We hear it all the time, and most FEA codes spit it out as if it’s the holy grail of field outputs - but what actually is von Mises stress? Hopefully this post will help you understand that a little bit better, and to realize how constitutive FEA material models are taking some liberties with the actual physics at hand.
Where Did It Originate?
It is believed that the first mention of the idea that would later be termed ‘von
Mises stress’ was contained within a letter from James Clerk Maxwell to William Thomson in 1865, although this only described the general conditions. Tytus Maksymillian Huber (1) then anticipated the criterion in some more detail in 1904, in a paper that suggested separation of hydrostatic and distortion strain energy, rather than relying on total strain energy as his predecessors had.
It wasn’t until 1913 that Richard Elder von Mises (2) rigorously formulated the scalar representation of stress based on the second invariant of the deviatoric stress tensor. It should be noted that Heinrich Hencky (3) also formulated the same criterion as von Mises independently in 1924, and it was he that also recognized that it is actually directly related to deviatoric strain energy, hence it is sometimes known as Maxwell-Huber-Hencky-von Mises theory.
Hydrostatic and Deviatoric Components of Stress
Before we get into the yield criterion and von Mises stress, first we must do some homework and remind ourselves the difference between hydrostatic and deviatoric stress.
Simply put, hydrostatic stress is the stress that causes change in volume of the material and deviatoric stress is that which causes change in shape. Any stress tensor can be decomposed as follows:
Where the first part of the equation is the hydrostatic stress (simply the average of the three normal stresses), and deviatoric stress, σ'ij, makes up the rest of the tensor.
The same can be done to the strain tensor as shown below:
Yield Criteria
So what is von Mises stress? Firstly, let’s try to understand the yield criterion and work backwards from there. The goal of the yield criterion was to develop a method whereby the ductile behavior of materials could be predicted for any complex, 3D loading condition, rather than just for typical nominal laboratory test loading. To do this, we must boil the stress state down into a single scalar value that is compared to a material’s yield strength and can be measured by a simple tensile test.
The yield criterion must, therefore, relate the full stress tensor to the deformation strain energy density in some way. We can do this by first noting that total strain energy, which has units of Energy/Volume, W, is:
And for linear elastic materials, this equals:
Since we know from our homework that σ = σHyd + σ’ and e = eHyd + e’, then we can rearrange this equation to be:
It should be noted that the double dot product of any hydrostatic tensor with a deviatoric tensor is always zero, hence the missing expansion terms.
Now if only we could find a representative scalar stress that is directly related to this deviatoric strain energy, W’… that would be a suitable ‘yield criterion’.
With some manipulation of Hooke’s law, which we won’t go into here, but which can be found in this full derivation, we know that the deviatoric stress and strain tensors are directly related to each other like so (where G is shear modulus, but we’ll cancel that out later anyway):
Combining this with the deviatoric part of the partitioned strain energy equation, we get:
This confirms our hope that deviatoric strain energy can be directly related to the deviatoric stress tensor. Now comes the interesting part… If we were to replace the double dot product of the deviatoric stress tensor with a representative stress value (let’s call it σRep for now) we can start to see how the von Mises stress takes its form… a scalar stress that provides the same deviatoric strain energy as the full 3D stress tensor:
Cancelling the 4G from both sides, we get to:
Now what if we use this formula to calculate σRep for the simplest laboratory loading case; uniaxial tension? Well, the full 3D matrix looks like this:
Which is as simple as it gets, right? Removing the hydrostatic stress, σ/3, from each normal stress leaves a deviatoric stress tensor that looks like this:
And finally, a σRep that looks like this:
von Mises Stress
Irritatingly, the ‘representative’ stress, σRep, for meeting the yield criteria that we’ve derived above doesn’t add up to the uniaxial stress, σ, that we have in pure tension, σ. It is actually √2/3 times that as we can see above. In order to overcome this potentially confusing issue, we simply scale the reference stress by √3/2 in order to get a representative scalar stress value that is equivalent to the uniaxial tensile stress from the test. We’re allowed to do that in this case because the resulting σVM is still proportional to the deviatoric strain energy:
And there we have it… the von Mises stress! Here it is in it’s most recognizable form:
And if we’ve manipulated our stress tensor such that we know the principal stresses, we can simplify the equation down to:
One thing that we must mention here is that the yield criterion derived above, even though it is related directly to the deviatoric strain energy, is not a law of nature per se. When materials deform, there are mechanisms occurring on micro scales that define true yielding behavior… what we have here is a happy or convenient coincidence that allows us to mathematically represent that chaotic behavior in an efficient and elegant manner.
The Yield Surface
If we take a second to think about what hydrostatic stress (stresses equal in all three principal directions) is and how it looks in 3D, we can visualize a line emanating from zero and extending equidistantly from all of the principal axes to form the σ1 = σ2 = σ3 axis as shown in the image below.
Now, remember that when we talk about deviatoric stress we’re really talking about the distance from this line in a plane perpendicular to it. With that in mind, we can draw the above figure a little differently, as if we were looking down the line of hydrostatic stress (σ1 = σ2 = σ3 axis) and, in this way, we only see deviatoric stresses on our plot; the magnitude of hydrostatic stress is irrelevant (remember, this is really only true in metals). This might seem like a pointless exercise until we acknowledge that it is deviatoric stresses alone that result in yielding, as discussed prior. If we were to put a point on each principal axis where yielding occurs in a tensile test, and then join them up with a circle, we now have a nice 2D visualization of our von Mises yield surface.
In 3D, this looks like the famous cylinder that is concentric with the hydrostatic stress condition. I don’t have the skills, nor the time to draw that out, so I’ll borrow the excellent schematic provided by Wikipedia.
Final Thoughts
The von Mises stress is the default stress output of most, if not all, commercial FEA codes because it conveniently describes, at a glance, the entire stress tensor (or the ‘state of stress’) at any point within the model. Yet, until you dig a little deeper, it can be somewhat of a ‘black box’ value.
We hope this article has shed some light onto what it actually is and where it came from, such that next time you look at an FEA output database, you can visualize the image above and how it relates to the stress states within your analysis.
If you need more help with FEA or just want to get in touch to discuss simulation, don’t hesitate to reach out! And please look forward to more simulation related content here at the Fidelis Blog.
References
1. Huber, M.T. (1904) Czasopismo Techniczne, Lemberg, Austria, Vol. 22, pp. 181.
2. Von Mises, R. (1913) "Mechanik der Festen Korper im Plastisch Deformablen Zustand," Nachr. Ges. Wiss. Gottingen, pp. 582.
3. Hencky, H.Z. (1924) "Zur Theorie Plasticher Deformationen und der Hierdurch im Material Hervorgerufenen Nachspannungen," Z. Angerw. Math. Mech., Vol. 4, pp. 323.