COMPOSITES SCIENCE AND TECHNOLOGY ELSEVIER Composites Science and Technology 61 (2001)461-47 73 www.elsevier.com/locate/compscitech The creep response of uni-directional fiber-reinforced ceramic composites: a theoretical study Susmit Kumar,rajn. Singh* Department of Materials Science and Engineering, University of Cincinnati, Cincinnati, OH 45221-0012,USA Received 21 July 1999; received in revised form 2 May 2000; accepted August 2000 Abstract .Both the finite-element technique and the elastic-viscoelastic correspondence principle (in conjunction with the Hashin ar n Aboudi methods of calculating the effective elastic properties)methods have been used to calculate the effective creep properties of SCS6 (fiber)/SiC(matrix)composites and the results have been compared. In the finite-element technique, three different arrange- ments of fibers were used to study the effects of fiber arrangements on the creep properties of the composite. Effects of the fiber compliance, (i.e. the compliance along the fiber axis), calculated from the five models. But, it was found that the fiber ansverse compliance, S22, and transverse shear compliance, Sa4. For S22, the corre spondence principle method based on Aboudi method and the square array of fibers in the finite-element technique gave the upper and lower bounds, respectively. For S44, the square array of fibers in the finite-element technique and the correspondence principle metho Hashin (for low volume fraction of fiber)/Aboudi (for high volume fraction of fiber) method gave the upper and lower bounds, respectively. The effects of fiber arrangements on S22 and S44 are explained by the strain energy and the energy dissipation due to creep. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Creep; Fiber-reinforced materials; Ceramic composites; Finite element method 1.Introduction the metal-matrix composites having elastic fibers and creeping matrix and showed that even when the creep Although the creep behavior of metal-matrix composites behavior of the constituent phases may individually be reinforced with short fibers [1-4], continuous fibers [5] described by relatively simple steady-state equations and pearticle rein forced composites (6 ave bee e composite will respond in a more complex manner e stress is distributed between the creep behavior of continuous-fiber-reinforced cerar constituent phases, on the modes of deformation o matrix composites. Owing to the need for a each phase, and on interactions between them. Conse- of structural materials for various engineering applica-quently the composite may not exhibit a steady-state tions at high temperatures, engineers/researchers have creep rate. shown interest in ceramics for these applications Several researchers have used the finite-element tech ligh strength and oxidation resistance nique to calculate the creep response of continuous Hence, there is a need to study the creep behavior of fiber-reinforced composites [8-10]. For a square ceramic composites arrangement of fibers, Park and Holmes [8] used the By using the shear-lag approach, Mileiko ] McL ean finite element technique to study the creep behavior of a [2], Lilholt [3] and Pachalis et al. [5] have studied the unidirectional SiC(fiber)/Si3N4 (matrix) composite creep of aligned short-fiber-reinforced composites. d the radial stresses developed at the McLean [4] discussed the creep deformation of both fiber/matrix interface. According to their analyses, the short-fiber and continuous fiber composites He analyzed simultaneous addition of a transverse load, while main- taining the axial load constant, decreases the overall axial creep rate of the composite. By using the finite- *Corresponding author. element technique and a hexagonal fiber array, Kondo 0266-3538/01/s- see front matter2001 Elsevier Science Ltd. All rights reserved PII:S0266-353800)00156-1
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S. Kumar, R.N. Singh/ Composites Science and Technology 61(2001)461-473 et al. [9]studied the creep behavior of continuous carbon -(which is hereafter referred to as square-D)and square fiber-reinforced unidirectional composites because of the arrays of the fibers [as shown in Fig. 1(aHc)] to study viscoelasticity of the resin matrix. In their analyses, the the effects of fiber arrangements on the creep behavior matrix had non-linear viscoelastic properties and the fiber of the comp had linear elastic properties. As a consequence of the scatter in the experimental data, they were unable to verify eir theoretical prediction that the creep compliances decrease with increasing fibre volume fraction. By using ne finite-element technique, Aravas et al. [10] solved a number of unit cell problems with periodic boundary conditions, consistent with the requirements of homo- )Matrix genization theory, to calculate the creep response of the fiber-reinforced metal-matrix composites. Their finite- element results were in good agreement with the pre- dictions of the analytical model [11] developed by one of the authors in Aravas et al. [10] Yancey and Pindera [12] used the elastic-viscoelastic correspondence principle in conjunction with the Aboudi's micro-mechanics model [13, 14 to calculate the creep response of unidirectional composites consisting of linearly viscoelastic resin matrix and elastic carbon fibers. Their theoretical results were in good agreement Fiber with the experimental creep response of T300/934 carbon/ epoxy unidirectional COmDa In the present study, five different methods were used to calculate the creep behavior of the SCS6(fiber)/Sic (matrix) ceramic composite. Out of five methods, two were analytical methods based on the elastic-viscoelastic correspondence principle(using the Aboudi and Hashin micromechanics models to calculate the effective elastic properties to be used in the correspondence principle method), and the rest three methods were based on the finite element technique. In the finite-element method three different fiber arrangements were used to study the effects of the fiber arrangements on the creep properties of the composite. The creep compliances, S1(long itudinal compliance), Sx2(transverse compliance), and ○○○○ S44 (transverse shear compliance) were calculated from all the five methods and compared. The effects of the Fiber fiber arrangements on S22 and S44 were explained by the strain energy and energy dissipation due to creep Effects of the fiber volume fraction on the creep beha vior of the composite were also studied 2. Theory Five theoretical methods were used to calculate the ffective creep properties of the unidirectional fiber reinforced composites. Out of these five methods, two were based he correspondence principle and the other three were based on the finite-element technique Fig. 1. Hexagonal (a), squa (b)and square(c) In the correspondence principle method, we have used models used in the finite-element techniqu denoted by ABCD the hashin and aboudi methods for the calculation of of the smallest possible parallelepiped periodic cell.For nal and square-D the effective elastic properties. In the finite-element arrays of the fibers, ABJAC and 1, respectively. For the square method, we have used hexagonal, square-displace array of fibers, AB=AC
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S. Kumar, R.N. Singh/ Composites Science and Technolog y 61(2001)461-473 2. 1. Correspondence principle prescribed) and the boundar (where the surface displacements are given) is independent of time [16] The most general form of the linear stress-strain If the problem of determination of internal fields in a elation for a material is given by viscoelastic composite subjected to homogeneous bound ary conditions is formulated and the laplace transform is 0i(0)=Cik/(t dekt) (1) applied to all equations, the Laplace transform problem is elastic composite. Elastic phase moduli C (or elastic where the tensor CikAt) denotes the relaxation moduli moduli of the component) are replaced by transform tensor of the material, and aio) and Ekdo) denote the domain moduli sC(s). According to Hashin [17]"The stress and strain tensor, respectively at time t. These effective transform domain moduli/complianc describe the basic time dependent properties of the viscoelastic composite are obtained by replacement of material and as such are the counterparts of the elastic phase elastic moduli by corresponding phase transform moduli. In fact, if the relaxation function tensor CiikAt) domain moduli in the expressions for effective elastic were independent of time, relations denoted by Eq (1) moduli compliances of an elastic composite with iden- could be integrated directly to give elasticity type relations tical phase geometry. "In the notational form, if C is the [15]. Eq (1)can be written as, elastic stifness/compliance matrix of the composite, ) C(t-) dE(t) (2)C()=C(Cm(,C(, where superscript m and f denote the corresponding where o(), C(o) and E(r) denote the stress tensor, properties for matrix and fiber, respectively, and E relaxation moduli tensor, and strain tensor, respectively. denotes the volume fraction of the fiber, we get the fol An alternative form of the viscoelastic stress-strain lowing equations by the application of the correspon relations can be obtained by expressing strain as a time dence principle, dependent function of stress, through C()=(1/s)C(sC"(s),sC(s),V) Now, by inverting Eq ( 8)to the time domain, we can where S(a)is the compliance tensor. By taking the get the effective relaxation moduli of the composite. Laplace transforms of Eqs. (2)and (3), and using the Also, we can get the corresponding effective creep com- convolution theorem, we get pliance by using Eq (6) We have used two micromechanics models -aboudi (s)=C(s)(s) (4) Model [13, 14] [Fig. 2(a)] and Hashin's Composite [8-22][Fig.2(b) with the correspondence principle to calculate the effective creep properties of SCS6(fiber )SiC (matrix) composite (s)=sS(3)(5 (5) The expressions for the effective relaxation moduli or creep compliances in the transform domain given by where s is the transform variable, and bar denotes the these two models are quite complicated and it is very Laplace transform. From Eqs. (4)and(5), we get the Laplace tranny closed form solutions for inverting sC(s)sS(s)=l (6) developed by Bellman et al. [23], can be used to invert the Laplace transform of a function assuming that the One point worth noting is that the Laplace transformed time domain function is smooth. Since the viscoelastic viscoelastic stress-strain relations [Eqs. (4)and (5)] are of response of a material under creep or relaxation loading he same form as the corresponding elastic results if is ooth function of time. this method can be identify sC(s) with the elastic moduli C (or sS(s)with S). employed to generate the relaxation moduli in the time This identification is the basis of the elastic-viscoelastic domain from Eq (8), or the corresponding compliance correspondence principle. It allows determination of the from Eq(6). Bellman [23] used Legendre polynomials stresses and strains of a viscoelastic problem if the to invert a known function of s to its corresponding stresses and strains of an equivalent elastic problem are function of time. The result of this formulation is a known. The correspondence principle can be applied matrix that, when multiplied by the value of the Laplace only to the boundary problems where the interface transform of the function at given values of s, yields the between the boundary Tp(where the external forces are desired function of time
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S. Kumar, R.N. Singh/ Composites Science and Technology 61(2001)461-473 inside ABCD in the square model [Fig. 1(c)]. In the longitudinal direction, the axis of the fibers was assumed to be parallel and of equal lengths. The axis I is along the fiber's longitudinal axis. The numbers of elements inside the fiber and in the matrix were varied Matrix depending on the fiber volume fraction. Perfect bonding between the fiber and the matrix was assumed at the fiber-matrix interface ABAQUS finite element software version 5.6-I was used for the calculation purposes As discussed in Appendix B of Aravas et al. [10], the generalized plane strain conditions were assumed. Peri odic boundary conditions were used between AB and CD. and between ac and BD In order to calculate the effective creep compliances SIl, S2 and S44, the following boundary conditions were appli 1. for the calculation of S1, 011#0, and all other Matri 2. for the calculation of S22, 022#0, and all other 3. for the calculation of transverse shear compliance S44, 012=021. and all other =0 We verified the finite-element model by comparing the calculated creep compliances S1l, S22 and S44, of fiber only with those calculated from the correspondence principle. There were significant differences for other Sis calculated from these two methods. Most probably this is because Su, S22 and 44, were one dimensional properties calculated by applying one dimensional Fig. 2.(a) Aboudi model;(b) Hashin model stresses, and the generalized plain strain model is not a good approximation for the other creep compliances like S12, for material having a non-linear time dependent One important point worth noting is that the elastic- property [24] viscoelastic correspondence principle is valid only for Although the elastic-viscoelastic correspondence those materials for which the strain rate is linearly principle can be used only for those materials for which dependent on the stress, i.e. the creep property of the the strain rate is linearly proportional to the stress [E material can be described by the following equation (9)1. the finite element technique can be used for any type of material properties. Finite-element modeling of e= Aot (9) composites having constituents with non-linear stress dependent creep properties will be subject of another where, E=dE/dt, and A and m are constant paper[25] 2. 2. Finite-element method 3. Results and discussion In the present study, three types of fiber arrangements hexagon D, and square arrangements as We have calculated the creep compliances S1l(com shown in Fig. 1(ac)], have been used to model the pliance along the longitudinal axis of the fiber), S22(the fiber-reinforced composites. In each type of array, the compliance in the transverse plane of the fiber)and S44 smallest possible parallelepiped periodic cell, ABCD (the shear compliance in the transverse plane of the [Fig. 1(ac)], has been used. For hexagonal, square-D fiber)of SCS6(fiber)-Sic (matrix) ceramic composite and square arrangements of fibers, the ratio AB: AC are The material properties of the fiber and matrix at al to and 1: 1, respectively. As shown in 1200 C are listed in Table 1. The shear modulus(G)of Fig. 1(a)and(b), there is one fiber at the center of the Sic (matrix)at Tk was calculated from the following hexagonal and square-D models, and there is no fiber equation [26]
? , , , * ! , I2 ! % % 5 ) 4 :3 * H * , 7 "+/9+/; 4 ()0H :7 "+/9+/; 7 3 * H * ()(0 * A" "" "" ( , 7 "+/ +/ 3 * H ()0H * :7 "+/; 4 3 , * 3 " F 3 3 , B , 3 , 9 3 ()(MN , < 1 " , ( ( 3 ) ( :"'; C , B , , () 0H , (0 )H 4 . "" !! 66 , , " "" ""' 2I' ! !! !!' 2I' A 66 "!I!" 2I' J "" !! 66 , , . 2 , & "" !! 66 , C 3 @ "! :!6; ( , :5* +$/; * 7 , , :!<; J "" + 3 / !! + / 66 + / 01 +/9 0 + 3/ 3 "!'' 0 " +6/ 0 + 3/ + , , * :!1; 7 ! +/ ( % +/ - 616 % "& '%(% ) * +) , - . , /01
S. Kumar, R.N. Singh/Composites Science and Technolog y 61(2001)461-473 Table I differential rates of strain would necessitate the creation A summary of material properties at 1200.C used in modelling of holes/voids in one phase. Fig. 3(a)and(b) shows the SCS6(fiber)r8) Sic(matrix)(26.34 plot of longitudinal creep compliance Su as a function of time for composites with various volume fractions of Youngs modulus, E(GPa) 367.0 the fiber. The creep compliance Su was calculated for Vr=0.10, 0.20, 0.40 and 0.60. Fig 3(a)and(b) displays mmr[Eq(9)] -0.667 the Su values from the Aboudi, Hexagonal, and rule of mixtures models for each fiber volume fraction. For a For a and t in Pa and second, respectively particular Vf, the values of the time-dependent Su for all the three analytical models, namely Aboudi, Hashin △G)T (10) and rule of mixtures, are nearly the same [the values of Su calculated for Hashin model are also nearly the same as those plotted for Aboudi model shown in Fig. where Go and AG are the shear modulus at temperature 3(a)and(b) for various Vr]. This is due to the fact that 0 K and decrease in shear modulus per unit K. For Sic although the geometrical arrangements of the fiber in matrix, Go and AG are 160 GPa and 0.023 GPa/K, the two-dimensional cross-section of the Aboudi and respectively [26] Hashin models are different. their behaviors in the For this composite system, matrix creep rate is much longitudinal direction of the fiber, i.e. in the XI-direc lower than for the fiber. The composite is assumed to be tion, are the same. When a load is applied on the com- defect free, i.e. has no defects like crack and porosity. posite in the X-direction, the elongation of the We have calculated the creep compliances for several composite in the X-direction(or longitudinal direction) fiber volume fractions, Vr=0.10. 0.20, 0.40, and 0.60 does not depend on the geometrical arrangement of The strain energy per unit volume and energy dissipa- the fiber (or the matrix)for different models, but tion due to creep per unit volume are also calculate For convenience, we will use only the terms strain (a) energy and energy dissipation instead of strain energy per unit volume and energy dissipation due to creep per unit volume The elastic properties of a transverse material can be described by five independe meters. Using the composite cylinder model, [18-20] was able to get exact equations for four para vr=0.10 meters, namely El(the Youngs modulus along the fiber axis), v12(the Poissons ratio in 1-2 plane), K23(the 当- Hexagona,v=.20 plane strain bulk modulus), and u1(shear modulus in ule of Mixture, v=0 the fiber direction). The remaining fifth parameter A23 (the transverse shear modulus)cannot be exactly calcu lated from Hashin's composite cylinder model. But, one The et bounds for u23 from composite cylinder model. (b)5. The displacement boundary condition in the composite cylinder model can be used to calculate the upper bound for A23, whereas the stress boundary condition can be used to get the lower bound for u23 Hashin and Rosen 27] showed that the bounds do not coincide, except at very low and very high volume fractions. But, using the Christensen's three phase cylinder model [15], one can get an exact value for 423. In the present study, both *-Hexagonal, V,-O60 Hashin's composite cylinder model and Christensens o- Rule of Mixture, V,=0.60 three phase cylinder model were used to calculate var ious elastic parameters 3.1. Compliance along the longitudinal axis of the fiber, Su Fig 3. Dependence of the longitudinal creep compliance Su on time for the various models. The values for the hashin model are the same When a stress is applied along the longitudinal axis of as those for the Aboudi model. Similarly, the values for square and the fiber both fiber and matrix deform at the same rate denotes the volume fraction of fiber(a)Vr=0.10 and 0.20, and(b) for This is necessary to maintain material continuity since Vr=0.40 and 0.60
6 6' 6+ , 6' 6 ' 7 0 3 6' 6 "1' OB ' '!A OB2 :!1; 7 3 , @ @ J 4 I' "' ' !' ' 6' ' 1' 7 , , N - :">9!'; , 3 * "" + G F 3 / "! + B F "9! / "!A + @ / "! + / !A + / 3 - F ) !A !A , , !A - :!=; , 3 , ) 0 F :"; 0 + 3/ :!1A6; G F +OB/ A1= ' A'' ' B ' "= ' "$ ! :5* +$/; = ! "'"< " "'1 "'!< :5* +$/; ' 11= ' 7 B 7 A H "" - ( * * H 3 4 +/ 4 I' "' ' !' +/ 4 I' 6' ' 1' % "& '%(% ) * +) , - . , /01 61<
S. Kumar, R.N. Singh/ Composites Science and Technology 61(2001)461-473 epends solely on the volume fraction of fiber, Er(the V=0.10 modulus of the fiber), and Em (the modulus of the -V=0.20 matrix). For the Hashin's composite cylinder model [15, 19, 20, El is given by the equation El= vrer +(l-vrem 4V/(1-(w-)km (1-V)m/(kr+p/3)+[rm/(km+m/3)+1 where v, k and u are the Poissons ratio, bulk modulus, ind shear modulus, respectively, and the subscripts f and m denote fiber and matrix, respectively The third term on the right-hand side of Eq. (11)is Time(Hour) egligible as compared to the first two terms as the Fig 4. Dependence of the longitudinal stress (i.e. the stress along the numerator contains five terms out of which four terms fiber axis) in the matrix on time for various fiber volume fractions(Vr are less than 1, and the denominator is more than1 under a 200 MPa stress applied to composite in the longitudinal which makes the third term very small as compared to decon the first two terms. Hence. the third term can be neglected. If we neglect the third term on right-hand sustain higher longitudinal stress as time progresses side of Eq. (11), the equation for El becomes the Fig. 4 shows that after 1500 h the longitudinal stress in equation based on the rule of mixtures the matrix is about 210 and 282 MPa, respectively, for Similarly, all the three FEM models(square, sat the re- the fiber volume fractions of 0. 10 and 0.60, respectively D and hexagonal) gave nearly the same values For a given Vr and time, the calculated strain energy time-dependent Su for a particular V. Although there per unit volume was same for all the three finite-element are some differences between the values of Su calcu- models subjected to a 200 MPa stress applied along lated from the analytical and finite element techniques, longitudinal axis of the fiber and the same was true for the differences are negligible as compared to their abso- the energy dissipation per unit volume. These observa lute values tions are in agreement with the calculation of Sn from As shown in Fig 3(a)and (b), the creep compliance the finite-element models, i. e. for a given Vr and time, Su increases with an increase in the fiber volume frac- the calculated value of Su is same for all the three finite- tion, Ve. This is due to the fact that the fiber is more element models. Fig. 5(a)and(b) shows the variation of compliant than the matrix. The parameter A in Eq (9) the strain energy per unit volume and energy dissipation is directly proportional to compliance and A for the per unit volume, respectively, with respect to time for fiber is much higher than that for the matrix the finite-element model and various Ve. As shown in Fig 4 shows the variation of longitudinal stress in the Fig. 5(a), in the beginning the strain energy per unit matrix with respect to time for various fiber volume volume is inversely proportional to the fiber volume fractions when a 200 MPa longitudinal stress is applie fraction, i.e. it is higher for low Vf than for high Vf. But, As shown in Fig 4, although initially there is not much as time progresses the strain energy becomes propor- difference in the longitudinal stress in the matrix for tional to Vf. This can be explained with the help of the ifferent fiber volume fractions, i.e. the initial long- following three equations [Eqs. (12a)and(12b) are valid itudinal stress in the matrix does not depend very much at time, t=0, whereas Eq (12c)is valid for any 1]. on the fiber volume fraction. But, as the fiber(sCS6)is much more creepy than the matrix(SiC), the stress in of= Erer (12a) the fi ber decreases with time which results in an increase in the longitudinal stress in the matrix because them (12b) applied longitudinal stress on the composite is constant, i.e. 200 MPa and any decrease in the longitudinal stress orVr+om(I-Vr=capp on the fiber creates additional longitudinal stress in the matrix. Hence it is the fiber volume fraction which where aapp denotes the applied longitudinal stress, and determines the final or long-term longitudinal stress in Ef diately after the application of a long- the matrix. Higher the fiber volume fraction, larger will itudinal he composite follows Eqs.(12a)and be the force which would be relieved by the fiber(12b), but later on stress in the fiber is relieved due to because of the higher creep of the fiber. Therefore, for creep and it causes an increase in the matrix stress(as higher the fiber volume fraction, the matrix will have to explained earlier for Fig. 4). For high Vf, the initial
+ / + 3/ 7 - F :"! &B ' "' ' 1' 7 4 , !'' &B 3 , , "" 4 "" 7 <+/ +/ , , 4 ( , 7 <+/ , 4 4 ) 4 3 , , * :5* +"!/ +"!/ I' , 5* +"!/ ; 4 " 4 , I 4 , 5* +"!/ +"!/ 3 + 3 7 6/ 7 4 7 6 H + 3 / 3 +4 / !'' &B 611 % "& '%(% ) * +) , - . , /01
S. Kumar, R N. Singh/Composites Science and Technology 61(2001)461-473 (a)75 3.2. Compliance in the transverse plane of the omposite, S22 自z3 ig. 6(a-d) shows the dependence of the transverse creep compliance of the composite, S22, on the time for various Vr and for each of the five models. For a given Vr and time, the creep compliance S22 is always higher for the square-D model than the hexagonal and square E6.0 models. This is because for a given Vf, the average di tance between two fibers in the square-D model is less than the other two models. Table 2 gives a summary of 1500 the distances of the neighboring fibers from a given fiber in the various finite-element models the distances are calculated as the minimum distance between the fiber and its nearest neighbor fibres in any direction. a is the length of the side Ab in the square array of fiber. For E2382525品 small V the distances of the neighboring fibers from a given fiber for the various fiber arrangements can b summarized as following 1. Hexagonal array [Fig. 1(a)]-for the central fiber small Vf. there are four fibers each at a dis- tance of a/V3, and for a fiber having center at one of the corners and small V, there are two 00 fibers each at a distance of a/yv3, and one at a distance of energy dissipation per unit volum pplied in the longitudinal directior or all three finite-element model 二 2. Square-D array [Fig. I(b)]-for the central fiber and small V, there are four fibers each at a dis tance of a/v2 and for a fibers having center at one of the corners and small Ve, there are two fibers models)are the same each at a distance of a and one fiber at a distance of a/v2 3. Square array [Fig. 1(c)]-for each fiber and small longitudinal stress in fiber is low as compared to that for Ve, there are two fibers each at a distance of a and low Vr[Eq (12c)for the same applied stress. Hence, for one at a distance of a√2 igh Vr the strain energy, which is given by eq .(13),is ower than for low V Hence, the interaction between fibers is higher in the case of square-D model than the other two models. As Strain energy= (1-V (13) fiber is more compliant than the matrix, the compliance S22 for the composite is higher for the square-D model than the other two models. Similarly, for a given V, the As strain energy is inversely proportional to the com- average distance between the fibers is less in the case of pliance, the plots in Fig. 5(a)are in agreement with the the hexagonal model than the square model. For this variation of compliance along the longitudinal axis of very reason, the creep compliance S22 is higher for the fiber, Su, as shown in Fig 3(a)and(b). In the begin- hexagonal model than the square model. Also, as shown time progresses, Su for high Vr becomes higher than for because the fiber is more compliant than the mall c ning, SuI for low Vr is higher than for high Vf. but as in Fig. 6(ahd), S22 increases with an increase in ow Vr. But, as shown in Fig. 5(b) the energy dissipation One important point worth noting is that although per unit volume is proportional to Vr and the rate of with a change in the fiber volume fraction, the magni- increase in the energy dissipation is very high initially, tude of S22 changes for all the three models, the relative but as time progresses the rate of increase in energy changes among these three models remain the same, i.e. dissipation decreases. As energy dissipation is propor- the curve for the hexagonal model always remains at the tional to the compliance, the plots in Fig. 5(b) are in same position with respect to the other two curves, i.e greement with the variation of Si shown in Fig 3(a) to the curves corresponding to square-D and square and (b) models. This is due to the fact that although the distance
, , 4 :5* +"!/; - 4 , 5* +"A/ , , 4 ! 4 ! " 4 ( 7 <+/ , 3 "" , 7 A+/ +/ 4 "" , 4 4 "" 4 , 4 ) , 7 <+/ 4 ( 7 <+/ , "" , 7 A+/ +/ 1% % ) ) !! 7 1+/9+/ , !! 4 7 4 !! , * H 3 * 4 , , * H , ! , () * 7 4 C , " -3 :7 "+/; K 4 A 4 , A A ! * H :7 "+/; K 4 2 ! 4 , 2 ! A * :7 "+/; K 4 , ! - , * H , ( 3 !! * H , 4 , 3 * 7 !! 3 * ( , 7 1+/9+/ !! , 4 3 ? , , !! 3 , , , * H * 7 < H +/ +/ !'' &B 7 4 +3 * H * / % "& '%(% ) * +) , - . , /01 61=
S. Kumar, R.N. Singh/Composites Science and Technology 61(2001)461-473 (a)3.7 (b)40 36 3.8 3.6 34 Square-D 3.3 l200 1500 1200 15 Time(Hour (c)4.5 V=0.40 -z--mm-m:(d)5.5 V1=060 4.5 4.0 Square-D 3.0 30 0 1200 1200 Fig. 6. Dependence of the transverse creep compliance S22 on time for the various models: (a)for Vr=0.10, (b) for Vr=0.20. (c)for Vr=0.40, and (d) for Vr=( Table 2 between two fibers in a particular model changes with a Interfiber spacings for various finite-element models change in V the relative change in these distances in all Model Number of these models remains more or less the same neighbors As shown in Fig. 6(a)d), in the short time Hexagonal is higher for low Vf than for high Vf, but in the long time region the opposite is true. This is because of the fact that in the short time region the elastic moduli of Corner a( 5 the materials determine the deform tion of the cor site and in the long time region, the creep properties of the materials determine the deformation as the mod Square- Center ulus of the fiber is greater than the matrix (Table 1), the compliance S22 is inversely proportional to Vr in the short time region. But as the fiber is more creepy than the matrix, S22 is proportional to Vf in the long time region Fig. 7(a-d)shows the variation of strain energy and Corne energy dissipation with respect to time for the three finite-element models and for Ve=0.10 and 0.40 sub- jected to 200 MPa stress applied in the transverse plane Length of side aB in the square array of fiber(Fig. 1). The areas As shown in Fig. 6(a-(d), for a given Vr and time, the of the hexagonal, square-D, and square models are taken to be the sa upper and lower bounds for S22 in the three finite-ele
, , , 4 ( , 7 1+/9+/ !! , 4 4 ( 3 + "/ !! 4 ) 3 !! 4 7 =+/9+/ , , 4 I' "' ' 6' !'' &B ( , 7 1+/9+/ 4 , !! 7 1 H !! % +/ 4 I' "' +/ 4 I' !' +/ 4 I' 6' +/ 4 I1' ! 4 & B H -3 0 6 " A ! 4 ! 0 ! " A ! 4 ! " A ! 4 ! * H 0 6 " ! ! 4 ! 0 ! " ! 4 ! " " ! ! 4 ! * 0 ! " ! 4 " ! ! 4 I8 () * +7 "/ 3 * H * @ 61> % "& '%(% ) * +) , - . , /01
S. Kumar, R N. Singh/Composites Science and Technology 61(2001)461-473 V;=0.10 V=0.4 Hexagonal Hexagonal Square-D 65 60 1000 0.10 V;=0.40 D Square 0.2 quare 0.0 1500 1000 1500 Time (Hour) Time(Hour) c) 7. Dependence of the (a and b) strain energy per unit volume, and (c and d)energy dissipation per unit volume on time for various finite. nt models subjected to a stress of 200 MPa applied in the transverse plane ment models are given by square-D and square models, dissipation for square-D and square models are the respectively, and S22 for hexagonal model is in the middle highest and lowest, respectively, and for hexagonal of these two values. As the strain energy is inversely model, it is in the middle of the other two models. As proportional to the compliance, the plots of S22 shown energy dissipation is proportional to the compliance, in Fig. 6(a)and 6(c)are in agreement with Fig. 7(a)and these plots are also in agreement with the plots of S22 (b), where for a given Vr and time, the upper and lower shown in Fig. 7(a-d), where the square-D and square bounds of the strain energies are given by square and models give the upper and lower bounds of S22 among square-D models, respectively, and the strain energy for the three finite-element models and S22 for the hex the hexagonal model is in the middle of these two agonal model is in the middle of those for the other two values. Also for a given time, the difference between the models strain en ergies for square and square-D models fo As shown in Fig. 6(aHd), the creep compliance S22 Ve=0.40 is much higher than for V=0.10. This is in calculated from the Aboudi model is always higher than agreement with the plots of S22 shown in Fig. 6(a-(d), the Hashin model For small Vf[Fig. 6(a)], the difference where the difference between the values of S22 for between the S22 values calculated from these two ana square-D and square models at a given time is propor- lytical methods is relatively large(in terms of the actual tional to V. It is also observed that in the initial stages magnitude), but for high V [Fig. 6(d)], the difference the strain energy for high Vr at a given time is lower becomes small. On the other hand, the difference than for low Ve which are in agreement with the plots of between the S22 values for any of the two finite-element S22[Fig. 6(a)(d) models increases with an increase in Ve Yancey [271 Fig. 7(c)and(d)show the variation of energy dis- used the Hashin's composite cylinder model which gives sipation per unit volume with respect to time for the upper and lower limits of the creep parameter S22 Vr=0.10 and 0.40. For a given Vr and time, the energy and not a unique value
* H * !! 3 , ( !! , 7 1+/ 1+/ , 7 =+/ +/ , 4 , * * H 3 , ( . , * * H 4 I' 6' 4 I' "' , !! , 7 1+/9+/ , . , !! * H * 4 4 4 , , 4 , , !! :7 1+/9+/; 7 =+/ +/ , , 4 I' "' ' 6' 7 4 * H * , 3 , ( , !! , 7 =+/9+/ , * H * , !! !! 3 , ( , 7 1+/9+/ !! ( , - 7 4 :7 1+/; . , !! , + / 4 :7 1+/; . ? . , !! , , 4 G :!=; - F , , !! * 7 = H + / + / !'' &B % "& '%(% ) * +) , - . , /01 61$
S. Kumar, R.N. Singh/Composites Science and Technology 61(2001)461-473 3.3 compliance along the longitudinal axis of th model. But, there is no central fiber in the square model and hence, there would not be any stress build up in this case in the middle except at the fiber-matrix interfaces Fig &(aHd)show the variation of shear compliance at each corners, which are there in the other two models along longitudinal axis of the fiber, S44, with respect to too. Consequently, the square model displays the highest time for various Ve and for all the five models For a hear compliance S44. given Vf and time, the creep compliance S44 is always Similar to S22, although the magnitude of S44 changes lower for the square-D model than the square and hex- for all the three models with a change in Vf, the relative agonal models, and S4 for square model is higher than changes among these three models remain the same, i.e the other two finite-element models. The relative position the curve for the hexagonal model always remains at the of the S44 curves for these three models are opposite to same position with respect to the other two curves. This those of the S22 curves. For S22, the curve for the is because although the distance between two fibers in a square-D was on the top and the curve for the square particular model changes with a change in Vf, the relative model was at the bottom. This can be explained from change in these distances in all these models remains the fact that in the case of square-D model, there is a more or less the same central fiber which causes a stress build up at the fiber- As shown in Fig. &(ahd), the differences between S44 matrix interface and hence, when a shear stress is (in terms of the actual magnitude) calculated from these applied, the shear resistance of the model increases three models increase with an increase in Vr. With an which results in a lower shear compliance. Although increase in Vf, the total area of the fiber-matrix inter- there is a central fiber in the hexagonal model as well. it face in the finite element models increases. Hence with is at a larger distance from the other four fibers(one an increase in Vf, the area where the stress build up each at the four corners). Hence, the shear compliance occurs due to an applied shear stress, increases. There- S44 for the hexagonal model is higher than the square-D fore, the differences between the values of S44 calculated a)88 (b)100 Vr=020 Harbin o- Hash -Square-D (c)120 Vt=0.60 110 Fig 8. Dependence of the transverse shear creep compliance S44 on time for different models with(a)Vr=0.10, (b)Vr=0.20.(c)Vr=0.40, and(d)
1%1% ) ) & 7 ) 89 66 7 >+/9+/ , 3 66 , 4 7 4 66 , , * H * 3 66 * , 66 !! 7 !! * H , * , 3 * H , 9 3 , , , ( 3 , + / - 66 3 * H ) * , 3 9 3 , , 0 * * 66 !! 66 , 4 3 , , , , , , 4 ( , 7 >+/9+/ . , 66 + / , 4 J 4 9 3 - , 4 , . , 66 7 > H 66 . , +/ 4 I' "' +/ 4 I' !' +/ 4 I' 6' +/ 4 I1' 6=' % "& '%(% ) * +) , - . , /01