Relations Among Some Elastic Constants

In the previous sections, some elastic constants were mentioned, namely, E, G, v, 2 and K. The constants E and G, the Young s and the shear modules, respectively, relate to stress-strain relations represented as  

Poisson s ratio, Lame s constant and the bulk modulus were given earlier, respectively, as  

These three constants are not independent, but if a solid is isotropic, these constants are sufficient for describing the mode of deformation, as given below. 

This shear strain is also given as stress per the shear modulus, namely  

There is a right-angle triangle, CC E, where CE is perpendicular to AC. Two angles in this right-angle triangle are 45°. 

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With few exceptions, we shall idealize the elasticity of solids as isotropic, as stated earlier, so as not to burden the discussion of the physical mechanisms with inessential operational detail. We note here, however, that many cubic crystals are quite anisotropic. Tungsten, W, which is often cited as being isotropic, is so only at room temperature. Thus, we shall make use principally of the elastic relations in eqs. (4.15) and (4.16), unless we are specifically interested in anisotropic solids such as some polymer product that had undergone deformation processing. The relationships among various combinations of elastic constants of isotropic elasticity are listed in Table 4.1 for ready reference. 

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