Calculating the Effective Shear Stress: Understanding Stress Invariants

[hexa]

please help: solving a matrix problem

Hello,

can anyone help me with solving the following question? I guess I don't quiet understand what this is about, partly as my english is not particularly good, partly because some information is missing in my course note eventhough some text (I deleted) from this question refers to the course notes.

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The state of stress at some point in a 2-dimensional Cartesian space is defined by the stress
tensor σij, for example:

σij=
[
σ11 σ12
σ21 σ22
]
=
[
5 3
3 2
]

(this is meant to be a matrix σij=[]=[]

(a) Calculate the numerical value of the “effective shear stress” σ'E at this point. Remember
that σ'E is the square root of the “second stress invariant”:
σ'E = (½σ'ijσ'ij)½
where σ'ij is the deviatoric stress tensor. State also what is meant by stress “invariant”.

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Thanks a lot,

hexa

 

[quantumdude]

The state of stress at some point in a 2-dimensional Cartesian space is defined by the stress
tensor σij, for example:

σij=
[
σ11 σ12
σ21 σ22
]
=
[
5 3
3 2
]

(this is meant to be a matrix σij=[]=[]

Let's do this in LaTeX. To see the code just click on the image. A tutorial is available here:

https://www.physicsforums.com/showthread.php?t=8997

$$\sigma_{ij} = \left[\begin{array}{cc}\sigma_{11} & \sigma_{12}\\ \sigma_{21} & \sigma_{22}\end{array}\right]=\left[\begin{array}{cc}5 & 3\\ 3 & 2\end{array}\right]$$

(a) Calculate the numerical value of the “effective shear stress” σ'E at this point. Remember
that σ'E is the square root of the “second stress invariant”:
σ'E = (½σ'ijσ'ij)½

You need to do 2 things:

1.) Define $\sigma^{\prime}_{ij}$. I can't tell what it is in terms of $\sigma_{ij}$.
2.) Show how you started this problem, and where you got stuck.