# VON MISES STRESS LÀ GÌ

The idea of von Mises stress was first proposed by MaksymilianHuber in 1904. However, it only received real attention in 1913 when Richard von Mises proposed it again. While both only proposed a math equation, it was HeinrichHencky who developed the idea of “von Mises stress” as a reasonable physical interpretation.

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Let’s start by considering a simple uniaxial tensile kiểm tra on an isotropic & ductile specimen.

Fig. 01: Stress-strain curve from a uniaxial tensile test

As shown in Fig. 01, the material starts lớn dekhung elastically up to the elastic (or yield) limit, followed by some “yielding”, “necking” & finally breaking at the ultimate bít tất tay.

This point (or stress) at which the material behavior transforms from elastic khổng lồ plastic behavior is known as “yield stress”. We often say that the material yields if the bức xúc is greater than the yield strength. However, it is important lớn note that the bao tay is a tensor & not a single number (or scalar). Let’s say the material was being pulled along the x-x direction. It is technically accurate khổng lồ say that the material starts to lớn yield when the x-x component of căng thẳng is greater than the yield áp lực.

However, in real life applications, the bít tất tay tensors are more generic và not essentially uniaxial. It is likely that each component of the bao tay tensor is non-zero. In such a case, how can one say that the material has started to lớn 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 áp lực found experimentally?

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## von Mises găng tay Important Terminology in Plasticity and Inelastic Modeling

To proceed further, it is necessary to understvà some essential & frequently used terminology in the area of plasticity and inelastic modeling. The căng thẳng tensor has six independent components and can be decomposed inkhổng lồ volumetric (or hydrostatic) & deviatoric parts. Similarly, the strain tensor can also be decomposed inkhổng lồ the analog strains.

Mathematically, the volumetric strain và stress can be defined as one-third of the trace of the strain & ức chế tensor. The difference yields the deviatoric ức chế.

The volumetric strain purely corresponds to lớn 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 khổng lồ the shearing and distortion effects observed.

### Distortion Energy and von Mises Stress

Now that we understand the idea of volumetric and deviatoric strains, we can go ahead and define thedistortion energy.

We should always rethành viên 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 toàn thân. Strain energydensityis defined as:

In other words, this is the total strain energy stored in each differential volume of the body. If this strain energy is summedover 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, a part goes into changing the volume of the material (or volumetric strain) và is otherwise known as volumetric energy. The rest of the energy is used khổng lồ distort the shape of the material & is otherwise known as deviatoric energy. The von Mises áp lực is related lớn this total găng component going into the distortion energy. Or in mathematical terms:

where subscripts v & d represent the volumetric & deviatoric parts respectively. However, the product of any volumetric và deviatoric tensor is always zero. Thus, the strain energy mật độ trùng lặp từ khóa reduces to:

where the total energy can be written in terms of volumetric và deviatoric parts. Now, we can rewrite the deviatoric strain energy through a “scalar representative stress” as:

The representative bức xúc here is the von Mises bức xúc. Taking a leaf out of the 1-D áp lực state, the von Mises găng can be rewritten as:

### Principal Stress

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

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## von Mises ức chế Von Mises Yield Criterion

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

Fig. 02 illustrates the yield criterion in the principal bao tay space. Any bít tất tay state can be converted lớn the three principal stresses, which,if considered the three coordinates, the von Mises găng tay for different combinations leads to lớn a cylindrical surface as shown in Fig. 02.

Fig. 02: The von Mises and Tresca yield surfaces in the principal áp lực coordinates, including the Deviatoric Plane & 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, the Tresca yield criterion is defined based on the maximum possible normal and shear stresses that the material can withstvà.

## von Mises áp lực Conclusion

Most often, structures consist of materials lượt thích steel that show a plastic deformation & yielding before undergoing fracture. It is always preferred khổng lồ design structures so that they are within the elastic limit & bởi vì not yield. While most of the experiments are simple loading conditions (like uniaxial tensile), designers are often in a quandary as lớn how this can be related to lớn generic loading conditions observed in reality.

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The von Mises găng tay, though it sounds fancy, is just a metric of measurement khổng lồ determine whether the structure has started lớn yield at any point. The stresses calculated at any point can be mathematically written into a scalar quantity known as von Mises găng, which can then be compared with experimentally observed yield points.

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