Correlation between Young's modulus and porosity in porous materials

Abstract

where E is the effective Young’s modulus of porous material with porosity p, E0 is Young’s modulus of solid material, pc is the porosity at which the effective Young’s modulus becomes zero and f is the parameter dependent on the grain morphology and pore geometry of porous material [1]. As was noted by Wagh et al. [2], fittings of the experimental data to Equation 1 often give pc= 1 [1, 3] and do not explain the data accurately. In recent experimental works, either pc≡ 1 is preferably used [4–6] or a linearized model ( f ≡ 1) by Lam et al. [7] is used, where pc is considered to be an initial powder porosity. In this letter, it will be shown that the empirical relationship shown in Equation 1 is identical with the percolation theory equation for the behavior of Young’s and shear modulus with porosity. Further, the applicability of the percolation model for Young’s modulus of porous materials will be demonstrated and the results will be discussed. The porous materials are preferably prepared from powders, the particle size and shape of which can vary significantly. During the powder consolidation, various porosities can be achieved by varying the technological parameters such as temperature, external pressure or time. Compacting starts from just touching powder particles and goes to the lower porosity by the creation and growth of the necks between particles. The subsequent closure of the pore channels leads to the elimination of the pores. Three various porosity ranges can be usually identified, e.g., Danninger et al. [8] observed for sintered iron the following porosity ranges: 1. porosity≤3%: fully isolated pores of nearly spherical or elliptical shape 2. porosity ≥20%: fully interconnected pores of complex shape 3. porosity between 3% and 20%: both isolated and interconnected pores are present in various amounts. This indicates that the powder consolidation is in general a connectivity problem, which is studied by the percolation theory [9]. According to the percolation theory, there exists a critical volume fraction nc, called a percolation threshold, at which a solid phase forms a continual network spanning the whole system. At and above the percolation threshold, the geometrical, physical and mechanical properties of the system behave as