Section 1.3.3.4. Particular elastic constants

Let us apply a hydrostatic pressure (Section 1.3.2.5.2). The medium undergoes a relative variation of volume (the cubic dilatation, Section 1.3.1.3.2). If one replaces in (1.3.3.8) the stress distribution by a hydrostatic pressure, one obtains for the components of the strain tensorFrom this, we deduce the volume compressibility, , which is the inverse of the bulk modulus, κ: This expression reduces for a cubic or isotropic medium to

Under the action of a hydrostatic pressure, each vector assumes a different elongation. This elongation is given by equation (1.3.1.6): where the 's are the direction cosines of r. The coefficient of linear compressibility is, by definition, . Replacing by its value , we obtain for the coefficient of linear compressibility In the case of a cubic or isotropic medium, this expression reduces to

The coefficient of linear compressibility is then equal to one third of the coefficient of volume compressibility. We note that the quadric of elongations is a sphere.

If the applied stress reduces to a uniaxial stress, , the strain tensor is of the form In particular, We deduce from this that Young's modulus (equation 1.3.3.1) is

The elongation of a bar under the action of a uniaxial stress is characterized by and the diminution of the cross section is characterized by and . For a cubic material, the relative diminution of the diameter is One deduces from this that is necessarily of opposite sign to and one calls the ratio Poisson's ratio.

Putting this value into expression (1.3.3.12) for the coefficient of compressibility in cubic or isotropic materials gives

As the coefficient of compressibility, by definition, is always positive, we have

In practice, Poisson's ratio is always close to 0.3. It is a dimensionless number. The quantity represents the departure from isotropy of the material and is the anisotropy factor. It is to be noted that cubic materials are not isotropic for elastic properties. Table 1.3.3.2 gives the values of , , , ν and for a few cubic materials.

Table 1.3.3.2 | top | pdf | Elastic compliances of some cubic materials in (GPa)−1 (after Landoldt-Börnstein, 1979) Material ν Ag 22.9 −9.8 22.1 0.428 0.338 Al 16.0 −5.8 35.3 0.362 0.810 C (diamond) 1.12 −0.14 1.83 0.125 0.726 Cu 15.0 −6.3 13.3 0.42 0.312 Fe 7.67 −2.83 8.57 0.369 0.408 Ge 9.73 −2.64 14.9 0.271 0.602 Mo 2.71 −0.74 9.00 0.273 1.304 Ni 7.67 −2.93 8.23 0.382 0.388 Pb 93.7 −43.04 68.0 0.459 0.249 Si 7.74 −2.16 12.60 0.279 0.636 W 2.49 −0.70 6.35 0.281 0.995 LiF 11.6 −3.35 15.8 0.289 0.528 MgO 4.01 −0.96 6.47 0.239 0.651 NaCl 23.9 −3.20 78.7 0.133 1.452 GaAs 11.75 −3.66 16.8 0.311 0.545 ZnS 19.7 −7.6 22.6 0.386 0.414 BaTiO3 8.33 −2.68 9.24 0.322 0.420 Adamantane 240 −79 295 0.329 0.462 Almandine (Fe3Al2Si3O12) 4.036 −1.093 10.77 0.271 1.050 Spinel (MgAl2O4) 5.80 −2.05 6.49 0.353 0.413

It is interesting to calculate Young's modulus in any direction. For this it is sufficient to change the axes of the tensor . If A is the matrix associated with the change of axes, leading to the direction changing to the direction , then Young's modulus in this new direction is with The matrix coefficients are the direction cosines of with respect to the axes , and . In spherical coordinates, they are given by (Fig. 1.3.3.3) where θ is the angle between and , and φ is the angle between and . Using the reduction of for the various crystal classes (Section 1.1.4.9.9 ), we find, in terms of the reduced two-index components, the following.

Figure 1.3.3.3 | top | pdf | Spherical coordinates.

The representation surface of , the inverse of Young's modulus, is illustrated in Figure 1.3.3.4 for crystals of different symmetries. As predicted by the Neumann principle, the representation surface is invariant with respect to the symmetry elements of the point group of the crystal but, as stated by the Curie laws, its symmetry can be larger. In the examples of Fig. 1.3.3.4, the symmetry of the surface is the same as that of the point group for sodium chloride (Fig. 1.3.3.4a), tungsten (Fig. 1.3.3.4b) and aluminium (Fig. 1.3.3.4c), which have as point group, for tin (Fig. 1.3.3.4e, ) and for calcite (Fig. 1.3.3.4f, ). But in the case of zinc (Fig. 1.3.3.4d, ), the surface is of revolution and has a larger symmetry. It is interesting to compare the differences in shapes of the representation surfaces for the three cubic crystals, depending on the value of the anisotropy factor, which is larger than 1 for sodium chloride, smaller than 1 for aluminium and close to 1 for tungsten (see Table 1.3.3.2). In this latter case, the crystal is pseudo-isotropic and the surface is practically a sphere.

Figure 1.3.3.4 | top | pdf | Representation surface of the inverse of Young's modulus. (a) NaCl, cubic, anisotropy factor ; (b) W, cubic, anisotropy factor ; (c) Al, cubic, anisotropy factor ; (d) Zn, hexagonal; (e) Sn, tetragonal; (f) calcite, trigonal.