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 century1

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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 ether2

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.

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 19133

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. 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 Sy

${S}_{y}$

.

**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.