PSI - Issue 26

E. Solfiti et al. / Procedia Structural Integrity 26 (2020) 187–198 E. Solfiti and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

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a-axis (trough the graphene layers) also in the out-of-plane direction of FG. This would mean a phonon-based heat transfer mechanism in all the directions. The mixing concept described above has often been applied when modeling of the material conductivity has been attempted. The approaches present in literature are various but show often the common starting point: the observation of micromechanism of material formation at very low densities. The material is assumed to be a binary mixtures made by air and a solid part and some hypothesis or deductions [as in Bonnissel et al. (2001) and Celzard et al. (2005)] are made about the pore shape. At higher densities, corresponding to rolled commercial FG range, the conclusions are obtained by regression and compared with experimental data which some times result scarce. Celzard et al. (2005) observed the conductivity threshold at very low densities, that is the critical density corresponding to a sudden increase of the inherent conductivity. Around such threshold the percolation theory was found to be adequate on modeling the conductivity and elasticity behavior using power laws of density such as λ ∝ ( ρ − ρ c ) t (3) where ρ c is the critical density corresponding to the threshold and t is a fitting exponent. Another attempt of modeling has been carried out by Bonnissel et al. (2001) by mean of the Hashin-Shtrikman model for two phases compound [upper bound, Hashin and Shtrikman (1963)]. The thermal conductivity of the overall compact can be expressed as a function of the thermal conductivity of the solid content λ s and the porosity P = 1 − ρ/ρ G , being ρ G = 2.26 g / cm 3 the density of crystalline graphite: λ = λ s 1 − P 2 + P (4) λ s is considered as a function mixing the contributions to the thermal conductivity from the in-plane direction and out-of-plane direction of the solid content. This mixing approach seems to fit the experimental data with a low error in a wide range of density, up to the density of commercial FG foils. Finally, Chen and Chung (2014) compared the two approaches of Hashin-Shtrikman and the Rule of Mixtures: λ = v s λ s + v a λ a (5) in the out-of-plane direction. v s and v a are the volume fractions of the solid part and air, whereas λ a ≈ 0 is the conductivity of the air. The Rule of Mixtures does not consider the solid content orientation changes during the compaction considering the conduction path as uniformly oriented in the same direction but still captures the trend with increasing density. The specific heat capacity c p (T) at constant pressure is dependent on the temperature but, for example, usually a constant reference value is employed to calculate the thermal conductivity from the thermal di ff usivity α when using the flash method [Parker et al. (1961)], by mean of the following equation: λ ( T ) = ρ · ˆ c p · α ( T ) (6) No data are available in literature about the FG specific heat against temperature trend except data from datasheet (see figure 6). Only one measured valued at room temperature is given by Bonnissel et al. (2001) and it results c p = 850 J / kgK. Grafoil ® declares a value of 711 J / kgK at 21 ◦ C which is quite similar to that of polycrystalline graphite [704 to 720 J / kgK when T = 19 - 25 ◦ C, Picard et al. (2006)]. The increase of graphite c p around room temperature can be considered as linear whereas a polynomial fit is proposed when the range of T is extended such as in Butland and Maddison (1973). The tendency of the copper and aluminum curve is markedly di ff erent from that of graphitic materials, with a milder increment for copper and a positive concavity for aluminum. 3.2. Specific heat capacity

3.3. Coe ffi cient of thermal expansion

About the coe ffi cient of linear thermal expansion α T , no data are available in literature. Only data from datasheets are shown in figure 7 and compared with thermal expansion of copper and aluminum. Due to the manufacturing process, FG foils result on a very low expansion, or even negative, along the in-plane direction which has been already