On modifications of Newton's second law and linear continuum elastodynamics
On modifications of Newton's second law and linear continuum elastodynamics
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DOI：
10.1098/rspa.2006.1795
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发表时间：
2007-03-08
影响因子：
3.5
通讯作者：
Willis, John R.
中科院分区：
文献类型：
--
作者：
Milton, Graeme W.;Willis, John R.
在本文中，我们提出了一种新的观点，即牛顿第二运动定律被一个更一般的定律所取代，这个定律对于描述看似刚体的宏观物体的运动是一个更好的近似。我们证实了威利斯的一个发现，即在给定的振荡频率下，物体的密度可以是各向异性的。力和加速度之间的关系在时间上是非局部的（而是因果的）。反之，对于每一个满足这些性质且具有适当高频极限的响应函数，都存在一个实现该响应函数的模型。在许多情况下，牛顿第二定律和新定律之间的差异很小，但在某些情况下，例如在特殊设计的复合材料中，差异是巨大的。对于非表面刚体，弹性动力学的连续统方程控制其行为，也需要修改。这里提出的这些方程的修改版本是威利斯提出的描述复合材料弹性动力学方程的推广。有人认为，这些新的方程组可以适用于所有的物理材料，而不仅仅是复合材料。威利斯方程控制平均位移场的行为，而一组新方程控制平均加权位移场的行为，其中加权位移场可能对材料中无法观察到或未定义位移的“隐藏”区域附加零权重。从平均加权位移场的知识，可以得到系综平均能量密度的近似公式。另外两组新方程控制微观结构具有微惯性时的行为，即在连续体模型选择的尺度以下存在内部旋转质量。在第一个集合中，假设平均位移场是可观测的，而在第二个集合中，假设平均加权位移场是可观测的。
In this paper, we suggest a new perspective, where Newton's second law of motion is replaced by a more general law which is a better approximation for describing the motion of seemingly rigid macroscopic bodies. We confirm a finding of Willis that the density of a body at a given frequency of oscillation can be anisotropic. The relation between the force and the acceleration is non-local ( but causal) in time. Conversely, for every response function satisfying these properties, and having the appropriate high-frequency limit, there is a model which realizes that response function. In many circumstances, the differences between Newton's second law and the new law are small, but there are circumstances, such as in specially designed composite materials, where the difference is enormous. For bodies which are not seemingly rigid, the continuum equations of elastodynamics govern behaviour and also need to be modified. The modified versions of these equations presented here are a generalization of the equations proposed by Willis to describe elastodynamics in composite materials. It is argued that these new sets of equations may apply to all physical materials, not just composites. The Willis equations govern the behaviour of the average displacement field whereas one set of new equations governs the behaviour of the average-weighted displacement field, where the weighted displacement field may attach zero weight to 'hidden' areas in the material where the displacement may be unobservable or not defined. From knowledge of the average-weighted displacement field, one obtains an approximate formula for the ensemble averaged energy density. Two other sets of new equations govern the behaviour when the microstructure has microinertia, i.e. where there are internal spinning masses below the chosen scale of continuum modelling. In the first set, the average displacement field is assumed to be observable, while in the second set an average-weighted displacement field is assumed to be observable.