An Isogeometric Layerwise Approach for the Buckling Analysis of Delaminated Laminates with Contact

1. An Isogeometric Layerwise Approach for the Buckling Analysis of Delaminated Laminates with Contact PhD Candidate: Yujie Guo Department: ASM Section: Aerospace Structures and Computational Mechanics Supervisor: Martin Ruess Promoter: Zafer Gürdal Start date: 18-9-2011 Funding: CSC 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. • 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. Fig 3: Schematic representation of a delamination model Isogeometric Contact Iteration Strategy Inadmissibledelaminat-ionstatesconsist of non-physicaloverlaps of laminateplys, (Fig 6, reddomain). Wefollowthe iterative contactanalysis (Fig 4) toremovethecriticaloverlap and toregaincontrolover a reliablebucklinganalysis. The stiffnessrepresentation of delaminatedstructureisstepwisecorrectedby a contactstiffnesscon-tributionbased on a penaltyapproach. The freescalability of modeshapesrequires in addition a mode check toremain in eachiterat-ionstepconsistentwiththe original problem. 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 composite’s thickness. Correspondingly, the displacement field for each layer is interpolated by a separate C0-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. Fig 7: Iteration history Aerospace Engineering Revised buckling state results The first three buckling modes of a [00/900] laminate with α=0.3 and α=0.5 are considered. Various positions βof 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. Fig 4: Contact iteration loop Fig 1: Isogeometric refinement schemes of a two layer laminate model Numerical Example - Contact Study A two layered [00/900] 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. Fig 8: Effect of contact constraint on the buckling load of [00/900] laminate with α=0.3 Fig 5: Delamination model for a two layer composite plate Original Iteration 3 Final Fig 2: Multi-patch connection Fig 9: Effect of contact constraint on the buckling load of [00/900] laminate with α=0.5 Fig 6: Changes of the buckling mode shapes 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 87-140 [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 4135-4195