An extended Hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution

authored by: Philipp Junker, Daniel Balzani
Abstract: An established strategy for material modeling is provided by energy-based principles such that evolution equations in terms of ordinary differential equations can be derived. However, there exist a variety of material models that also need to take into account non-local effects to capture microstructure evolution. In this case, the evolution of microstructure is described by a partial differential equation. In this contribution, we present how Hamilton’s principle provides a physically sound strategy for the derivation of transient field equations for all state variables. Therefore, we begin with a demonstration how Hamilton’s principle generalizes the principle of stationary action for rigid bodies. Furthermore, we show that the basic idea behind Hamilton’s principle is not restricted to isothermal mechanical processes. In contrast, we propose an extended Hamilton principle which is applicable to coupled problems and dissipative microstructure evolution. As example, we demonstrate how the field equations for all state variables for thermo-mechanically coupled problems, i.e., displacements, temperature, and internal variables, result from the stationarity of the extended Hamilton functional. The relation to other principles, as the principle of virtual work and Onsager’s principle, is given. Finally, exemplary material models demonstrate how to use the extended Hamilton principle for thermo-mechanically coupled elastic, gradient-enhanced, rate-dependent, and rate-independent materials.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Ruhr-Universität Bochum
Type: Article
Journal: Continuum Mechanics and Thermodynamics
Volume: 33
Pages: 1931-1956
No. of pages: 26
ISSN: 0935-1175
Publication date: 07.2021
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Materials Science(all), Mechanics of Materials, Physics and Astronomy(all)
Electronic version(s): https://doi.org/10.1007/s00161-021-01017-z (Access: Open)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{01b3961052f24ef295e03a86ce4aebef,
title = "An extended Hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution",
abstract = "An established strategy for material modeling is provided by energy-based principles such that evolution equations in terms of ordinary differential equations can be derived. However, there exist a variety of material models that also need to take into account non-local effects to capture microstructure evolution. In this case, the evolution of microstructure is described by a partial differential equation. In this contribution, we present how Hamilton{\textquoteright}s principle provides a physically sound strategy for the derivation of transient field equations for all state variables. Therefore, we begin with a demonstration how Hamilton{\textquoteright}s principle generalizes the principle of stationary action for rigid bodies. Furthermore, we show that the basic idea behind Hamilton{\textquoteright}s principle is not restricted to isothermal mechanical processes. In contrast, we propose an extended Hamilton principle which is applicable to coupled problems and dissipative microstructure evolution. As example, we demonstrate how the field equations for all state variables for thermo-mechanically coupled problems, i.e., displacements, temperature, and internal variables, result from the stationarity of the extended Hamilton functional. The relation to other principles, as the principle of virtual work and Onsager{\textquoteright}s principle, is given. Finally, exemplary material models demonstrate how to use the extended Hamilton principle for thermo-mechanically coupled elastic, gradient-enhanced, rate-dependent, and rate-independent materials.",
keywords = "Coupled processes, Local and non-local effects, Multi-physics, Variational modeling",
author = "Philipp Junker and Daniel Balzani",
year = "2021",
month = jul,
doi = "10.1007/s00161-021-01017-z",
language = "English",
volume = "33",
pages = "1931--1956",
journal = "Continuum Mechanics and Thermodynamics",
issn = "0935-1175",
publisher = "Springer New York",
number = "4",
}