An advanced shell model for the analysis of geometrical and material nonlinear shells
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ORIGINAL PAPER
An advanced shell model for the analysis of geometrical and material nonlinear shells F. Gruttmann1 · W. Wagner2 Received: 9 July 2020 / Accepted: 6 August 2020 © The Author(s) 2020
Abstract In this paper layered shells subjected to static loading are considered. The displacements of the Reissner–Mindlin theory are enriched by a an additional part. These so-called fluctuation displacements include warping displacements and thickness changes. They lead to additional terms for the material deformation gradient and the Green–Lagrangian strain tensor. Within a nonlinear multi-field variational formulation the weak form of the boundary value problem accounts for the equilibrium of stress resultants and couple resultants, the local equilibrium of stresses, the geometrical field equations and the constitutive equations. For the independent shell strains an ansatz with quadratic shape functions is chosen. This leads to a significant improved convergence behaviour especially for distorted meshes. Elimination of a set of parameters on element level by static condensation yields an element stiffness matrix and residual vector of a quadrilateral shell element with the usual 5 or 6 nodal degrees of freedom. The developed model yields the complicated three-dimensional stress state in layered shells for elasticity and elasto-plasticity considering geometrical nonlinearity. In comparison with fully 3D solutions present 2D shell model requires only a fractional amount of computing time. Keywords Layered plates and shells · Coupled global local boundary value problems · Interface to 3D material law · Four-node shell element · Standard nodal degrees of freedom · Fast computation of the load deflection behaviour
1 Introduction Shell elements which account for the layer sequence of a laminated structure are able to predict the deformation behaviour of the reference surface in an accurate way. Also the assumption of a linear shape of the in-plane strains through the thickness is sufficient, if the shell is not too thick. In contrast to that only constant transverse shear strains through the thickness are obtained within the Reissner–Mindlin theory. As a consequence only the average of the transverse shear stresses is accurate. Neither the shape of the stresses is correct nor the boundary conditions at the outer surfaces are ful-
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W. Wagner [email protected] F. Gruttmann [email protected]
1
Fachgebiet Festkörpermechanik, Technische Universität Darmstadt, Franziska-Braun-Str. 7, 64287 Darmstadt, Germany
2
Institut für Baustatik, Karlsruher Institut für Technologie, Kaiserstr. 12, 76131 Karlsruhe, Germany
filled. Within the Kirchhoff theory the transverse shear strains are set to zero. By assumption the thickness normal stresses are neglected in a standard shell theory. This is necessary to avoid unphysical stresses due to inextensibility assumptions in thickness direction. In several publications the equilibrium equations are exploited within post-processing procedures to obtain the interlami
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