这篇论文介绍了弹性中协方差概念的一些进展。简要回顾了连续介质力学中的几何观点。在此基础上,将参考位形和环境空间视为具有度量的黎曼流形,建立了演化参考位形弹性体的拉格朗日场理论。结果表明,即使在这种一般情况下,由水平(参考)变化引起的欧拉-拉格朗日方程也等效于由垂直(空间)变化引起的方程。经典的Green-Naghdi-Rivilin定理被重新审视,并讨论了它的物质版本。它表明,能量平衡,在一般情况下,在参考构型的等距下不可能是不变的,在这种情况下,它是由R^3的子集确定的。得到了参考构型在刚性平移和旋转条件下能量平衡的变换性质。弹性的空间协变理论也被重述。给出了参考构型任意异胚态下能量平衡的变换,并证明了变换后的能量平衡中出现了一些非标准项。然后给出了能量平衡具有物质协变的条件。 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 Doyle-Ericksen 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.
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