Potential energy release rate for solids with bounded domains during evolutions involving changes to boundaries and boundary data
Abstract
We investigate the rate of change of equilibrium potential energy for nonlinear elastic solids undergoing evolutions that may involve changes in body geometry, material properties, applied tractions, and prescribed displacements. We represent such an evolution by a one-parameter family of variational problems and identify a class of these families for which the potential-energy release rate is well defined. Particular attention is given to the formulation of the potential energy at each evolution state: each constituent variational problem must yield the appropriate equilibrium state, while its equilibrium value must also correctly represent the physical change in potential energy between states. Within this framework, we derive a closed-form expression for the potential-energy release rate for solids with bounded domains, allowing for spatially nonuniform material evolution, sharp material interfaces, and evolving boundary geometry and prescribed boundary data. The expression is determined entirely by the equilibrium solution at the state under consideration and by the quantities prescribing the local evolution, so neighboring equilibrium problems need not be solved. Finally, we apply the result to the extension of a notch in a two-dimensional elastic body and show that the resulting potential-energy release rate reduces to the classical J-integral.
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BibTeX
@misc{kothari2026potential,
title = {{Potential energy release rate for solids with bounded domains during evolutions involving changes to boundaries and boundary data}},
author = {Mrityunjay Kothari and Sayaka Kochiyama and Haneesh Kesari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13874},
eprint = {2610.13874},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13874},
}
RIS
TY - UNPB
TI - Potential energy release rate for solids with bounded domains during evolutions involving changes to boundaries and boundary data
AU - Mrityunjay Kothari
AU - Sayaka Kochiyama
AU - Haneesh Kesari
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13874
UR - https://arxiv.org/abs/2610.13874
AB - We investigate the rate of change of equilibrium potential energy for nonlinear elastic solids undergoing evolutions that may involve changes in body geometry, material properties, applied tractions, and prescribed displacements. We represent such an evolution by a one-parameter family of variational problems and identify a class of these families for which the potential-energy release rate is well defined. Particular attention is given to the formulation of the potential energy at each evolution state: each constituent variational problem must yield the appropriate equilibrium state, while its equilibrium value must also correctly represent the physical change in potential energy between states. Within this framework, we derive a closed-form expression for the potential-energy release rate for solids with bounded domains, allowing for spatially nonuniform material evolution, sharp material interfaces, and evolving boundary geometry and prescribed boundary data. The expression is determined entirely by the equilibrium solution at the state under consideration and by the quantities prescribing the local evolution, so neighboring equilibrium problems need not be solved. Finally, we apply the result to the extension of a notch in a two-dimensional elastic body and show that the resulting potential-energy release rate reduces to the classical J-integral.
ER -