Hamiltonian/Stroh formalism for anisotropic media with microstructure

Moving 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's) theory of couple stress elasticity. The resulting canonical formalism appears in the form of a Differential Algebraic system of Equations (DAE), 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 plain and anti-plane deformations are also developed and they both lead to a 7-dimensional coupled linear system of differential equations. In particular, the antiplane problem shows remarkable similarity with 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.