线性弹性连续介质中非线性重塑的后验可控性
A posteriori controllability of non-linear remodelling in linear elastic continua
- University “Aldo Moro” of Bari（巴里阿尔多·莫罗大学）
- Politecnico di Torino（都灵理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文基于几何控制理论，研究线性弹性连续介质中非线性重塑的后验可控性，通过有限维控制模型分析发现三维下可实现完全控制，而平面情形则丧失可控性。
AI中文摘要：
我们在连续介质系统的几何控制理论框架内研究了重塑的后验可控性。重塑通过等容的Bilby--Kroner--Lee分解来描述，其中内部结构转变由重塑张量表示，其演化由应力驱动的本构定律控制。假设无限小变形和重塑拉伸，同时保留主重塑方向的有限旋转，我们推导出以主重塑拉伸和方向变量表示的非线性状态空间表示。该公式允许将重塑过程解释为由材料位置参数化的有限维非线性控制系统。对平衡构型进行了表征，并利用几何控制理论的工具研究了欠驱动可控性问题。三维设置表明，重塑拉伸与方向之间的非线性耦合使得通过减少数量的独立控制能够完全操纵内部状态，而平面简化则揭示了尽管存在非线性，可控性仍然丧失。这些结果为重塑可控性提供了首次系统性研究，并进一步支持了连续介质控制理论作为分析和设计连续介质中受控内部结构转变框架的发展。
英文摘要：
We investigate the a posteriori controllability of remodelling within the framework of Geometric Control Theory applied to continuum systems. Remodelling is described through an isochoric Bilby--Kroner--Lee decomposition, where the internal structural transformation is represented by a remodelling tensor and its evolution is governed by a stress-driven constitutive law. Assuming infinitesimal deformation and remodelling stretches, while retaining finite rotations of the principal remodelling directions, we derive a nonlinear state-space representation in terms of principal remodelling stretches and orientation variables. This formulation allows the remodelling process to be interpreted as a finite-dimensional nonlinear control system parametrised by the material position. Equilibrium configurations are characterised, and under-actuated controllability problems are studied by means of the tools of Geometric Control Theory. The three-dimensional setting shows that the nonlinear coupling between remodelling stretches and orientations enables complete steering of the internal state through a reduced number of independent controls, while the planar reduction reveals a loss of controllability despite the presence of nonlinearity. The results provide a first systematic investigation of remodelling controllability and further support the development of Continuum Control Theory as a framework for the analysis and design of controlled internal structural transformations in continuous media.
