Numerical Solution of Static and Dynamic Problems of Nanobeams and Nanoplates

By: Behera, LaxmiContributor(s): Chakraverty, S [Supervisor] | Department of MathematicsMaterial type: TextTextLanguage: English Publisher: 2015Description: 185 pSubject(s): Mathematics and Statistics | Applied MathematicsOnline resources: Click here to access online Dissertation note: Thesis (Ph.D) National Institute of Technology, Rourkela Summary: Recently nanotechnology has become a challenging area of research. Accordingly, a new class of materials with revolutionary properties and devices with enhanced functionality has been developed by various researchers. Structural elements such as beams, membranes and plates in micro or nanoscale have a vast range of applications. Conducting experiments at nanoscale size is quite difficult and so development of appropriate mathematical models plays an important role. Among various size dependent theories, nonlocal elasticity theory pioneered by Eringen is being increasingly used for reliable and better analysis of nanostructures. Finding solutions for governing partial differential equations are the key factor in static and dynamic analyses of nanostructures. It is sometimes difficult to find exact or closed-form solutions for these differential equations. As such, few approximate methods have been developed by other researchers. But, the existing methods may not handle all sets of boundary conditions and sometimes those are problem dependent. Accordingly, computationally efficient numerical methods have been developed here for better understanding of static and dynamic behaviors of nanostructures. Also these numerical methods can handle all classical boundary conditions of the static and dynamic problems of nanobeams and nanoplates with ease. It may be noted that application of numerical methods converts bending problem to system of equations while buckling and vibration problems to generalized eigen value problems. The present thesis first investigates bending of nanobeams and nanoplates. Next buckling and vibration of the above nanostructural members are studied by solving the corresponding partial differential equations. In the above regard, various beam and plate theories are considered for the analysis and corresponding results are reported after the convergence study and validation in special cases wherever possible. Finally, few complicating effects are also considered in some of the problems. As regards, structural members (nanobeams and nanoplates) with variable material properties are frequently used in engineering applications to satisfy various requirements. For efficient design of nanostructures, sometimes non-uniform material properties of the nano-components should also be studied. As such, we have considered here non-uniform material properties of nanobeams and nanoplates and investigated the deflection in static problems and vibration characteristics in vibration problems. Similarly, other complicating effects such as surrounding medium and temperature are important in the nanotechnology applications too. Accordingly, the effect of these complicating effects on the nanobeams and nanoplates have also been investigated in detail. It is worth mentioning that Rayleigh-Ritz and differential quadrature methods have been used to solve the above said problems. In the Rayleigh-Ritz method, simple and boundary characteristic orthogonal polynomials have been used as shape functions. Use of boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method has some advantages over other shape functions. This is because of the fact that some of the entries of stiffness, mass and buckling matrices become either one or zero. On the other hand, Differential Quadrature (DQ) method is also a computationally efficient method which can be used to solve higher order partial differential equations that may handle all sets of classical boundary conditions. Accordingly DQ method has also been used in solving the problems of nanobeams with complicating effects. In view of the above, systematic study of static and dynamic problems of nanobeams and nanoplates are done after reviewing the existing ones. Various new results of the above problems are reported in term of Figures and Tables. The new results obtained through the above mathematical models may serve as bench mark and those may certainly be used by design engineers and practitioners to validate their experimental work for better design of the related nanostructures.
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Thesis (Ph.D) National Institute of Technology, Rourkela

Recently nanotechnology has become a challenging area of research. Accordingly, a new class of materials with revolutionary properties and devices with enhanced functionality has been developed by various researchers. Structural elements such as beams, membranes and plates in micro or nanoscale have a vast range of applications. Conducting experiments at nanoscale size is quite difficult and so development of appropriate mathematical models plays an important role. Among various size dependent theories, nonlocal elasticity theory pioneered by Eringen is being increasingly used for reliable and better analysis of nanostructures. Finding solutions for governing partial differential equations are the key factor in static and dynamic analyses of nanostructures. It is sometimes difficult to find exact or closed-form solutions for these differential equations. As such, few approximate methods have been developed by other researchers. But, the existing methods may not handle all sets of boundary conditions and sometimes those are problem dependent. Accordingly, computationally efficient numerical methods have been developed here for better understanding of static and dynamic behaviors of nanostructures. Also these numerical methods can handle all classical boundary conditions of the static and dynamic problems of nanobeams and nanoplates with ease.
It may be noted that application of numerical methods converts bending problem to system of equations while buckling and vibration problems to generalized eigen value problems. The present thesis first investigates bending of nanobeams and nanoplates. Next buckling and vibration of the above nanostructural members are studied by solving the corresponding partial differential equations. In the above regard, various beam and plate theories are considered for the analysis and corresponding results are reported after the convergence study and validation in special cases wherever possible. Finally, few complicating effects are also considered in some of the problems. As regards, structural members (nanobeams and nanoplates) with variable material properties are frequently used in engineering applications to satisfy various requirements. For efficient design of nanostructures, sometimes non-uniform material properties of the nano-components should also be studied. As such, we have considered here non-uniform material properties of nanobeams and nanoplates and investigated the deflection in static problems and vibration characteristics in vibration problems. Similarly, other complicating effects such as surrounding medium and temperature are important in the nanotechnology applications too. Accordingly, the effect of these complicating effects on the nanobeams and nanoplates have also been investigated in detail.
It is worth mentioning that Rayleigh-Ritz and differential quadrature methods have been used to solve the above said problems. In the Rayleigh-Ritz method, simple and boundary characteristic orthogonal polynomials have been used as shape functions. Use of boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method has some advantages over other shape functions. This is because of the fact that some of the entries of stiffness, mass and buckling matrices become either one or zero. On the other hand, Differential Quadrature (DQ) method is also a computationally efficient method which can be used to solve higher order partial differential equations that may handle all sets of classical boundary conditions. Accordingly DQ method has also been used in solving the problems of nanobeams with complicating effects. In view of the above, systematic study of static and dynamic problems of nanobeams and nanoplates are done after reviewing the existing ones. Various new results of the above problems are reported in term of Figures and Tables. The new results obtained through the above mathematical models may serve as bench mark and those may certainly be used by design engineers and practitioners to validate their experimental work for better design of the related nanostructures.

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