Help with understanding stress tensors

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    Stress Tensors
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Discussion Overview

The discussion revolves around understanding the components of the stress tensor in continuum mechanics, specifically the meanings of various terms such as normal and shear stresses. The scope includes conceptual clarification and technical explanation related to the application of stress tensors.

Discussion Character

  • Technical explanation, Conceptual clarification

Main Points Raised

  • One participant seeks clarification on the meaning of components of the stress tensor, particularly Pn, Pnn, and Pnt.
  • Another participant explains that for a stress tensor \sigma, \sigma_{11} represents normal stress in the 1-direction, while \sigma_{12} indicates shear stress in the 1-2 plane.
  • A further contribution clarifies that the component \sigma_{ij} represents the force per unit area in the i-direction on a surface with a normal in the j-direction, distinguishing between normal and shear stresses based on the indices.
  • A participant attempts to relate the terms Pn, Pnn, and Pnt to the definitions provided, suggesting that Pn is the force per unit area on the normal surface, Pnn is the normal force on the normal surface, and Pnt is the force in the normal direction on the tangent surface.

Areas of Agreement / Disagreement

Participants provide different explanations and interpretations of the stress tensor components, indicating that there is no consensus on the precise meanings of Pn, Pnn, and Pnt, and the discussion remains unresolved.

Contextual Notes

Some assumptions about the definitions of normal and shear stresses may not be explicitly stated, and the discussion includes varying interpretations of the terms used in the context of stress tensors.

MaxManus
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I'm taking a continuum mechanics course and we use the 3*3 stress tensor a lot. The problem is that I do not understand what each component mean.

What does the tension(Pn normal(Pnn and tangent(Pnt tension mean, or just Pxy?
 
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For a stress tensor [itex]\sigma[/itex], [itex]\sigma_{11}[/itex] is the normal stress in the 1-direction and [itex]\sigma_{12}[/itex] is the shear stress in the 1-2 plane. Does this answer your question?
 
The component [itex]\sigma_{ij}[/itex] of the stress tensor [itex]\sigma[/itex] is the force per unit area in the i-direction on a surface with normal in the j-direction. When the indices are the same, the force is normal to the surface so that it represents a pressure. When the direction of the first index is normal to that of the second, it represents a shear.
 
Thanks, does that mean that the Pn is the force per unit area in the i-direction on the normal surface and Pnn is the the normal force on the normal surface and Pnt is the the force in the normal direction on the tangent surface?

Pn = P dot n
pnn = P dot n dot n
Where P is the stress tensor and n is the normal.
 

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