The Isotropic Cosserat Shell Model Including Terms up to O ( h 5 ) $O(h^{5})$ . Part II: Existence of Minimizers
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The Isotropic Cosserat Shell Model Including Terms up to O(h5 ). Part II: Existence of Minimizers Ionel-Dumitrel Ghiba1,2 · Mircea Bîrsan3,1 · Peter Lewintan3 · Patrizio Neff3
Received: 19 March 2020 / Accepted: 17 September 2020 © Springer Nature B.V. 2020
Abstract We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solution for the theory including O(h5 ) terms. The form of the energy allows us to show the coercivity for terms up to order O(h5 ) and the convexity of the energy. Secondly, we consider only that part of the energy including O(h3 ) terms. In this case the obtained minimization problem is not the same as those previously considered in the literature, since the influence of the curved initial shell configuration appears explicitly in the expression of the coefficients of the energies for the reduced two-dimensional variational problem and additional mixed bending-curvature and curvature terms are present. While in the theory including O(h5 ) the conditions on the thickness h are those considered in the modelling process and they are independent of the constitutive parameter, in the O(h3 )-case the coercivity is proven under some more restrictive conditions on the thickness h. Keywords Geometrically nonlinear Cosserat shell · 6-parameter resultant shell · In-plane drill rotations · Thin structures · Dimensional reduction · Wryness tensor ·
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
Ia¸si Branch, Octav Mayer Institute of Mathematics of the Romanian Academy, 700505 Ia¸si, Romania
3
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.
Dislocation density tensor · Isotropy · Calculus of variations · Uniform convexity · Existence of minimizers Mathematics Subject Classification 49J10 · 74K25 · 74K20 · 49S05 · 74A60 · 74B20 · 74G10 · 46N20
1 Introduction Shell and plate theories are intended for the study of thin bodies, i.e., bodies in which the thickness in one direction is much smaller than the dimensions in the other two orthogonal directions. In this follow up paper we investigate the existence of minimizers to a recently developped isotropic Cosserat shell model [6, 21], including higher order terms. The Cosserat shell model naturally includes an independent triad of rigid directors, which are coupled to the shell-deformation. From an engineering point of view, such models are preferred, since the independent rotation field allows for transparent coupling between shell
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