Silicon nitride is a material with high potential for engineering applications. Accurate prediction of elastic modulus of such a material is of broad scientific as well as technological importance. Several investigators [1][2][3] have reported that the elastic modulus of Si3N4 is a linear function of porosity having the form E = E0(1 -hP)where E and E0 are the elastic moduli (Young's or shear) at volume fraction porosity P and zero, respectively, and h is a material constant. Moulson [4] has fitted the Young's modulus-porosity data of reactionbonded silicon nitride (RBSN) over the porosity range 0 < P < 0.3, to a relationship of the Ryshkewitch-assuming E0 = 300 GPa and b = 3. has proposed a modified equationwhere b and c are non-negative material constants for correlating data over a relatively wider range of porosity, 0 ~< P ~< 0.38. The equations mentioned above have some limitations. These are not usually found to be valid over a wide range of porosity and applicable to all polycrystalline brittle solids irrespective of the nature of transition of the pore structure that occurs due to densification of a material [10,15]. The evaluation of elastic modulus at zero porosity by extrapolation following these equations may, therefore, yield inaccurate or unrealistic values. With a view to resolving the problem, the present authors [15] have derived semiempirically a new equation of the formwhere a and n are material constants. The material constant a is defined as a "packing geometry factor" and is equal to 1/P~r~t where Pent. is the critical porosity at which elastic modulus becomes zero. The equation, therefore, satisfies the boundary conditions E = E0 at P = 0 and E = 0 at P ~< 1. The minimum value of a = 1 corresponds to the maximum value of PeAt. = 1. For materials composed of uniform spherical particles packed in cubic, orthorhombic and rhombohedral array, the values of Pent are 0.476, 0.397 and 0.26, respectively [16]. Hence, for cubic, orthorhombic and rhombohedral packing, the values of a are 2.1, 2.52 and 3.85, respectively. The other material constant, n, depends on pore geometry and grain morphology. The equation satisfies well the exact theoretical solution for cubic packing (both ideal and non-ideal) over the entire range of porosity, 0 ~< P ~< 0.476. The equation has also been found to agree well with the data on many polycrystalline brittle solids over a wide range of porosity [15,17,18] Fig. 1 shows the plots of Young's and shear moduli of Si3N 4 against porosity along with the corresponding regression lines following Equation 6. Different processing can lead to different microstructures and this is reflected in the scatter of the values of Young's modulus. The equations obtained are E = 308(1 -P)Z58GPa (7) and E = 122(1 -P)2-94GPa (8) for Young's and shear modulus, respectively. 511
Many equations have been previously proposed to describe the elastic modulusporosity relation of brittle solids. The equations are not applicable to all materials over a wide range of porosity. A new equation representing elastic modulusporosity relation of brittle solids has been proposed. It has been shown that the proposed equation describes best the Young's modulus-porosity data of six rareearth oxides over a wide range of porosity reported by previous investigators. The equation suggests that there was some kind of ordered packing in all the oxides. HE elastic modulus of a brittle solid is Tknown to depend on its porosity as well as the pore structure. The pore structure undergoes a continuous change during densification of a material. Since the shape and size of the particles and the densification mechanism determine the nature of transition of the pore structure, the entire process of transition of the pore structure is basically a very complex one. The effect of porosity on the elastic modulus of brittle solids has, therefore, been the subject of considerable theoretical as well as experimental investigations. Numerous equations have been proposed to relate elastic modulus to porosity and most of these have been summarized by Wachtman. ' The frequently used equations are as follows:where E and Eo are the elastic moduli at volume fraction porosity P and zero, respectively, and h, b, A,fi andf' are empirical constants. Equation (1) is a linear approximation of all other equations at low porosities. Equation (2) is an empirical equation given by Spriggs.' The equation has been criticized3 for not satisfying the boundary condition E =O at P = 1. Soroka and Sereda4 have studied the porosity dependence of gypsum over two different porosity ranges, 0.11sPCO.3 and 0 . 4 9 s P C0.7, using Eq. (2) and obtained values of E, differing by 1 order of magnitude. They have, therefore, concluded that Eq. (2) is not valid over a wide range of porosity. Wang5 has shown that Eq. (2) holds good only up to a porosity range of 20% and proposed a modified equation with a quadratic exponent E=Eo exp[-(bP+cP2)1 ( 5 ) (where c is an empirical constant) for extending its validity up to a porosity range of 38%. Equation (3) is a semiempirical equation derived by Hasselman3 from the theoretical work of Hashin.' Wan2 and the present authors7-lo have shown that Eq. (3), though it satisfies the boundary condition E =O at P = I , yields unrealisticvalues of Eo when fitted over a wide range of porosity. This has been attributed to the transition of the pore structure from interconnected to c l o~e d .~ The present authors'" have analyzed the data on reaction-bonded silicon nitride in terms of the quadratic Eq. (4) and found that the equation does not obey the boundary condition E=O at P C 1 and is not valid over a wide range of porosity. Similar analysis of empirical dependence of elastic moduli on porosity for ceramic materials has been made by Dean and Lopez.'' PROPOSED EQUATIONThe equations mentioned above are not applicable to all bri...
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