Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies
Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies
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DOI：
10.1007/s00205-013-0621-0
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发表时间：
2013-04
影响因子：
2.5
通讯作者：
A. Yavari
中科院分区：
文献类型：
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作者：
A. Yavari
弹性力学的协调方程已经有将近150年的历史了。有趣的是，到目前为止，对于非单连通体，它们似乎还没有得到严格的研究。本文导出了任意非单连通体在欧氏空间中非线性弹性力学的协调方程的充要条件。对于一个非单连通体，即使满足标准的相容性方程（“体”相容性方程），应变的度量也可能是不相容的。事实证明，可能有拓扑障碍的兼容性，本文旨在了解他们的变形梯度F和正确的柯西-格林应变C = FTF。我们证明了变形梯度F相容的充要条件是它的外导数和它的所有周期为零，即它在物质流形的第一同调群的生成元上的积分为零.我们将证明，不是每一个非零同伦路径需要补充的F和线性化应变的协调方程。然后，我们找到了必要和充分的兼容性条件的权利柯西-格林应变张量C为任意非单连通体时，材料和周围的空间流形具有相同的尺寸。讨论了线性化条件下的必要相容性方程和Cesàro-Volterra路径积分。然后，我们得到的充分条件，协调的线性化应变时，身体不是简单连接。总而言之，兼容性问题归结为两个问题：i）可积性条件，对于变形梯度为d（FdX）= 0，对于C为曲率消失条件，以及ii）拓扑条件。对于Fdx，这是一个同调条件，因为试图求解的方程采用dφ= Fdx的形式。然而，对于C，涉及并行传输，这意味着需要求解dR/ ds =RK形式的方程，其中R取正交群中的值。因此，这是一个关于基本群的正交表示的问题，由于正交群不是交换的，因此一般不能被简化为同调问题。
Compatibility equations of elasticity are almost 150 years old. Interestingly, they do not seem to have been rigorously studied, to date, for non-simply-connected bodies. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible, even if the standard compatibility equations (“bulk” compatibility equations) are satisfied. It turns out that there may be topological obstructions to compatibility; this paper aims to understand them for both deformation gradientFand the right Cauchy-Green strainC = FTF. We show that the necessary and sufficient conditions for compatibility of deformation gradientFare the vanishing of its exterior derivative and all its periods, that is, its integral over generators of the first homology group of the material manifold. We will show that not every non-null-homotopic path requires supplementary compatibility equations forFand linearized straine. We then find both necessary and sufficient compatibility conditions for the right Cauchy-Green strain tensorCfor arbitrary non-simply-connected bodies when the material and ambient space manifolds have the same dimensions. We discuss the well-known necessary compatibility equations in the linearized setting and the Cesàro-Volterra path integral. We then obtain the sufficient conditions of compatibility for the linearized strain when the body is not simply-connected. To summarize, the question of compatibility reduces to two issues: i) an integrability condition, which is d(FdX) = 0for the deformation gradient and a curvature vanishing condition forC, and ii) a topological condition. ForFdxthis is a homological condition because the equation one is trying to solve takes the form dφ= FdX. ForC, however, parallel transport is involved, which means that one needs to solve an equation of the form dR/ ds =RK, whereRtakes values in the orthogonal group. This is, therefore, a question about an orthogonal representation of the fundamental group, which, as the orthogonal group is not commutative, cannot, in general, be reduced to a homological question.