# Decomposing Uniaxial Stresses

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1. Mar 1, 2016

### muskie25

1. The problem statement, all variables and given/known data
I am having trouble decomposing a uniaxial compressive stress into hydrostatic and pure shear components.

2. Relevant equations

3. 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?

2. Mar 2, 2016

### Staff: Mentor

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

3. Mar 2, 2016

### muskie25

Chestermiller,

Do you mean the hydrostatic pressure, $p$ ?

$p = -\sigma/3$

4. Mar 2, 2016

### muskie25

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.

5. Mar 2, 2016

### Staff: Mentor

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