Analytically finding the stress tensor field

[Ataman]

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

 

[aperception]

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 textbook - for example in the x direction the equilibrium condition is:

$$\frac{\partial\sigma_{xx}}{\partial x} + \frac{\partial\tau_{yx}}{\partial y} + \frac{\partial\tau_{zx}}{\partial z} + F_x =0$$

where $$\sigma_{ij}$$ and $$\tau_{ij}$$ are the principal and shear stresses respectively and $$F_x$$ 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!

 

[aperception]

http://ocw.mit.edu/OcwWeb/Materials-Science-and-Engineering/index.htm" 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!

 

[nvn]

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?