非线性磁弹性静力学的变分原理及其对应关系
Variational principles of nonlinear magnetoelastostatics and their correspondences
AI总结：
本文推导了非线性磁弹性静力学的多个变分公式，证明其等价性，并指出能量项选择对麦克斯韦应力张量的影响，强调选择变分原理和未知场需谨慎以保持一致性。
摘要：
我们使用几种涉及机械变形和代表磁性分量的独立场的变分公式，推导了非线性磁弹性静力学的方程。我们还讨论了这些公式之间的等价性，在模去某些边界积分或常数积分的情况下，利用勒让德变换和麦克斯韦方程的性质。对于增量场以及所有五个变分原理，都陈述了基于二阶变分的分岔方程。当总势能定义在物体周围无限空间上时，我们发现能量原理中包含与外部施加磁场相关的某一项会导致麦克斯韦应力张量和相关边界条件发生微小变化。另一方面，当磁场中包含的能量限制在有限体积内时，我们发现所讨论的公式与物理实体的相关表达式之间存在对应关系。鉴于文献中研究的问题中边界数据和外部施加控制的性质多种多样，以及建模中使用的变分原理同样多样，我们的分析强调在选择变分原理和未知场时要谨慎，以便与其他选择的一致性也得到满足。
英文摘要：
We derive the equations of nonlinear magnetoelastostatics using several variational formulations involving the mechanical deformation and an independent field representing the magnetic component. An equivalence is also discussed, modulo certain boundary integrals or constant integrals, between these formulations using the Legendre transform and properties of Maxwell's equations. The second variation based bifurcation equations are stated for the incremental fields as well for all five variational principles. When the total potential energy is defined over the infinite space surrounding the body, we find that the inclusion of certain term in the energy principle, associated with the externally applied magnetic field, leads to slight changes in the Maxwell stress tensor and associated boundary conditions. On the other hand, when the energy contained in the magnetic field is restricted to finite volumes, we find that there is a correspondence between the discussed formulations and associated expressions of physical entities. In view of a diverse set of boundary data and nature of externally applied controls in the problems studied in the literature, along with a equally diverse list of variational principles employed in modeling, our analysis emphasizes care in the choice of variational principle and unknown fields so that consistency with other choices is also satisfied.
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