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Analytically finding the stress tensor field

  1. Jan 21, 2009 #1
    This hasn't been asked before, and I am more or less new to this subject. Therefore, I haven't done an attempt on the solution.

    Say we have a 2 dimensional square of sides "a". 2 forces "F" of equal magnitude and opposite direction act on the opposite ends of the square such that the square is in static equilibrium.

    How would I determine what the stress components are at any point x and y within the square? Let's assume this problem to be completely 2 dimensional.

    -Ataman
     
  2. jcsd
  3. Feb 10, 2009 #2
    I also need to know about a related problem although not for homework... I'm trying to do an order of magnitude calculation for a photoelasticity experiment.

    If I have a cube/cuboid with a point force applied normal to one of the surfaces in the center and an equal and opposite force applied on the opposite surface (also in the center) - is there a way to analytically calculate the resulting stress distribution within the volume of the material?

    I was reading about the stress equations of equilibrium in a photoelasticity text book - for example in the x direction the equilibrium condition is:

    [tex]\frac{\partial\sigma_{xx}}{\partial x} + \frac{\partial\tau_{yx}}{\partial y} + \frac{\partial\tau_{zx}}{\partial z} + F_x =0[/tex]

    where [tex]\sigma_{ij}[/tex] and [tex]\tau_{ij}[/tex] are the principal and shear stresses respectively and [tex]F_x[/tex] is the force in the x-direction. There are two similar equations for forces in the y and z directions which I don't think there's much point in posting.

    From my google/google scholar search I haven't been able to find an analytical solution to these (or find out if one exists) for the relatively simple case I outlined above. I'd be very grateful if anyone can help me out or point to a good book or paper.

    Thanks in advance!
     
  4. Feb 10, 2009 #3
    this was quite useful.... and there's a reference in one of the modules to a book called "Roark's Formulas for Stress and Strain" which gives solutions to most of the problems for which analytical solutions exist.

    From a skim read of the Roark's Formulas book - I think the answer is that analytical solutions can be found for one dimensional beams, or circular plates... most other geometries are solved numerically using not just the equilibrium equations I previously alluded to but also the constitutive equations, and kinematic equations, as well as the boundary conditions of course.

    hope that helps anyone who stumbles onto this thread.

    And if I got anything wrong then by all means, correct me on it!
     
  5. Feb 11, 2009 #4

    nvn

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    aperception: I think you gave an excellent answer. Putting all those things you mentioned together is sometimes called continuum mechanics. Or sometimes it might be called elasticity. There is a good book on continuum mechanics by Y. C. Fung, but I have not read much of it yet. It would be interesting if someone could show how to start solving the problem given by Ataman.

    Ataman: The applied force F, on each end of your problem, is a concentrated load at the side midpoint, directed outward and perpendicular to the side, right?
     
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