal tow architecture); b geometry model of RVC shown from fill tow direction 7 Representative volume cell model o multiphase and multilayer woven Xu et al. Prediction of effective elastic modulus of CMC–SiC Materials Science and Technology 2008 VOL 24 NO 4 439

Xu et aL. Prediction of effective elastic modulus of CMc-Sic T300 fiber silicon carbide 11 Model of Rvc on tow/matrix/coat scale □T3o0fber Results and discussions ■■ pyrolytic carbon Three kinds of plain weave multiphase and multilayer MC prepared and the in plane extensional modulus were measured experimentally The fibre/interface/matrix RVC model of composite A in Fig. 10a, of one layer of pyrolytic carbon interface and three layers of matrixes made up of alternately infiltrated silicon carbide and pyrolytic carbon. The fibre/interface/matrix RVC model of composite C, as depicted in Fig. 10b, just consists of a with one layer of interface and three layer matrixes; b with one layer of interface one layer of pyrolytic carbon interface The tow/matrix/coat RVC model of composite A o Representative volume cell model on fibre/interface and C, depicted in Fig. 11, consist of two layers of natrix scale matrixes made up of alternately infiltrated pyrolytic the tow material coordinate system with respect to the carbon and silicon carbide and one layer of silicon RVC coordinate system. Mechanical properties of each material phase are The tow/matrix/coat finite element model of the RvC listed in Table 3, and the units of the modulus are all in is depicted in Fig 9. Here, tetrahedron elements are GPa. Geometrical parameters evaluated according to the given infiltration time and volume fractions of Table 3 Constituent elastic properties constituent materials in Ref. 16 are listed in Table 4 In Table 5(the units of the modulus are all in gPa) T300 carbon fibre Pyrolytic carbon Silicon carbide the elastic modulus obtained by the energy based method using the present model are compared with experimental results given in Ref. 16. The agreement is satisfactory. However, it should be pointed out that n425 because rudimental cavities generated within the com- posite for emission of large infiltrated byproducts during the infiltration are not considered in the present paper, 0-42 the in plane extensional modulus computed numerically 02 0-12 0-12 0-2 by the present method are a little bigger than the experimental results Table 4 Geometric values of microstructures on two scales Composite Geometrical parameters onfibre/interface/matrix scale Geometrical parameters ontow/matrix/coat scale A m,L,=15 um aw= af =40 mm hw= hf =4 m Pr =10 um, Li=15 um oy 40 mm d1=091ur hw= hf=4 mm a2=0.35um =30° t1=133mr t2=027m C pr=12. 5 um, L=15 um hw= hf=4 mm t1=115mm t2=045mm Materials Science and Technology 2008 VOL 24 No 4 44

the tow material coordinate system with respect to the RVC coordinate system. The tow/matrix/coat finite element model of the RVC is depicted in Fig. 9. Here, tetrahedron elements are used due to the complexity of the model. Results and discussions Three kinds of plain weave multiphase and multilayer CMC–SiC samples (composite A, B and C) were prepared and the in plane extensional modulus were measured experimentally.16 The fibre/interface/matrix RVC model of composite A and B, depicted in Fig. 10a, consists of one layer of pyrolytic carbon interface and three layers of matrixes made up of alternately infiltrated silicon carbide and pyrolytic carbon. The fibre/interface/matrix RVC model of composite C, as depicted in Fig. 10b, just consists of one layer of pyrolytic carbon interface. The tow/matrix/coat RVC model of composite A, B and C, depicted in Fig. 11, consist of two layers of matrixes made up of alternately infiltrated pyrolytic carbon and silicon carbide, and one layer of silicon carbide coat. Mechanical properties of each material phase are listed in Table 3, and the units of the modulus are all in GPa. Geometrical parameters evaluated according to the given infiltration time and volume fractions of constituent materials in Ref. 16 are listed in Table 4. In Table 5 (the units of the modulus are all in GPa), the elastic modulus obtained by the energy based method using the present model are compared with experimental results given in Ref. 16. The agreement is satisfactory. However, it should be pointed out that because rudimental cavities generated within the composite for emission of large infiltrated byproducts during the infiltration are not considered in the present paper, the in plane extensional modulus computed numerically by the present method are a little bigger than the experimental results. (a) (b) a with one layer of interface and three layers of matrixes; b with one layer of interface 10 Representative volume cell model on fibre/interface/ matrix scale 11 Model of RVC on tow/matrix/coat scale Table 3 Constituent elastic properties T300 carbon fibre Pyrolytic carbon Silicon carbide E11 22 12 350 E22 22 12 350 E33 220 30 350 G12 7. 75 4.3 145. 8 G23 4. 8 2.0 145. 8 G13 4. 8 2.0 145. 8 n12 0. 42 0. 4 0. 2 n23 0. 12 0.12 0. 2 n13 0. 12 0. 12 0. 2 Table 4 Geometric values of microstructures on two scales Composite Geometrical parameters onfibre/interface/matrix scale Geometrical parameters ontow/matrix/coat scale A wf510 um, Lf 515 um d150. 95 um d250.66 um d350.75 um aw5 af 540 mm hw 5 hf 54 mm h530u t151. 2 mm t250. 4 mm B wf 510 um, Lf515 um d150.91 um d250.35 um d351.10 um aw 5 af540 mm hw 5 hf54 mm h5 30u t151. 33 mm t250. 27 mm C wf 512. 5 um, Lf 515 um aw 5 af540 mm hw 5 hf54 mm h530u t151. 15 mm t250. 45 mm Xu et al. Prediction of effective elastic modulus of CMC–SiC Materials Science and Technology 2008 VOL 24 NO 4 441

Xu et al. Prediction of effective elastic modulus of cmc-sic Table 5 Comparison of numerical results with experimental ones In plane extensional modulus Through thickness extensional modulus Share modulus a Present method 132.9 32912612 Experimental results 109.3 109-3 b Present method c Present method 2020 120-4 533 199132132 129-4 562 207116116 1081 Conclusions Province Research Prog no.2006K05- G25), Doctorate For Northwestern The plain weave multiphase and multilayer CMC-SiC is Polytechnical University X200610)and udied in the present paper. Finite element modelling of Xi'an Applied Materials Innovation Fund(grant the multiphase and multilayer CMC-SiC is carried out o.XA-AM-200705) for the first time. The strain energy based method is and effective elastic properties are obtained based on the References sequential homogenisation from the fibre/interface/ I.L. T Zhang L F Cheng Y D Xu, Y.S.Li,QFZengNDong matrix scale to the tow/matrix/coat scale. The compar and X. G. Luan:J. Aeron. Mater. 2006. 26. 226-232 ison with the experimental results shows the validity and V. Carvelli and C. Poggi: Composites A. 2001, 32A, 1425- rationality of the present method. Further studies will be focused on the following important issues: influence of 3. X. D. Tang, J. Whitcomb, A D. Kelkar and J. Tate: Compos. Sci. Technol,2006,66,2580-2590 the thickness and material composition of layers upon 4. J M. Guedes and N. Kikuchi: Comput. Methods. App. Mech.Eng he effective properties, e.g. t multiphase and multilayer CMC-SiC, design optimisa 5. O. Sigmund: Int. J. Solids. Struct, 1994, 3117, 2313-2329 tion of the effective properties and formulation of the 6. B Hassani and E. Hinton: Comput. 69,707-717 7. B Hassani and E. Hinton: Comput. design criteria of the multiphase and multilayer CMC- 8. J. Whitcomb, K. Srirengan and C 1995,31,137-149 9. J. Whitcomb, K. Srirengan and C. Chapman: Compos. Struct Acknowledgements 1997.39.145-156 10. K. Woo and J. Whitcomb: Compos. Struct, 1997, 37, 343-355 The present work is supported by the Natural Science A. Dasgupta, R.K. Agarwal and S M. Bhandarkar: Compos. Sci TechnoL,1996,56,209-22 Foundation of China(grant no. 90405016, 50775184), 12. Z M. Huang: Compos. Sci. TechnoL, 2000, 60, 479 973 Programme (grant no. 2006CB601205), 863 13. I Ivanov and A. Tab Struct,2001,54,489-49 Programme (grant no. 2006AA04Z122), Specialised 14. A Tabiei and w Struct.,2002,58 Research Fund for the Doctoral Program of Higher 15. A Tabiei and S B Ication(grant no. 20060699006), The Aeronautical 16. X F Han, L. T Zhang, L F Cheng and Y D. Xu: J Chn Ceram. ce Foundation (grant no. 2006ZA53006), Shaanxi Soc.2006.34.871874 442 Materials science and technol 2008voL24

Conclusions The plain weave multiphase and multilayer CMC–SiC is studied in the present paper. Finite element modelling of the multiphase and multilayer CMC–SiC is carried out for the first time. The strain energy based method is adopted as an efficient approach for the homogenisation and effective elastic properties are obtained based on the sequential homogenisation from the fibre/interface/ matrix scale to the tow/matrix/coat scale. The comparison with the experimental results shows the validity and rationality of the present method. Further studies will be focused on the following important issues: influence of the thickness and material composition of layers upon the effective properties, e.g. the toughness, of the multiphase and multilayer CMC–SiC, design optimisation of the effective properties and formulation of the design criteria of the multiphase and multilayer CMC– SiC. Acknowledgements The present work is supported by the Natural Science Foundation of China (grant no. 90405016,50775184), 973 Programme (grant no. 2006CB601205), 863 Programme (grant no. 2006AA04Z122), Specialised Research Fund for the Doctoral Program of Higher Education (grant no. 20060699006), The Aeronautical Science Foundation (grant no. 2006ZA53006), Shaanxi Province Research Programme (grant no. 2006K05- G25), Doctorate Foundation of Northwestern Polytechnical University (grant no. CX200610) and Xi’an Applied Materials Innovation Fund (grant no. XA-AM-200705). References 1. L. T. Zhang, L. F. Cheng, Y. D. Xu, Y. S. Li, Q. F. Zeng, N. Dong and X. G. Luan: J. Aeron. Mater., 2006, 26, 226–232. 2. V. Carvelli and C. Poggi: Composites A, 2001, 32A, 1425– 1432. 3. X. D. Tang, J. Whitcomb, A. D. Kelkar and J. Tate: Compos. Sci. Technol., 2006, 66, 2580–2590. 4. J. M. Guedes and N. Kikuchi: Comput. Methods. Appl. Mech. Eng., 1990, 83, 143–198. 5. O. Sigmund: Int. J. Solids. Struct., 1994, 3117, 2313–2329. 6. B. Hassani and E. Hinton: Comput. Struct., 1998, 69, 707–717. 7. B. Hassani and E. Hinton: Comput. Struct., 1998, 69, 719–738. 8. J. Whitcomb, K. Srirengan and C. Chapman: Compos. Struct., 1995, 31, 137–149. 9. J. Whitcomb, K. Srirengan and C. Chapman: Compos. Struct., 1997, 39, 145–156. 10. K. Woo and J. Whitcomb: Compos. Struct., 1997, 37, 343–355. 11. A. Dasgupta, R. K. Agarwal and S. M. Bhandarkar: Compos. Sci. Technol., 1996, 56, 209–223. 12. Z. M. Huang: Compos. Sci. Technol., 2000, 60, 479–498. 13. I. Ivanov and A. Tabiei: Compos. Struct., 2001, 54, 489–496. 14. A. Tabiei and W. T. Yi: Compos. Struct., 2002, 58, 149–164. 15. A. Tabiei and S. B. Aminjikarai: Compos. Struct., 2007, 81, 407– 418. 16. X. F. Han, L. T. Zhang, L. F. Cheng and Y. D. Xu: J. Chn Ceram. Soc., 2006, 34, 871–874. Table 5 Comparison of numerical results with experimental ones Composite In plane extensional modulus Through thickness extensional modulus Share modulus E11 E22 E33 G12 G23 G31 A Present method 132. 9 132. 9 76. 7 32. 9 12. 6 12. 6 Experimental results 109.3 109. 3 B Present method 120. 4 120. 4 53. 3 19. 9 13. 2 13. 2 Experimental results 101. 4 101. 4 C Present method 129. 4 129. 4 56. 2 20. 7 11. 6 11. 6 Experimental results 108. 1 108. 1 Xu et al. Prediction of effective elastic modulus of CMC–SiC 442 Materials Science and Technology 2008 VOL 24 NO 4

《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_Prediction of effective elastic modulus of plain

《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_Prediction of effective elastic modulus of plain