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What is the von Mises Stress and Yield Criterion?

The idea of von Mises stress was first proposed by Huber in 1904 but received more attention only in 1913 when Richard von Mises proposed it again. While both of them only proposed a math equation, it was Hencky who gave the idea of “von Mises stress” a reasonable physical interpretation.

Let us start by considering a simple uniaxial tensile test on an isotropic and ductile specimen.

uniaxial tensile test von mises stress

Fig 01: Stress-strain curve from uniaxial tensile test (Source)

As shown in Fig. 01, the material starts to deform elastically up to the elastic (or yield) limit, followed by some “yielding”, “necking” and finally breaking at the ultimate stress.

This point (or stress) at which the material behavior transforms from elastic to plastic behavior is known as “yield stress”. Here we often say that the material yields if the stress is greater than the yield strength. However, it is always important to note that the stress is a tensor and not a single number (or scalar). Let’s say the material was being pulled along the x-x direction. 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 all the 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 such that one is certain that we are within the yield limit? What is that scalar number that we can compare with the yield stress found experimentally?

Important Terminology

To proceed further, it is necessary to understand some major terminology used in the area of plasticity and inelastic modeling. 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 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.

Distortional Energy and von Mises Stress

Further on, now that we understand the idea of volumetric and deviatoric strains, one can go ahead to define distortional energy.

We should always remember 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 but 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 done on the body. Now this energy is being stored in as strain energy in the body. Strain energy density is defined as:

streneg01

Or in other words, this is the total strain energy stored in each differential volume of the body. Now if this strain energy is summed over all the differential volumes (or otherwise called integration over the entire volume), we can obtain the total strain energy stored in the body.

Out of this total energy, some part of the energy goes into changing the volume of the material (or volumetric strain) or otherwise known as volumetric energy. The rest of the energy is used to distort the shape of the material or otherwise known as deviatoric energy. The von Mises stress is related to this total stress component going into the distortional energy. Or in mathematical terms:

StrainEnergy

where subscripts v and d represent the volumetric and deviatoric parts respectively. However, the product of any volumetric and deviatoric tensor is always zero. Thus the strain energy density reduces to:

streneg03

where the total energy can be written in terms of volumetric and deviatoric parts. Now, re-writing the deviatoric strain energy through a “scalar representative stress” as:

von Mises Derivation 01

The representative stress here is the von Mises stress. Taking a leaf out of the 1-D stress state, the von Mises stress can be re-written as:

von Mises Derivation 02

Principal Stress

The next important issue to consider is the idea of principal stresses. In a generic situation, the stress is a full symmetric matrix. In this situation, it is difficult to make design decisions considering data from simple uniaxial experiments. However, in any situation, there will exist a plane that is subjected to pure volumetric loading. Rotating a general stress tensor leads to a diagonal matrix. The diagonal elements are known as principal stresses.

von Mises Yield Criterion

The term derived above, with the square root of 2/3, for the representative or “von Mises” stress, looks familiar! The three principal stresses can be treated as coordinates and the resulting von Mises stress can be plotted.

Fig 02 pictorially expresses the yield criterion in the principal stress space. Any stress state can be converted to the three principal stresses, which if are considered the three coordinates, the von Mises stress for different combinations leads to a cylindrical surface as shown in Fig. 02.

The von Mises yield surfaces in principal stress coordinates circumscribes a cylinder with radius 2 3 σ y {\sqrt {{\tfrac {2}{3}}}}\sigma _{y} around the hydrostatic axis. Also shown is Tresca's hexagonal yield surface.

Fig 02: the Von mises and Tresca yield surfaces in the principle stress coordinates, including the Deviatoric Plane and the Hydrostatic axis (Source)

In other words, this means that if the stress state at any point is on the cylinder, then the material has started to yield at this point in the structure. Similarly, Tresca yield criterion is defined based on maximum possible normal and shear stresses that the material can withstand.

Conclusion

Most often, structures are made of materials like steel that show a plastic deformation and yielding before undergoing fracture. It is always preferred to design structures such that they are within the elastic limit and do not yield. While most of the experiments are simple loading conditions (like uniaxial tensile etc), designers are often in a quandry as to how this can be related to generic loading conditions observed in reality.

The von Mises stress, though sounds fancy, is just a metric of measurement to determine if the structure has started to yield at any point. The stresses calculated at any point can be mathematically written into a scalar quantity known as von Mises stress that can be compared with experimentally observed yield points.

Have fun designing structures! Here’s a static stress analysis of Toyota Hilux bumper that you can copy and use as template for your own FEA simulation. You just need to create a free SimScale Community account here.

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