欧拉型泊松括号形式下的弹性动力学：在手性奇异性固体中的应用
Elastodynamics from Eulerian Poisson-bracket formalism: application to chiral odd solids
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中文总结 AI 辅助
本文开发应用于弹性系统的欧拉型泊松括号形式，阐明其与拉格朗日形式的一致性，利用该形式推导出手性活性固体的奇异性弹性模量，证明其可捕捉受驱活性固体的非线性弹性行为。
中文摘要 AI 辅助
泊松括号（PB）形式被广泛用于推导粗粒化（CG）场的动力学，以捕捉大尺度物理现象，将PB在经典粒子力学中的作用扩展到宏观场，已被应用于临界现象的涨落、液晶流体动力学、液晶弹性体、组织，以及自旋粒子产生的奇异性粘度。PB形式可在拉格朗日框架（采用参考空间）或欧拉框架（采用真实空间）中构建，传统上拉格朗日形式用于弹性固体，欧拉形式用于流体。然而，由于真实空间中自然定义的现象（如粘弹性响应、移动界面、粒子的场诱导结构变化），人们对固体的欧拉描述的兴趣日益增长。本文开发了一种系统的形式化方法，将欧拉PB形式应用于通常以拉格朗日空间编写的弹性系统，并阐明其与拉格朗日对应形式的一致性。研究表明，欧拉形式会产生拉格朗日框架中不存在的额外非线性，这些非线性源于坐标变换下的CG体积变化，以及粒子流跨相邻CG体积，当非线性效应重要时必须保留。作为示例，本文研究了有限尺寸粒子组成的手性活性固体，其中主动扭矩驱动粒子内部旋转并产生几何非线性，这些非线性导致奇异性弹性模量，该模量在应力-应变响应中非互耦合两种不同的剪切模式。通过直接从欧拉PB形式中恢复该模量，本文证明了其捕捉受驱活性固体中涌现的非线性弹性行为的能力，这类固体的应力可在真实空间中自然测量。
英文摘要
The Poisson-bracket (PB) formalism is widely used to derive dynamics of coarse-grained (CG) fields to capture large-scale physics, extending the role of PBs in classical particle mechanics to macroscopic fields. It has been applied to fluctuations in critical phenomena, hydrodynamics of liquid crystals, liquid crystal elastomers, tissues, and the emergence of odd viscosity from spinning particles. The PB formalism can be formulated in either the Lagrangian framework, using reference space, or the Eulerian framework, using real space. Conventionally, the Lagrangian formulation is used for elastic solids, and the Eulerian one for fluids. However, growing interest in Eulerian descriptions of solids has emerged for phenomena naturally defined in real space, such as viscoelastic responses, moving interfaces, and field-induced structural changes in particles. Here we develop a systematic formulation for applying the Eulerian PB formalism to elastic systems with potentials typically written in Lagrangian space, and clarify its consistency with the Lagrangian counterpart. We show that the Eulerian formulation generates additional nonlinearities absent in the Lagrangian framework. Such nonlinearities originate from CG volume changes under coordinate transformation and from particle flow across neighboring CG volumes. They must be retained when nonlinear effects are important. To illustrate, we study chiral active solids of finite-sized particles, where active torques drive internal particle rotations and generate geometric nonlinearities. These nonlinearities give rise to the odd elastic modulus, which non-reciprocally couples two different shear modes in stress-strain response. By recovering this modulus directly from the Eulerian PB formalism, we demonstrate its ability to capture emergent nonlinear elastic behavior in driven active solids, whose stresses are naturally measured in real space.
