On Spatial and Material Covariant Balance Laws in Elasticity
On Spatial and Material Covariant Balance Laws in Elasticity
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弹性空间与材料协变平衡定律
DOI：
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发表时间：
2006
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通讯作者：
M. Ortiz
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作者：
A. Yavari;J. Marsden;M. Ortiz
本文介绍了一些发展有关的协方差弹性的想法。简要回顾了连续介质力学的几何观点。在此基础上，将参考位形和周围空间看作具有各自度量的黎曼流形，发展了具有演化参考位形的弹性体的拉格朗日场论.结果表明，即使在这种一般的设置，欧拉-拉格朗日方程所产生的水平参考变化是等效的垂直空间变化所产生的。经典的Green-Naghdi-Rivilin定理被重新审视，并讨论了它的一个具体版本。它示出的能量平衡，在一般情况下，不能是不变的参考配置的等距下，在这种情况下，确定与R 3的一个子集。得到了参考位形在刚体平移和转动情况下能量平衡的变换性质。空间协变弹性理论也重新审视。得到了参考构型任意双态性下的能量平衡变换，并指出变换后的能量平衡中出现了一些非标准项。然后得到了能量平衡实质协变的条件。可以看出，能量平衡的物质协变性等价于质量守恒、各向同性、物质Doyle-Ericksen公式和一个我们称之为构型无粘的额外条件。在文章的最后一部分，我们讨论了Noether定理和协方差之间的联系。它表明，DoyleEricksen公式可以得到的拉格朗日密度的空间协方差的后果。同样，它表明，材料的Doyle-Ericksen公式可以得到的材料协方差的拉格朗日密度。© 2006美国物理研究所。DOI：10.1063/1.2190827
This paper presents some developments related to the idea of covariance in elasticity. The geometric point of view in continuum mechanics is briefly reviewed. Building on this, regarding the reference configuration and the ambient space as Riemannian manifolds with their own metrics, a Lagrangian field theory of elastic bodies with evolving reference configurations is developed. It is shown that even in this general setting, the Euler-Lagrange equations resulting from horizontal referential variations are equivalent to those resulting from vertical spatial variations. The classical Green-Naghdi-Rivilin theorem is revisited and a material version of it is discussed. It is shown that energy balance, in general, cannot be invariant under isometries of the reference configuration, which in this case is identified with a subset of R 3 . Transformation properties of balance of energy under rigid translations and rotations of the reference configuration is obtained. The spatial covariant theory of elasticity is also revisited. The transformation of balance of energy under an arbitrary diffeomorphism of the reference configuration is obtained and it is shown that some nonstandard terms appear in the transformed balance of energy. Then conditions under which energy balance is materially covariant are obtained. It is seen that material covariance of energy balance is equivalent to conservation of mass, isotropy, material Doyle-Ericksen formula and an extra condition that we call configurational inviscidity. In the last part of the paper, the connection between Noether’s theorem and covariance is investigated. It is shown that the DoyleEricksen formula can be obtained as a consequence of spatial covariance of Lagrangian density. Similarly, it is shown that the material Doyle-Ericksen formula can be obtained from material covariance of Lagrangian density. © 2006 American Institute of Physics. DOI: 10.1063/1.2190827