Hodge duality between stress space and strain space in anisotropic media

Duality in a linear elasticity theory is studied based on a theory of differential form. A generalized expression of a Hodge star operator with an index is introduced. The index in the Hodge star operator means a superposition of ordinary Hodge star operators. By using the superposed Hodge star operator, a linear constitutive relation in anisotropic media can be expressed as a duality between a stress and strain spaces. Then, a basic elastodynamic equation is derived from the conservation law and the linear constitutive relation in the stress and strain spaces. Moreover, a geometric expression of equation of stress function and displacement function is derived by using the differential form and the dual relation of linear constitutive relation. These geometric results imply that the approach of differential forms is applicable to an analysis of deformation in anisotropic media. The superposed Hodge star operator discussed in this study has similar properties of a discrete Hodge operator, which represents the summation of differential forms on the decomposed regions. This means that the superposed Hodge star operator and the discrete Hodge operator express the constitutive relation in which physical quantities act on each other.