Hamiltonian/Stroh formalism for anisotropic media with microstructure

Abstract

From variational principles we develop the Hamiltonian formalism for generally anisotropic microstructured materials, in an attempt to extend the celebrated Stroh formulation. Microstructure is expressed through the indeterminate (or Mindlin–Tiersten) theory of couple-stress elasticity. The resulting canonical formalism appears in the form of a differential algebraic system of equations, which is then recast in purely differential form. This structure is due to the internal constraint that relates the micro- to the macro-rotation. The special situations of plane and antiplane deformations are also considered, and they both lead to a seven-dimensional coupled linear system of differential equations. In particular, the antiplane problem shows remarkable similarity to the theory of anisotropic plates, with which it shares the Lagrangian. Yet, unlike for plates, a classical Stroh formulation cannot be obtained, owing to the difference in the constitutive assumptions. Nonetheless, the canonical formalism brings new insight into the problem’s structure and highlights important symmetry properties. This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’.