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5 Mechanical Properties
5.1 Elasticity
Alumina shows the deformation behavior of a typical brittle solid, which is linear elasticity to failure. Elastic deformation is instantaneous when stress is applied, and is completely reversible when it is removed. In a tensile test of a rod or bar of alumina, the strain is linear with stress to failure; the slope of thisstress–strain curve is the Young’s modulus. Values of various moduli and Poisson’s ratio for pure, dense polycrystalline alumina are given in Table 4 (from [15]). These values are considerably higher than those for most other oxides, as a result of the strong (high energy) aluminum–oxygen bonds in alumina. The various elastic constants of single crystal alumina are given in Table 5. As the temperatureincreases the elastic moduli decrease (as shown in Table 6) because of the increase in atomic displacements as the temperature increases, and consequent reduced bond strength.

5.2 Strength
The mechanical strength of a brittle material such as alumina depends on flaws (cracks) in the alumina surface. When a tensile stress is applied perpendicular to a deep, thin crack, the stress at the tip of thecrack is greatly magnified above the applied stress. Thus, the surface condition of a brittle solid determines it strength. Surface flaws develop from abrasion, so the higher the abrasion resistance of a brittle solid the greater its practical strength. Strengths of alumina are given in Table 7. If a solid has no surface or internal flaws (a “perfect” lattice), it should have very high strength.Various theoretical equations for this ultimate strength S of a brittle solid have been proposed; one is [16]
S2 = Eg / 4b (4)
in which E is Young’s modulus, g is the surface energy, and b the lattice parameter. With E = 403 GPa (Table 4), g = 6.0 J m−2 [17, 18], and b = 0.177 nm, the ultimate strength S of alumina is about 58 GPa. This value is very high because of the high bond strength ofalumina; for example, silicate glasses and quartz have theoretical strength values of 18 GPa or lower. Practical strengths of brittle materials vary over wide ranges depending on their surface condition and history. For alumina, tensile or bonding strengths vary over a wide range of values because of different surface conditions, resulting in different flaw depths and flaw distributions. See [19] for adiscussion of flaw distribution functions. The strength values for alumina are higher than for most other oxides; of course all of these strengths are far smaller than the theoretical strength, and depend strongly on the history and treatment of the samples. As the temperature increases, the strength of alumina decreases (as shown in Table 7) because of the increase of atomic vibrations andreduction in bond strength, just as for the reduction in elastic modulus with temperature. The strength of polycrystalline alumina depends strongly on its grain size, as shown by one set of strength values from [2]. See also [20] for strengths of alumina machined and annealed at different temperatures. The strength also decreases as the alumina becomes more porous, as shown in Table 8; isolated poresincrease the applied stress on their surfaces, and open porosity means much more surface for flaw development.

5.3 Fatigue
The strengths of crystalline and glassy oxides decrease with time under a constant applied load. This static fatigue is usually modeled with a power law equation between times to failure t when a sample is subjected to an applied stress s:
logt = c −nlogs (5)
in which cis a constant and the stress exponent n is a measure of the susceptibility of the material to fatigue. The larger the n value the more resistant the material is to fatigue. Typical values of n for silicate glasses are 13 or lower [21]; for alumina an n value of about 35 was found [21], showing that alumina has much better fatigue resistance than most other oxides under ambient conditions....
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