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 thermallyouter boundary and the pseudo-RVE. Periodic bound￾ary 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 effec￾tive response of an RVE even in cases where statistical￾homogeneity 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,respec￾tively). 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 inhomo￾geneity 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 temperature￾induced boundary conditions. Thus,rather than just a global averaged response local homogeneity is achieved. Further decreasing of the Biot number past the homo￾genization point continues to decouple the conductivity contribution of the fibers. The differences between Eq. (1) and the FEM predictions again increase as the per￾turbation due to the thermal decoupling of the fibers from the matrix becomes stronger. In Fig. 4(a,b),a non-dimensional plot of the differ￾ence (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 pseudo￾RVE 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 fiber￾matrix 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