Seminars

(Post)Buckling Analysis of Isogeometric Kirchhoff-Love Shells with Applications to Wrinkling of Thin Sheets

In this talk, a model to predict wrinkling behaviour of thin sheets on a foundation is presented. In particular, the model is based on isogeometric analysis motivated by the advantage of describing geometries exactly and representing solutions with higher continuity (Hughes et al., 2005). As wrinkling and folding are governed by (a series of ) structural instabilities, continuation methods or arc-length methods have been implemented. As wrinkling is a phenomena that occurs for thin sheets, a Kirchhoff-Love shell model is used in this research, neglecting shear deformations of the cross-section. The model is based on the implementation of (Goyal, 2015) in the Geometry + Simulation toolbox (i.e. G+smo) (Jüttler et al., 2014) and relevant additions that have been made to this model are a buckling solver and an extra term in the tangential stiffness matrix which takes into account follower pressures. Furthermore, the arc-length method based on Crisfield’s formulation (Crisfield, 1981) was implemented to model post-buckling behaviour with snap-through phenomena. This arc-length method was extended with the extended arc-length method (Wriggers et al., 1988) and the bisection method (Wagner and Wriggers, 1988) to approach bifurcation or limit points where the tangential stiffness matrix is singular and hence no solutions can be found. In this way, limit points can be passed by the method and branching can be done without using initial deformations. Detection of limit and bifurcation points is done based on monitorring of the lowest diagonal value of the diagonal matrix of the Cholesky Decomposition (de Borst et al., 2012). Furthermore, the work of Feng et al. (1996) and Zhou and Murray (1995) was used to make the code more robust. Benchmarks studies have been performed based on the works of Pagani and Carrera (2018) and Zhou et al. (2015). The results of the benchmark studies show on the one hand that the code is able to accurately predict theoretical buckling loads, despite the presence of axial stiffness and hence extensibility. Secondly, the benchmarks show that unstable branches (negative determinant of the tangential stiffness matrix) and post-buckling behaviour show excellent agreements with the results of Pagani and Carrera. The model has been applied to the Lamé problem and to a thin sheet under tension. Similar studies have been performed by Taylor et al. (2015) and Nayyar et al. (2011) using finite element models and either dynamic relaxation or arc length methods, both with a priori applied initial deformations. Experiments of the thin sheet under tension were performed by (Cerda et al., 2002). Although not (yet) reproduced with similar parameters, the model results show agreements in the formation and propagation of the wrinkles. The wrinkling pattern of the thin sheet under tension is depicted in figure 1. Concluding the present study shows the combination of continuation or arc length methods on structural problems using the isogeometric Kirchhoff-Love shells. Benchmark problems have shown that the model is capable of accurately capturing bifurcation points as well as post-buckling behaviour. Furthermore, applications on the Lamé problem and the thin sheet under tension show that the method is also capable of capturing wrinkling behaviour in sheets. Next steps in this study are to combine the thin sheets with elastic foundations to resemble a floating sheet on calm waters.

Figure 1: Wrinkles in the thin sheet under tension. The sides are loaded by a uniform line load.

References

Cerda, E., Ravi-Chandar, K., and Mahadevan, L. (2002). Wrinkling of an elastic sheet under tension. Nature, 419(6907):579–580.

Crisfield, M. (1981). A Fast Incremental/Iterative Solution Procedure That Handles “Snap-Through”. In Computational Methods in Nonlinear Structural and Solid Mechanics, pages 55–62. Pergamon.

de Borst, R., Crisfield, M. A., Remmers, J. J., and Verhoosel, C. V. (2012). Non-Linear Finite Element Analysis of Solids and Structures: Second Edition. Wiley.

Feng, Y., Perić, D., and Owen, D. (1996). A new criterion for determination of initial loading parameter in arc-length methods. Computers & Structures, 58(3):479–485.

Goyal, A. (2015). Isogeometric Shell Discretizations for Flexible Multibody Dynamics. doctoralthesis, Technische Uni- versität Kaiserslautern.

Hughes, T., Cottrell, J., and Bazilevs, Y. (2005). Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194(39-41):4135–4195.

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Nayyar, V., Ravi-Chandar, K., and Huang, R. (2011). Stretch-induced stress patterns and wrinkles in hyperelastic thin sheets. International Journal of Solids and Structures, 48(25-26):3471–3483.

Pagani, A. and Carrera, E. (2018). Unified formulation of geometrically nonlinear refined beam theories. Mechanics of Advanced Materials and Structures, 25(1):15–31.

Taylor, M., Davidovitch, B., Qiu, Z., and Bertoldi, K. (2015). A comparative analysis of numerical approaches to the mechanics of elastic sheets. Journal of the Mechanics and Physics of Solids, 79:92–107.

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Zhou, Y., Stanciulescu, I., Eason, T., and Spottswood, M. (2015). Nonlinear elastic buckling and postbuckling analysis of cylindrical panels. Finite Elements in Analysis and Design, 96:41–50.

Zhou, Z. and Murray, D. (1995). An incremental solution technique for unstable equilibrium paths of shell structures. Computers & Structures, 55(5):749–759