On Gradient Theory: What is Next?

点击翻译按钮获取中文摘要

Gradient Theory Phase I: Fundamental AspectsE.C. Aifantis: Center for the Mechanics of Material Instabilities and Manufacturing Processes, Michigan Technological University, Houghton, MI 49931, USAAbstract: Gradient theory, i.e. a theory equipped with an internal length in the form of higher-order gradients for the constitutive variables, has been introduced by Aifantis and co-workers in the early 1980's in order to address the problem of dislocation patterning, to provide estimates of shear band thickness, and to settle the issue of mesh-size dependence in finite element calculations in the strain softening regime. An additional problem that was possible to address within the setting of gradient theory was concerned with the bridging of length scales by providing a mechanism for capturing the occurrence of material instabilities at the microscale and investigating their effect on the macroscopic response. Various authors have made significant contributions on the subject along the original second-order "symmetric stress" theory mentioned above (e.g. Zbib, Muhlhaus, Vardoulakis, de Borst, Tomita, etc.) or along the first-order "asymmetric stress" theory introduced by Fleck and Hutchinson in the early 1990's (e.g. Shu, Stolken, Huang, Gao, Nix, etc.). The main objective of this project is to revisit previously employed gradient models of elasticity, dislocation dynamics, and plasticity with respect to their physical origin, the nature of the gradient coefficients, and the form of the pertinent boundary conditions. A comparison between the aforementioned "symmetric stress" strain gradient theory and the "asymmetric stress" strain gradient theory on all these aspects will be provided and an assessment will be made on their advantages and shortcomings. Particular examples selected for this comparison will be simple shear / pure bending configurations, nanoidentation problems, and solid interfaces.