Interlaminar Stresses In Composite Sandwich Panels Using Variational Asymptotic Method (VAM)
Abstract
In aerospace applications, use of laminates made of composite materials as face sheets in sandwich panels are on the rise. These composite laminates have low transverse shear and transverse normal moduli compared to the inplane moduli. It is also seen that the corresponding transverse strength values are very low compared to the inplane strength leading to delaminations. Further, in sandwich structures, the core is subjected to significant transverse shear stresses. Therefore the interlaminar stresses (i.e., transverse shear and normal) can govern the design of sandwich structures. As a consequence, the first step in achieving efficient designs is to develop the ability to reliably estimate interlaminar stresses.
Stress analysis of the composite sandwich structures can be carried out using 3D finite elements for each layer. Owing to the enormous computational time and resource requirements for such a model, this process of analysis is rendered inefficient. On the other hand, existing plate/shell finite elements, when appropriately chosen, can help quickly predict the 2D displacements with reasonable accuracy. However, their ability to calculate the thicknesswise distributions of interlaminar shear and normal stresses and 3D displacements remains as a research goal. Frequently, incremental refinements are offered over existing solutions. In this scenario, an asymptotically correct dimensional reduction from 3D to 2D, if possible, would serve to benchmark any ongoing research. The employment of a mathematical technique called the Variational Asymptotic Method (VAM) ensures the asymptotical correctness for this purpose.
In plates and sandwich structures, it is typically possible to identify (purely from the defined material distributions and geometry) certain parameters as small compared to others. These characteristics are invoked by VAM to derive an asymptotically correct theory. Hence, the 3D problem of plates is automatically decomposed into two separate problems (namely 1D+2D), which then exchange relevant information between each other in both ways. The throughthethickness analysis of the plate, which is a 1D analysis, provides asymptotic closed form solutions for the 2D stiffness as well as the recovery relations (3D warping field and displacements in terms of standard plate variables). This is followed by a 2D plate analysis using the results of the 1D analysis. Finally, the recovery relations regenerate all the required 3D results. Thus, this method of developing reduced models involves neither ad hoc kinematic assumptions nor any need for shear correction factors as postprocessing or curvefitting measures. The results are most general and can be made as accurate as desired, while the procedure is computationally efficient.
In the present work, an asymptotically correct plate theory is formulated for composite sandwich structures. In developing this theory, in addition to the small parameters (such as small strains, small thicknesstowavelength ratios etc.,) pertaining to the general plate theory, additional small parameters characterizing (and specific to) sandwich structures (viz., smallness of the thickness of facial layers compared to that of the core and smallness of elastic material stiffness of the core in relation to that of the facesheets) are used in the formulation. The present approach also satisfies the interlaminar displacement continuity and transverse equilibrium requirements as demanded by the exact 3D formulation. Based on the derived theory, numerical codes are developed inhouse. The results are obtained for a typical sandwich panel subjected to mechanical loading. The 3D displacements, interlaminar normal and shear stress distributions are obtained. The results are compared with 3D elasticity solutions as well as with the results obtained using 3D finite elements in MSC NASTRAN®. The results show good agreement in spite of the major reduction in computational effort. The formulation is then extended for thermoelastic deformations of a sandwich panel.
This thesis is organized chronologically in terms of the objectives accomplished during the current research. The thesis is organized into six chapters. A brief organization of the thesis is presented below.
Chapter1 briefly reviews the motivation for the stress analysis of sandwich structures with composite facesheets. It provides a literature survey on the stress analysis of composite laminates and sandwich plate structures. The drawbacks of the existing anlaytical approaches as opposed to that of the VAM are brought out. Finally, it concludes by listing the main contributions of this research.
Chapter2 is dedicated to an overview of the 3D elasticity formulation of composite sandwich structures. It starts with the 3D description of a material point on a structural plate in the undeformed and deformed configurations. Further, the development of the associated 3D strain field is also described. It ends with the formulation of the potential energy of the sandwich plate structure.
Chapter3 develops the asymptotically correct theory for composite sandwich plate structure. The mathematical description of VAM and the procedure involved in developing the dimensionally reduciable structural models from 3D elasticity functional is first described. The 1D throughthethickness analysis procedure followed in developing the 2D plate model of the composite sandwich structure is then presented. Finally, the recovery relations (which are one of the important results from 1D throughthethickness analysis) to extract 3D responses of the structure are obtained.
The developed formulation is applied to various problems listed in chapter
4. The first section of this chapter presents the validation study of the present formulation with available 3D elasticity solutions. Here, composite sandwich plates for various length to depth ratios are correlated with available 3D elasticity solutions as given in [23]. Lastly, the distributions of 3D strains, stresses and displacements along the thickness for various loadings of a typical sandwich plate structure are correlated with corresponding solutions using well established 3D finite elements of MSC NASTRAN® commerical FE software.
The developed and validated formulation of composite sandwich structure for mechanical loading is extended for thermoelastic deformations. The first sections of this chapter describes the seamless inclusion of thermoelastic strains into the 3D elasticity formulation. This is followed by the 1D throughthethickness analysis in developing the 2D plate model. Finally, it concludes with the validation of the present formulation for a very general thermal loading (having variation in all the three coordinate axes) by correlating the results from the present theory with that of the corresponding solutions of 3D finite elements of MSC NASTRAN® FE commercial software.
Chapter6 summarises the conclusions of this thesis and recommendations for future work.
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