Elasticity Constants of Isotropic Materials

The elasticity theory of isotropic media involves two elasticity constants characterizing the elastic properties of the material which thus is sometimes called a Fig. 1.48 Torsion of a beam of square cross-section in the large deformation theory (from La Torsion des Prismes by Saint-Venant )

bi-constant theory. However, four different constants are usually measured from various experimental tests, so that any one of the four can be represented as a function of two others.

Whatever the sign convention involved in their measurement, the four elasticity constants may be determined as follows:

• Young modulus E: Considering a rod elongated of 8z along a z-axis by a uniform stress az applied on each end, the Young modulus is the ratio E = \oz/(8z/z)\. Its dimension is [F][L—2].

• Shear modulus G: Considering a rod of diameter 2a distorted of the angle t per unit length by a torsion moment M, the shear modulus is the ratio G = \2M/nTa4\. Its dimension is [F][L-2].

• Poisson ratio v: Considering a rod elongated of 8z/z along a z-axis by a uniform stress and whose cross-section is then changed of 8r/r, the Poisson ratio is the ratio v = —(8r/r)/(8z/z).

• Isotropic compression modulus: Considering a sphere of radius a compressed by a pressure p > 0, the isotropic compression modulus is the ratio K = -1 p/(8a/a). Its dimension is [F][L-2].

These quantities are linked by the formulas

The two Lamé coefficients , X = vE/[(1 — 2v)(1 + v)] and ^ = G, are sometimes used.

Although from these quantities, the Poisson ratio may have values such as —1 < v < 1/2, we do not know in nature a continuous and isotropic material with v < 0, i.e. which would laterally expand when longitudinally elongated. For all materials, the Poisson ratio is in the domain 0 < v < 1/2. For rubber, the Poisson ratio is close to 1/2, hence the isotropic compression modulus K is close to infinity; this means that for rubber the volume change in the deformation is quasi null.

Glass or vitro-ceramic materials show perfectly invariant quantities E, G, v, and K up to the rupture. For instance in the measure of E, the axial stress oz is exactly proportional to the strain Sz/z up to rupture which corresponds to the ultimate strength cult. However many substances, such as most metal alloys, show a deviation from linearity between stress and strain. Because of this deviation, the experimental measures of these elastic constants by dedicated testing machines are generally carried out in the low stress level of their elastic domain.

As is customary in elasticity analysis, the elastic constants E and v are generally used instead of G and K. For some typically linear materials used in active optics, the values of these constants and the tensile maximum stress, or max, are given in Table 1.10.