The Isotropic Cosserat Shell Model Including Terms up to O ( h 5 ) $O(h^{5})$ . Part I: Derivation in Matrix Notation
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The Isotropic Cosserat Shell Model Including Terms up to O(h5 ). Part I: Derivation in Matrix Notation Ionel-Dumitrel Ghiba1,2 · Mircea Bîrsan3,1 · Peter Lewintan3 · Patrizio Neff4
Received: 13 March 2020 / Accepted: 17 September 2020 © Springer Nature B.V. 2020
Abstract We present a new geometrically nonlinear Cosserat shell model incorporating effects up to order O(h5 ) in the shell thickness h. The method that we follow is an educated 8-parameter ansatz for the three-dimensional elastic shell deformation with attendant analytical thickness integration, which leads us to obtain completely two-dimensional sets of equations in variational form. We give an explicit form of the curvature energy using the orthogonal Cartan-decomposition of the wryness tensor. Moreover, we consider the matrix representation of all tensors in the derivation of the variational formulation, because this is convenient when the problem of existence is considered, and it is also preferential for numerical simulations. The step by step construction allows us to give a transparent approximation of the three-dimensional parental problem. The resulting 6-parameter isotropic shell model combines membrane, bending and curvature effects at the same time. The Cosserat shell model naturally includes a frame of orthogonal directors, the last of which does not necessarily coincide with the normal of the surface. This rotation-field is coupled to the shell-deformation and augments the well-known Reissner-Mindlin kinematics (one independent director) with so-called in-plane drill rotations, the inclusion of which is decisive
B I.-D. Ghiba
[email protected] M. Bîrsan [email protected] P. Lewintan [email protected] P. Neff [email protected]
1
Department of Mathematics, Alexandru Ioan Cuza University of Ia¸si, Blvd. Carol I, no. 11, 700506 Ia¸si, Romania
2
Octav Mayer Institute of Mathematics of the Romanian Academy, Ia¸si Branch, 700505 Ia¸si, Romania
3
Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen, Germany
4
Head of Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen, Germany
I.-D. Ghiba et al.
for subsequent numerical treatment and existence proofs. As a major novelty, we determine the constitutive coefficients of the Cosserat shell model in dependence on the geometry of the shell which are otherwise difficult to guess. Mathematics Subject Classification 74K25 · 74K20 · 49S05 · 74A60 · 74B20 · 74G10 Keywords Geometrically nonlinear Cosserat shell · 6-Parameter resultant shell · In-plane drill rotations · Thin structures · Dimensional reduction · Cosserat elasticity · Wryness tensor · Dislocation density tensor · Isotropy
1 Introduction The theory of shells is an important branch of the theory of deformable solids. Its importance resides in the multitude of applications that can be investigated using shell models. In general, the shel
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