Configurational forces in magnetism with application to the dynamics of a small-scale ferromagnetic shape memory cantilever
Configurational forces in magnetism with application to the dynamics of a small-scale ferromagnetic shape memory cantilever
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DOI：
10.1007/s001610100072
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发表时间：
2002-02
影响因子：
2.6
通讯作者：
R. James
中科院分区：
文献类型：
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作者：
R. James
本文的第一部分（第1节）涉及磁弹性力学中的真实的力和构形力的公式，后者是在Eshelby意义下理解的。在第2节中，我们使用这些公式推导运动方程的MEMS（微机电系统）悬臂梁的一个不寻常的概念。第1节的主要结果是变形固体中界面上的构形力的公式（92），其中界面是变形梯度和磁化强度的不连续表面。考虑这种奇异性的主要动机是铁磁形状记忆（及相关）材料的出现[26]。这些材料具有表现出磁化和变形梯度两者的大跳跃的界面。这些界面的运动可以引起比超磁致伸缩材料大50倍的大的宏观形状变化。正如目前所理解的，这些材料的形状变化主要来自内部界面上的构型力（而不是真实的电磁力）。根据微磁学的通常公式（没有变形），磁化强度为H 1，因此不允许磁化强度在平面上不连续。点奇异是允许的，并且这些点奇异在墙体子结构的某些研究中起着重要的作用。然而，交换常数相对较小，并且常见的情况是具有由相对尖锐的畴壁分隔的几乎恒定的磁化区域（参见DeSimone [12]，用于讨论微磁学预测的交换常数变为零时的极限变分原理）。因此，为了简化和便于模拟，最好将界面视为尖锐的，并将壁结构的所有复杂性归纳为理想化尖锐界面的运动和能量定律。基本思想可以追溯到吉布斯[19]，并且Eshelby [16]，[17]认识到在相关情况下需要构型力。最重要的是，正如Abeyaratne和Knowles [1]，[2]所讨论的，构型力的公式可以用来制定动态运动定律，“关闭方程”并在能量有利的方向上移动畴壁，同时避免确定壁结构的复杂性。当引入变形时，关于磁弹性总能量的最小值的平滑性的情况不太清楚。在已知的铁磁形状记忆材料中，变形梯度的跳跃不连续性被认为是原子级尖锐的。类似的情况在材料科学中反复出现断裂力学，相变，位错力学，晶粒生长，并且有各种各样的方法来制定动态方程。上面讨论的方法依赖于奇点上的构型力f的公式（遵循Eshelby）。我们可以写出形式为v =-μ f的动力学定律，其中v是奇点速度的适当度量，μ是迁移率。尽管这一方法被广泛使用，或者说是它的明显的推广，但很少有直接的实验或其他方法来检验这样的动力学定律（由于磁构形力可以以许多不同的方式应用，人们希望磁性最终可以提供这样的检验）。另一种方法是制定一个额外的动力学定律的基础上的微观物理。在没有形变的情况下，有一个著名的磁性动力学定律：朗道-利夫希茨-吉尔伯特定律
The first part of this paper (Sect. 1) concerns the formulas in magnetoelasticity for real forces and for configurational forces, the latter understood in the sense of Eshelby. In Sect. 2 we use these formulas in the derivation of equations of motion for an unusual concept for a MEMS (micro-electro-mechanical systems) cantilever. The main result of Sect. 1 is the formula (92) for the configurational force on an interface in a deforming solid, where the interface is a surface of discontinuity of deformation gradient and magnetization. The main motivation for considering such singularities is the emergence of ferromagnetic shape memory (and related) materials [26]. These materials have interfaces that exhibit large jumps of both magnetization and deformation gradient. The motion of these interfaces can give rise to large macroscopic shape changes that are some 50 times larger then those of giant magnetostrictive materials. As these materials are currently understood, these shape changes arise primarily from configurational forces on the internal interfaces (rather than real electromagnetic forces).According to the usual formulation of micromagnetics (without deformation), the magnetization is in H 1 and therefore does not admit discontinuities of magnetization across a plane. Point singularities are allowed, and these play an important role in some studies of wall substructure. However, exchange constants are relatively small, and a common situation is to have nearly constant regions of magnetization separated by relatively sharp domain walls (see DeSimone [12] for a discussion of the limiting variational principle predicted by micromagnetics as the exchange constant goes to zero). Thus for reasons of simplicity and accessibility to simulation, it is desirable to treat the interfaces as sharp, and to lump all the complexity of wall structure into laws for the motion and energy of idealized sharp interfaces. The basic idea goes back to Gibbs [19], and the need for configurational forces in related situations was recognized by Eshelby [16],[17]. Most importantly, as discussed by Abeyaratne and Knowles [1],[2], formulas for configurational force can then be used to formulate dynamic laws of motion that ‘close the equations’ and move domain walls in an energetically favorable direction, while avoiding the complexity of determining wall structure. When deformation is introduced, the situation with regard to the smoothness of minimizers of the total energy of magnetoelasticity is less clear. In the known ferromagnetic shape memory materials, it is believed that the jump discontinuities of deformation gradient are atomically sharp. A similar situation recurs throughout materials science–fracture mechanics, phase transformations, dislocation mechanics, grain growth–and there are various approaches to the formulation of dynamic equations. The approach discussed above relies on a formula for the configurational force f on the singularity (following Eshelby). One writes a kinetic law of the form v=-µf, where v is an appropriate measure of the velocity of the singularity and µ is the mobility. Despite the widespread use of this recipe, or its obvious generalizations, there are few direct experimental or other tests of such kinetic laws (Since magnetic configurational forces can be applied in many different ways, it is hoped that magnetism might finally provide such tests). An alternative approach is the formulation of an additional kinetic law based on the microscopic physics. In the case without deformation there is a reputable dynamic law for magnetism: the Landau-Lifschitz-Gilbert