# Homework Help: Definition of stress and usage of normal vector

1. Feb 18, 2017

### fisher garry

1. The problem statement, all variables and given/known data

The texts are taken from

http://ingforum.haninge.kth.se/armin/fluid/exer/deriv_navier_stokes.pdf

and

https://simple.wikipedia.org/wiki/Stress_(mechanics)

2. Relevant equations

3. The attempt at a solution

The formula for stress is $\sigma=\frac{F}{A}$ (I). From the document above it is also seen that

$$T_{x}=(\sigma_{xx},\sigma_{xy},\sigma_{xz})$$

If one looks at the drawing for $$T_{x}$$ and for simplicity sets $$\sigma_{xy}=0,\sigma_{xz}=0$$ so that $$T_{x}$$ is normal to the zy-plane. Then from

$$F_x=T_{x}n_{1}$$

and (I) one should obtain that

$$F_x=T_{x}n_{1}=\sigma A$$

But I don't get how this is correct. Can someone show a derivation

2. Feb 18, 2017

### Staff: Mentor

The stress vectors in those figures do not look (to me) like they are perpendicular to the three planes of interest. It looks to me like there are shear components in all three cases.

3. Feb 18, 2017

### fisher garry

I tried to assume that it was normal to make my question easier. I believed it would not hurt the discussion.

4. Feb 18, 2017

### Staff: Mentor

Then I don't understand what you are asking.

5. Feb 18, 2017

### fisher garry

I will try one more time.

If I do not set $$\sigma_{xy}=0,\sigma_{xz}=0$$ can you then derive why

$$F_x=\sigma A$$ and
$$F_x=n \cdot T_{x}=n \cdot(\sigma_{xx},\sigma_{xy},\sigma_{xz})$$

are the same values

6. Feb 18, 2017

### Staff: Mentor

The stress vector an a plane perpendicular to the x axis is given by:$$\vec{T}_x=\sigma_{xx}\vec{i}_x+\sigma_{xy}\vec{i}_y+\sigma_{xz}\vec{i_z}$$A unit normal to this plane is $\vec{n}=\vec{i}_x$. What do you get when you dot the stress vector on the plane with the unit normal?

7. Feb 19, 2017

### fisher garry

You would get $$\sigma_{xx}\vec{i}_x=\sigma_{xx}$$

but in the text they get

$$\vec{T}_x \cdot \vec{i}_x=\sigma_{xx}\vec{i}_x=F_{x}$$

8. Feb 19, 2017

### Staff: Mentor

That's not what I see them getting. Of course, they did leave out the area A. I see them getting the following:

$$\vec{T}_x\centerdot \vec{i}_x=(\sigma_{xx}\vec{i}_x+\sigma_{xy}\vec{i}_y+\sigma_{xz}\vec{i_z})\centerdot \vec{i}_x=\sigma_{xx}$$

9. Feb 19, 2017

### fisher garry

I assumed thy did get this in the text

$$\vec{T}_x \cdot \vec{i}_x=\sigma_{xx}\vec{i}_x=F_{x}$$

because of the $$\vec{F}$$ that is used in the text that I uploaded above

10. Feb 19, 2017

### Staff: Mentor

The reason you are so confused is because their notation sucks. The relationships I wrote were correct.