A modified Lekhnitskii formalism a la Stroh for anisotropic elasticity and classifications of the 6 * 6 matrix N

摘要

The Stroh formalism for two-dimensional deformations of anisotropic elastic materials computes the eigenvalue p and the eigenvector a from an eigenrelation. The vector b is then determined from a. Depending on the number of repeated eigenvalues p and the number of independent eigenvectors (a,b) the system has, it can be classified into six groups. The Lekhnitskii formalism has no eigenrelation to speak of. We present a modified Lekhnitskii formalism that computes the eigenvalue p and the eigenvector b from an eigenrelation. The vector a is then determined from b. Thus the modified Lekhnitskii formalism is a dual to the Stroh formalism. Not only does the modified formalism enable us to do the classifications, it is much simpler than using the Stroh formalism. The six groups are the SP, SS, D1, D2, ED and ES groups. The ES group does not exist for a real material. The SS group (that has p_1 = p_2 not= p_3) and the D2 group (that has p_1 = p_2 = p_3) can be identified without computing the eigenvalues p and the eigenvectors (a,b). We show that the repeated eigenvalue p_1 = p_2 in the SS and D2 groups is simply a root of the quadratic equation l_2 = 0. We present an explicit expression of p_3 for the SS group. The ED group that has three identical p can also be identified without computing p and (a,b); however, we do present an explicit expression of p. We show that monoclinic materials with the symmetry plane at x_3 = 0 cannot belong to the ED group. The identification of the SP and D1 groups is the only one that requires computation of p but not (a,b). For special classes of materials, however, they can be identified without computing p. In all cases, the eigenvectors and the generalized eigenvectors are obtained explicitly.