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A note on the Hill-Ogden generalised strains.
1Department of Mechanical & Manufacturing Engineering, University of Calgary, Calgary, AB, Canada.
Mathematics and Mechanics of Solids : MMS
|September 26, 2024
Summary
This study reviews Hill-Ogden generalised strain tensors and their representation in curvilinear coordinates. It explores covariant and contravariant components for specific geometric contexts.
Area of Science:
- Continuum Mechanics
- Differential Geometry
Background:
- Generalised strain tensors are crucial for describing material deformation.
- Representing these tensors in curvilinear coordinates presents unique challenges.
Purpose of the Study:
- To provide an overview of the Hill-Ogden generalised strain tensors.
- To discuss their representation in generalised (curvilinear) coordinates using a covariant formalism.
- To explore the adaptability of this formalism to Riemannian manifolds.
Main Methods:
- Review of the Hill-Ogden generalised strain tensor formulation.
- Analysis of tensor representation in generalised (curvilinear) coordinates.
- Application of a fully covariant formalism.
Main Results:
- The Hill-Ogden generalised strain tensors can be naturally defined with covariant or contravariant components.
- Each component type (covariant/contravariant) is best suited to a specific geometrical context.
- The covariant formalism is adaptable to more general theories on Riemannian manifolds.
Conclusions:
- Understanding the dual nature of covariant and contravariant components is key for selecting the appropriate representation.
- The covariant formalism offers a flexible framework for advanced continuum mechanics on curved spaces.
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