COMPOSITES SCIENCE AND TECHNOLOGY ELSEⅤIER Composites Science and Technology 62(2002 www.elsevier.com/locate/compscitech The transverse thermal conductivity of 2D-SiCr/sic composites G.E. Youngblood a ,*. David J Senora, R.H. Jones. Samuel Graham Pacific Northwest National Laboratory, PO Box 999, MSIN K2-44, Richland, WA99352, US.A Sandia National Laboratories. Livermore. CA 94550. USA Received 21 June 2001; received in revised form 7 February 2002: accepted 7 February 2002 Abstract The Hasselman-Johnson(H-)model for predicting the effective transverse thermal conductivity(Kefr) of a 2D-SiCr/Sic com- posite with a fiber-matrix thermal barrier was assessed experimentally and by comparison to numerical FEM predictions. Agree- ment within 5% was predicted for composites with simple unidirectional or cross-ply architectures with fiber volume fractions of 0.5 or less and with fiber-to-matrix conductivity ratios less than 10. For a woven 2D-SiCdSic composite, inhomogeneous fiber packing d numerous direct fiber-fiber contacts would introduce deviations from model predictions. However, the analytic model should be very appropriate to examine the degradation in Kefr in 2D-woven composites due to neutron irradiation or due to other mechanical or environmental treatments. To test this possibility, expected effects of irradiation on Ker were predicted by the H-J model for a hypothetical 2D-SiCrSic composite made with a high conductivity fiber and a cvi-Sic matrix. Before irradiation, redicted Kefr for this composite would range from 34 down to 26 W/m K)at 200 and 1000C, respectively. After irradiation to saturation doses at 200 or 1000C, the respective Kefr-values are predicted to decrease to 6 or 10 w/(m-K).C 2002 Elsevier Science Ltd. All rights reserved Keywords: A. Ceramic-matrix composites(CMCs); B Modeling: B. Thermal conductivity; C Computational simulatio 1. ntroduction for possible applications in advanced nuclear fission Silicon carbide(SiC) exhibits favorable mechanical A major issue to be considered when using SiC/ SiC in ld chemical properties at high temperatures for many a high-temperature neutron radiation environment, or applications. Desirable properties of SiC that also make in other non-radiation environments where components it attractive for use in fusion reactor applications in a or structures are subjected to a high heat flux, is the neutron radiation environment are its dimensional sta- expected in-service behavior of its effective transverse bility in the 800-1000oC temperature range, low thermal conductivity, Keff. Knowledge about the expec induced radioactivity and low afterheat [1]. However, ted range of Kefr is necessary to optimize SiC/SiC con the brittle nature of SiC limits its use as a structural figurations for their intended uses. Several modeling material. As compared to monolithic SiC, continuous studies have shown how Kefr depends upon constituent fiber-reinforced SiC-matrix composites (SiCr/SiC) exhi- fiber and matrix thermal conductivity values, and their bit improved toughness with a high, non-catastrophic volume fractions and distributions [6-8]. However strain-to-failure [2]. For these reasons, SiC in the form many experimental measurements have indicated that of SiCr/SiC is being considered as a structural material interfaces between fibers and matrices in a composite for first wall or breeder blanket applications in introduce a thermal barrier that may affect Keff [9-12] advanced fusion power plant concepts in the US [3] Furthermore, Kefr may be affected by physical changes and in international programs [4, 5]. In the US, the of the interface and even the surrounding atmosphere DOE-sponsored Nuclear Energy Research Initiative As with mechanical behavior, to attain desired thermal (NERD) program also is examining SiC composites behavior of SiC SiC proper attention needs to be given to the design of the interphase and to the control of Corresponding author. Tel:+1-509. interfacial thermal effects Classical composite models recently have been up- dated to include the effect of interfacial thermal barrier 0266-3538/02/S. see front matter C 2002 Elsevier Science Ltd. All rights reserved. PII:S0266-3538(02)00069-6
The transverse thermal conductivity of 2D-SiCf/SiC composites G.E. Youngblooda,*,David J. Senora ,R.H. Jonesa ,Samuel Grahamb a Pacific Northwest National Laboratory, PO Box 999, MSIN K2-44, Richland, WA 99352, USA bSandia National Laboratories, Livermore, CA 94550, USA Received 21 June 2001; received in revised form 7 February 2002; accepted 7 February 2002 Abstract The Hasselman–Johnson (H–J) model for predicting the effective transverse thermal conductivity (Keff) of a 2D-SiCf/SiC composite with a fiber-matrix thermal barrier was assessed experimentally and by comparison to numerical FEM predictions. Agreement within 5% was predicted for composites with simple unidirectional or cross-ply architectures with fiber volume fractions of 0.5 or less and with fiber-to-matrix conductivity ratios less than 10. For a woven 2D-SiCf/SiC composite,inhomogeneous fiber packing and numerous direct fiber–fiber contacts would introduce deviations from model predictions. However,the analytic model should be very appropriate to examine the degradation in Keff in 2D-woven composites due to neutron irradiation or due to other mechanical or environmental treatments. To test this possibility,expected effects of irradiation on Keff were predicted by the H–J model for a hypothetical 2D-SiCf/SiC composite made with a high conductivity fiber and a CVI-SiC matrix. Before irradiation, predicted Keff for this composite would range from 34 down to 26 W/(m K) at 200 and 1000 C,respectively. After irradiation to saturation doses at 200 or 1000 C,the respective Keff-values are predicted to decrease to 6 or 10 W/(m–K). # 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Ceramic-matrix composites (CMCs); B. Modeling; B. Thermal conductivity; C. Computational simulation 1. Introduction Silicon carbide (SiC) exhibits favorable mechanical and chemical properties at high temperatures for many applications. Desirable properties of SiC that also make it attractive for use in fusion reactor applications in a neutron radiation environment are its dimensional stability in the 800–1000 C temperature range,low induced radioactivity and low afterheat [1]. However, the brittle nature of SiC limits its use as a structural material. As compared to monolithic SiC,continuous fiber-reinforced SiC-matrix composites (SiCf/SiC) exhibit improved toughness with a high,non-catastrophic strain-to-failure [2]. For these reasons,SiC in the form of SiCf/SiC is being considered as a structural material for first wall or breeder blanket applications in advanced fusion power plant concepts in the US [3] and in international programs [4,5]. In the US, the DOE-sponsored Nuclear Energy Research Initiative (NERI) program also is examining SiC composites for possible applications in advanced nuclear fission reactors. A major issue to be considered when using SiCf/SiC in a high-temperature neutron radiation environment,or in other non-radiation environments where components or structures are subjected to a high heat flux,is the expected in-service behavior of its effective transverse thermal conductivity, Keff. Knowledge about the expected range of Keff is necessary to optimize SiCf/SiC con- figurations for their intended uses. Several modeling studies have shown how Keff depends upon constituent fiber and matrix thermal conductivity values,and their volume fractions and distributions [6–8]. However, many experimental measurements have indicated that interfaces between fibers and matrices in a composite introduce a thermal barrier that may affect Keff [9–12]. Furthermore, Keff may be affected by physical changes of the interface and even the surrounding atmosphere. As with mechanical behavior,to attain desired thermal behavior of SiCf/SiC proper attention needs to be given to the design of the interphase and to the control of interfacial thermal effects. Classical composite models recently have been updated to include the effect of interfacial thermal barriers 0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(02)00069-6 Composites Science and Technology 62 (2002) 1127–1139 www.elsevier.com/locate/compscitech * Corresponding author. Tel.: +1-509-375-2314; fax: +1-509-375- 2186. E-mail address: ge.youngblood@pnl.gov (G.E. Youngblood)
G E. Youngblood et al. /Composites Science and Technology 62(2002)1127-1139 [13]. Interfacial thermal barriers are quantitatively where h is the effective interfacial conductance; Km and characterized by a value called the interfacial con- Kr are the thermal conductivity values of the matrix and ductance, which includes the effect of imperfect match- fiber constituents, respectively; and Vr and a are the ing of surfaces at an interface as well as the effect of fiber volume fraction and radius, respectively. Exami interfacial gaps brought about by debonding of the fiber nation of Eq. (1)indicates that the value of the non from the matrix or microcracking within the fiber coat- dimensional paramete 0, K p fah, relative to the fiber-to- ing [13]. In particular, interfacial fiber-matrix debonding matrix conductivity rat f Km, controls the overall may develop in service due to thermal expansion mis- effect of interfacial barrier resistances on Keff match or thermomechanical fatigue: or in a radiation For analysis, Eq(1) can be written in a simpler non- environment, due to differential swelling/shrinkage dimensional form by making the substitutions: characteristics of the irradiated fiber-matrix constituents The overall interfacial conductance in this paper will be Ker/Km=R interpreted in the broadest sense to also include the Kr/Km =r effective heat transfer coefficient of a thin fiber coating, K/ah=x which may or may not act as a thermal barrier [14, 15]. Vr=f For instance, thin (100-500 nm)pyrocarbon(PyC) or boron-nitride fiber coatings are commonly used to pro- then, using the algebraic substitutions A=(1 x+ r) vide protection of Sic-type fibers during fabrication of and B=(1+ x-r), Eq (1)becomes toughening. Thus, the interfacial conductance, and there- R=[l-(B/A)//[+(B/A)/T (3) fore Keff, will depend upon the fib and the thermal, mechanical and neutron radiation expo- In Eq. 3),lAl is always greater than B), A is always sure histories, and perhaps the surrounding atmosphere positive and b can be positive or negative. Therefore, that can permeate any interfacial gaps [16] he thermal conductivity ratio R is less than or greater The purposes of this study are: first, to assess the than 1 for b being positive or negative, respectively validity limitations for using a particularly simple, but Also, Keff=Km (i.e. R=1) for B=0, or equivalently for very useful model derived by Hasselman and Johnson x=r-l. An explicit solution for x, which is the reci- [13] to predict Kefr for a two-dimensional (2D)SiCr/Sic procal of the Biot number for heat transfer at the fiber composite; and second, to examine the expected effects surface, is given in terms of measurable quantities R, r of temperature and atmosphere on Keff for an example and f by combination of currently available SiC fiber and matrix types. Only the effects of changes in the interfacial con x={(R+1)r-1)+(r+1)(1-R){R+1)+(R-1) ductance on Kefr in the transverse direction are exam- ined. although interfacial conductance effects on in- lane Kefr also are important and expected For In Fig. 1(a, b), the relative thermal conductivity R is instance, the in-plane Keff can be significantly affected plotted as a function of h for fiber volume fractions by microcracking within the composite matrix and by f=0. 1, 0.4, 0.5 and 0.6 for two different fiber /matrix debonding of the fiber from the matrix, as shown by Lu thermal conductivity ratios, r=5 and r=0.2, respec- and Hutchinson [17]. However, the thickness and the tively. To easily compare the effects of r and h on Keff, transverse Kefr of a structural wall govern, in part, ther- the same size fiber (a=5 um) and the same matrix ther mal management in a system, which is the primary focus mal conductivity [Km=20 W/(m K)] were assigned for here. Finally, the expected effects of radiation exposure this example. In these figures the units for h were selec on the interfacial conductance and ultimately on Kefr are ted to be 104 w/(m'K). The numerical labels on the considered for a hypothetical SiC/SiC composite that plot cover a range 0. 1-105 W/( For composites was designed to have a high thermal conductivity containing Sic-type fibers, h-values ranging from I to 400 W/cmK) have been reported [12] L. The hasselman-Johnson model The following observations are made: In 1987, Hasselman and Johnson [13] derived an 1)Asf→0(e.g.f=0.1),R→ I for all values of h. expression for the transverse Keff of dispersed uniaxial (2) For r>l, there is a common crossover point at fibers in a matrix with thermal barriers(thin, insulating R=l for all values of f when x=r-1 type fiber coatings or fiber/matrix debonds) given by (3)For r<l, there is no crossover point and R <I for all values off and h Kerr=Km[(Kr/Km-1-kr/ah)Vr+(1+Kr/Km+kr/ah) (4)For h-0(complete fiber-matrix thermal decou pling), R attains its minimum possible value inde- (1-Kr/Km+ Krahvr+(1+Kr/Km+ krah (1) ndently of r and is given by Rmin=(1-f/(1+f
[13]. Interfacial thermal barriers are quantitatively characterized by a value called the interfacial conductance,which includes the effect of imperfect matching of surfaces at an interface as well as the effect of interfacial gaps brought about by debonding of the fiber from the matrix or microcracking within the fiber coating [13]. In particular,interfacial fiber-matrix debonding may develop in service due to thermal expansion mismatch or thermomechanical fatigue; or in a radiation environment,due to differential swelling/shrinkage characteristics of the irradiated fiber-matrix constituents. The overall interfacial conductance in this paper will be interpreted in the broadest sense to also include the effective heat transfer coefficient of a thin fiber coating, which may or may not act as a thermal barrier [14,15]. For instance,thin (100–500 nm) pyrocarbon (PyC) or boron-nitride fiber coatings are commonly used to provide protection of SiC-type fibers during fabrication of SiCf/SiC and to provide a compliant layer for composite toughening. Thus,the interfacial conductance,and therefore Keff,will depend upon the fiber coating characteristics and the thermal,mechanical and neutron radiation exposure histories,and perhaps the surrounding atmosphere that can permeate any interfacial gaps [16]. The purposes of this study are: first,to assess the validity limitations for using a particularly simple,but very useful model derived by Hasselman and Johnson [13] to predict Keff for a two-dimensional (2D) SiCf/SiC composite; and second,to examine the expected effects of temperature and atmosphere on Keff for an example combination of currently available SiC fiber and matrix types. Only the effects of changes in the interfacial conductance on Keff in the transverse direction are examined,although interfacial conductance effects on inplane Keff also are important and expected. For instance,the in-plane Keff can be significantly affected by microcracking within the composite matrix and by debonding of the fiber from the matrix,as shown by Lu and Hutchinson [17]. However,the thickness and the transverse Keff of a structural wall govern,in part,thermal management in a system,which is the primary focus here. Finally,the expected effects of radiation exposure on the interfacial conductance and ultimately on Keff are considered for a hypothetical SiCf/SiC composite that was designed to have a high thermal conductivity. 1.1. The Hasselman–Johnson model In 1987,Hasselman and Johnson [13] derived an expression for the transverse Keff of dispersed uniaxial fibers in a matrix with thermal barriers (thin,insulatingtype fiber coatings or fiber/matrix debonds) given by: Keff ¼Km½ ð Þ Kf=Km1Kf =ah Vf þ ð Þ 1þKf =KmþKf =ah ½ ð Þ 1Kf =Km þ Kf =ah Vf þ ð Þ 1 þ Kf =Km þ Kf =ah 1 ð1Þ where h is the effective interfacial conductance; Km and Kf are the thermal conductivity values of the matrix and fiber constituents,respectively; and Vf and a are the fiber volume fraction and radius,respectively. Examination of Eq. (1) indicates that the value of the nondimensional parameter, Kf/ah,relative to the fiber-tomatrix conductivity ratio, Kf/Km,controls the overall effect of interfacial barrier resistances on Keff. For analysis,Eq. (1) can be written in a simpler nondimensional form by making the substitutions: Keff=Km ¼ R Kf=Km ¼ r Kf=ah ¼ x Vf ¼ f ð2Þ then,using the algebraic substitutions A=(1 + x + r) and B=(1 + xr),Eq. (1) becomes: R ¼ ½ 1ð Þ B=A f =½ 1 þ ð Þ B=A f : ð3Þ In Eq. (3),|A| is always greater than |B|, A is always positive and B can be positive or negative. Therefore, the thermal conductivity ratio R is less than or greater than 1 for B being positive or negative,respectively. Also, Keff=Km (i.e. R=1) for B=0,or equivalently for x=r1. An explicit solution for x,which is the reciprocal of the Biot number for heat transfer at the fiber surface,is given in terms of measurable quantities R, r and f by: x¼ f R½ þ ð Þ þ1 ð Þ r1 ð Þ rþ1 ð Þ 1R = f Rð Þþ þ 1 ð Þ R1 : ð4Þ In Fig. 1(a,b),the relative thermal conductivity R is plotted as a function of h for fiber volume fractions f=0.1,0.4,0.5 and 0.6 for two different fiber/matrix thermal conductivity ratios, r=5 and r=0.2,respectively. To easily compare the effects of r and h on Keff, the same size fiber (a=5 mm) and the same matrix thermal conductivity [Km=20 W/(m K)] were assigned for this example. In these figures the units for h were selected to be 104 W/(m2 K). The numerical labels on the plot cover a range 0.1–105 W/(cm2 K). For composites containing SiC-type fibers, h-values ranging from 1 to 400 W/(cm2 K) have been reported [12]. The following observations are made: (1) As f ! 0 (e.g. f=0.1), R ! 1 for all values of h. (2) For r >1,there is a common crossover point at R=1 for all values of f when x=r 1. (3) For r <1,there is no crossover point and R <1 for all values of f and h. (4) For h!0 (complete fiber-matrix thermal decoupling), R attains its minimum possible value independently of r and is given by Rmin=(1f)/(1 + f). 1128 G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139
G E. Youngblood et al./Composites Science and Technology 62(2002)1127-1139 5) For h-oo(perfect fiber-matrix thermal cou- fibers become thermally decoupled from the matrix and pling), R approaches a maximum value which is Rmin effectively represents the relative thermal con given by ductivity for the limiting case of dispersed cylindrical pores with a volume fraction /. This latter point has Rmax=(1-f)+r(1+O)/(1+f)+r(1-/) important consequences for SiC/Sic designed to have a high thermal conductivity value, i.e. for a composite (6) For r>l (r<1), as r increases the transition made from matrix and fibers with individually high region where R passes through its maximum rate K-values as well as with high conductance interfaces of change occurs for lower(higher) values of h Due to mechanical thermal or environmental stress the interface alone may be sufficiently degraded for Kefr/K Overall, for dispersed fibers in a matrix Kefr clearly is to be reduced from 2.3(at h=oo) down to 0.42(at controlled primarily by the thermal conductivity of the h=0), a possible 81% reduction! Independent degrada continuous matrix phase, Km. To attain a Kefr value tion of Km would further reduce Keff greater than Km, both relatively high Ke and h-values Interestingly, when h+0, r depends on the size of the are necessary. Even when r=100, by observation(5)for fibers through the x-term. For the same fiber volume a typical SiCr/SiC fiber packing fraction=0. 4, Rmax fraction and interfacial conductance larger R-values are would be less than 2.3 for perfect (h-oo) fiber-matrix obtained with a few large-diameter fibers rather than thermal coupling. At the other extreme when h-0, the numerous small-diameter fibers. Finally, it is apparent that for r<l, Kef is not very sensitive to the values of h. In contrast, Ke is quite sensitive to the values of h for r>l and more so for higher fiber volume fractions K,=100W(mK) K=20 2. Validity of the Hasselman-Johnson model The preceding observations about the effect of inter facial conductance on Kefr are all based on the H-J model. The model, a modification of the rayleigh Maxwell equations [18, 19], is considered a dilute con- centration model where a single fiber is surrounded by matrix material and is subjected to periodic boundary 05 conditions. The effects of adjacent fibers on the local field response are not considered (non-interacting fibers). Ho microstructural constituents are expected for real com- Interfacial thermal conductance, h [w/(cm. K) posite materials. Therefore, the conditions for making reliable predictions of Kef for actual composites using the h-j model need to be assessed In this section, for composites containing a random 08 distribution of fibers having an interfacial thermal resistance the predictions based on the H-J model are red to numerical pi derived from a fi element model(FEM). Briefly, a square region of material containing a random distribution of uniaxial fibers was modeled (pseudo-representative volume ele ment or pseudo-RVE). Based on extensive analysis [20], the scale of the region was chosen such that the length of the side of the window(Lw) was seven times the dia- meter of the fibers within the window To simulate a representative volume element of material, pseudo- random-periodic boundary conditions were employed nterfacial thermal conductance, h [wr(cm .. K)] using the commercial finite element code ABAQUS Fig. 1.(a, b) Comparison of analytical solutions of the Hasselman- This was accomplished by creating another square Johnson Eq.(1)as a function of h for fiber volume fractions up to region whose sides were 1.5 L and concentric with the f=0.6 and for two different K/ Km ratios(r=5 and 0.2, respectively). pseudo-RVE(Fig. 2). Doubly periodic reflections of the For each case, the fiber radius a=5 um and Km=20 w/(m K) random geometry were made in the space between the
(5) For h!1 (perfect fiber-matrix thermal coupling), R approaches a maximum value which is given by Rmax ¼ ½ ð Þþ 1f rð Þ 1 þ f =½ ð Þþ 1 þ f rð Þ 1f : (6) For r>1 (r<1),as r increases the transition region where R passes through its maximum rate of change occurs for lower (higher) values of h. Overall,for dispersed fibers in a matrix Keff clearly is controlled primarily by the thermal conductivity of the continuous matrix phase, Km. To attain a Keff- value greater than Km,both relatively high Kf- and h-values are necessary. Even when r=100,by observation (5) for a typical SiCf/SiC fiber packing fraction f=0.4, Rmax would be less than 2.3 for perfect (h!1) fiber-matrix thermal coupling. At the other extreme when h!0,the fibers become thermally decoupled from the matrix and Rmin effectively represents the relative thermal conductivity for the limiting case of dispersed cylindrical pores with a volume fraction f. This latter point has important consequences for SiCf/SiC designed to have a high thermal conductivity value,i.e. for a composite made from matrix and fibers with individually high K-values as well as with high conductance interfaces. Due to mechanical,thermal or environmental stress,the interface alone may be sufficiently degraded for Keff/Km to be reduced from 2.3 (at h=1) down to 0.42 (at h=0),a possible 81% reduction! Independent degradation of Km would further reduce Keff. Interestingly,when h6¼0, R depends on the size of the fibers through the x-term. For the same fiber volume fraction and interfacial conductance,larger R-values are obtained with a few large-diameter fibers rather than numerous small-diameter fibers. Finally,it is apparent that for r<1, Keff is not very sensitive to the values of h. In contrast, Keff is quite sensitive to the values of h for r>1,and more so for higher fiber volume fractions. 2. Validity of the Hasselman–Johnson model The preceding observations about the effect of interfacial conductance on Keff are all based on the H–J model. The model,a modification of the Rayleigh– Maxwell equations [18,19], is considered a dilute concentration model where a single fiber is surrounded by matrix material and is subjected to periodic boundary conditions. The effects of adjacent fibers on the local field response are not considered (non-interacting fibers). However,interactions between neighboring microstructural constituents are expected for real composite materials. Therefore,the conditions for making reliable predictions of Keff for actual composites using the H–J model need to be assessed. In this section,for composites containing a random distribution of fibers having an interfacial thermal resistance the predictions based on the H–J model are compared to numerical predictions derived from a finite element model (FEM). Briefly,a square region of material containing a random distribution of uniaxial fibers was modeled (pseudo-representative volume element or pseudo-RVE). Based on extensive analysis [20], the scale of the region was chosen such that the length of the side of the window (Lw) was seven times the diameter of the fibers within the window. To simulate a representative volume element of material,pseudorandom-periodic boundary conditions were employed using the commercial finite element code ABAQUS. This was accomplished by creating another square region whose sides were 1.5 Lw and concentric with the pseudo-RVE (Fig. 2). Doubly periodic reflections of the random geometry were made in the space between the Fig. 1. (a,b) Comparison of analytical solutions of the Hasselman– Johnson Eq. (1) as a function of h for fiber volume fractions up to f=0.6 and for two different Kf/Km ratios (r=5 and 0.2,respectively). For each case,the fiber radius a=5 mm and Km=20 W/(m K). G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139 1129
G E. Youngblood et al./Composites Science and Technology 62(2002)1127-1139 Pseudo-RVE 25F K,/Km 1.5 c 0.5 想d h [w/(cmK) 3.5 K/Km=100 T1 1.5L RVE of dimension Lw and imposed periodic structural boundary 1.5 conditions nested within an outer window of dimension 1.5 Lw outer boundary and the pseudo-RVE. Periodic bound 0.5 ary conditions were then imposed on the outer region that induced random-periodic boundary conditions on he inner pseudo-RVE. Random-periodic boundary h [w/(cm2.K conditions have been shown to approximate the effec- tive response of an RVE even in cases where statistical- matrix composite with a normalized effective thermal conductivity homogeneity of microstructural quantities is not satis- R(=Ken/ Km)calculated by Eq(1). The results are presented as a fied [20-22]. The effective thermal conductivity(Ke) function of h for Vr values up to 0.5 and fiber-to-matrix conductivity then was determined by dividing the average heat flux ratios of(a)r=10, and(b)r=100 by the average temperature gradient determined between the boundaries of the pseudo-RVE. Further interface. The fiber perturbation is eventually cancelled details of the analysis are presented elsewhere [23] out as the Biot number (ah/K=1/ x)decreases, and the In Fig 3(a-c), the predictions of R(=Kcfr/Km)and the material behaves as a homogeneous continuum. This is numerical FEM results for fiber packing fractions he reason for the common crossover point for all fiber Ve=0.1-0.5 are compared for two Ko/Km ratios, r=10 volume fractions predicted by the H-J model as well as and 100, respectively. It is quite clear that differences the FEM method when r>1. This point, called the between the H-J model and the numerical results occur homogenization point, results in the local values of heat for Ve>0.3, but only for conditions approaching perfect flux being equal in the fiber and matrix for temperature- thermal coupling or decoupling(hoo or 0, respec- induced boundary conditions. Thus, rather than just a tively). Also, the differences between the analytical H-J global averaged response local homogeneity is achieved model and numerical FEM results increase with Further decreasing of the Biot number past the homo- increasing volume fraction genization point continues to decouple the conductivity The differences between the analytical model and the contribution of the fibers. The differences between Eq numerical FEM predictions are due to the inhomo- 1)and the fem predictions again increase as the per- geneity created in the microstructure accounted for by turbation due to the thermal decoupling of the fibers he FEM net, but not by the H-J model. When r>l, from the matrix becomes stronger. fiber interaction cannot be ignored as larger fiber In Fig. 4(a, b), a non-dimensional plot of the differ volume fractions introduce larger perturbations in local ence(error) between the H-J model and the numerical material response. However, as an interfacial thermal FEM predictions are presented for Vr=0.4 and 0.5. resistance is introduced, this perturbation in material respectively. For the range of parameters studied, the response is dampened and fiber interaction effects are H-J model predictions deviate from the numerical reduced. Deviations from the H-J model are then results by a maximum of 9% for the perfect thermal reduced as a result of introducing an imperfect thermal coupling case and a15% for the condition of thermally
outer boundary and the pseudo-RVE. Periodic boundary conditions were then imposed on the outer region that induced random-periodic boundary conditions on the inner pseudo-RVE. Random-periodic boundary conditions have been shown to approximate the effective response of an RVE even in cases where statisticalhomogeneity of microstructural quantities is not satis- fied [20–22]. The effective thermal conductivity (Keff) then was determined by dividing the average heat flux by the average temperature gradient determined between the boundaries of the pseudo-RVE. Further details of the analysis are presented elsewhere [23]. In Fig. 3(a–c),the predictions of R(=Keff/Km) and the numerical FEM results for fiber packing fractions Vf=0.1–0.5 are compared for two Kf/Km ratios, r=10 and 100,respectively. It is quite clear that differences between the H–J model and the numerical results occur for Vf>0.3,but only for conditions approaching perfect thermal coupling or decoupling (h!1 or 0,respectively). Also,the differences between the analytical H–J model and numerical FEM results increase with increasing volume fraction. The differences between the analytical model and the numerical FEM predictions are due to the inhomogeneity created in the microstructure accounted for by the FEM net,but not by the H–J model. When r>1, fiber interaction cannot be ignored as larger fiber volume fractions introduce larger perturbations in local material response. However,as an interfacial thermal resistance is introduced,this perturbation in material response is dampened and fiber interaction effects are reduced. Deviations from the H–J model are then reduced as a result of introducing an imperfect thermal interface. The fiber perturbation is eventually cancelled out as the Biot number (ah/Kf=1/x) decreases,and the material behaves as a homogeneous continuum. This is the reason for the common crossover point for all fiber volume fractions predicted by the H–J model as well as the FEM method when r>1. This point,called the homogenization point,results in the local values of heat flux being equal in the fiber and matrix for temperatureinduced boundary conditions. Thus,rather than just a global averaged response local homogeneity is achieved. Further decreasing of the Biot number past the homogenization point continues to decouple the conductivity contribution of the fibers. The differences between Eq. (1) and the FEM predictions again increase as the perturbation due to the thermal decoupling of the fibers from the matrix becomes stronger. In Fig. 4(a,b),a non-dimensional plot of the difference (error) between the H–J model and the numerical FEM predictions are presented for Vf=0.4 and 0.5, respectively. For the range of parameters studied,the H–J model predictions deviate from the numerical results by a maximum of 9% for the perfect thermal coupling case and 15% for the condition of thermally Fig. 2. Example finite element mesh for f=0.5 shown with a pseudoRVE of dimension Lw and imposed periodic structural boundary conditions nested within an outer window of dimension 1.5 Lw. Fig. 3. Comparison of the finite element results for a uniaxial fibermatrix composite with a normalized effective thermal conductivity R (=Keff/Km) calculated by Eq. (1). The results are presented as a function of h for Vf values up to 0.5 and fiber-to-matrix conductivity ratios of (a) r=10,and (b) r=100. 1130 G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139
G E. Youngblood et al. /Composites Science and Technology 62(2002)1127-1139 vf=0.4 口 Kf/Km= 100 o KI/Km=1 1000 Biot number b) 山 O Ki/Km= 0.001 10 1000 Biot number Fig 4. Nondimensional plot of the error between the finite element results and Eq (1)for Kd Km=1, 10 or 100 and (a)Vr=0. 4 and (b)vr=0.5 decoupled fibers. As expected, the differences for the Based on values reported in the literature, h-values Ve=0.5 case are somewhat larger than for the Vf=0.4 between 0. 1 and 105 W/(cm2K)are possible [12, 24, 25 case. However, for r< 10 the differences are less than The H-J model should satisfactorily describe Keff of 5%. Thus, for composite systems with V<0.5 and many ceramic matrix composites, including SiC/SiC r-values between I and 10, the H-J model is applicable with simple unidirectional or cross-ply laminate fiber over a wide range of h-values. This criterion is often met architectures. More complicated theories that take into by typical commercial ceramic matrix composites. account fiber interaction may only be necessary when
decoupled fibers. As expected,the differences for the Vf=0.5 case are somewhat larger than for the Vf=0.4 case. However,for r410 the differences are less than 5%. Thus,for composite systems with Vf40.5 and r-values between 1 and 10,the H–J model is applicable over a wide range of h-values. This criterion is often met by typical commercial ceramic matrix composites. Based on values reported in the literature, h-values between 0.1 and 105 W/(cm2 K) are possible [12,24,25]. The H–J model should satisfactorily describe Keff of many ceramic matrix composites,including SiCf/SiC with simple unidirectional or cross-ply laminate fiber architectures. More complicated theories that take into account fiber interaction may only be necessary when Fig. 4. Nondimensional plot of the error between the finite element results and Eq. (1) for Kf/Km=1,10 or 100 and (a) Vf=0.4 and (b) Vf=0.5. G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139 1131