Abstract
The present dissertation develops a finite element method for the static, stability and dynamic analysis of gradient elastic beam structures, which find applications in modern microelectronic and nanoelectronic devices. These linear elastic structures have extremely small dimensions, which are comparable to their microstructural lengths and thus their static, stability or dynamic response to applied mechanical loading depends on their microstructure. Classical linear elasticity theories cannot take into account microstructural effects and generalized or higher-order elasticity theories have to be employed. These theories are characterized by non-locality of stress and internal length parameters and can take into account microstructural effects in a macroscopic manner. The general theory of elasticity with microstructure due to Mindlin (1964) in its simplified form with just one elastic constant in addition to the classical ones, known as the gradient theory of elasticity, is adopted here. Even though many simple problems concerning gradient elastic beams, plates and shells under static or dynamic loads have been solved analytically, for structures of complicated geometry and loading, use of numerical methods of solution is imperative. Here the powerful and very popular finite element method is used to study static, stability and dynamic problems of gradient elastic beam structures. In static or stability analysis, the exact solution of the equilibrium governing equation of the problem is used as the displacement function for the development of the stiffness matrix of a beam-column element and hence the resulting matrix is also exact and leads to the exact solution of the problem. For a gradient elastic Bernoulli-Euler beam element with two nodal points of three degrees of freedom each (deflection, slope and curvature) the element stiffness matrix is of the 6×6 order. The buckling or critical load is obtained by taking the determinant of the stiffness matrix equal to zero. In dynamic analysis, the governing equation of flexural motion is brought first in the frequency domain and thus its exact solution can be easily obtained. Using this exact solution as the displacement function an exact 6×6 finite element stiffness matrix is constructed in the frequency domain. Use of this matrix in a finite element framework can lead to the exact solution of either the free or the forced vibration problem. Solution of the free vibration problem leads to natural frequencies and modal shapes. Solution of the forced vibration problem provides the response in the frequency domain, which after a numerical inversion leads to the time domain response. Here the frequency domain is the complex one or the Laplace transform domain and the inversion concerns the Laplace transform. Numerical examples are presented in detail to illustrate the method and demonstrate its advantages. Comparisons with analytical methods are also presented for verification purposes. Finally, parametric studies are conducted in order to assess the effect of microstructure on the response of structures composed of gradient elastic beams to mechanical loading.
Advisory committee
Supervisor: Hatzigeorgiou George, Associate Professor HOU.
Karampalis Dimitris, Professor, University of Patras.
Polyzos Demosthenes, Professor, University of Patras.