A Naghdi Type Nonlinear Model for Shells with Little Regularity
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A Naghdi Type Nonlinear Model for Shells with Little Regularity Matko Ljulj1 · Josip Tambaˇca1
Received: 8 April 2020 / Accepted: 24 October 2020 / Published online: 27 November 2020 © Springer Nature B.V. 2020
Abstract In this paper a new nonlinear two-dimensional shell model of Naghdi’s type is formulated for shells which middle surface is parameterized by a W 1,∞ function. Therefore the model inherently contains undeformed geometries with corners, so the model also includes models of junctions of nonlinear shells. Deformation of the shell is described by a pair (ψ, R) of independent unknowns, where ψ is the deformation of the middle surface and R is a function with value in rotations that describes rotation of the shell cross-section. The model is formulated as the minimization problem for the total energy functional that includes flexural, membrane, shear and drill energies differently scaled with respect to the thickness of the shell. We relate the new model for smooth enough undeformed geometry to the known shell models in two ways. First we restrict the proposed model on two particular subsets of admissible functions and obtain exactly the flexural shell model and a perturbation of the Koiter shell model. More important, we consider asymptotics, using –convergence, of the proposed model with respect to the thickness as a small parameter in the membrane and flexural regime and obtain exactly the nonlinear membrane and flexural shell model obtained as –limits starting from nonlinear three–dimensional elasticity. In that way we link rigorously the proposed model with the nonlinear three–dimensional elasticity. Keywords Shell model · Nonlinear elasticity · Naghdi’s model · –convergence · Cosserat model · Membrane model · Flexural model · Koiter model Mathematics Subject Classification (2010) 74K25 · 49J45 · 74B20 · 74K30 · 74G65
B J. Tambaˇca
[email protected] M. Ljulj [email protected]
1
Department of Mathematics, Faculty of Science, University of Zagreb, Bijeniˇcka 30, 10000 Zagreb, Croatia
448
M. Ljulj, J. Tambaˇca
1 Introduction The purpose of this article is to formulate a new two-dimensional nonlinear shell model that will be applicable in all situations, irrespective of the geometry, boundary conditions or scaling order of energy. The model will be formulated in terms of two mutually independent unknown functions ψ, R, where ψ parameterizes the middle surface of the deformed shell and R is a function with values in rotations that describe the rotation of the cross-section of the shell. The cross-sections will be allowed to shear with respect to the deformed middle surface, which is typical to the Naghdi type models. Thus the kinematics of the body is described as usual in 6-parameter shell theories, see [8, 29] for more details, or [4] for the intrinsic theory of special Cosserat shells. This research is the continuation of the research in the linear case and the formulation of a two-dimensional linear shell model of Naghdi type from [12, 71], and the previous work for Koiter type models from [3, 10
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