Definition of stress and usage of normal vector

[fisher garry]

Homework Statement

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)

Homework Equations

The Attempt at a Solution

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

$$T_{x}=(\sigma_{xx},\sigma_{xy},\sigma_{xz})$$[/B]

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

 

[Chestermiller]

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.

 

[fisher garry]

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.

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

 

[Chestermiller]

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

Then I don't understand what you are asking.

 

[fisher garry]

Homework Statement

View attachment 113386

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)

Homework Equations

The Attempt at a Solution

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

$$T_{x}=(\sigma_{xx},\sigma_{xy},\sigma_{xz})$$[/B]

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

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

 

[Chestermiller]

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?

 

[fisher garry]

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?

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

but in the text they get

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

 

[Chestermiller]

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

but in the text they get

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

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

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

 

[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

 

[Chestermiller]

View attachment 113424

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

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