Cosserat-Type Shells
In this chapter we discuss the Cosserat-type theories of plates and shells. We call Cosserat-type shell theories various theories of shells based on the consideration of a shell base surface as a deformable directed surface, that is the surface with attac
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and Victor A. Eremeyev
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Lehrstuhl Technische Mechanik, Institut f¨ ur Mechanik, Fakult¨ at f¨ ur Maschinenbau, Otto-von-Guericke-Universit¨ at Magdeburg, Germany South Scientific Center of RASci and South Federal University, Rostov on Don, Russia Abstract In this chapter we discuss the Cosserat-type theories of plates and shells. We call Cosserat-type shell theories various theories of shells based on the consideration of a shell base surface as a deformable directed surface, that is the surface with attached deformable or non-deformable (rigid) vectors (directors), or based on the derivation of two-dimensional (2D) shell equations from the three-dimensional (3D) micropolar (Cosserat) continuum equations. Originally the first approach of such a kind belongs to Cosserat brothers who considered a shell as a deformable surface with attached three unit orthogonal directors. In the literature are known theories of shells which kinematics is described by introduction of the translation vector and additionally p deformable directors or one deformable director or three unit orthogonal to each other directors. Additional vector fields of directors describe the rotational (in some special cases additional) degrees of freedom of the shell. The most popular theories use the one deformable director or three unit directors. In both cases the so-called direct approach is applied. Another approach is based on the 3D-to-2D reduction procedure applied to the 3D motion or equilibrium equations of the micropolar shell-like body. In the literature the various reduction methods are known using for example asymptotic methods, the-through-the thickness integration procedure, expansion in series, etc. The aim of the chapter is to present the various Cosserat-type theories of plates of shells and discuss the peculiarities and differences between these approaches.
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Introduction
The mechanics of the Cosserat continuum is based on the introduction of translations and rotations as kinematically independent quantities. The H. Altenbach, V. A. Eremeyev (Eds.), Generalized Continua from the Theory to Engineering Applications, DOI 10.1007/978-3-7091-1371-4_3, © CISM, Udine 2013
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H. Altenbach and V.A. Eremeyev
two- and one-dimensional analogues of the Cosserat continuum were presented by Cosserat and Cosserat (1909). On the other hand in the theory of plates and shells the independence of rotations was recognized by Reissner (1944, 1945, 1947, 1985). Since the paper by Ericksen and Truesdell (1958) the Cosserat model has found applications in construction of various generalized models for beams, rods, plates, and shells. Within the framework of the direct approach applied by Ericksen and Truesdell (1958), the shell is modeled as a deformable surface at each point of which a set of deformable directors is attached. Hence, the deformation of a shell is described by the position vector r and p directors di , i = 1, . . . p. This approach is developed in the original papers by Green et al. (1965); Green and Naghdi (1967a,b, 1968, 197
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