发表机构
Mathematics Institute, University of Warwick（华威大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出小畸变几何模型，基于小畸变假设与线张力近似，结合时空框架，实现由位错运动直接驱动的单晶弹塑性建模，并连接离散位错与连续介质理论。
AI 中文摘要
非线性弹塑性理论的一个核心问题是，构建一个几何非线性、大应变的单晶弹塑性模型，在该模型中塑性流动直接由位错运动驱动，并且允许从离散位错线到位错密度的均匀化过程。本文基于两个假设引入了这样一个模型：（1）小畸变假设认为，总塑性畸变可以任意大，但其主导部分可表示为梯度形式，这意味着位错的畸变效应相对于试样尺寸较小。具体而言，该假设通过“乘法型亥姆霍兹分解”实现，该分解将任意矩阵场分解为一个梯度场与一个旋度受控的矩阵场的乘积。（2）线张力近似将所有位错视为无限细的线，其局部应力场相对于试样尺度应力可单独忽略，仅在集合中起作用。由此产生的模型称为小畸变几何模型，进一步采用了所谓的时空框架，该框架提供了一种几何语言来精确描述位错线的平流。因此，它在理想化的介观尺度上提供了晶体塑性的物理基础描述，弥合了位错力学与连续弹塑性之间的鸿沟。
英文摘要
A central problem in nonlinear elasto-plasticity theory is to formulate a geometrically nonlinear, large-strain model of elasto-plasticity in single crystals in which plastic flow is driven directly by the motion of dislocations and which allows for a homogenization procedure from discrete dislocation lines to dislocation densities. In this work, such a model is introduced based on two hypotheses: (1) The Small-Distortion Hypothesis posits that the total plastic distortion may be arbitrarily large but its dominant part admits a representation as a gradient, meaning that the distorting effect of dislocations is small relative to the specimen size. Concretely, this hypothesis is realized here through a "multiplicative Helmholtz-type decomposition", which splits an arbitrary matrix field into the product of a gradient and a matrix field with controlled curl. (2) The Line-Tension Approximation treats all dislocations as infinitesimally thin lines whose local stress fields are individually negligible compared to specimen-scale stresses and only matter in the aggregate. The resulting model, termed the Small-Distortion Geometric Model, furthermore employs the so-called Space-Time Framework, which furnishes a geometric language to precisely describe the advection of dislocation lines. As such, it provides a physically grounded account of crystal plasticity in an idealized mesoscopic setting that bridges dislocation mechanics and continuum elasto-plasticity.
Comments71 pages
