Noncanonical Poisson brackets for elastic and micromorphic solids
Resource
International Journal of Solids and Structures 44 (24): 7715-7730
Journal
International Journal of Solids and Structures
Journal Volume
44
Journal Issue
24
Pages
7715-7730
Date Issued
2007
Date
2007
Author(s)
Abstract
This paper investigates the Lagrangian-to-Eulerian transformation approach to the construction of noncanonical Poisson brackets for the conservative part of elastic solids and micromorphic elastic solids. The Dirac delta function links Lagrangian canonical variables and Eulerian state variables, producing noncanonical Poisson brackets from the corresponding canonical brackets. Specifying the Hamiltonian functionals generates the evolution equations for these state variables from the Poisson brackets. Different elastic strain tensors, such as the Green deformation tensor, the Cauchy deformation tensor, and the higher-order deformation tensor, are appropriate state variables in Poisson bracket formalism since they are quantities composed of the deformation gradient. This paper also considers deformable directors to comprise the three elastic strain density measures for micromorphic solids. Furthermore, the technique of variable transformation is also discussed when a state variable is not conserved along with the motion of the body. © 2007 Elsevier Ltd. All rights reserved.
Subjects
Continuum mechanics; Lagrangian-to-Eulerian transformation; Microcontinuum; Noncanonical Poisson bracket
SDGs
Other Subjects
Density measurement (specific gravity); Euler equations; Function evaluation; Green's function; Hamiltonians; Lagrange multipliers; Lagrangian-to-Eulerian transformation; Micromorphic solids; Poisson brackets; Poisson distribution
Type
journal article
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