Graduation Year

2007

Document Type

Thesis

Degree

M.S.C.E.

Degree Granting Department

Civil Engineering

Major Professor

Daniel Simkins, Ph.D.

Keywords

Couple stress, Bimaterial, Higher order diffrential equation, Messless, Isogeometric

Abstract

The Reproducing Kernel Element Method (RKEM) implementation of the Fleck-Hutchinson phenomenological strain gradient theory in 1D, 2D and 3D is implemented in this research. Fleck-Hutchinson theory fits within the framework of Touplin- Mindlin theories and deals with first order strain gradients and associated work conjugate higher-order stress. Theories at the intrinsic or material length scales find applications in size dependent phenomena. In elasticity, length scale enters the constitutive equation through the elastic strain energy function which depends on both strain as well as the gradient of rotation and stress. The displacement formulation of the Touplin Mindlin theory involve diffrential equations of the fourth order, in conventional finite element method C1 elements are required to solve such equations, however C1 elements are cumbersome in 2D and unknown in 3D. The high computational cost and large number of degrees of freedom soon place such a formulation beyond the realm of practicality. Recently, some mixed and hybrid formulations have developed which require only C0 continuity but none of these elements solve complicated geometry problems in 2D and there is no problem yet solved in 3D. The large number of degrees of freedom is still inevitable for these formulation. As has been demonstrated earlier RKEM has the potential to solve higher-order problems, due to its global smoothness and interpolation properties. This method has the promise to solve important problems formulated with higher order derivatives, such as the strain gradient theory.

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