Von mises stress что это

What is von Mises Stress?

Von Mises stress is a value used to determine if a given material will yield or fracture. It is mostly used for ductile materials, such as metals. The von Mises yield criterion states that if the von Mises stress of a material under load is equal or greater than the yield limit of the same material under simple tension then the material will yield.

History of von Mises Stress

It is commonly accepted that the history of elasticity theory began with the studies of Robert Hooke in the 17th century\(^1\) who explored the fundamental concepts of deformation of spring and the displacement of a beam. However, engineering wasn’t the only reason for the study of elasticity theory, as that research was also linked to the attempt of interpreting the nature and theory of ether\(^2\).

It was only in the 19th century that the quantitative and mathematical theory of the elasticity of bodies was born, together with the continuum mechanics, which allowed the use of integral and differential calculus when modeling elastic phenomena. The continuum mechanics supposes what is called a homogenization of the medium, such that microscopic fluctuations are averaged and a continuous field that models the medium can be obtained. Therefore, it assumes that for every instant of time and for every point in space occupied by the medium, there is a punctual particle.

Many theories and concepts have been derived from the basic concept of continuum mechanics. One of those is the maximum distortion energy theory, which is applied in many fields such as rubber bearings and applications with other ductile materials. It was initially proposed by Hubert in 1904 and further developed by von Mises in 1913\(^3\). According to it, yielding occurs when the distortion energy reaches a critical value. This critical value, which is specific for each material, can easily be obtained by performing a simple tension test.

Introduction

When a body, in an initial state of equilibrium or undeformed state, is subjected to a body force or a surface force, the body deforms correspondingly until it reaches a new state of mechanical equilibrium or deformed state. The inner body forces are the result of a force field such as gravity, while the surface forces are forces applied on the body through contact with other bodies.

The relations between external forces — which characterize what is called the stress — and the deformation of the body, which characterizes strain, are called Stress-Strain relations. These relations represent properties of the material that compose the body and are also known as constitutive equations.

The figures below (adapted from [4]) illustrate the curve obtained when studying the strain response of the uniaxial tension of a mild steel beam. The description of each emphasized point is as follows:

This diagram is commonly approximated for many materials as is shown in the picture below:

Von Mises Yield Criterion

The elastic limits discussed before are based on simple tension or uniaxial stress experiments. The maximum distortion energy theory, however, originated when it was observed that materials, especially ductile ones, behaved differently when a non-simple tension or non-uniaxial stress was applied, exhibiting resistance values that are much larger than the ones observed during simple tension experiments. A theory involving the full stress tensor was therefore developed.

The von Mises stress is a criterion for yielding, widely used for metals and other ductile materials. It states that yielding will occur in a body if the components of stress acting on it are greater than the criterion\(^4\):

The constant \(k\) is defined through experiment and \(\tau\) is the stress tensor. Common experiments for defining \(k\) are made from uniaxial stress, where the above expression reduces to:

If \(\tau_y\) reaches the simple tension elastic limit, \(S_y\), then the above expression becomes:

Which can be substituted into the first expression:

The von Mises stress, \(\tau_v\), is defined as:

Therefore, the von Mises yield criterion is also commonly rewritten as:

That is, if the von Mises stress is greater than the simple tension yield limit stress, then the material is expected to yield.

The von Mises stress is not a true stress. It is a theoretical value that allows the comparison between the general tridimensional stress with the uniaxial stress yield limit.

The von Mises yield criterion is also known as the octahedral yield criterion\(^5\). This is due to the fact that the shearing stress acting on the octahedral planes (i.e. eight planes forming an octahedron, whose normals form equal angles with the coordinate system) can be written as:

Which, for the case of uniaxial or simple tension, simplifies to:

Again, if \(\tau_y\) reaches the simple tension elastic limit, \(S_y\), then the above expression becomes:

And, by applying this result in the octahedral stress expression:

Similar to the result obtained for the von Mises stress, this defines a criterion based on the octahedral stress. Consequently, if the octahedral stress is greater than the simple stress yield limit, then yield is expected to occur.

The von Mises stress can, for example, be applied in fields such as drilling of hydrocarbon reservoirs, where pipes are expected to be under high pressure and combined loading conditions. In this case, the von Mises stress can be written as\(^5\):

Where \(z\), \(r\), and \(t\) are the axial, radial and tangential stresses. The criterion is the same as before, that is, if the von Mises stress obtained from the above expression is equal or greater than the simple tension yield stress of the material, then yielding is expected to occur.

Tresca Yield Criterion

The Tresca yield criterion is another example of a common criterion used for determining the maximum stress of material before yielding. Calculating yielding with Tresca’s method always results in a lower result compared to the von Mises method. It is commonly known as a more conservative estimate on failure within the science community. Also, it is known as the maximum shearing stress yield criterion\(^4\). The most general expression for the maximum shearing stress is:

This criterion can be simplified when the ordering of the magnitude of the stress components are known. The above expression then reduces to:

The Tresca yield criterion is piecewise linear, while the von Mises yield criterion is non-linear. However, the Tresca yield surface can involve singularities. The differences in predictions between the two conditions are considerably small.

Von Mises Stress on SimScale

There are many fields that benefit from the von Mises yield criterion. There are SimScale public projects that can help to get a more practical grasp of the von Mises stress theory. For example, the picture below shows a study of the von Mises stress on a mounting plate subjected to a certain load.

The picture below is taken from a step-by-step tutorial that shows a structural and plasticity analysis for the burst of a gas tank and is an interesting resource for beginners.

Become a SimScale member!

Try our structural mechanics tutorials and observe the von Mises stresses yourself. Join SimScale today and experience cloud-based simulation like no other.

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Напряжения в ANSYS которые можно получить в стандартном интерфейсе

Вот это ещё? :

я за пивом и хелп буду читать

Слушай OXOTHuK спасибо! без твоей бы ссылки не нашёл!

PS Зачем пишут «Mechanical Application»

Для корректного прогнозирования прочности и долговечности конструкции по условию образования макроразрушения необходимо введение некоторого конечного минимального объёма материала, повреждение которого однозначно описывается с помощью локальных критериев, сформулированных в терминах механики сплошной деформируемой среды. Иными словами, при рассмотрении НДС не в материальной точке, а в некотором объёме материала со своими реологическими свойствами прогноз образования макроразрушения на основании локальных критериев будет адекватным; при анализе НДС в меньшем объеме локальные критерии не описывают реального разрушения материала. Очевидно, что свойства и размер такого характерного объема, так называемого структурного элемента, могут зависеть от особенностей механизма деформирования и процессов разрушения материала.

Ведене структурного элемента как параметра, являющегося связующим звеном между микро- и макропроцессами разрушения, даёт возможность подойти к вопросу о масштабе зарождения макроразрушения или, что то же самое, о размере зародышевой макротрещины. Поскольку прогноз зарождения макротрещины ведется с помощью локальных критериев, использование которых правомочно при анализе деформирования и разрушения в объеме, не меньшем чем структурный элемент, то очевидно, что минимальную длину зародышевой макротрещины можно принять равной линейному размеру этого элемента.

Анализ зарождения и развития разрушения в элементе конструкции в значительной степени зависит от универсальности тех или иных локальных критериев разрушения. При формулировке критериев эмпирическим путём – только на основе непосредственных механических испытаний – возникает опасность неадекватной оценки разрушения конструкции при нагружении, отличном от нагружения при проведённых экспериментах. Повысить степень универсальности локальных критериев можно, опираясь на физические механизмы, протекающие на микроуровне. Одним из путей решения данного вопроса является создание физико-механических моделей разрушения материала, на основании которых могут быть даны формулировки локальных критериев разрушения в терминах механики сплошной среды на базе физических и структурных процессов деформирования и повреждения материала.

При больших упругопластических деформациях возможно значительное изменение формы конструкции, что ведет к необходимости учёта геометрической нелинейности. Учёт изменения геометрии тела в процессе деформирования можно реализовать : После каждой итерации пересчитываются координаты узлов всех конечных элементов в соответствии с полученными значениями приращений перемещений узлов. Таким образом, по завершении итерационного процесса условия равновесия и текучести будут выполнены применительно к телу, геометрия которого отвечает полученным при решении деформациям.

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Von Mises Stress

Introduction

The von Mises stress is often used in determining whether an isotropic and ductile metal will yield when subjected to a complex loading condition. This is accomplished by calculating the von Mises stress and comparing it to the material’s yield stress, which constitutes the von Mises Yield Criterion.

The objective is to develop a yield criterion for ductile metals that works for any complex 3-D loading condition, regardless of the mix of normal and shear stresses. The von Mises stress does this by boiling the complex stress state down into a single scalar number that is compared to a metal’s yield strength, also a single scalar numerical value determined from a uniaxial tension test (because that’s the easiest) on the material in a lab.

It should be noted that this is not an exact science like, say \(F = m\,a\). It is an empirical process, with inherent error and deviations. In fact, there is no hard & fast rule saying that metals must yield according to von Mises yield criteria. It is as much a coincidence as anything. Nevertheless, it does work very well and remains the method of choice a full century after it was first proposed.

History

The defining equation for the von Mises stress was first proposed by Huber [1] in 1904, but apparently received little attention until von Mises [2] proposed it again in 1913. However, Huber and von Mises’ definition was little more than a math equation without physical interpretation until 1924 when Hencky [3] recognized that it is actually related to deviatoric strain energy.

In 1931, Taylor and Quinney [4] published results of tests on copper, aluminum, and mild steel demonstrating that the von Mises stress is a more accurate predictor of the onset of metal yielding than the maximum shear stress criterion, which had been proposed by Tresca [5] in 1864 and was the best predictor of metal yielding to date. Today, the von Mises stress is sometimes referred to as the Huber-Mises stress in recognition of Huber’s contribution to its development. It is also called Mises effective stress and simply effective stress.

Technical Background

A complete understanding of the von Mises stress requires an understanding of hydrostatic and deviatoric components of stress and strain tensors, Hooke’s Law, and strain energy density. The hydrostatic and deviatoric stresses and strains have already been reviewed. And Hooke’s Law has already been touched on here and here, but will need to be discussed in additional detail on this page as well. Strain energy density will also be introduced here.

Hydrostatic and Deviatoric Components

Recall that any stress tensor can be decomposed into the sum of hydrostatic and deviatoric stresses as follows

\[ \sigma_ = <1 \over 3>\delta_ \sigma_ + \sigma’\!_ \]
where \( <1 \over 3>\delta_ \sigma_ \) is the hydrostatic term and \( \sigma’ \) is the deviatoric stress.

The same is true for strain.

\[ \epsilon_ = <1 \over 3>\delta_ \epsilon_ + \epsilon’\!_ \]
where \( <1 \over 3>\delta_ \epsilon_ \) is the hydrostatic term and \( \epsilon’ \) is the deviatoric strain.

These two will be multiplied together farther down the page.

Hooke’s Law

We’ve seen that Hooke’s Law can be written as

\[ \epsilon_ = <1 \over E>\left[ (1 + \nu) \sigma_ — \nu \, \delta_ \sigma_ \right] \]
which is shorthand for

\[ \epsilon_ = <1 \over E>\big[ \sigma_ — \nu \, ( \sigma_ + \sigma_ ) \big] \] \[ \epsilon_ = <1 \over E>\big[ \sigma_ — \nu \, ( \sigma_ + \sigma_ ) \big] \] \[ \epsilon_ = <1 \over E>\big[ \sigma_ — \nu \, ( \sigma_ + \sigma_ ) \big] \]
for the normal terms, and

\[ \epsilon_ = < 1 + \nu \over E >\sigma_ \qquad \epsilon_ = < 1 + \nu \over E >\sigma_ \qquad \epsilon_ = < 1 + \nu \over E >\sigma_ \]
for the shear terms. The shear terms are more commonly written as

\[ \gamma_ = < \tau_\over G> \qquad \quad \gamma_ = < \tau_\over G> \qquad \quad \gamma_ = < \tau_\over G> \]
where

\[ \gamma_ = 2 \epsilon_ \qquad \gamma_ = 2 \epsilon_ \qquad \gamma_ = 2 \epsilon_ \qquad <\rm and>\qquad G = \]
Return now to Hooke’s Law in tensor form

\[ \epsilon_ = <1 \over E>\left[ (1 + \nu) \sigma_ — \nu \, \delta_ \sigma_ \right] \]
and multiply both sides by \(\delta_\).

\[ \delta_ \epsilon_ = <1 \over E>\left[ (1 + \nu) \sigma_ — \nu \, \delta_ \sigma_ \right] \delta_ \]
This simplifies to

Now subtract the above equation from the original Hooke’s Law equation to get

But \(< (1 + \nu) \over E >\) is \(<1 \over 2G>\), so the equation can be further simplified to

\[ \epsilon’\!_ = < 1 \over 2 G>\sigma’\!_ \]
So the deviatoric stress and strain are directly proportional to each other. The amazing thing here is that this is always true for Hooke’s Law, always, even for the normal strain components.

For what it’s worth, the equation can also be written as

Deviatoric Example with Hooke’s Law

Suppose you have a material with Poisson’s ratio, \(\nu = 0.5\), and elastic modulus, \(E = 15\;MPa\).

For the stress tensor below, use Hooke’s Law to calculate the strain state. Then get the deviatoric stress and strain tensors and show that they are proportional to each other by the factor \(2G\).

\[ \boldsymbol <\sigma>\; = \; \left[ \matrix < 8 & 2 & 4 \\ 2 & 6 & 6 \\ 4 & 6 & 4 >\right] \]
Note that this stress tensor clearly has a significant amount of hydrostatic stress. It is

So the question becomes, «Will (\(2 \, G \, \boldsymbol<\epsilon>‘\)) give \(\boldsymbol<\sigma>‘\)?»

To answer this, first compute \(G\).

Strain Energy Density

Strain energy density, W, has units of Energy / Volume and is

\[ W = \int \boldsymbol <\sigma>: d \boldsymbol <\epsilon>\]
For linear elastic materials, this equals

\[ W = <1 \over 2>\boldsymbol <\sigma>: \boldsymbol <\epsilon>\]
which expands out to give

\[ <1 \over 2>\boldsymbol <\sigma>: \boldsymbol <\epsilon>= <1 \over 2>[ \sigma_ \epsilon_ + \sigma_ \epsilon_ + \sigma_ \epsilon_ + 2 ( \sigma_ \epsilon_ + \sigma_ \epsilon_ + \sigma_ \epsilon_ ) ] \]
But since \( \boldsymbol <\sigma>= \boldsymbol<\sigma>_\text + \boldsymbol<\sigma>‘ \) and \( \boldsymbol <\epsilon>= \boldsymbol<\epsilon>_\text + \boldsymbol<\epsilon>‘ \), these identities can be substituted into the equation to obtain

\[ W = <1 \over 2>\boldsymbol <\sigma>: \boldsymbol <\epsilon>= <1 \over 2>(\boldsymbol<\sigma>_\text + \boldsymbol<\sigma>‘) : (\boldsymbol<\epsilon>_\text + \boldsymbol<\epsilon>‘) \]
and expanding the multiplication out gives

\[ W = <1 \over 2>\boldsymbol <\sigma>: \boldsymbol <\epsilon>= <1 \over 2>\boldsymbol<\sigma>_\text : \boldsymbol<\epsilon>_\text + <1 \over 2>\boldsymbol<\sigma>_\text : \boldsymbol<\epsilon>‘ + <1 \over 2>\boldsymbol<\sigma>‘ : \boldsymbol<\epsilon>_\text + <1 \over 2>\boldsymbol<\sigma>‘ : \boldsymbol<\epsilon>‘ \]
But (\(\boldsymbol<\sigma>_\text : \boldsymbol<\epsilon>‘\)) and (\(\boldsymbol<\sigma>‘ : \boldsymbol<\epsilon>_\text\)) are zero! This is because the double-dot product of any hydrostatic tensor with a deviatoric tensor is always zero. So the equation reduces to

\[ W = <1 \over 2>\boldsymbol <\sigma>: \boldsymbol <\epsilon>= \underbrace < <1 \over 2>\boldsymbol<\sigma>_\text : \boldsymbol<\epsilon>_\text >_ + \underbrace < <1 \over 2>\boldsymbol<\sigma>‘ : \boldsymbol<\epsilon>‘ >_ \]
This shows that strain energy can be partitioned into hydrostatic and deviatoric components.

Von Mises Stress

The von Mises stress is directly related to the deviatoric strain energy term in the above equation.

\[ W’ = <1 \over 2>\boldsymbol<\sigma>‘ : \boldsymbol<\epsilon>‘ \]
Recall from the section on Hooke’s Law that

\[ \boldsymbol<\epsilon>‘ = <1 \over 2\, G>\boldsymbol<\sigma>‘ \]
Combining the two gives

\[ W’ = <1 \over 4 \, G>\boldsymbol<\sigma>‘ : \boldsymbol<\sigma>‘ \]
So the deviatoric part of the strain energy density is directly related to the double dot product of the deviatoric stress with itself. Note the similarity to Kinetic Energy, \(KE = <1 \over 2>M v^2\), a spring’s internal energy, \(E = <1 \over 2>K x^2\), electrical power, \(P = R I^2\), and any other form one can think of.

It is finally time to introduce an equivalent or effective stress that will turn out to be proportional to the von Mises stress, though about 20% low. Use the symbol \( \sigma_ \) for representative stress to represent this stress value. And it is a scalar stress value, not a tensor! The defining equation for \( \sigma_ \) is

\[ W’ = <1 \over 4\,G>(\sigma_)^2 \]
The form of the equation is deliberately chosen to be the scalar equivalent of the one above. Setting them equal to each other (since both are equal to W’) gives

\[ W’ = <1 \over 4\,G>(\sigma_)^2 = <1 \over 4\,G>\boldsymbol<\sigma>‘ : \boldsymbol<\sigma>‘ \]
Clearly \( \sigma_ \) is intended to be the scalar stress value that gives the same deviatoric strain energy as the actual 3-D stress tensor. Cancelling \(4\,G\) from both sides gives

\[ \sigma_ = \sqrt < \boldsymbol<\sigma>‘ : \boldsymbol<\sigma>‘ > \]
The final step is one of simple convenience. It is motivated by the simplest straight-forward case of uniaxial tension. To see it, calculate \(\sigma_\text\) for this case. The stress state for uniaxial tension is

\[ \boldsymbol <\sigma>= \left[ \matrix < \sigma & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 >\right] \]
The hydrostatic stress is \( <1 \over 3>\sigma\), and the deviatoric stress tensor is

\[ \boldsymbol<\sigma>‘ = \left[ \matrix < <2 \sigma \over 3>& 0 & 0 \\ 0 & <-\sigma \over 3>& 0 \\ 0 & 0 & <-\sigma \over 3>> \right] \]
So \(\boldsymbol<\sigma>‘ : \boldsymbol<\sigma>‘\) equals \(2 \sigma^2 / 3\). And therefore

\[ \sigma_\text = \sqrt < \boldsymbol<\sigma>‘ : \boldsymbol<\sigma>‘ > = \sqrt < 2 \over 3>\; \sigma \]
And therein lies the frustration. The representative stress for uniaxial tension is not equal to the uniaxial tension stress, \(\sigma\), but is instead about 82% of it. This is terribly inconvenient, but the fix is simple. Simply scale the representative stress up until it equals the uniaxial tension stress. This is done by simply multiplying \(\sigma_\text\) by \(\sqrt<3>\).

This is acceptable because anything proportional to \( \sqrt < \boldsymbol<\sigma>‘ : \boldsymbol<\sigma>‘ > \) will still reflect the relationship to deviatoric strain energy. It will just be scaled up some. The final result is the von Mises stress.

\[ \sigma_\text = \sqrt < <3 \over 2>\boldsymbol<\sigma>‘ : \boldsymbol<\sigma>‘ > \]
And this is the defining equation for it.

Alternate Forms

Algebraic manipulation of the above equation gives many other equivalent forms. They are summarized here.

\[ \sigma_\text = \sqrt<<1\over 2>\left[\left(\sigma_ — \sigma_\right)^2 + \left(\sigma_ — \sigma_\right)^2 + \left(\sigma_ — \sigma_\right)^2 \right] + 3 \left(\tau^2_ + \tau^2_ + \tau^2_\right) > \]
\[ \sigma_\text = \sqrt <\sigma^2_+ \sigma^2_ + \sigma^2_ — \sigma_ \sigma_ — \sigma_ \sigma_ — \sigma_ \sigma_ + 3 \left(\tau^2_ + \tau^2_ + \tau^2_\right) > \]
\[ \sigma_\text = \sqrt<<3\over 2>\sigma_\sigma_ — <1\over 2>( \sigma_ )^2> \quad \quad \quad \sigma_\text = \sqrt<<3\over 2>\sigma’\!_\sigma’\!_> \]

In 2-D applications, \( \sigma_ = \tau_ = \tau_ = 0\). This leaves

\[ \sigma_\text = \sqrt <\sigma^2_+ \sigma^2_ — \sigma_ \sigma_ + 3 \, \tau^2_ > \]

Tensor Manipulation of Von Mises Equation

One can (relatively) easily obtain other equations for von Mises stress thru tensor manipulations of the equation based on deviatoric values. Starting with

\[ \sigma_\text = \sqrt<<3\over 2>\sigma’\!_\sigma’\!_> \]
and expressing \(\sigma’\!_\) in terms of the full stress tensor as

\[ \sigma’\!_ = \sigma_ — <1 \over 3>\delta_ \sigma_ \]
gives the following form.

\[ \sigma_\text = \sqrt <<3\over 2>\left( \sigma_ — <1 \over 3>\delta_ \sigma_ \right) \left( \sigma_ — <1 \over 3>\delta_ \sigma_ \right) > \]
Multiplying this out gives

\[ \sigma_\text = \sqrt <<3\over 2>\left( \sigma_ \sigma_ — <2 \over 3>\delta_ \sigma_ \sigma_ + <1 \over 9>\delta_ \delta_ ( \sigma_ )^2 \right) > \]
which simplifies down to

\[ \sigma_\text = \sqrt<<3\over 2>\sigma_\sigma_ — <1\over 2>( \sigma_ )^2> \]
The other forms listed above can be obtained by expressing this explicitly in terms of \(\sigma_\), \(\sigma_\), \(\sigma_\), etc.

Specific Loading Cases

We’ve already seen during the derivation above that for uniaxial tension, the von Mises stress equals the uniaxial tension stress. But this is also (almost) true for compression as well. The only issue is that for compression, the numerical value of the compressive stress will be negative, but the von Mises stress is always positive because it is a square-root of a sum of stress values squared. So when one is reading a von Mises stress of say, 10 MPa, it is impossible to know from this alone if the object is undergoing tension or compression. One can look at the principal stress values to determine this.

Actually, some FEA post-processors will make color stress contours of a quantity call signed von Mises stress. This has the same absolute value as the conventional von Mises stress, but the +/- sign is determined by checking the sign of the hydrostatic stress. If it is negative, then the signed von Mises stress is also negative.

The case of pure shear stress is most interesting. One can see from the equations above that for a pure shear stress, \(\tau_\), the von Mises stress is

\[ \sigma_\text = \sqrt <3>\, \tau \]
So if a metal yields in uniaxial tension (or compression) at \(\sigma = \pm 500 \text< MPa>\), then it will also yield in shear at a stress that is only 58% of this, or \(\tau = \pm 290 \text< MPa>\).

Graphical Representations

Here again is the sketch at the top of the page. It shows a bounding surface in a 3-D principal stress coordinate system where the von Mises stress is a constant value. (This is the so called High-Westerguard Space.) It is based on the fact that any stress state can be converted into its principal values and compared to this sketch. If the resulting principal stress point in the coordinate system is within the cylinder, then the material has not yielded. If it is on the surface, then the material has yielded. And if it is outside the cylinder, it means that you did an elastic analysis of a situation that cannot in fact be correct because yielding would have long since taken place.

The remarkable result is that if you look down the \(\sigma_1 = \sigma_2 = \sigma_3\) axis, the cross-section of the cylinder is a perfect circle. Note that the hydrostatic stress in this situation does not show up at all.

Experimental Data

The figure here presents experimental data confirming that ductile metals yield much more consistently at prescribed von Mises stress levels regardless of the the loading state than at any other criteria.

Recall that the shear stress criterion was first proposed by Tresca in 1864, and this act is considered to represent the birth of the field of metal plasticity research.

The one exception here is the cast iron metal. It yields, fractures in fact, at a constant maximum principal stress criterion. This signifies that the iron is brittle and behaves more like glass than a ductile metal.

Reference: Dowling, N.E., Mechanical Behavior of Materials, Prentice Hall, 1993.

Note that the correlation here is not perfect. This is a consequence of the fact that the so-called von Mises Yield Criterion is NOT a law of nature. It is more of a convenient coincidence. It is a consequence of dislocation movement on millions and billions of planes of atoms sliding over each other at the atomic scale. Those planes of atoms are all randomly oriented, and the resulting response at the macroscale is. the von Mises yield criterion.

Contrasting Stress and Strain

We’ve seen how the von Mises stress is «the stress» when worrying about metal yielding and plasticity. Recall that it is

\[ \sigma_\text = \sqrt < <3 \over 2>\boldsymbol<\sigma>‘ : \boldsymbol<\sigma>‘ > \]
The next question is, «Is there a strain analog to the von Mises stress?» The answer is yes. It is the effective strain, or sometimes the Mises effective strain. It is

\[ \epsilon_\text = \sqrt < <2 \over 3>\boldsymbol<\epsilon>‘ : \boldsymbol<\epsilon>‘ > \]
Note that it is \(2/3\), not \(3/2\). This arises because the strain tensor for uniaxial tension of an incompressible material (which includes the plastic part of the total deformation of a metal) is

This makes it possible to more fairly compare the stress and strain states of two different deformation modes, say tension versus shear. In fact, in a perfectly isotropic metal, plots of effective stress versus effective strain will be indistinguishable in the plastic region regardless of the deformation mode. Although in reality, metals usually become increasingly anisotropic after yielding.

References

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