The idea of von Mises stress was first proposed by Maksymilian Huber in However, it only received real attention in when Richard von Mises proposed it again. As shown in Fig. We often say that the material yields if the stress is greater than the yield strength.
However, it is important to note that the stress is a tensor and not a single number or scalar. It is technically accurate to say that the material starts to yield when the x-x component of stress is greater than the yield stress. However, in real life applications, the stress tensors are more generic and not essentially uniaxial. It is likely that each component of the stress tensor is non-zero. In such a case, how can one say that the material has started to yield? Or how can we design components so that one is certain that we are within the yield limit?
What is that scalar number that we can use to compare with the yield stress found experimentally? The purpose of a helmet is to protect the person who wears it from a head injury during impact. In this project, the impact of a human skull with and without a helmet was simulated with a nonlinear dynamic analysis.
Download this case study for free. The stress tensor has six independent components and can be decomposed into volumetric or hydrostatic and deviatoric parts. Similarly, the strain tensor can also be decomposed into the analog strains. Mathematically, the volumetric strain and stress can be defined as one-third of the trace of the strain and stress tensor.
The difference yields the deviatoric stress. The volumetric strain purely corresponds to a change in volume of the object without any changes in the overall shape.
This is like scaling an object. In contrast, deviatoric strain corresponds to the shearing and distortion effects observed. Now that we understand the idea of volumetric and deviatoric strains, we can go ahead and define the distortion energy. We should always remember that the mechanical behavior of materials is also governed by the two laws of thermodynamics.
As per the first law of thermodynamics, energy is neither created nor destroyed. It is only converted from one form to another. So, when a mechanical force acts on a body or upon application of a prescribed displacement , some work is being placed on the body. This energy is stored in as strain energy in the body.
Which is as simple as it gets, right? And there we have it… the von Mises stress! 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.
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. 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.
And please look forward to more simulation related content here at the Fidelis Blog. Huber, M. Von Mises, R. Gottingen , pp. The figures below adapted from [4] illustrate the curve obtained when studying the strain response of the uniaxial tension of a mild steel beam.
The description of each emphasized point is as follows:. The elastic limits discussed before are based on simple tension or uniaxial stress experiments. The maximum distortion energy theory, however, originated when it was observed that materials, especially ductile ones, behaved differently when a non-simple tension or non-uniaxial stress was applied, exhibiting resistance values that are much larger than the ones observed during simple tension experiments.
A theory involving the full stress tensor was therefore developed. The von Mises stress is a criterion for yielding, widely used for metals and other ductile materials. That is, if the von Mises stress is greater than the simple tension yield limit stress, then the material is expected to yield. The von Mises stress is not a true stress. It is a theoretical value that allows the comparison between the general tridimensional stress with the uniaxial stress yield limit.
This is due to the fact that the shearing stress acting on the octahedral planes i. Similar to the result obtained for the von Mises stress, this defines a criterion based on the octahedral stress.
Consequently, if the octahedral stress is greater than the simple stress yield limit, then yield is expected to occur. The von Mises stress can, for example, be applied in fields such as drilling of hydrocarbon reservoirs, where pipes are expected to be under high pressure and combined loading conditions.
The criterion is the same as before, that is, if the von Mises stress obtained from the above expression is equal or greater than the simple tension yield stress of the material, then yielding is expected to occur. The Tresca yield criterion is another example of a common criterion used for determining the maximum stress of material before yielding. It is commonly known as a more conservative estimate on failure within the science community. The most general expression for the maximum shearing stress is:.
This criterion can be simplified when the ordering of the magnitude of the stress components are known. The above expression then reduces to:. The Tresca yield criterion is piecewise linear, while the von Mises yield criterion is non-linear. However, the Tresca yield surface can involve singularities. The differences in predictions between the two conditions are considerably small.
There are many fields that benefit from the von Mises yield criterion.
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