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Analysis of a Two Membrane Thermoelastic Problem

  • Nathaniel J. Fink-Humes
  • Carlos N. Rautenberg
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Abstract

We formulate and study a mathematical model of a thermoforming process involving two interdependent thermoelastic membranes coupled with equations governing heat transfer. These membranes interact with each other through means of elastic forces and thermal transfer which determines changes in elastic properties and further deform the membranes. The two thermoelastic membranes lead to a pair of implicit obstacle problems which are modeled both as a 2-membrane problem or a pair of quasi-variational inequalities. Mathematically, the primary model couples an elliptic two membrane problem with associated nonlinear elliptic equations describing temperature distribution. We show, under physically relevant conditions, the existence of solutions via a fixed point approach that involves reformulation of the problem to improve regularity of solutions.

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Cite

BibTeXRIS
BibTeX
@misc{finkhumes2026analysis,
  title = {{Analysis of a Two Membrane Thermoelastic Problem}},
  author = {Nathaniel J. Fink-Humes and Carlos N. Rautenberg},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05819},
  eprint = {2611.05819},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05819},
}
RIS
TY  - UNPB
TI  - Analysis of a Two Membrane Thermoelastic Problem
AU  - Nathaniel J. Fink-Humes
AU  - Carlos N. Rautenberg
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.05819
UR  - https://arxiv.org/abs/2611.05819
AB  - We formulate and study a mathematical model of a thermoforming process involving two interdependent thermoelastic membranes coupled with equations governing heat transfer. These membranes interact with each other through means of elastic forces and thermal transfer which determines changes in elastic properties and further deform the membranes. The two thermoelastic membranes lead to a pair of implicit obstacle problems which are modeled both as a 2-membrane problem or a pair of quasi-variational inequalities. Mathematically, the primary model couples an elliptic two membrane problem with associated nonlinear elliptic equations describing temperature distribution. We show, under physically relevant conditions, the existence of solutions via a fixed point approach that involves reformulation of the problem to improve regularity of solutions.
ER  -