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Definition of stress and usage of normal vector

  1. Feb 18, 2017 #1
    1. The problem statement, all variables and given/known data
    Uten navn.png

    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. jcsd
  3. Feb 18, 2017 #2
    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.
     
  4. Feb 18, 2017 #3
    I tried to assume that it was normal to make my question easier. I believed it would not hurt the discussion.
     
  5. Feb 18, 2017 #4
    Then I don't understand what you are asking.
     
  6. Feb 18, 2017 #5
    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
     
  7. Feb 18, 2017 #6
    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?
     
  8. Feb 19, 2017 #7
    You would get $$\sigma_{xx}\vec{i}_x=\sigma_{xx}$$

    but in the text they get

    $$\vec{T}_x \cdot \vec{i}_x=\sigma_{xx}\vec{i}_x=F_{x}$$
     
  9. Feb 19, 2017 #8
    That's not what I see them getting. Of course, they did leave out the area A. I see them getting the following:

    $$\vec{T}_x\centerdot \vec{i}_x=(\sigma_{xx}\vec{i}_x+\sigma_{xy}\vec{i}_y+\sigma_{xz}\vec{i_z})\centerdot \vec{i}_x=\sigma_{xx}$$
     
  10. Feb 19, 2017 #9
    Uten navn.png

    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
     
  11. Feb 19, 2017 #10
    The reason you are so confused is because their notation sucks. The relationships I wrote were correct.
     
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