Complex Eigenvalues of the Deformation Gradient

Abstract

Thomson and Tait pointed out that the deformation gradient F at a point X in a continuous medium has at least one real positive eigenvalue and showed that material line elements along N, the corresponding right eigenvector of F, retain their direction in the deformation. Here, we consider the case when the two remaining eigenvalues of F are complex conjugate, with corresponding right eigenvectors c ± id (say). It is seen that the "directional ellipse" corresponding to the bivector c + id is preserved in the deformation, the ellipse associated with the bivector c + id being transformed into a similar and similarly situated ellipse. Any two material line elements at X along a pair of conjugate radii of the directional ellipse are deformed into material line elements along another pair of conjugate radii of this ellipse. They are stretched by the same amount which is the modulus of the complex eigenvalue. Key Words: continuum kinematics, deformation, material line element, bivector