Tensor derivation of acoustic waves in solids

In tensor notation the three Cartesian directions x, y, and z are designated by suffixed variables i,j, k, l, etc. (Landau and Lifshitz 1970 Auld 1973). Thus the force acting per unit area on a surface may be described as a traction vector with components rj j = x, y, z. The stress in an infinitesimal cube volume element may then be described by the tractions on three of the faces, giving nine elements of stress cry (i, j = x, y, z), where the first suffix denotes the normal to the plane on which a given traction operates, and the second suffix denotes the direction of a traction component. 

A kinetic argument shows that ay - oy always. Any imbalance between these two would lead to an angular acceleration of a volume element. If this volume element were shrunk, then the torque would reduce in proportion to the linear dimension cubed, but the moment of inertia would reduce in proportion to the fifth power of the linear dimension, so that the angular acceleration would increase as the reciprocal of the square of the size of the volume element, becoming infinite in the limit. Thus reductio ad absur-dum, ay = cry. Hence there are only six independent components of the stress tensor. 

Strain is expressed in terms of the derivatives of the displacements, Uk, of points in a solid, in such a way that a rigid rotation gives zero strain. The 

The stiffness tensor Cy is a tensor of fourth rank, with 81 elements. To enable it to be written as a matrix, a reduced notation for the independent elements of stress and strain is used, 

The abbreviated suffixes are obtained by counting along the diagonal and then either way around two sides of the matrix thus a — Oyy — — Oyz —  

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