The porosity dependence of sound velocities in ceramic materials
Introduction
It is well known that sound velocities are intimately connected to the elastic behavior of materials [1], [2], [3], [4], [5], [6], [7] and to some degree correlated to other properties such as thermal conductivity [8], strength [9] and hardness [10]. For this reason ultrasonic wave propagation techniques can be used to determine elastic properties [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], and for the same reason the elastic properties of rocks play a key role e.g. in seismology. In particular, in statistically isotropic materials two independent elastic constants can be uniquely determined from a measurement of the velocities of longitudinal and transverse sound waves (and vice versa). Since the sound velocity in solids is always higher than in gases (or common liquids), the introduction of porosity (pores or pore space) always reduces the sound velocity of a porous material. This is a natural consequence of the well-known fact that the effective elastic moduli of a porous material are always lower than those of its dense solid counterpart. However, while the porosity dependence of elastic moduli has been a mainstay in materials science for decades [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], the porosity dependence of the sound velocities has been treated only occasionally (mainly by Phani [22], [23], [24], whose work amounts essentially to curve fitting, however), and to the best of our knowledge a systematic treatment of this problem is not available to date. In particular, no work seems to be available so far that would provide a systematic overview of the relations available for parameter-free sound velocity predictions for porous ceramics. It is the aim of the present paper to fill this gap. More than that, we will take the opportunity to extend the available predictions by three relations that have never been used in the context of sound velocities before. These are based on the (Pabst-Gregorová-type) exponential relation [25], a percolation relation proposed by Pabst and Gregorová [26] and a numerical benchmark relation recently obtained by Pabst and Uhlířová [27] via computer-based modeling of partially sintered random packings of monosized spheres (similar modeling results are available for duplex structures of grains with bidisperse size distribution [28]). In this paper these model predictions will be critically scrutinized and compared to literature data.
Section snippets
Theory
The elastic behavior of isotropic materials is completely characterized by two elastic constants. These can be the Lamé constants [29] or any other pair from the set containing the tensile modulus (Young’s modulus) , the shear modulus , the bulk modulus and the Poisson ratio . While the porosity dependence of the elastic moduli (, , ) is bounded from above the upper Hashin-Shtrikman bound and several predictions are available for their (tentative) prediction [30], the possibilities to
Results and discussion
Fig. 5 shows the predictions of the porosity dependence of the relative sound velocity of transverse waves for porous materials with solid Poisson ratios 0.35, 0.2, 0 and –0.1. All these predictions have been calculated according to the relations listed in Table 1 in combination with Eq. 8. It is evident that all models correctly predict the decrease of the transverse wave velocity with porosity and that the predicted values decrease in the order MMT (Maxwell / Mori-Tanaka), differential,
Summary and conclusions
In this work the porosity dependence of transverse and longitudinal sound wave velocities has been studied for porous ceramics with statistically isotropic microstructure. It has been found that six model relations for the porosity dependence of these velocities can be constructed from the model predictions for elastic moduli, namely the Maxwell / Mori-Tanaka / MMT model relation, the differential relation, the exponential relation, the self-consistent relation, as well as the numerical
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
This work is part of the project “Impulse excitation as an unconventional method for monitoring phase changes and microstructure evolution during thermal loading of materials” (GA22–25562S), supported by the Czech Science Foundation (GAČR). P. Š. also acknowledges that this work was supported from the grant of specific university research – grant No A2_FCHT_2022_004.
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