Decomposing Uniaxial Stresses

[muskie25]

Homework Statement

I am having trouble decomposing a uniaxial compressive stress into hydrostatic and pure shear components.

Homework Equations

The Attempt at a Solution

I am starting with

$\begin{pmatrix} -\sigma & 0 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}$

I then do
$\begin{pmatrix} -\sigma & 0 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} -p & 0 & 0 \\ 0 & -p & 0\\ 0 & 0 & -p \end{pmatrix} + \begin{pmatrix} -2p & 0 & 0 \\ 0 & p & 0\\ 0 & 0 & p \end{pmatrix}$

where $p$ is the hydrostatic pressure. I don't think that this looks correct. Any thoughts?

 

[Chestermiller]

What is 1/3 of the trace of the stress tensor?

 

[muskie25]

Chestermiller,

Do you mean the hydrostatic pressure, $p$ ?

$p = -\sigma/3$

 

[muskie25]

What is 1/3 of the trace of the stress tensor?

Chestermiller,

Do you mean the hydrostatic pressure, $p$ ?

$p = −\sigma/3$

I know that the deviatoric tensor is indeed pure shear, because the sum of the diagonal = 0, but my assignment says that there are two states of pure shear. I am either misunderstanding the wording of the problem or I am misunderstanding how to decompose a stress tensor.

 

[Chestermiller]

Chestermiller,

Do you mean the hydrostatic pressure, $p$ ?

$p = −\sigma/3$

I know that the deviatoric tensor is indeed pure shear, because the sum of the diagonal = 0, but my assignment says that there are two states of pure shear. I am either misunderstanding the wording of the problem or I am misunderstanding how to decompose a stress tensor.

Just replace the p's in your post #1 by $\sigma/3$, and you'll have the right answer.