# Why are 3 planes needed to define stress at a point?

• Sirius
In summary: Stress is not exactly like a flux. The analogy might be helpful in understanding the concept, but it is not an accurate representation of the stress tensor. Stress is not exactly like a flux. The analogy might be helpful in understanding the concept, but it is not an accurate representation of the stress tensor.
Sirius
My question is simple. Why do we need 9 different quantities, ie 1 normal stress and 2 shear stresses on 3 different planes, to define stress at a point?

example: http://www.geosci.usyd.edu.au/users/prey/Teaching/Geol-3101/Strain/stress.html

I think it should be enough to define the 3 stresses (normal and 2 shear) on a single plane through the point. For any other plane just use the projection of the 3 above to get the new values. Shouldn't this work?

TIA :-)

Neither is correct. You need six independent quantities, the principal stresses (3 independent) and their directions (2 independent for the first direction, 1 for the second, 0 for the last). The stress tensor is a rank-2 symmetric tensor, which generally has 6 independent components.

Orodruin said:
Neither is correct. You need six independent quantities, the principal stresses (3 independent) and their directions (2 independent for the first direction, 1 for the second, 0 for the last). The stress tensor is a rank-2 symmetric tensor, which generally has 6 independent components.

Equilibrium renders the matrix symmetric reducing the need for 9 to 6 dependants right? But mathematically didn't we need 9 quantities to define stress and be able to express stresses in any plane/direction?

Can u please elaborate what you mean by "their directions"? I understand the principle stresses are the normal stresses. Thanks is advance!

The principal stresses are not the normal stresses (except in the principal directions).

One normal stress and two shear stresses on a specified plane is not sufficient to determine the stresses on a plane of arbitrary orientation. To do that, you need to specify the entire stress tensor. Even though, in terms of arbitrary orthogonal directions, there are nine components, the symmetry of the stress tensor reduces this to 6 independent components (3 normal stresses and 3 shear stresses). It is always possible to find 3 principal directions in which the stress tensor has only normal stress components (and no shear stress components).

Chestermiller said:
One normal stress and two shear stresses on a specified plane is not sufficient to determine the stresses on a plane of arbitrary orientation. To do that, you need to specify the entire stress tensor.

Yes, this is exactly what I want to know. Why is this so, why is one normal and two shear not enough?

Thank you for the reply sir.

Sirius said:
Yes, this is exactly what I want to know. Why is this so, why is one normal and two shear not enough?

Thank you for the reply sir.
To better idea of why, you need to review the derivation of the stress tensor.

Chestermiller said:
To better idea of why, you need to review the derivation of the stress tensor.

Can you link me to a good source? I have tried many videos, websites but they all go like ' we need 3 planes to define stress at a point' or 'consider a infinitesimal cube at a point and its faces' assuming it as an axiom.

Stephen Tashi
Sirius said:
Can you link me to a good source? I have tried many videos, websites but they all go like ' we need 3 planes to define stress at a point' or 'consider a infinitesimal cube at a point and its faces' assuming it as an axiom.
I think it would be very worthwhile for you to learn about the stress tensor (and other 2nd order tensors) on the basis of Dyadic tensor notation. This is a very simple approach, and, for me and many many others, cleared up all my confusion about the essence of the stress tensor (as well as how to work with it). An excellent treatment of this is presented in Appendix A.3 of the famous textbook Transport Phenomena by Bird, Stewart, and Lightfoot (Wiley, New York).

Chestermiller said:
I think it would be very worthwhile for you to learn about the stress tensor (and other 2nd order tensors) on the basis of Dyadic tensor notation. This is a very simple approach, and, for me and many many others, cleared up all my confusion about the essence of the stress tensor (as well as how to work with it). An excellent treatment of this is presented in Appendix A.3 of the famous textbook Transport Phenomena by Bird, Stewart, and Lightfoot (Wiley, New York).

Thank you Sir. I had a look at the reference and the closest thing I could find is this sentence "(In physical problems we often work with quantities that require the simultaneous specification of two directions. For example, the flux of x-momentum across a unit area of surface perpendicular to the у direction is a quantity of this type. Since this quantity is sometimes not the same as the flux of y-momentum perpendicular to the x direction, it is evident that specifying the two directions is not sufficient; we must also agree on the order in which the directions are given.)"

I suppose stress is like a flux quantity but I am still not sure why it requires knowing information of 3 planes (not 1,2 (too less) and not 4 (redundant)) to define it completely and in every plane possible

Sirius said:
Thank you Sir. I had a look at the reference and the closest thing I could find is this sentence "(In physical problems we often work with quantities that require the simultaneous specification of two directions. For example, the flux of x-momentum across a unit area of surface perpendicular to the у direction is a quantity of this type. Since this quantity is sometimes not the same as the flux of y-momentum perpendicular to the x direction, it is evident that specifying the two directions is not sufficient; we must also agree on the order in which the directions are given.)"

I suppose stress is like a flux quantity but I am still not sure why it requires knowing information of 3 planes (not 1,2 (too less) and not 4 (redundant)) to define it completely and in every plane possible
In Bird et al, the treat stress as a momentum flux, which I don't really like.

In my judgment, you need to get some experience working some problems before you can better feel for how all this works. Have you seen any description of Mohr's Circle yet? This also can help you with your insight.

Chestermiller said:
In Bird et al, the treat stress as a momentum flux, which I don't really like.

In my judgment, you need to get some experience working some problems before you can better feel for how all this works. Have you seen any description of Mohr's Circle yet? This also can help you with your insight.

I gave what you said a thought, starting with a simpler case of 2D. I realized that the same question I asked is valid for 2D too! Why 2 planes/square at a point is considered instead of one line.

I am, in some sense, confusing shear stress and normal stress. I will give it some more thinking a hopefully will reach a valid explanation for the same.

Thanks a lot sir.

Chestermiller

## 1. Why are 3 planes needed to define stress at a point?

The concept of stress at a point is a result of the Cauchy's stress principle, which states that the stress at a point in a body is independent of the orientation of the plane passing through that point. This means that in order to fully define the stress state at a point, we need to consider three mutually perpendicular planes passing through that point.

## 2. Can't we just use two planes to define stress at a point?

No, two planes are not sufficient to define the stress state at a point because they do not provide enough information about the stress components in all three dimensions. In order to fully understand the stress at a point, we need to know the stress components in all three orthogonal directions.

## 3. Why do the three planes have to be perpendicular?

The three planes used to define stress at a point must be perpendicular in order to satisfy the Cauchy's stress principle. This principle states that the stress at a point is independent of the orientation of the plane passing through that point. If the planes were not perpendicular, the stress state at a point would vary depending on the orientation of the plane, which goes against this principle.

## 4. How are the three planes selected for defining stress at a point?

The three planes used to define stress at a point are typically chosen as the x-y, y-z, and x-z planes, which form a right-handed coordinate system. This is the most common convention used in engineering and allows for a clear and consistent understanding of the stress state at a point.

## 5. Is the use of three planes to define stress at a point universal?

Yes, the use of three planes to define stress at a point is universal and is used in all branches of science and engineering. This is because it follows the fundamental principle of Cauchy's stress principle and provides a complete understanding of the stress state at a point in three dimensions.

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