# Analytically finding the stress tensor field

• Ataman
In summary: So the stress equation would look something like this:\frac{\partial\sigma_{xx}}{\partial x} + \frac{\partial\tau_{yx}}{\partial y} + \frac{\partial\tau_{zx}}{\partial z} = F_xwhere $$\sigma_{ij}$$ are the stress components at point $$x$$ and $$y$$, and $$\tau_{ij}$$ are the stresses at point $$i$$ in direction $$j$$.
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

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.

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!

Last edited by a moderator:
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?

## What is the stress tensor field?

The stress tensor field is a mathematical representation of the stress distribution in a physical system. It describes the internal forces and stresses that act on each point within a material or structure.

## How is the stress tensor field calculated?

The stress tensor field is calculated using analytical methods, which involve solving equations that govern the behavior of the system. These equations take into account factors such as material properties, geometry, and external forces.

## What information does the stress tensor field provide?

The stress tensor field provides information about the magnitude and direction of stress at every point within a system. This information can be used to analyze the stability, strength, and behavior of materials and structures under different conditions.

## What are some applications of the stress tensor field?

The stress tensor field has many applications in engineering, physics, and materials science. It is used to design and analyze structures such as bridges and buildings, understand the behavior of materials under stress, and predict the failure of mechanical systems.

## What are the limitations of the stress tensor field?

The stress tensor field is a simplified model that does not take into account factors such as material heterogeneity, nonlinear behavior, and time-dependent effects. It also assumes that the material is homogeneous and isotropic, which may not always be the case in real-world situations.

• Mechanical Engineering
Replies
18
Views
2K
• Special and General Relativity
Replies
10
Views
1K
• Electromagnetism
Replies
6
Views
1K
Replies
3
Views
3K
• Differential Geometry
Replies
3
Views
1K
• Mechanical Engineering
Replies
7
Views
2K
• Special and General Relativity
Replies
57
Views
2K
• Electromagnetism
Replies
1
Views
1K
• Atomic and Condensed Matter
Replies
3
Views
2K
• Special and General Relativity
Replies
3
Views
2K