弹性动力学中的因果性和被动性
Causality and Passivity in Elastodynamics
摘要：
考虑的线性弹性动力学系统既有因果关系（效应接替原因）又有被动关系（系统不产生能量），这一要求对弹性动力学的频率相关本构张量施加了什么约束？与电磁学相比，弹性动力学由于其不可对角化的本构张量而变得复杂。在本文中，我们澄清了因果性和被动性对非常一般形式的弹性动力学本构关系的约束。具体来说，我们表明，被动性（和因果关系）的满意度直接要求，埃尔米特部分，如后所定义的，弹性动力学本构张量的时间导数的傅立叶变换是半正定在所有频率。此外，我们表明，随后的条件要求的非厄米的本构张量的傅立叶变换的部分，当除以虚数$i$，是半正定的频率为正值和负半定的频率为负值。我们表明，当主要的对称性假设这些明确的关系简单地适用于实部和虚部的相关张量。对于对角和一维问题，这些正半定关系简化为简单的不等式关系。最后，我们将结果推广到高度一般的本构关系，其中包括威利斯非齐次关系作为一种特殊情况。
英文摘要：
What are the constraints placed on the frequency dependent constitutive tensors of elastodynamics by the requirements that the linear elastodynamic system under consideration be both causal (effects succeed causes) and passive (system doesn't produce energy)? Compared to electromagnetism, elastodynamics is complicated by its generally non diagonalizable constitutive tensors. In this paper we clarify the constraints that causality and passivity place on very general forms of elastodynamic constitutive relations. Specifically we show that the satisfaction of passivity (and causality) directly requires that the hermitian parts, as defined later, of the Fourier transforms of the time derivatives of the elastodynamic constitutive tensors be positive semi-definite at all frequencies. Additionally, we show that the conditions subsequently require that the non-hermitian parts of the Fourier transforms of the constitutive tensors, when divided by the imaginary number $i$, be positive semi-definite for positive values of frequency and negative semi-definite for negative values of frequency. We show that when major symmetries are assumed these definiteness relations apply simply to the real and imaginary parts of the relevant tensors. For diagonal and one-dimensional problems, these positive semi-definiteness relationships reduce to simple inequality relations. Finally we extend the results to highly general constitutive relations which include the Willis inhomogeneous relations as a special case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
