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首页> 外文期刊>International Journal of Solids and Structures >Conservation laws from any conformal transformations and the parameters for a sharp V-notch in plane elasticity
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Conservation laws from any conformal transformations and the parameters for a sharp V-notch in plane elasticity

机译：任何保形变换的守恒定律和平面弹性中V形缺口的参数

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摘要

Based on the well known complex Kolosov-Muskhelishvili potentials, two new independent Lagrangian functions are presented and their variational problems lead to two independent harmonic equations, which are also the Navier's displacement equations in plane elasticity. By applying Noether's theorem to these Lagrangian functions, it is found that their symmetry-transformation in material space is a conformal transformation in planar Euclidean space. Since any analytic function is a conformal transformation in planar Euclidean space, the conservation law obtained from this kind of symmetry-transformation possesses universality and leads to a path-independent integral. By adjusting the conformal transformation or analytic function, a finite value can be obtained from calculating this kind of path-independent integral around a material point with any order singularity. By applying this path-independent integral to the tip of a sharp V-notch, unlike Rice's J-integral, the parameters of Mode I and II problems are found, which remain invariant because of path independence for a fixed notch opening angle. That is, these two parameters are equivalent to the notch stress intensity factors (NSIFs), and two examples are presented to show the application.
机译：基于众所周知的复杂Kolosov-Muskhelishvili势，提出了两个新的独立拉格朗日函数，其变分问题导致两个独立的谐波方程，这也是平面弹性中的Navier位移方程。通过将Noether定理应用于这些拉格朗日函数，发现它们在材料空间中的对称变换是平面欧氏空间中的保角变换。由于任何解析函数都是平面欧几里德空间中的保角变换，因此从这种对称变换获得的守恒律具有普遍性，并导致与路径无关的积分。通过调整保形变换或解析函数，可以通过以任意阶的奇点计算围绕物质点的这种与路径无关的积分来获得有限值。通过将这个与路径无关的积分应用于尖锐的V形缺口的尖端，这与Rice的J积分不同，可以找到模式I和II问题的参数，这些参数仍然是不变的，因为对于固定的缺口打开角度而言，路径是独立的。也就是说，这两个参数等效于缺口应力强度因子（NSIF），并给出两个示例来说明其应用。

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