Two higher order elasticity theories: their variational formulations and applications

Classical elasticity cannot be used to explain effects related to material microstructures due to its lack of a material length scale parameter. To mitigate this deficiency, higher order elasticity theories have been developed. Two simple higher order theories and their applications are studied in this research. One is a modified couple stress theory and the other is a simplified strain gradient theory, each of which contains only one material length scale parameter in addition to the classical elastic constants. Variational formulations are provided for these two theories by using the principle of minimum total potential energy. In both cases, the governing equations and complete boundary conditions are determined simultaneously for the first time. Also, the displacement form is explicitly derived for each theory for the first time. The modified couple stress theory is applied to solve a simple shear problem, to develop a new Bernoulli-Euler beam model, and to derive the constitutive relations for hexagonal honeycomb structures, while the simplified strain gradient theory is used to solve the pressurized thick-walled cylinder problem. All these models/solutions are obtained for the first time and supplement their counterparts in classical elasticity. Numerical results obtained from the newly developed models and derived solutions and their comparisons with their counterpart results in classical elasticity reveal that the higher order theory based models and solutions have the capacity to account for microstructural effects; their counterparts in classical elasticity do not have the same capability. Nevertheless, the former are shown to recover the latter if the microstructural effects are suppressed or ignored.