Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics
Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics
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非线性位错力学的黎曼-嘉当几何
DOI：
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发表时间：
2012
期刊：
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通讯作者：
A. Goriely
中科院分区：
文献类型：
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作者：
A. Yavari;A. Goriely
本文提出了一种含分布位错的非线性固体的几何理论。在这个理论中，物质流形--物体无应力的地方--是一个Weitzenböck流形，也就是说，一个具有挠率但非度量性消失的平坦仿射连接的流形。用非线性位错力学的位错密度张量来确定材料流形的挠率。使用嘉当的移动标架，我们构造的材料流形的几个例子的机构分布位错。我们还提出了非平凡的零应力位错分布的例子。更重要的是，在这个几何框架中，我们能够计算残余应力场，假设非线性弹性体是不可压缩的。利用能量平衡及其协变性导出了非线性位错力学的协变性控制方程。
We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold—where the body is stress free—is a Weitzenböck manifold, that is, a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan’s moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields, assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance.