Is stress tensor a 3x3 or 3x3x3 tensor?

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Discussion Overview

The discussion revolves around the nature of the mechanical stress tensor, specifically whether it is a 3x3 tensor or a 3x3x3 tensor. Participants explore the representation of stress in diagrams and the implications of these representations on the understanding of the tensor's structure.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether the stress tensor is a 3x3 tensor with scalar entries or a 3x3x3 tensor of scalars, suggesting that nonzero entries might be ignored in the latter case.
  • Another participant asserts that stress and strain are indeed 3x3 tensors, clarifying that diagrams with arrows represent the direction of forces acting on faces of a block.
  • A participant elaborates on the meaning of the notation σ11 in diagrams, indicating it represents the magnitude of force per unit area, raising a point about the need for multiplication by surface area for full understanding.
  • One participant confirms the previous statement regarding the interpretation of σ11 and its relation to force components.

Areas of Agreement / Disagreement

Participants express differing views on the dimensionality of the stress tensor, with some asserting it is a 3x3 tensor while others consider the possibility of a 3x3x3 tensor structure. The discussion remains unresolved regarding the nature of the stress tensor.

Contextual Notes

Participants reference diagrams and notation that may depend on specific definitions and interpretations, which are not fully explored in the discussion.

Hassan2
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Hi all,

I have a fundamental question about ( mechanical) stress tensor. Stress tensor a 3x3 tensor whose 9 entries looks "scalars" but in figures, the stress is illustrated by nine "vectors". Does it mean the stress tensors is in fact a 3x3x3 tensor of scalars whose nonzero entries are ignored?
 
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Stress and strain are definitely 3x3 tensors.

If you mean diagrams of a rectangular block with arrows attached to the faces, the arrows are showing the direction of the forces acting on the face, i.e. the components of ##\sigma \cdot \bf{n}## where ##\bf{n}## is the normal vector to the face.

If you meant some other diagram, can you post a link to it?
 
Yes those diagrams that you mentioned. So when they write σ11 beside one of the arrows, the arrow is the force component in that defined direction andσ11 tells us about "magnitude" of the force. However it seems to be the force per unit area, otherwise the stress must be multiplied by the surface.

Thanks.
 
Hassan2 said:
So when they write σ11 beside one of the arrows, the arrow is the force component in that defined direction andσ11 tells us about "magnitude" of the force. However it seems to be the force per unit area, otherwise the stress must be multiplied by the surface.
Correct.
 

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