The Element-Free kp-Ritz Method for Analysis of Functionally Graded Carbon Nanotube-Reinforced Composite Plates and Cylindrical Panels


Student thesis: Doctoral Thesis

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  • Zuxiang LEI


Awarding Institution
Award date12 Jun 2014


Carbon nanotubes (CNTs) are new advanced materials with high strength, high stiffness and high aspect ratio, but low density. Thus, CNTs have attracted significant research attention. Experimental, analytical, and theoretical investigations on the mechanical, electrical, and thermal performances of CNTs are continuously being explored. Studies have shown that CNTs have excellent mechanical, electrical, and thermal properties that are superior to those of conventional carbon fibers. CNTs exist in the form of single- and multi-walled structures with a theoretical strength of 100 times higher than that of steel. Moreover, their elastic strain can reach 0.12, and their specific gravity is only one-sixth that of steel. Given these remarkable properties, CNTs can be considered as an excellent reinforcement of polymer composites that may significantly improve the mechanical, electrical, and thermal properties of the resulting nanocomposites.

The load transfer between nanotubes and the matrix is imperfect. Thus, the first and most important issue before the mechanical analysis of CNT-reinforced composite is to determine the effective material properties of the resulting nanocomposites. Several micromechanical models such as the Eshelby-Mori-Tanaka scheme and the extended rule of mixture have been successfully developed to predict the effective material properties of CNT-reinforced nanocomposites. The Eshelby-Mori-Tanaka approach, known as the equivalent inclusion-average stress method, is based on the equivalent elastic inclusion idea of Eshelby and the concept of average stress in the matrix from the Mori-Tanaka method. Compared with the Mori-Tanaka scheme that is applicable to microparticles, the extended rule of mixture is simple and convenient to obtain the overall material properties and responses of CNT-reinforced nanocomposites. In the extended rule of mixture, a parameter which is called CNT efficiency parameter, is introduced to account for the load transfer between the nanotubes and polymeric phases (e.g., the surface effect, strain gradient effect, and intermolecular coupling effect), as well as the other factors that affect the effective material properties.

Functionally graded materials (FGMs), in which material properties spatially vary according to a certain non-uniform distribution of one of the constituents, offer significant potential for use in various engineering applications. Stimulated by the concept of FGMs, the functionally graded (FG) distribution pattern of reinforcement has been successfully applied for CNT-reinforced nanocomposites. In actual structural applications, CNT-reinforced composites, as a type of advanced material, may be incorporated in the form of beams, plates, or shells as structural components. Therefore, exploring the mechanical responses of the structures is important. Performance of the structure system is highly dependent on the mechanical behaviors of the structural components, such as bending, free vibration, buckling, large deformation, and postbuckling. Given that the geometric and material properties can significantly affect the mechanical responses, the influence of some geometric and material parameters is studied. Furthermore, the effect of CNT distribution in the CNT-reinforced nanocomposite structure and boundary conditions is examined in detail. The dynamic stability analysis of CNT-reinforced nanocomposite cylindrical panels is studied by subjecting the CNTreinforced nanocomposite structure to static and periodic axial forces.

Subsequently, with the construction of constitutive relationship, a computational framework based on the element-free kernel particle (kp)-Ritz method is implemented to study mechanical behaviors of CNT-reinforced nanocomposites. Given that the shape function is a critical factor in the development of an element-free method, the construction of the shape functions and their derivatives are described based on the kp concept. Based on kp approximations for the field variables, the Ritz method is employed to obtain the discretized governing equations. Using the convergence and comparison studies, the accuracy and efficiency of this method are validated by comparisons with the available published results.

This research is a systematic theoretical and numerical study of the mechanical behaviors of CNT-reinforced nanocomposite plate and cylindrical panel. A comprehensive study of the mechanical behaviors of CNT-reinforced nanocomposite plate and cylindrical panel with various loadings and boundary conditions is provided for static and dynamic problems. The plate and cylindrical panel are reinforced by singlewalled CNTs, which are assumed to be graded through the thickness direction with different distribution types. The effective material properties of the resulting nanocomposites are estimated through a micromechanical model based on the extended rule of mixture or the Eshelby-Mori-Tanaka scheme. To eliminate shear locking for a very thin plate or cylindrical panel, the system’s bending stiffness is evaluated using a stabilized conforming nodal integration scheme and the membrane and shear terms are calculated using the direct nodal integration method. Several numerical examples are employed to validate the present element-free kp-Ritz approach. Computational results are compared with those obtained from the finite element method and existing data. In general, a good agreement is obtained. With the free choice of nodes in the element- free computational framework, the degrees of freedom of the system can be largely reduced, thereby, saving a large amount of computational resources. Therefore, some typical mechanical behaviors such as bending, free vibration, buckling, large deformation, and postbuckling of CNT-reinforced nanocomposite plate and cylindrical panel with various constraints are numerically simulated using the proposed element-free computational framework. Detailed case studies are provided to investigate the effects of some geometric and material parameters on the mechanical responses. Finally, this approach is extended to the study of dynamic stability analysis of CNT-reinforced FG cylindrical panels under static and periodic axial force.

    Research areas

  • Element-free method, Composite, Functionally-graded, Nonlinear, Large Deformation, Postbuckling, Dynamic Stability