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Hamiltonian/Stroh formalism for anisotropic media with microstructure.

Andrea Nobili1,2, Enrico Radi3,2

  • 1Department of Engineering Enzo Ferrari, University of Modena and Reggio Emilia, via Vivarelli 10, 41125 Modena, Italy.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|July 20, 2022
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Researchers developed a Hamiltonian formalism for anisotropic microstructured materials, extending the Stroh formulation using couple-stress elasticity. This new approach reveals insights into material behavior and symmetry properties.

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Area of Science:

  • Solid Mechanics
  • Materials Science
  • Continuum Mechanics

Background:

  • The Stroh formulation is a powerful tool for analyzing anisotropic elastic materials.
  • Microstructured materials exhibit complex behaviors due to their internal structure.
  • Couple-stress elasticity accounts for microstructural effects beyond classical elasticity.

Purpose of the Study:

  • To extend the Stroh formulation to anisotropic microstructured materials.
  • To develop a Hamiltonian formalism for these materials.
  • To analyze the structure and symmetry properties of the derived equations.

Main Methods:

  • Application of variational principles to derive the Hamiltonian formalism.
  • Inclusion of the indeterminate (Mindlin-Tiersten) theory of couple-stress elasticity.
  • Recasting the differential algebraic system into a purely differential form.
  • Analysis of plane and antiplane deformation cases.

Main Results:

  • A canonical Hamiltonian formalism for anisotropic microstructured materials was developed.
  • The formalism results in a differential algebraic system, reducible to a differential form due to micro-macro rotation constraints.
  • Plane and antiplane deformations lead to seven-dimensional coupled linear differential equations.
  • The antiplane problem shares a Lagrangian with anisotropic plates but differs in constitutive assumptions, preventing a classical Stroh formulation.

Conclusions:

  • The developed canonical formalism offers new insights into the structure of problems involving anisotropic microstructured materials.
  • Important symmetry properties of these materials are highlighted by the formalism.
  • The approach provides a framework for analyzing wave generation and transmission in such complex media.