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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 — which is easy to determine experimentally —, 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 concepts that are fundamental for engineering today, such as the deformation of a spring and the displacement of a beam. However, it’s important not to think that engineering was the only reason for the study of elasticity theory, as that research was also linked to the attempt to interpret 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, and 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 an uniaxial tension of compression of a mild steel beam. The description of each emphasized point is as follows:

  • Elastic Limit: The elastic limit defines the region where energy is not lost during the process of stressing and straining. That is, the processes that do not exceed the elastic limit are reversible. This limit is also called yield stress. Above that limit, the deformations stop being elastic and start being plastic, and the deformation includes an irreversible part. The stress value of the elastic limit is used here as \(S_y\).
  • Upper yield and lower yield: When mild steel is in the plastic range and reaches a critical point — called the upper yield limit —, it will drop quickly to the lower yield limit, from which deformation happens at constant stress, until it starts resisting deformation again.
  • Rupture stress: Rupture, or fracture, is the separation of an object caused by stress. Therefore, at this point, the fracture of the body is expected. Materials such as mild steel — which have the property of fracturing only after large plastic deformations — are called ductile. The fracture illustrated here is called ductile fracture. You can recognize a ductile fracture when the diagram has a curve like the one shown below. This means that as the material gets thinner, the more pressure is applied until it suddenly breaks when it reaches the rupture stress.
Stress-strain diagram of mild steel

Figure 1: Stress-strain diagram of mild steel

This diagramm is commonly approximated for amny materials as is shown in the picture below:

Commonly approximated stress-strain plot

Figure 2: Common approximation for the stress-strain plot

When a given particle of the body is under a given stress that satisfies a certain relation referred to as a yield criterion, plastic components of strain are produced (deformation). One such yield criterion is the von Mises yield criterion. The definition of the von Mises yield criterion states that the von Mises stress of a material under load should not be equal nor greater than the yield limit of the material under uniaxial stress.

Von Mises Yield Criterion

The elastic limits discussed before are based on simple tension or uniaxial stress experiments. The maximum distortion energy theory originated from the observation that materials, especially ductile materials, behaved differently when a non-simple tension or non-uniaxial stress experiment was conducted, 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\(^4\):

\[\frac{1}{6}[(\tau_{11}-\tau_{22})^2 + (\tau_{22}-\tau_{33})^2 + (\tau_{33}-\tau_{11})^2 + 6(\tau^2_{12}+\tau^2_{23}+\tau^2_{13})]=k^2 \tag{1}\]

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

\[\frac{\tau^2_y}{3}=k^2 \tag{2}\]

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

\[\frac{S_y^2}{3}=k^2 \tag{3}\]

Which can be substituted into the first expression:

\[\frac{1}{6}[(\tau_{11}-\tau_{22})^2 + (\tau_{22}-\tau_{33})^2 + (\tau_{33}-\tau_{11})^2 + 6(\tau^2_{12}+\tau^2_{23}+\tau^2_{13})]=\frac{S_y^2}{3} \tag{4}\]

or, finally

\[\sqrt{\frac{(\tau_{11}-\tau_{22})^2 + (\tau_{22}-\tau_{33})^2 + (\tau_{33}-\tau_{11})^2 + 6(\tau^2_{12}+\tau^2_{23}+\tau^2_{13})}{2}}=S_y \tag{5}\]

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

\[\tau_v^2=3k^2 \tag{6}\]

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

\[\tau_v\geq{S_y} \tag{7}\]

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 that form an octahedron, whose normals form equal angles with the coordinate system) can be written as:

\[\frac{1}{3}\sqrt{(\tau_1-\tau_2)^2 + (\tau_2-\tau_3)^2 + (\tau_3-\tau_1)^2}=\tau_{oct} \tag{8}\]

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

\[\frac{\sqrt2}{3}\tau_y=\tau_{oct} \tag{9}\]

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

\[\frac{\sqrt2}{3}S_y=\tau_{oct} \tag{10}\]

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

\[\sqrt{\frac{(\tau_1-\tau_2)^2 + (\tau_2-\tau_3)^2 + (\tau_3-\tau_1)^2}{2}}=S_y \tag{11}\]

Similarly 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\(^5\):

\[\sqrt{\frac{(\tau_z-\tau_t)^2 + (\tau_t-\tau_r)^2 + (\tau_r-\tau_z)^2}{2}}=\tau_v \tag{12}\]

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 a material before yielding. Calculating yielding with Trescas 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:

\[[(\tau_1-\tau_2)^2-(S_y)^2][(\tau_2-\tau_3)^2-(S_y)^2][(\tau_3-\tau_1)^2-(S_y)^2]=0 \tag{13}\]

This criterion can be simplified when the ordering of the magnitude of the components of stress are known. When the expression for the maximum stress reduces to:

\[(\tau_1-\tau_3)^2-(S_y)^2=0 \tag{14}\]

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 & SimScale

There are many fields that benefit from the von Mises yield criterion. There are SimScale public projects that can help getting 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 plate subjected to a certain load.

Commonly approximated stress-strain plot

Figure 3: Von Mises stess of a simulated plate under 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.

Commonly approximated stress-strain plot

Figure 4: Von Mises stress of a gas tank under pressure burst

Resources

\(^1\): Hooke, R., “Lectures de potentia restitutiva or of spring explaining the power of springing bodies”, 1678

\(^2\): Capecchi, D., Ruta, G., “Strength of Materials and Theory of Elasticity in 19th Century Italy”, 2015

\(^3\): Bourgoyne Jr, A. T., Millheim, K. K., Chenevert, M. E., Young Jr, F. S., “Applied Drilling Engineering”, 1986

\(^4\): Armenàkas, A. E., “Advanced Mechanics of Materials and Applied Elasticity”, 2006

\(^5\): Shigley, J. E., Mischke, C. R., Budynas, R. G., “Mechanical Engineering Design”