Theories of Crystal Elasticity

Abstract

FROM the time of the publication of the “Lehrbuch der Kristallphysik”¹ until 1951 there was no challenge to the classical theory of the elastic properties of crystals. At the Solvay Conference of that year, Laval² published an account of a theory which rejected some of the fundamental assumptions made in classical theory. Further work leading to the same conclusions was published later by Le Corre³, Viswanathan⁴, Raman⁵ and Krishnamurti⁶. In non-mathematical language the difference between the old and new theories may be expressed as follows. On the classical theory, the equilibrium of a cube within a body at rest, which is subjected to any given set of mechanical stresses, is maintained solely by the forces acting normally or tangentially to its surfaces. If one edge of the cube is supposed vertical then tangential forces act on the vertical faces and, in equilibrium, the couples due to the tangential shear stresses acting on these faces just balance. The new theory denies the adequacy of this statement and asserts that ‘volume-couples’ can exist which must be balanced by the difference between the couples which were assumed to be equal and opposite in the old theory. In more formal language, the old theory asserts that the stress tensor is symmetric while the new theory asserts that it is in general non-symmetric. A corresponding difference exists in the treatment of the deformation experienced by the cube. The old theory assumes that all net rotations of the whole body may be regarded as irrelevant, so far as its elastic deformation is concerned. The new theory denies this and asserts that the deformation can only be adequately described by taking into account any net rotation which may be caused by the stresses. More formally, the ol d theory assumes a symmetric and the new theory a non-symmetric tensor for the strain. One of the consequences of the new theory is that the number of elastic constants is increased, for example, in the triclinic system from twenty-one on Voigt's theory to forty-five on the new theory.