Elastic properties of lotus-type porous iron: acoustic measurement and extended effective-mean-field theory
Introduction
Artificial inhomogeneous materials, such as composites or porous materials, combine inherent physical properties of constituents. In general, however, their effective or macroscopic physical properties (e.g., mechanical properties, electrical conductivity, thermal conductivity, thermal expansion, and so forth) frequently do not follow a simple rule of mixture using only volume fractions. In practice, effects of inclusion’s shape, orientation, arrangement and so forth must be considered for a better understanding of them. Thus, computation or prediction of the effective properties is scientific interest of long standing.
Our primary concern in this paper lies on the macroscopic elastic properties of porous materials. Several sophisticated theories for predicting them have been established to date [1], [2], [3], [4], [5]. A famous model called “unit cell model” proposed by Gibson and Ashby [1] or micromechanics calculation method [2] based on Eshelby equivalent inclusion theory [6] and Mori–Tanaka’s mean-field (MTMF) theory [7] are frequently used for predicting/calculating the effective elastic constants of the porous materials. The former model exhibits a high performance in an extremely high porosity region. The latter has a great advantage that it can take analytically account of shape, volume fraction and orientation of inclusions, and it enables us to calculate whole porosity dependence relatively easily. It is, however, not capable of calculating the elastic constants for high porosity. In order to overcome this shortcoming, Tane and Ichitsubo [8] have proposed the effective-mean-field (EMF) theory, which is constructed based on the effective medium approximation (EMA) [9] and MTMF theories. It has been demonstrated that the results obtained from the EMF theory agree well with the measurement data even in a high porosity range and furthermore the experimentally observed power-law behavior of effective modulus with regard to porosity can be successfully reproduced [10].
This paper intends to be devoted to experimental and theoretical study on the elastic properties of lotus-type porous metals (lotus metals) fabricated recently by Nakajima et al. [11], [12], [13], [14], [15], [16]. Conventional porous metals have non-uniform pores that are spatially dispersed irregularly [17], [18], [19], whereas lotus metals have cylindrical pores that are aligned unidirectionally (see Fig. 1). To clarify the effects of such characteristic pores on the effective elastic constants is an attractive and important research subject. In this work, we measure the anisotropic elastic constants of lotus iron newly fabricated using the continuous zone-melting technique by Nakajima et al. [20]. In addition, we extend the EMF theory so as to take account of the pore-orientation effect on the effective elastic constants, and applied the extended EMF theory to lotus-type porous iron to validate the theory as a prediction method of the elastic properties of lotus-type porous metals.
Section snippets
Experimental
The mould-casting method is useful for fabrication of lotus-type porous metals consisting of metals with a high thermal conductivity, e.g., copper [15], [16] or magnesium [10], in which the numerous pores are formed by using the solubility gap between solid and liquid, and the nucleated pores grow upward from the bottom during the unidirectional solidification. For metals possessing a low thermal conductivity, however, the mould-casting method is unsuitable, because the movement velocity of
Results
Fig. 3 shows the porosity dependence of Young’s moduli E⊥ and E∥ and the elastic stiffness c11, c33, c13, c44 and c66 of lotus iron with hydrogen or nitrogen pores, where E∥ and E⊥ indicate Young’s moduli in the directions parallel and perpendicular to the pore direction, respectively. As well as the porosity dependence of the other lotus metals [10], [21], E∥ decreases linearly, while E⊥ drops steeply in the small porosity region. On the other hand, Nakajima and coworkers [26] have found that
Micromechanics modeling
We extend the EMF theory recently proposed by some of the present authors [8] so as to consider the effects of pore orientation on the effective elastic constants of lotus metals. The extension will be made following Dunn–Ledbetter–Heyliger’s extended MTMF theory [5].
Conclusions
We studied the elastic properties of lotus-type porous iron possessing pores unidirectionally aligned through the acoustic-resonance measurement and the EMF theory. The following conclusions can be drawn from this work.
- (1)
We have measured the elastic constants of two kinds of lotus iron that include nitrogen pores and hydrogen pores. The values of them are virtually identical to each other, which indicates that the solute nitrogen does not modify the elastic properties, being different from the
Acknowledgement
The authors express their gratitude to Dr. H. Ogi and Mr. N. Katsumoto for experimental help and advice.
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