Analysis of Dissipative Dynamical Systems by Geometrical and Variational Methods with Application to Shock-wave Loaded Viscoplastic Structures

耗散动力系统的力学处理非光滑能量函数，其分析导致非线性演化问题，需要更复杂的模型与非光滑变分法。为了建立一致的模型，需要先进的数学工具，如测度理论，几何测度理论和抛物型偏微分方程理论。 然而，从实际应用的角度来看，需要突破性的方法来有效地、低耗时地模拟这种情况，因此，该提案旨在建立在一致的几何框架内模拟耗散动力系统的理论和实验方法，但数值方法在经典动力学的背景下并不是很远。将设想用经典动力学的几何方法来模拟耗散系统。特别地，将辛Brezis-Ekeland-Nayroles变分原理和模型降阶技术用于求解整体演化问题，并通过冲击波加载铜板的实验和数值模拟验证了所提方法的有效性。通过对哈密顿量的增强，建立了梯度损伤模型。为了考虑空穴的成核和生长，引入非局部损伤项来增强哈密顿量。这种增强引起的子结构相关的内在长度尺度和非局部和局部损伤变量之间的关系的梯度参数的引入。一个实验方法来调查原位生热和应变局部化之间的关系，在冲击波载荷下的粘塑性变形的设想，并将使用泰勒-昆尼系数的方法进行比较。

The mechanics of dissipative dynamical systems deals with non-smooth energy functions, whose analysis leads to nonlinear evolutionary problems that require more sophisticated models with non-smooth calculus of variations. To establish consistent models advanced mathematical tools, such as measure theory, geometric measure theory and also theory of parabolic partial differential equations, are required. For the practical purposes, breakthrough methods however are needed to model such situations in an efficient and lower time-consuming way.The proposal therefore aims to build up theoretical and experimental methods to model dissipative dynamical systems in a consistent geometrical framework, but the numerical approaches are not very far in the background of classical dynamics. The modelling of dissipative systems by using the geometrical methods of classical dynamics will be envisaged. In particular, the symplectic Brezis-Ekeland-Nayroles variational principle and model reduction techniques will be used to solve global evolution problems.The efficiency of the proposed method is then verified with the experimental and numerical simulation of shock-wave loaded copper plates. The gradient damage model will be developed by the enhancement of the Hamiltonian. To take void nucleation and growth into account, the Hamiltonian will be enhanced by introducing a non-local damage term. This enhancement gives rise to an introduction of gradient parameters in terms of a substructure-related intrinsic length-scale and a relationship between non-local and local damage variables. An experimental method to investigate a relationship between in situ heat generation and strain localisation during the viscoplastic deformation under shock-wave loading is envisaged and will be compared with approaches using the Taylor-Quinney coefficient.