On a holistic variational formulation for material modeling including dissipative evolution

authored by: Philipp Junker, Tobias Bode, Klaus Hackl
Abstract: Based on Hamilton's principle of stationary action, we present a holistic variational formulation for material modeling including dissipative evolution. To this end, we recall the definition of the action as path integral of the momentum vector. Reformulation of the action and inserting the 1st and 2nd Law of Thermodynamics yield an extended Hamilton functional. We show that the stationarity conditions yield well-known expressions as well as new conditions in an extended nested time domain. Introducing an asymptotic two-scale approach transforms the expressions in the nested time domain back to the physical time. Hereby, we receive usual differential equations, e.g., heat conductivity equation, diffusion equation, and Biot equation, and the constitutive laws for, e.g., temperature, entropy, and chemical potential, all from one holistic stationarity principle. Moreover, the formulation in the nested time domain produces additional, virtual conditions that naturally lead to the concept of dissipation distances. Due to its variational origin, our approach yields in a consistent manner a coupled space–time formulation.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Ruhr-Universität Bochum
Type: Article
Journal: Journal of the Mechanics and Physics of Solids
Volume: 200
ISSN: 0022-5096
Publication date: 07.04.2025
Publication status: E-pub ahead of print
Peer reviewed: Yes
ASJC Scopus subject areas: Condensed Matter Physics, Mechanics of Materials, Mechanical Engineering
Electronic version(s): https://doi.org/10.1016/j.jmps.2025.106133 (Access: Open)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{4d472f65cee741c78c1e16887771895c,
title = "On a holistic variational formulation for material modeling including dissipative evolution",
abstract = "Based on Hamilton's principle of stationary action, we present a holistic variational formulation for material modeling including dissipative evolution. To this end, we recall the definition of the action as path integral of the momentum vector. Reformulation of the action and inserting the 1st and 2nd Law of Thermodynamics yield an extended Hamilton functional. We show that the stationarity conditions yield well-known expressions as well as new conditions in an extended nested time domain. Introducing an asymptotic two-scale approach transforms the expressions in the nested time domain back to the physical time. Hereby, we receive usual differential equations, e.g., heat conductivity equation, diffusion equation, and Biot equation, and the constitutive laws for, e.g., temperature, entropy, and chemical potential, all from one holistic stationarity principle. Moreover, the formulation in the nested time domain produces additional, virtual conditions that naturally lead to the concept of dissipation distances. Due to its variational origin, our approach yields in a consistent manner a coupled space–time formulation.",
keywords = "Coupled processes, dissipation, Hamilton's principle of stationary action, Variational material modeling",
author = "Philipp Junker and Tobias Bode and Klaus Hackl",
note = "Publisher Copyright: {\textcopyright} 2025 The Authors",
year = "2025",
month = apr,
day = "7",
doi = "10.1016/j.jmps.2025.106133",
language = "English",
volume = "200",
journal = "Journal of the Mechanics and Physics of Solids",
issn = "0022-5096",
publisher = "Elsevier Ltd.",
}