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

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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.