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 cou￾pling), 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 con￾ductivity 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 degrada￾tion 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 inter￾facial 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 con￾centration 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 com￾posite 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 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 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