Anisotropy of Young’s modulus

Katz, J.L. 1980. Anisotropy of Young s modulus of bone. Nature, 283,106-107. 

Figure 8.15 Comparison of the observed variation in modulus Eo with angle 0 to draw direction and the theoretical relation (i.e. full curve) calculated from Eo, E4S and Ego for low-density polyethylene sheet drawn to a draw ratio of 4.65. (Reproduced from Raumann, G. and Saunders, D.W. (1961) Ehe anisotropy of Young s modulus in drawn polyethylene. Proc. Phys. Soc., 77, 1028. Copyright (1961).)...

A AH increase with increasing draw ratio k has been found for solid-state extruded and highly drawn pol5miers, and a correlation between indentation anisotropy and Young s modulus has been foimd (4,31). Indentation anisotropy values of several carbon-fiber composites have been reported (32). Recently, the indentation anisotropy of cold-drawn PET, annealed at different temperatures, has also been examined (33). 

Table 2.3 contains an overview of the elastic constants for some metals and ceramics. As can be seen, the anisotropy factor of tungsten is 1.0, so it is (almost) isotropic even as a single crystal. For most other materials, almost isotropic properties can only be found in a polycrystalline state. The direction dependence of Young s modulus for selected materials is plotted in figure 2.10. 

Fig. 2.16. Change of Young s modulus E33 = a 3 3/63 3 as a function of draw ratio a and of the mechanical anisotropy S33/S11 = S3333/S1111 of the molecular domains (after [13], [85]).

Spruce soundboards have a Young s modulus anisotropy of about (11.6 GPa/0.71 GPa) = 16. A replacement material must therefore have a similar anisotropy. This requirement immediately narrows the choice down to composites (isotropic materials like metals or polymers will probably sound awful). 

The designer must be aware that as the degree of anisotropy increases, the number of constants or moduli required to describe the material increases with isotropic construction one could use the usual independent constants to describe the mechanical response of materials, namely, Young s modulus and Poisson s ratio (Chapter 2). With no prior experience or available data for a particular product design, uncertainty of material properties along with questionable applicability of the simple analysis techniques generally used require end use testing of molded products before final approval of its performance is determined. 

One must be aware that as the degree of anisotropy increases the number of constants or moduli required describing the material increases. With isotropic construction one could use the usual independent constants to describe the mechanical response of materials, namely, Young s modulus and Poisson s ratio. 

The method has also been applied to PAni [53]. The plasticizer is a solvent of the polymer N-methylpyrrolidone (NMP), and the film to be stretched is not, or is very poorly, crystalline and may contain up to 20 wt% NMP. Tg decreases as the NMP content increases, down to 110°C heating and stretching swollen EB films above that Tg value produces X > 6, increased crystallinity (Fig. 11), and sizable orientation and anisotropic properties [27,99,100]. Such films, doped with HC1, show a conductivity anisotropy = 25 [99-101]. Contrary to what is observed with PPV, the mechanical properties of stretched PAni remain modest the Young s modulus is below 0.5 GPa [53]. 

The condition for isotropic elasticity, as has been seen, is C44 = C55 = Cge = Cn C12). Cubic crystals, because of their high symmetry, almost satisfy this condition. Zener introduced the ratio 2C44/(cn - C12) as an elastic anisotropy factor for cubic crystals (Zener, 1948a). In a cubic crystal, if the Zener ratio is positive, the Young s modulus has a... 

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