THE STRUCTURE OF THE LINEAR ANISOTROPIC ELASTIC SYMMETRIES

AN INSIGHTFUL, STRUCTURALLY appealing and potentially utilitarian formulation of the anisotropic form of the linear Hooke's law due to Lord Kelvin was independently rediscovered by RYCHLEWSKI (1984, Prikl. Mat. Mekh. 48, 303) and MEHRABADI and COWIN (1990, Q. J. Mech. appl. Math. 43, 14). The eigenvectors of the three-dimensional fourth-rank anisotropic elasticity tensor, considered as a second-rank tensor in six-dimensional space, are called eigentensors when projected back into three-dimensional space. The maximum number of eigentensors for any elastic symmetry is therefore six. The concept of an eigentensor was introduced by KELVIN (1856, Phil. Trans. R. Soc. 166, 481) who called eigentensors "the principal types of stress or of strain". Kelvin determined the eigentensors for many elastic symmetries and gave a concise summary of his results in the 9th edition of the Encyclopaedia Britannica (1878). The eigentensors for a linear isotropic elastic material are familiar. They are the deviatoric second-rank tensor and a tensor proportional to the unit tensor, the spherical, hydrostatic or dilatational part of the tensor. MEHRABADI and COWIN (1990, Q. J. Mech. appl. Math. 43, 14) give explicit forms of the eigentensors for all of the linear elastic symmetries except monoclinic and triclinic symmetry. We discuss two approaches for the determination of eigentensors and illustrate these approaches by partially determining the eigentensors for monoclinic symmetry. With the nature of the eigentensors for monoclinic symmetry known, a rather complete table of the structural properties of all linear elastic symmetries can be constructed. The purpose of this communication is to give the most specifically detailed presentation of the eigenvalues and eigentensors of the Kelvin formulation to date.