Analytical and numerical study of the mechanical behaviour of materials and structures in the theory of gradient elasticity

Abstract

In this thesis, the gradient elasticity theory has been studied in relation to some commonly encountered structural configurations. In engineering design process these include: thin straight beams, thin curved beams, thin rings and tubes, thick walled cylindrical and spherical pressure vessels, long rotating cylinders and rotating discs and half plane problems. Closed form solutions have been derived for these configurations and examined for different boundary conditions. Consequently the difference between the gradient elasticity solution and its classical equivalent is revealed. It is well known that the microstructure of a material plays a very significant role in its deformation process. In this light, the gradient elasticity theory has been shown to provide additional information which the classical theory is incapable of providing because of the absence of an internal length parameter. Moreover, a relation is shown to exist between the characteristic size of a structure and its internal length parameter. With this relation, it is possible to reveal the well-known phenomenon of size and scale effect. Furthermore, two new numerical methods were developed: the gradient elastic microtruss model and the gradient elastic beam element model. In these models, bar and beam elements are introduced possessing gradient elastic parameters which enable micro scale behavior to be easily and readily accounted for new element. Stiffness matrices have been derived from fundamentals and applied to some simple configurations the simulation is compared with the classical elastic counterpart and reveals considerable differences with these stiffness matrices, it is easy to show that smaller micro structures dimensions of a material are stiffer than larger ones. Hence in a simple manner, it provides a platform to account for micro structural properties and behavior of a material.