S. Papargyri-Beskou

BENDING AND STABILITY ANALYSIS OF GRADIENT ELASTIC BEAMS

Abstract The problems of bending and stability of Bernoulli–Euler beams are solved analytically on the basis of a simple linear theory of gradient elasticity with surface energy. The governing equations of equilibrium are obtained by both a combination of the basic equations and a variational statement. The additional boundary conditions are obtained by both variational and weighted residual approaches. Two boundary value problems (one for bending and one for stability) are solved and the gradient elasticity effect on the beam bending response and its critical (buckling) load is assessed for both cases. It is found that beam deflections decrease and buckling load increases for increasing values of the gradient coefficient, while the surface energy effect is small and insignificant for bending and buckling, respectively.

Dynamic Analysis of Gradient Poroelastic Solids and Structures

The present chapter presents a review of previous works of the authors on the subject of the dynamic analysis of gradient poroelastic solids and structures. First, the governing equations of motion of a fluid-saturated poroelastic medium with microstructural (for both the solid and fluid) and microinertia (for the solid) effects are derived. These equations are of an order of two degrees higher than in the classical case and consist of seven equations with seven unknowns in three-dimensions. Second, the propagation of plane harmonic waves in an infinitely extended medium is studied analytically for the low and high frequency range. This is accomplished by separating the equations of motion in their dilatational and rotational parts for which wave dispersion curves can be constructed. Third, a simple one-dimensional boundary value problem, that of the transient behavior of a gradient poroelastic soil column, is solved analytically/ numerically with the aid of numerical Laplace transform. Finally, on the basis of the above, conclusions are drawn and suggestions for future research are made.

DYNAMIC ANALYSIS OF GRADIENT ELASTIC BEAMS BY FINITE ELEMENTS

The dynamic stiffness matrix of a gradient elastic flexural BernoulliEuler beam finite element is analytically constructed with the aid of the basic and governing equations of motion in the frequency domain. The flexural element has one node at every end with three degrees of freedom per node, i.e., the displacement, the slope and the curvature. Use of this dynamic stiffness matrix for a plane system of beams enables one through a finite element analysis to determine its dynamic response to harmonically varying with time external load or the natural frequencies and modal shapes of that system. The response to transient loading is obtained with the aid of Laplace transform with respect to time and the numerical inversion of the transformed solution. Because the exact solution of the governing equation of motion in the frequency domain is used as the displacement function, the resulting dynamic stiffness matrices and the obtained structural responses or natural frequencies and modal shapes are also exact. Two examples are presented to illustrate the method.

Static analysis of 3-D gradient elastic solids by BEM

Publisher Summary Classical linear elasticity cannot describe the mechanical behavior of linear elastic materials with microstructure. The microstructural effects can be successfully modeled in a macroscopic framework by defining the state of stress in a nonlocal manner with the aid of higher-order strain gradients. In this chapter, a boundary element methodology for the static analysis of three-dimensional elastic solids and structures characterized by microstructural effects is presented. These microstructural effects are taken into account with the aid of a simple strain-gradient elastic theory enhanced with surface energy terms. A representative example demonstrates the accuracy of the presented boundary element method. However, for realistic engineering problems characterized by complicated geometry and boundary conditions, analytical methods of solution are inadequate and to the solution lies with numerical methods, such as the finite element method (FEM) or the boundary element method (BEM). The FEM has already been successfully employed in two dimensional (2-D) gradient elastostatic problems.