# VON MISES STRESS LÀ GÌ

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The idea of von Mises bức xúc 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.quý khách vẫn xem: Von mises bao tay là gì

Let’s start by considering a simple uniaxial tensile demo on an isotropic and ductile specimen.

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

As shown in Fig. 01, the material starts khổng lồ desize elastically up lớn the elastic (or yield) limit, followed by some “yielding”, “necking” and finally breaking at the ultimate găng tay.

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

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 kiến thiết components so that one is certain that we are within the yield limit? What is that scalar number that we can use to lớn compare with the yield bức xúc found experimentally?

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## von Mises bức xúc Important Terminology in Plastiđô thị and Inelastic Modeling

To proceed further, it is necessary to understvà some essential & frequently used terminology in the area of plastiđô thị và inelastic modeling. The găng tay tensor has six independent components và can be decomposed inlớn volumetric (or hydrostatic) and deviatoric parts. Similarly, the strain tensor can also be decomposed into the analog strains.

Mathematically, the volumetric strain và bít tất tay can be defined as one-third of the trace of the strain and ức chế tensor. The difference yields the deviatoric bức xúc.

The volumetric strain purely corresponds to a change in volume of the object without any changes in the overall shape. This is lượt thích scaling an object. In contrast, deviatoric strain corresponds to lớn the shearing and distortion effects observed.

### Distortion Energy and von Mises Stress

Now that we understvà the idea of volumetric and deviatoric strains, we can go ahead và 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 size khổng lồ 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. Strain energydensityis defined as:

Out of this total energy, a part goes inlớn changing the volume of the material (or volumetric strain) and is otherwise known as volumetric energy. The rest of the energy is used lớn distort the shape of the material & is otherwise known as deviatoric energy. The von Mises bức xúc is related to lớn this total ức chế component going into the distortion energy. Or in mathematical terms:

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

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

The representative sầu bít tất tay here is the von Mises áp lực. Taking a leaf out of the 1-D căng thẳng 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 kiến thiết decisions considering data from simple uniaxial experiments. However, in any situation, there will exist a plane that is subjected to lớn pure volumetric loading. Rotating a general stress tensor leads lớn a diagonal matrix. The diagonal elements are known as principal stresses.

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## von Mises găng tay Von Mises Yield Criterion

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

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

Fig. 02: The von Mises and Tresca yield surfaces in the principal căng thẳng coordinates, including the Deviatoric Plane và the Hydrostatic axis (source)

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

## von Mises bức xúc Conclusion

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

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

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