On Gradient Theory: What is Next?

梯度理论第一阶段：基本方面。艾方提斯：密歇根理工大学材料不稳定性与制造过程力学研究中心，密歇根州霍顿49931梯度理论，即以本构变量的高阶梯度形式配备内部长度的理论，已由Aifantis及其同事在20世纪80年代初引入，以解决位错模式问题，提供剪切带厚度的估计，并解决应变软化制度中有限元计算中网格尺寸依赖的问题。在梯度理论的背景下可能解决的另一个问题与长度尺度的桥接有关，它提供了一种机制，可以捕捉微观尺度上材料不稳定性的发生，并研究它们对宏观响应的影响。许多作者沿着上述原始的二阶“对称应力”理论（如Zbib， Muhlhaus, Vardoulakis, de Borst， Tomita等）或沿着Fleck和Hutchinson在20世纪90年代初提出的一阶“非对称应力”理论（如Shu， Stolken, Huang, Gao， Nix等）在这一主题上做出了重大贡献。该项目的主要目标是重新审视以前使用的弹性、位错动力学和塑性梯度模型，包括它们的物理起源、梯度系数的性质和相关边界条件的形式。将上述“对称应力”应变梯度理论与“非对称应力”应变梯度理论在这些方面进行比较，并对其优缺点进行评价。这种比较选择的具体例子将是简单的剪切/纯弯曲结构、纳米识别问题和固体界面。

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.