# Why is stress a tensor

• I
Before I go any further, I do understand the ways that mechanical engineering textbooks explain why stress is a tensor.

But all of those explanations seem infused with geometry (which I do NOT mean in a negative way at all); and are demonsrtrations.

I am searching for a more concise/abstract reason why stress is a tensor.

However, my education in math had been deficient. So am teaching myself new things.
And now that I understand a bit more about what a tensor is, after going through many of these wonderful videos:

(and many others)

... I am hoping someone can explain to me, concisely, without drawing tetrahera or any of those ways, why is stress a tensor.

In other words, beginning with a understanding of what a tensor is, why is stress one of these things?
And by that: do not DEMONSTRATE it is by doing a transformatoin..... EXPLAIN why is IS a tensor.

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Orodruin
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(Linear) Stress is a concept that is a linear transformation from vectors (the surface normal direction) to another vector (the force on the surface element). This is an identifying property of a rank two tensor.

• vanhees71 and Dale
(Linear) Stress is a concept that is a linear transformation from vectors (the surface normal direction) to another vector (the force on the surface element). This is an identifying property of a rank two tensor.
I "feel" that this is along the lines of what I am looking for, but I lack your confidence. You say it so easily. Could you elaborate?

(Linear) Stress is a concept that is a linear transformation from vectors (the surface normal direction) to another vector (the force on the surface element). This is an identifying property of a rank two tensor.
I mean: "demonstrate"

Orodruin
Staff Emeritus
Homework Helper
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I "feel" that this is along the lines of what I am looking for, but I lack your confidence. You say it so easily. Could you elaborate?
A rank two tensor may be seen as a linear map from vectors to vectors. The quotient law for tensors also tells us that any such mapping must be a tensor. Since stress relates surface elements (a vector) to the force across the surface (also a vector), stress must be a tensor.

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• vanhees71
A rank two tensor may be seen as a linear mapping between from vectors to vectors. The quotient law for tensors also tells us that any such mapping must be a tensor. Since stress relates surface elements (a vector) to the force across the surface (also a vector), stress must be a tensor.
wow this makes a lot of sense....

May I finally ask: Did you understand what I meant when I said that the traditional "proofs" were, more or less, "demonstrations using the geometry of tetrahedra", and were.... loss of a word here to capture my feelings.... like cheating? In other words, I feel that only NOW that I have a stronger idea of a tensor from listening to those videos, do I feel I understand what stress is a tensor and it is reallyu simple.

Orodruin
Staff Emeritus