# Von mises stress là gì

The idea of von Mises bao tay 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 sẽ xem: Von mises bức xúc là gì

Let’s start by considering a simple uniaxial tensile test 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 to deform elastically up to the elastic (or yield) limit, followed by some “yielding”, “necking” & finally breaking at the ultimate găng.

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 găng is greater than the yield strength. However, it is important to note that the bức xúc 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 khổng lồ say that the material starts to lớn yield when the x-x component of bao tay is greater than the yield áp lực.

However, in real life applications, the bức xúc tensors are more generic & not essentially uniaxial. It is likely that each component of the bức xúc tensor is non-zero. In such a case, how can one say that the material has started 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 lớn compare with the yield căng thẳng found experimentally?

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## von Mises bít tất tay **Important Terminology in Plasticity và Inelastic Modeling**

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

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

### Distortion Energy & von Mises Stress

Now that we understvà the idea of volumetric & deviatoric strains, we can go ahead và define thedistortion 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 size 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 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 inkhổng lồ changing the volume of the material (or volumetric strain) và 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 găng tay is related to lớn this total căng thẳng component going inkhổng lồ the distortion energy. Or in mathematical terms:

where subscripts *v* and *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 and deviatoric parts. Now, we can rewrite the deviatoric strain energy through a “scalar representative sầu stress” as:

The representative găng tay here is the von Mises stress. 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 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 khổng lồ 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 áp lực tensor leads to lớn a diagonal matrix. The diagonal elements are known as principal stresses.

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## von Mises bít tất tay **Von Mises Yield Criterion**

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

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

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

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

## von Mises ức chế **Conclusion**

Most often, structures consist of materials like steel that show a plastic deformation & yielding before undergoing fracture. It is always preferred to lớn kiến thiết structures so that they are within the elastic limit và bởi not yield. While most of the experiments are simple loading conditions (like uniaxial tensile), designers are often in a quandary as khổng lồ 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 to lớn determine whether the structure has started lớn yield at any point. The stresses calculated at any point can be mathematically written inkhổng lồ a scalar quantity known as von Mises stress, which can then be compared with experimentally observed yield points.

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