International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 8-9
Section 1.1.3.5. Representation surface of a tensor^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France |
Let us consider a tensor represented in an orthonormal frame where variance is not important. The value of component in an arbitrary direction is given by where the , are the direction cosines of that direction with respect to the axes of the orthonormal frame.
The representation surface of the tensor is the polar plot of .
The representation surfaces of second-rank tensors are quadrics. The directions of their principal axes are obtained as follows. Let be a second-rank tensor and let be a vector with coordinates . The doubly contracted product, , is a scalar. The locus of points M such that is a quadric. Its principal axes are along the directions of the eigenvectors of the matrix with elements . They are solutions of the set of equations where the associated quantities λ are the eigenvalues.
Let us take as axes the principal axes. The equation of the quadric reduces to
If the eigenvalues are all of the same sign, the quadric is an ellipsoid; if two are positive and one is negative, the quadric is a hyperboloid with one sheet; if one is positive and two are negative, the quadric is a hyperboloid with two sheets (see Section 1.3.1 ).
Associated quadrics are very useful for the geometric representation of physical properties characterized by a tensor of rank 2, as shown by the following examples: