Non-classical laws with conserved quantities from the arbitrary functions included in general solution to elasticity and application
Non-classical laws with conserved quantities from the arbitrary functions included in general solution to elasticity and application
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DOI：
10.1016/j.apm.2018.08.022
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发表时间：
2019-02
影响因子：
5
通讯作者：
Weichen Shi
中科院分区：
文献类型：
--
作者：
Weichen Shi
静态和动态线弹性力学的一般解是位移与新的任意函数之间的变换，其保守性取决于新的任意函数所满足的独立偏微分方程。由于位移用两个新的任意函数表示，且这两个新的任意函数所满足的独立偏微分方程的最高阶导数之和与Navier-Cauchy方程的最高阶导数之和相同，因此张的通解在数学上是合适的。（1）在静态和动态通解中，独立的偏微分方程分别来自于作用在这两个新的任意函数上的拉普拉斯和D 'Alembert算子，并且发现这两个新的任意函数与转动、第一应变不变量和畸变有关;（ii）特别地，由函数的空间积分所满足的方程构造的守恒律仍然成立，尽管空间积分的某些任意函数已经被取消。在此基础上，利用Noether恒等式不仅可以应用于拉格朗日量，而且可以构造广义偏微分方程的泛函，利用任意整数阶空间导数或积分构造与转动、第一应变不变量和畸变有关的泛函，从而得到守恒律.这类非经典守恒律不是由弹性体的拉格朗日密度导出的，而是由标准方法导出的弹性场对称性的深层次性质。通过两个例子说明了该方法的有效性，并给出了弹性半空间表面垂直荷载的场强与伽利略变换运动坐标系中的路径无关积分的比较。
The general solution to static and/or dynamic linear elasticity is a transformation between the displacements and new arbitrary functions, whose conservativeness depends on some independent partial differential equations (PDEs) satisfied by the new arbitrary functions. Zhang's general solutions are mathematically appropriate since the displacements are expressed in terms of two new arbitrary functions, and the sum of the highest order derivative added together from the independent PDEs satisfied by the two new arbitrary functions is the same as that of Navier–Cauchy equations. Therefore, the following points should be emphasized: (i) the independent PDEs come from the Laplace and D'Alembert operators acting on the two new arbitrary functions in static and dynamic general solutions, respectively, and it is found that the two new arbitrary functions are related to the rotations, first strain invariant and distortion; (ii) especially, conservation laws constructed from the equations satisfied by the spatial integrals of functions hold true, although some arbitrary functions of the spatial integrals have been canceled. Based on these facts, since Noether's identity not only can be applied to a Lagrangian but also can be used to construct a functional for widespread PDEs, the functionals relating to the rotations, first strain invariant and distortion are constructed with arbitrary integer order spatial derivative or integral, and the conservation laws follow. This kind of non-classical conservation laws does not come from the Lagrangian density of an elastic body and belongs to the deep-level natures of symmetries of elastic field derived by standard techniques. Availability is shown by two examples, from which the field intensity of a vertical load applied to the surface of an elastic half-space and the path-independent integrals in a coordinate system moving with Galilean transformation are presented for comparison.