What is a stress tensor?

[trainguy]

For someone who makes design decisions based on other engineers' FEA, my curiousity still has me riveted to this forum.

Can anyone define "stress tensor " for me?

GA

 

[EdR]

Trainguy:

The stress tensor is generally considered to be all 6 stress components of stress as a unit (may be arranged as a 3x3 or a 6x1 matrix)...A tensor is just the next higher order operator in the progression from scalar, vector, tensor.....Just as vectors do not follow the rules of scalar math, tensors do not follow the rules of vector math...For example think of the differences when rotating vectors and rotating stresses....

Hope this answers your question

Ed.R.

 

[pjhype]

Stress tensors come from the mathematical theory of elasticity where the stress state is defined in indicial notation. That is, the notation for the stress field can be described in the form of a scalar (zero rank), vector (first rank) or a matrix (second rank). Tensors have special properties in that they must obey certain "rules" for transformation.

Based on my experience, it is necessary to speak in terms of "stress tensors" if you need to concisely describe the state of stress in strict mathematical terms. Usually, it is more common to describe the state of stress with an inherent description of the stress transformation (e.g. "principal", "deviatoric", etc.)

pj

 

[brad]

A "tensor" is a general term for a matrix.
A "first-order" tensor is a one-dimensional tensor, i.e. a "vector"
A "second-order tensor" is 2-dimensional (i x j) i.e. a "classic" matrix.

Recall that to transform one vector into another, we need a matrix (so a 1st order tensor times a 2nd order tensor yields a first order tensor).

There are third-order tensors which transform 1st order into 2nd order (not so common, but they have uses).

There are also fourth-order tensors which transform 2nd order tensors into other second-order tensors. An example is the elasticity tensor "C":
(STRESS) ij = C ijkl (STRAIN) kl

That's what tensors are. The "stress tensor" then is (in 3-space) the 3 x 3 symmetric matrix which describes the state of stress. From this, all other stress quantities can be derived (Von Mises, Principal stresses and directions, etc).

Brad

 

[brad]

I guess we all answered at the same time . . .
Brad