The purpose of this ongoing blog series is to make the Finite Element Analysis software within SolidWorks, more accessible to a wider audience of not-necessarily-engineering users.
Who/What is Von Mises stress, and why do we use it in FEA analysis?
I find distressing pleas for clarity out on the Interwebs discussion forums. Usually coupled with pages of blind-leading-the-blind responses, with maybe one or two calm, well-couched replies buried so deeply in the thread, one wonders how many actually probed deep enough to benefit. Then we have the danger of Wikipedia Engineering. I quote verbatim:
“The Von Mises yield criterion suggests that the yielding of materials begins when the second deviatoric stress invariant J_2 reaches a critical value.”
Oh. My. God bless you, Wikipedia. Sometimes you are written by humans. But sometimes you are written by mathematics doctoral candidates who care not for the utility of the sentence, as long as there are many beautiful words.
Materials Testing: The Uniaxial Tension Test
At the risk of repeating myself – I’m going to repeat myself. In several of the prior articles we referenced the same picture, below right, to illustrate how typical engineering materials are tested. The most common test is to machine a plate down to a thin, controlled-width sample of the material, capture it between a pair of machine jaws, and then pull it until it fails in tension.
The point at which the sample starts to take permanent stretching damage is called the Yield Point, (or the Yield Stress, or also the Yield Strength). If you stopped the test very shortly after reaching this stress level, you would find that the sample had suffered some level of permanent distortion. So engineers will typically try to design such that the peak stresses stay below that level. It is also sometimes called the Elastic Limit – below this stress level, we expect that un-loading the part will cause it to return elastically to its original shape. But above this level, we expect plasticity – permanent shape-change.
Note that passing the yield stress is not always fatal to the specimen, (or the overall design), because the sample is at least still in one piece. It has stretched, but probably not yet fractured. The second, higher stress level at which the sample finally ruptures is called the Ultimate Stress. For this article, let us first assume we are designing a product that will function for many uses over a long time, so we usually treat the lower of these two, the Yield Stress, as our design-limiting criterion.
Six Dimensions of Stress
The one-directional tug-test pictured above is simple. But many (most?) components in our highly engineered world are subjected to loading in other directions, and so could fail in other ways. Consider the image at right – a shaft designed to transmit mostly twisting, will fail in two different ways depending upon whether the twist is coupled with tension, or if the twisting is coupled with a sideways (shear) force. How do we take the numbers from our uniaxial tension-test, and apply them meaningfully to this, and to other, more complex loading patterns? Which combinations of loading patterns are the most dangerous?
In the most general case, a solid material could be loaded in as many as 6 ‘dimensions’ at once – three directions of Tension/Compression, and three directions of Shear, as depicted below. This is the primary difficulty – when you run an FEA program, and get a prediction, at each point, of all 6 dimensions of stress, how do you compare that meaningfully back to our database of material test-data, which, as we saw above, is collected only on the simplest loading-case possible?
The three directions of Tensile Stress are labelled here as sx, sy, and sz. By convention, tension is a positive number, (and so a negative number means compression). The three directions of Shear Stress are labelled using the symbol t. Shear does not act in a single simple direction, it acts within a plane; that’s why it takes two indicies to label each shear stress. txy is the same as tyx. This is also why there seem to be 6 shear stresses labeled, but there are really only three – the Red, Green, and Blue vectors are used to show which pairs of Shear are really just the symmetric members.
We need to be able to look at a complex, 6-dimensional state of stress, and somehow compare it to our reported Yield Strength in a simple, 1-dimensional state of stress, and ask: “Is this complex case worse – yes or no ?”
Consider the linkage element pictured below. This part is restrained on the large bore at the bottom, and undergoes a vertical (+Y) tug on the top face of 4600 lbs. (pink). There is a 450 lb. force acting in the –Z direction, (purple) on the front boss face, and a torque of 1200 in-lb. (red) on the two small horizontal arms, about the vertical axis. Finally, an 800 in-lb. torque (orange) acts on the central bore (the one with the keyway) about the –X direction.
The FEA must compute all 6 components of stress, at every element. How would it plot all 6 values at once, at every point? And, even if it could, how would you LOOK at them? Certainly, we can plot each component of stress one-at-a-time, and try to locate any weaknesses in our design that way.
Clearly we need some way of combining these 6 stress components into a single number that we can compare to our test of the Yield strength. That is exactly what the Von Mises stress calculation attempts to do.
You can think of the Von Mises stress as a sort of “credit score” for each finite-element in your part. Certainly, your own financial status has MANY dimensions – how much liquid cash in your pocket? How much in your bank accounts? How about your house – Own it? Above Water? Still paying on a car loan? And yet, somehow, with all of the complexity and subtly that could go into computing your net worth…. The credit card companies routinely boil you down to a single number. Less than 500? Misery. Greater than 720? Banks love you. Do you want to know the exact formula the credit card companies use to do this magic?
Me, neither. But for the purists out there, the formula for computing a single, scalar Von Mises stress, su from your 6 components is: (x=dir 11, y=dir22, z=dir33, and shear on xy plane is dir 12, etc).
OK, glad we got that out of the way. What does all that mean? Simply stated, Von Mises says that only the stresses which act to distort the shape of the part, will matter when checking for yield. If you were to mold a cube, like our stress-cube pictured 2 pages ago, but out of solid glass – and then you dropped it in the deepest trench in the ocean, 7 miles down – the pressure on each face of the glass would be enormous. But the Von Mises stress would be zero, because the pressure would act everywhere equally, in all directions, the cube would simply shrink, but would otherwise not flex. (Unless you had some molding defects, like air bubbles in the glass or variations in the density, the glass should survive any depth).
The Von Mises equation computes the net energy stored by element distortion, and then backs that value out as an equivalent stress. Unlike the 6 components of stress that feed into it, V.M stress has no direction, only a magnitude. We compare this number to our uni-axial Tension test, and there you have it – your yes-no criterion!
So let’s get back to that linkage arm for which we listed all the stress components – the plot of each element’s Von Mises stress is shown at right. Even though 3 other components of stress were all located in one (other) place, the most highly-distorted element is located inside the intersection of two holes, (which is also where the Y-axial stress was the greatest). Interestingly, the peak Y-component of stress was over 28,000 psi, but the Von Mises stress in the same area is less, about 27,600 psi.
This must mean that the other 5 components of stress in this immediate area must be acting such that they cancel out, to a small degree, some of the Y-axis distortion.
Other Yield Theories
It should be stated right here, that Richard Von Mises stress is not the only kid on the block. Other theories exist about how to evaluate the total state of stress. Especially, if you have a brittle material like glass, or concrete, then you should instead compare the material Yield Strength to the Mohr-Coulomb stress, or often just ‘Coulomb Stress’.
Coulomb stated that most things fail in shear. Once you transpose the coordinate system in which you view the part, to that orientation which produces the plane of worst-case shear, you just assume THAT shear is the value that is going to cause a crack to appear. It turns out that Coulomb’s theory is more conservative than Von Mises, and most engineers will use the Coulomb criterion for brittle materials, and Von Mises criterion for ductile materials like steel, aluminum, most plastics, etc.
Of course, material properties are not tabulated for the estimated engineering shear stress at which a test specimen fails, they are tabulated only for the engineering tensile stress. So if our material is brittle, we need to know the equivalent tensile-stress magnitude, corresponding to the worse-case shearing plane, as experienced by each finite-element. In Solidworks Simulation you do this by choosing the plot type:
“INT: Stress Intensity (P1-P3)”
And what, you might ask, are “P1” and “P3”? Good question! But not nearly as important a question as “what is Von Mises stress”. And the majority of engineering materials qualify as ‘ductile’ for our purposes, so that is why Solidworks creates a Nodal plot of Von Mises stress, as the default output, for every linear, static study. And this article has gotten too long already, so we’ll pick up the topic of other Directional stress components, for the next installment of KAP’s Corner.