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The generalized Hooke's Law is a broadened version of Hooke's Law, which extends to all types of stress and in every direction. Consider an isotropic material shaped into a cube subjected to multiaxial loading. In this scenario, normal stresses are exerted along the three coordinate axes. As a result of these stresses, the cubic shape deforms into a rectangular parallelepiped. Despite this deformation, the new shape maintains equal sides, and there is a normal strain in the direction of the...
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The shearing strain represents a cubic element's angular change when subjected to shearing stress. This type of stress can transform a cube into an oblique parallelepiped without influencing normal strains. The cubic element experiences a significant transformation when exposed solely to shearing stress. Its shape alters from a perfect cube into a rhomboid, clearly demonstrating the effect of shearing strain. The degree of this strain is considered positive if it reduces the angle between...
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Strain quantifies the deformation of a material under force, typically measured as normal strain, which represents the change in length when compared with the original length. Electrical strain gauges are used for enhanced accuracy. These devices consist of a conductive wire mounted on a paper backing that adheres to the material's surface. These gauges operate on the piezoresistive effect, where the wire's electrical resistance changes in response to mechanical deformation. The strain...
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Normal strain under axial loading is an important concept in the field of mechanics of materials. Axial loading implies the application of a force along the axis of a material, like a column or bar. This force can either compress or stretch the material. In the context of axial loading, normal strain is the deformation experienced by the material in the direction of the loading force. It's calculated as the change in length divided by the original length of the material. This unitless ratio...
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A note on the Hill-Ogden generalised strains.

Salvatore Federico1

  • 1Department of Mechanical & Manufacturing Engineering, University of Calgary, Calgary, AB, Canada.

Mathematics and Mechanics of Solids : MMS
|September 26, 2024
PubMed
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.

Keywords:
Metric tensorcovariant formalismgeneralised coordinatesgeneralised strainpolar decompositionstrain tensor

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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.