位错力学的几何建模与数值分析
Geometrical Modelling and Numerical Analysis of Dislocaion Mechanics
摘要：
本研究在微分几何的框架内进行位错的数学建模和数值分析。基本组态，即参考组态、中间组态和当前组态，被表示为黎曼-嘉当流形，它配备了黎曼度量和Weitzenböck连接。通过位错密度的Hodge对偶性得到了中间位形上的挠2-形式，并对相应的丛同构进行了Helmholtz分解。该分析引入了塑性变形的边界条件。利用弱变分法和等几何分析法求解了Cartan第一结构方程和应力平衡方程。本文的数值分析首次揭示了螺型位错和刃型位错周围塑性变形场的分布。它还演示了应力场周围的位错的远场显示完全符合经典的Volterra理论，而在同一时间消除奇异性，否则引入的位错由经典的方法。应力场包括几个特征，由于几何非线性包括在其中。我们还表明，自由表面影响塑性和弹性变形，但在不同的方式。本研究的数学框架适用于任意组态的位错。
英文摘要：
This study undertakes the mathematical modelling and numerical analysis of dislocations within the framework of differential geometry. The fundamental configurations, i.e. reference, intermediate and current configurations, are expressed as the Riemann-Cartan manifold, which equips the Riemannian metric and Weitzenböck connection. The torsion 2-form on the intermediate configuration is obtained through the Hodge duality of the dislocation density and the corresponding bundle isomorphism is subjected to the Helmholtz decomposition. This analysis introduces the boundary condition for plastic deformation. Cartan first structure equation and stress equilibrium equation are solved numerically using weak form variational expressions and isogeometric analysis. The numerical analysis carried out for this study reveals the distribution of plastic deformation fields around screw and edge dislocations for the first time. It also demonstrates stress fields around dislocations of which the distant fields show full agreement with the classical Volterra theory, while at the same time eliminating the singularity otherwise introduced at the dislocation by classical methods. The stress fields include several characteristic features due to the geometrical nonlinearity included therein. We also demonstrate that free surfaces affect both plastic and elastic deformation, but in different ways. The mathematical framework of this study is applicable to an arbitrary configuration of dislocations.
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