Anisotropic elastic constants of lotus-type porous copper: measurements and micromechanics modeling
Introduction
Porous metals attract much attention as one of the new industrial materials because they have noticeable features such as lightweight, high capability of impact-energy absorption, highly damping nature, low thermal conductivity, etc. Various fabrication techniques have been developed and improved [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], and the applications to innovative products are under way. However, when used as structural materials, conventional sphere-type porous materials are too much weakened by the stress concentration around the pores. Recently, to avoid this shortcoming, lotus-type porous metals with long straight pores aligned unidirectionally have been fabricated by unidirectional solidification from a melt in hydrogen and argon atmosphere [7], [8], [9], [10], [11], [12], [13], [14], [15]. Mechanical properties of such a lotus-type porous copper have been systematically studied, which reveal strong anisotropic tensile strengths; the ultimate strength in the solidification direction equals the specific strength determined from the relative density, but the strength of the normal direction decreases drastically with porosity [16]. The tensile strength can be summarized in the following empirical formula:where σ0 is the tensile strength for the nonporous (fully dense) material, p denotes the porosity determined from the relative density, and K is the stress concentration factor. Hyun et al. found experimentally that K≈1 when the stress is applied along the longitudinal direction of the pores (“pore direction”) and K≈3 when applied along the normal direction [16].
Similarly, the lotus-type porous copper is expected to be elastically anisotropic. For conventional porous materials the elastic properties have been studied theoretically [17], [18] and experimentally [19], [20], and various formulas appear for predicting them. According to Phani [20], the modulus-porosity correlation that is applicable over the entire porosity range is given in a similar form to Eq. (1):where M0 is the elastic modulus of the nonporous material. The stress concentration factor m depends on the geometry, shapes and orientations of pores like K in Eq. (1), and it is required for predicting the elastic property. However, as to the lotus-type porous metals, only Young’s modulus parallel to the pore direction has been measured by Simone et al. [6], [7]. No one has clarified the elastic anisotropy nor presented a complete set of elastic constants. This is probably because the existence of pores generally makes the ultrasonic measurement difficult; ultrasound scattering occurs and the conventional pulse-echo method is weak or inapplicable except for a few directions.
This paper studies the anisotropic elastic constants of lotus-type porous copper using accurate ultrasonic measurements and micromechanics modeling. The unidirectionally porous copper is regarded as a composite material with an overall hexagonal elastic symmetry with c-axis parallel to the longitudinal direction of the pores; there are five independent elastic constants to be determined: c11 (=c22), c33, c12, c13 (=c23), and c44 (=c55), where c66=(c11−c12)/2. For the simultaneous determination of all of them, we have used the combination of resonance ultrasound spectroscopy (RUS) and electromagnetic acoustic resonance (EMAR) methods. Moreover, we have demonstrated the validity of a micromechanics model, which considers the matrix elastic anisotropy and the pore shape to calculate the elastic constants of the lotus-type porous metals.
Section snippets
Specimen preparation
Porous-copper rods with long pores and various porosities have been prepared by unidirectional solidification in hydrogen and argon atmosphere; the detailed fabrication technique has been given elsewhere [8], [9], [10], [11], [16]. The specimens were cut and machined into the rectangular parallelepipeds with the surfaces normal to the growth (solidification) direction, as shown in Fig. 1. X-ray diffraction spectra indicated that the growth (x3) directions of all the specimens are virtually
Prediction for nonporous specimen
The RUS analysis needs an initial guess of the elastic constants. We then calculate the elastic constants of the nonporous material from those of the single-crystal copper through the Voigt, Reuss, and Hill approximations.
As shown in Fig. 1 (left), the nonporous specimen is a bundle of single-crystal rods with the transverse isotropy in the x3 plane. For such a case, the overall elastic constants are obtained from the single-crystal elastic constants by averaging with respect to the rotation
RUS and EMAR measurements
The RUS method was developed by Demarest [21] for cube specimens and was subsequently extended for rectangular parallelepiped specimens by Ohno [22]. When the material possesses orthorhombic symmetry, there are eight independent groups of free vibrations: OD (dilatation), EV (torsion), OX, OY, OZ (shear) and EX, EY, EZ (flexure) [21], [22]. The resonance frequencies of the vibration modes depend on the mass density, dimensions, and elastic constants of the specimen. With the known elastic
Results and discussion
Fig. 4(a) and (b) show the porosity dependence of the elastic constants, c11, c33, c12, c13, c44, and c66=(c11−c12)/2, and two Young’s moduli, E∥ and E⊥. The elastic constants of the nonporous specimen agree well with the results by the Hill approximation. (Small deviations are caused by its texture that slightly departs from an ideal [001] fiber texture.) Young’s modulus E∥ decreases linearly and c33 does slowly with p, while E⊥ and c11 drop considerably for small p. In the range of small
Micromechanics modeling
We consider a micromechanics model with the aim of predicting the elastic constants of the lotus-type porous metals. The model is based on Eshelby’s equivalent inclusion theory [24] and Mori-Tanaka’s mean field theory [25]. The mathematical treatment reformulated by Benveniste [26] is summarized below.
A composite material consists of a matrix and a type of inclusion, whose volume fractions are denoted by c0 and c1 (=1−c0), respectively. The average stress and strain of the composite are
Conclusions
We have performed elastic-constant measurements and micromechanics calculations for lotus-type porous copper made by unidirectional solidification. The present study aims at establishing the prediction method for the elastic constants of such porous metals. The combination of RUS and EMAR enabled us to determine all the independent elastic constants and Young’s moduli accurately. Lotus copper can be regarded as a composite possessing a hexagonal elastic symmetry with c-axis parallel to the
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