Part of the Advanced Structured Materials book series (STRUCTMAT, volume 7)

# Micropolar Shells as Two-dimensional Generalized Continua Models

• Holm Altenbach
• Victor A. Eremeyev
• Leonid P. Lebedev
Chapter

## Abstract

Using the direct approach the basic relations of the nonlinear micropolar shell theory are considered. Within the framework of this theory the shell can be considered as a deformable surface with attached three unit orthogonal vectors, so-called directors. In other words the micropolar shell is a two-dimensional (2D) Cosserat continuum or micropolar continuum. Each point of the micropolar shell has three translational and three rotational degrees of freedom as in the rigid body dynamics. In this theory the rotations are kinematically independent on translations. The interaction between of any two parts of the shell is described by the forces and moments only. So at the shell boundary six boundary conditions have to be given. In contrast to Kirchhoff-Love or Reissner’s models of shells the drilling moment acting on the shell surface can be taken into account. In the paper we derive the equilibrium equations of the shell theory using the principle of virtual work. The strain measures are introduced on the base of the principle of frame indifference. The boundary-value static and dynamic problems are formulated in Lagrangian and Eulerian coordinates. In addition, some variational principles are presented. For the general constitutive equations we formulate some constitutive restrictions, for example, the Coleman-Noll inequality, the Hadamard inequality, etc. Finally, we discuss the equilibrium of shells made of materials undergoing phase transformations, such as martensitic transformations, and formulate the compatibility conditions on the phase interface.

### Keywords

Micropolar shells 6-parametric theory of shells Variational principles Constitutive inequalities Coleman-Noll inequality Hadamard inequality Phase transformations

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## Authors and Affiliations

• Holm Altenbach
• 1
• Victor A. Eremeyev
• 1
• 2
• Leonid P. Lebedev
• 3
1. 1.Martin-Luther-Universität Halle-WittenbergHalleGermany
2. 2.South Scientific Center of RASci & South Federal UniversityRostov on DonRussia
3. 3.Universidad Nacional de ColombiaBogota D.C.Colombia