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.Quý Khách đang xem: Von mises ức chế là gì

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

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

As shown in Fig. 01, the material starts khổng lồ dekhung elastically up khổng lồ the elastic (or yield) limit, followed by some “yielding”, “necking” và finally breaking at the ultimate bức xúc.

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 căng thẳng is greater than the yield strength. However, it is important khổng lồ note that the bức xúc 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 lớn yield when the x-x component of áp lực is greater than the yield căng thẳng.

However, in real life applications, the áp lực tensors are more generic và 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 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 khổng lồ compare with the yield ức chế found experimentally?

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## von Mises stress **Important Terminology in Plasticity và Inelastic Modeling**

To proceed further, it is necessary khổng lồ understand some essential & frequently used terminology in the area of plasticity và inelastic modeling. The stress tensor has six independent components và can be decomposed inkhổng lồ volumetric (or hydrostatic) & deviatoric parts. Similarly, the strain tensor can also be decomposed into lớn the analog strains.Mathematically, the volumetric strain & stress can be defined as one-third of the trace of the strain & áp lực tensor. The difference yields the deviatoric bao tay.

The volumetric strain purely corresponds lớn 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 và distortion effects observed.

### Distortion Energy và von Mises Stress

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

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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 size to lớn another. So, when a mechanical force acts on a body toàn thân (or upon application of a prescribed displacement), some work is being placed on the body toàn thân. This energy is stored in as strain energy in the body toàn thân. Strain energydensityis defined as:

Out of this total energy, a part goes into changing the volume of the material (or volumetric strain) & 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 găng component going into lớn the distortion energy. Or in mathematical terms:

where subscripts *v* and *d* represent the volumetric and deviatoric parts respectively. However, the sản phẩm of any volumetric & 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 & deviatoric parts. Now, we can rewrite the deviatoric strain energy through a “scalar representative sầu stress” as:

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

### Principal Stress

The next important issue khổng lồ consider is the idea of principal stresses. In a generic situation, the bít tất tay is a full symmetric matrix. In this situation, it is difficult lớn make design 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 căng thẳng tensor leads lớn a diagonal matrix. The diagonal elements are known as principal stresses.

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## von Mises bức xúc **Von Mises Yield Criterion**

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

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

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

In other words, this means that if the găng 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 and shear stresses that the material can withstvà.

## von Mises găng **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 và vị not yield. While most of the experiments are simple loading conditions (like uniaxial tensile), designers are often in a quandary as to lớn how this can be related to lớn generic loading conditions observed in reality.

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The von Mises bít tất tay, though it sounds fancy, is just a metric of measurement lớn determine whether 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, which can then be compared with experimentally observed yield points.

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