Nonlocal nonlinear analysis of functionally graded plates

Kumar, Basant and Rajagopal, Amirtham (2019) Nonlocal nonlinear analysis of functionally graded plates. PhD thesis, Indian Institute of Technology Hyderabad.

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Abstract

The ever increasing demands from the aerospace industry has led to the development of tailored materials with high specific strength and high specific stiffness. Typical airframes in launch vehicles are long semi monocoque cylindrical shells of large diameters made up of high strength aluminum alloys. The stiffening is done by providing longerons riveted to the shell structure making it a complex 3-D structure and are subjected to thermo-mechanical environment. The coupled loading environments which a structural member is subjected to, during its operational life cycle requires it to be functionally capable of withstanding these loading conditions. This requires materials to have properties tailored in specific directions. Functionally graded materials (FGM’s) are the class of materials having continuous and smooth varying composition of properties in the required directions. These have been developed specially for high temperature applications with a combination of ceramic and metal to take care of the thermal and structural requirements, respectively. . In the present study, a small section of the large diameter cylindrical shell has been considered and idealized as a plate. The work considers moderately thick FGM plates subjected to transverse loads along with the thermal environment. Since the aerospace structures are complex in nature, conventional finite element treatment of the subject would be inadequate. To improvise on the conventional FE analysis, a meshless approach based on the natural neighbor Galerkin method has been proposed to analyze FGM plates subjected to transverse loads. A detailed literature survey has been carried out to have an understanding of FGM’s. The current trends and scope for further work was understood. The present research work has been formulated based on the requirement’s emanating from the airframe design for a typical launch vehicle environment. FGM’s are typically heterogeneous material’s at the macroscopic level and analysis of such mate- rials requires precise definition of the overall material property. Various homogenization techniques are used to estimate the equivalent properties for analysis of FGM plates. In the present work two different homogenization techniques have been used to estimate the overall effective property namely the Rule of Mixtures and the Mori-Tanaka scheme. Static flexural analysis has been carried out based on the shear deformation theory using linear strains for different power law indices of the FGM plate. A unique through thickness integration scheme has been proposed and formulated for FGM plates in this work. The effects of variation in the loads and boundary conditions have been presented and discussed. The results computed with the two homogenization schemes have been presented. It is observed that the effective material property depends on the homogenization scheme used and the difference in the material property through the thickness is of the order of 2-3% for graded plates. To consider the effect of thermal load, thermo-mechanical modeling of FGM plates have carried out. The plate formulations are based on FSDT and non-linear strains. Heat conduction in one dimensions has been considered to estimate the temperature distribution through the thickness of the FGM plate, imposing constant surface temperatures at the top and bottom surfaces. The results under the combined effect of pressure and temperature have been presented and show that effect of gradation on the deflections and stresses. It is observed that non-linearity is primarily driven by the temperature distribution and also the gradation index at low mechanical loads. At high loads the effect of large deformations govern the non-linearity. To take into account the geometric non-linearity and shear deformation, Reddys’ third order shear deformation theory (TSDT) as a specific case of generalized higher order shear deformation theory(GHSDT) has been implemented in the plate formulations. The non-linear finite element model of the governing equations have been developed and Newton’s iterative procedure is used to solve the nonlinear algebraic equations. The proposed method was found to be more accurate with similar models from literature. FGM’s are typically considered with a heat resistant material like ceramics on top and structural material like metals at the bottom. The thickness of the thermal layer depends on the severity of the thermal environment. For lower operating temperature cases, the thickness of the thermal layer required may be very small. In these cases the thermal layer is deposited in form of thin coatings with thicknesses ranging in microns (lower length scales). To incorporate this effect, a non-local non-linear model based on TSDT has been proposed and formulated. The non-local and nonlinear TSDT formulation has been implemented in the framework of FEM. Owing to the complex nature of the launch vehicle airframes, mesh based finite element models pose limitations in terms of accuracy of modeling. To overcome these problems, mesh-free analysis using natural neighbors Galerkin method have been carried for analysis of FGM plates subjected to transverse loads. The plate formulations are based on the non-local non-linear TSDT model. Natural neighbors Galerkin method has been used to estimate the C 1 shape functions. The complete model has been developed and parametric analysis has been carried out for static bending. The results from finite elements have been compared to those obtained from the proposed natural neighbor Galerkin method. The results considering local and non-local effects show better accuracy and equivalence with the numerical models and analytical results. Aerospace structures which undergo large deformations, would eventually need an adaptive discretization of nodes to obtain accurate results. The adaptive discretization of nodes in the critical domain can be done based on region by region based error estimates. The recovery stress is used as an error estimator and then relative energy norm of the error ηi used as a basis for refinement. The meshless approach used earlier is extended to incorporate the adaptive discretization of nodes based on error estimators. The adaptive natural neighbor Galerkin method is then used to analyze moderately thick plates. Various numerical examples are presented to demonstrate the efficiency of the proposed method. In the present work, the proposed plate formulations for the analysis of moderately thick FGM plates include: (a) Linear and non-linear analysis, (b) Non-local nonlinear analysis, (c) Thermo- mechanical analysis and (d) Adaptive meshless non-local analysis. The work also proposes a non- linear generalization of the higher order shear deformation theories with specific to Reddy’s TSDT which include the non-local model of Eringen. The work also proposes the non-local nonlinear model in the Galerkin framework to carry out flexural analysis of FGM plates subjected to transverse loads. An adaptive discretization of nodes based on natural neighbor Galerkin method has also been proposed for FGM plates and laminates. From the research work carried out, it is observed that FGM’s hold promise and are one of the candidate materials for applications in the aerospace sector. Driven by characteristic material non-linearity, the work was instrumental in understanding the behavior of FGM plates subjected to thermo-elastic environments. Meshfree methods are necessary for such kind of problems specifically with gradations where steep change in material properties is seen. The choice of gradation depends upon the application and insight of the designer.

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