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Extending the Transport Theorem to Rough Domains of Integration
Brian Seguin1, Denis F Hinz2, Eliot Fried3
1Postdoctoral Fellow Division of Mathematics, University of Dundee , Dundee DD1 4HN, Scotland , UK
Applied Mechanics Reviews
|December 18, 2014
Summary
Researchers present a new transport theorem for evolving, irregular domains in continuum physics, making complex theories accessible and demonstrating practical applications in phase transitions and fracture mechanics.
Area of Science:
- Continuum Physics
- Mathematical Physics
Background:
- Transport theorems are crucial in continuum physics for analyzing physical quantities over time.
- Existing theorems often struggle with domains that change shape or become irregular, limiting their applicability.
- Evolving irregular domains are common in phenomena like phase transitions and material fracture.
Approach:
- Presents Harrison's theory of differential chains, a mathematical framework for handling evolving domains.
- Explains the transport theorem developed by Seguin and Fried using this theory.
- Focuses on conceptual understanding for accessibility to continuum physics researchers.
Key Points:
- The new transport theorem accommodates domains that change shape and develop irregularities.
- Demonstrates the theorem's application through three concrete physical examples.
- Includes numerical approximations of the resulting terms from the transport theorem.
Conclusions:
- The presented transport theorem offers a more robust tool for continuum physics problems with complex geometries.
- Discusses potential to relax fundamental assumptions in continuum mechanics.
- Highlights challenges and future research directions for applying this generalized transport theorem.
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