Local FEM Analysis of Composite Beams and Plates: Free-Edge effect and Incompatible Kinematics Coupling

The use of composite laminated structures helped in the last two decades to reduce overall weight of transportation structures. As a consequence the energy needed to power those transportation means is reduced and hence fuel and monetary resources are economized and emissions are reduced. Especially the aerospace sector has a high need of a favourable weight to power ratio. Orthotropic laminated structures are able to provide a higher stiffness combined with a lower density compared to monolithic isotropic materials used in the past. It seems hence, that they are perfect for the use in even a wider spectrum of applications. However through the assembly of differently layers, it is more difficult to model and predict the structures mechanical response to outer loadings. In the recent past different computational methods were developed. Most of them under the scope of being capable to deliver very detailed results of the global behaviour of the structure but also of the interaction between the different layers of the laminate. As a major drawback, a detailed result comes with high computational costs. Hence a need for a good compromise between costs and accuracy has to be found. This benefits especially from the fact that stress concentrations in composites occur mainly in local domains of the structure. The use of detailed models only in those local domains of interest seems therefore straightforward. Examples for such local domains with stress concentrations are laminates with free edges. At the interface between two layers with different elastic properties the stresses have singular behaviour in the immediate vicinity of the free edge, assuming linear elastic material behaviour. This is due to the material discontinuity and the resulting mismatch of the elastic properties at the interface of the layers, the condition of traction-free edges and the equilibrium between the layers. Therefore they are critical to promote delamination. An adequate analysis method for this would be the use of a full three-dimensional analysis model. However it’s computational cost is significant. Composites are often rather thin planar structures, allowing the use of reduced dimensional models, which are also more attractive through their reduced computational cost. Therefore different reduced models with their appropriate hypotheses in the thickness direction are under consideration in this work. Via different thickness expansion functions suitable kinematical theories, are expressed. The Carrera’s Unified Formulation (CUF) is used to have a common base to build the models with the different kinematical theories. The CUF allowing not only purely displacement based models using the Principle of Virtual Displacements (PVD), but also mixed stress and displacement based models with the Reissner’s Mixed Variational Theorem (RMVT). In the first part of this work, the reduced dimensional modelling approaches are compared. Two main class are presented: Equivalent Single Layer (ESL) models treating the layered structure like one homogenous plate of equal mechanical properties, and the Layer Wise approach, treating each layer independently. Subsequently their capabilities to capture the appearing singularities are compared. In order to have a comparable measurement of those singularities, the obtained stress distributions will be expressed via a power law function, which has a priori a singular behaviour. Only two parameters fully describe therefore the singular stress components in the vicinity of the free edge. With the help of these two parameters not only the different models capabilities will be compared, but also the free edge effect itself will be measured and compared for different symmetrical laminates and the case of extensional and uniform bending load. The results for all laminates under both load cases confirm the before stated need for rather complex models in the vicinity of the free edge. However far from the free edges, in the composite plates centre, no significant difference can be noted for rather simple models. The second part of this work is therefore dedicated to the coupling of kinematically incompatible models. The use of costly expensive complex models is restricted to local domains of interest, while economic simple models will model the global domain. The Extended Variational Formulation (XVF) is identified as the most suitable way to couple the kinematically heterogenous but dimensional homogenous models. As it uses a configuration with one common interface without domain overlap, the additional efforts for establishing the coupling are limited. Further the XVF offers the possibility to adapt the conditions imposed at the interface using a single scalar parameter. It will be shown that for the homogenous dimensional problem under consideration only two different conditions can be imposed by this parameter. One matching the strong conditions imposed by the classical Multi Point Constrains (MPC) and a second one providing a weak condition. The last one is shown to provide the possibility to reduce further the domain using the complex kinematical model, without the loss of local precision. As this is the first application of the XVF towards composite structures, the need for a new coupling operator was identified. A new form is proposed, tested and its robustness will be evaluated.