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Numerical Example - Contact Study A two layered [0 0 /90 0 ] laminate with a pre-existing delamination at the ply interface is considered. The plate is in a state of plane strain and has a slenderness s=L/h=10, and a ratio of the delamination length α =0.4. The mode shape history with regard to the contact analysis is shown in Figure 6. In the isogeometric framework, discontinuities at the delamination interfaces can be ensured by knot repetition at the layer boundaries. Knots are a non-decreasing set of points subdividing the parameter space in which the NURBS Ansatz is specified into elements. The delaminated and undelaminated regions are modeled as separate patches and connected on the level of degrees of freedom. An Isogeometric Layerwise Approach for the Buckling Analysis of Delaminated Laminates with Contact Background Layerwise theories of composite laminates provide accurate predictions of the three-dimensional stress state which is in sharp contrast to the class of equivalent single layer theories that yield no or limited information about the transverse stress components [1-2]. Delaminations may cause obvious reduction of compressive load carrying capacity, for example, the buckling load etc. The two ways to deal with a buckling analysis are: (a) following the history of load-displacement variations in a non-linear analysis or (b) solving an eigenvalue problem for the critical buckling load and corresponding mode. The buckling modes of composite laminates with delaminations often accompanied with physically inadmissible mode shapes as addressed here (cf Fig 6). Isogeometric analysis [3], is a novel concept in computat- ional mechanics, which employs the basis functions used to describe the geometry also to approximate the physical response in an isoparametric sense. An isogeometric analysis specific refinement scheme (k-refinement) enables the adjustment of continuity across element boundaries. In the following we exploit the smoothness, the higher order continuity and refinement properties of isogeometric analysis for the buckling analysis of composites. Performance The iteration history of the buckling load and the maximum overlap for four different penalty factors are shown in Figure 7. References [1] Carrera E. (2002). Theories and finite elements for multilayered, anisotropic, composite plates and shells, Archives of Computational Methods in Engineering 9(2), pp [2] Reddy J.N. (2004). Mechanics of laminated composite plates and shells, CRC Press LLC, Boca Raton [3] Hughes T. J. R., Cottrell J.A., Bazilevs Y.,(2005). Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194, pp Isogeometric Layerwise Model Considering Delaminations The layerwise theory considers separate stiffness contributions for each layer, thus avoiding a homogenization of the elasticity properties through the composites thickness. Correspondingly, the displacement field for each layer is interpolated by a separate C 0 - continuous NURBS (non-uniform rational B-spline) Ansatz. The isogeometric layerwise refinement schemes of a two layer laminate model are shown in Figure 1. Using multiple NURBS-patches in isogeometric analysis allows to model the delaminated domains separately and independent of each other. In the following we summarize the strong coupling concept for isogeometric multi-patch models with respect to the multi-layer modeling in the framework of the delamination analysis of composite laminates. Exemplarily, the two NURBS-patches shown in Figure 2 are chosen to illustrate the basic principles of the coupling concept applying different basis functions through the thickness of a laminate. The proposed concept even allows the coupling of incompatible patches. PhD Candidate: Yujie Guo Department: ASM Section: Aerospace Structures and Computational Mechanics Supervisor: Martin Ruess Promoter: Zafer G ü rdal Start date: Funding: CSC Aerospace Engineering Fig 7: Iteration history Revised buckling state results The first three buckling modes of a [0 0 /90 0 ] laminate with α =0.3 and α =0.5 are considered. Various positions β of the delamination zone are investigated and shown in Figures 8-9. The variation of the buckling load for the second and third buckling mode is illustrated with respect to the different delamination models. Isogeometric Contact Iteration Strategy Fig 1: Isogeometric refinement schemes of a two layer laminate model Fig 2: Multi-patch connection Fig 4: Contact iteration loop Fig 6: Changes of the buckling mode shapes Original Iteration 3 Final Fig 3: Schematic representation of a delamination model Fig 5: Delamination model for a two layer composite plate Fig 8: Effect of contact constraint on the buckling load of [0 0 /90 0 ] laminate with α=0.3 Fig 9: Effect of contact constraint on the buckling load of [0 0 /90 0 ] laminate with α=0.5 Inadmissible delaminat- ion states consist of non-physical overlaps of laminate plys, (Fig 6, red domain). We follow the iterative contact analysis (Fig 4) to remove the critical overlap and to regain control over a reliable buckling analysis. The stiffness representation of delaminated structure is stepwise corrected by a contact stiffness con- tribution based on a penalty approach. The free scalability of mode shapes requires in addition a mode check to remain in each iterat- ion step consistent with the original problem.

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