It is the main task of engineers to ensure the stability of constructions according to standards, proving their price-performance outstanding efficiency. Nowadays, one can hardly imagine analyzing failure or fatigue in ductile materials without checking the von Mises stress values. **Von Mises criteria** are among the most commonly used criteria for checking yield conditions in aerospace engineering, civil engineering, oil and gas engineering, offshore and marine engineering, robotics, and heavy lifting.

Every engineer developing the mechanical design of elements has to know when to use von Mises stress (*σ _{v}*) and keep its value below the yield strength (

*σ*) of that material to make the design safe.

_{y}## Historical reference to von Mises theory

According to von Mises stress theory, material yields when a critical distortion value is reached. It is often called **Maxwell-Huber-Hencky-von Mises theory**, the distortion-energy theory, the shear-energy theory, or octahedral-shear-stress theory. It finds wide application in Finite Element Analysis. This critical and specific for each material value is easily obtained by performing a simple tension test. When we check the failure using the von Mises stress, which is not stress, but a number used as an index, we apply the von Mises yield criterion to determine yielding. Von Mises stress theory, which can be expressed in the formula *N = σ _{y} / σ’*, is suitable for computing the safety factor against failure. It is generally used for ductile materials – they have to be checked for fulfilling the von Mises criteria.

In 1865 James Clerk Maxwell mentioned the idea for the first time, describing its general conditions. Then Tytus Maksymillian Huber proposed it again in 1904 with a math equation, separating hydrostatic and distortion strain energy. In 1913 Richard Elder von Mises established the von Mises stress equation for scalar representation of stress based on the second invariant of the deviatoric stress tensor. Later in 1924, Heinrich Hencky independently gave the von Mises equations a reasonable physical interpretation, relating them to deviatoric strain energy. Hencky offered a von Mises criterion equation physical interpretation, suggesting that yielding begins when the elastic energy of distortion reaches a critical value. For this reason, the von Mises criterion is also known as the maximum distortion strain energy criterion.

Richard von Mises found that, even though none of the principal stresses exceeds the material yield stress, the combination of the stresses can still cause yielding. So he proposed a formula for combining the three principal stresses into equivalent stress. Equivalent stress must then be compared to the yield stress of the material to judge the failure condition of the material, which was called von Mises criteria.

Interestingly, the von Mises stress formula and definition are closely related to Lviv, Ukraine, one of the SDC Verifier offices locations. Not only was Richard von Mises born in Lviv, but also Maksymilian T. Huber studied and worked here, publishing his most essential works on deformation and strain of the material, which later became the basis for the following research on this topic.

**Components of the idea**

To make von Mises stress definition clearer, let us briefly look over the very important for understanding the concept of von Mises stress ideas below: hydrostatic and deviatoric components of stress and strain tensors, von Mises yield criterion, Hooke’s law, and strain energy density.

Let us find out that a tensor is a multidimensional array of numerical values helping to describe the material physical state or properties. Stress is a simple example of a geophysically relevant tensor. As in the case of pressure, it is defined as force per unit area. Though pressure is isotropic, a material can support different forces applied in different directions and has finite strength.

It is important to understand that the stress tensor is a field tensor depending on factors external to the material. The stress tensor must be symmetric for stress not to move the material: *σ _{ij}* =

*σ*– it has mirror symmetry about the diagonal.

_{ji}The general stress tensor has six independent components, and to exclude many calculations engineers can rotate it into the principal stress tensor, performing a suitable change of axes.

We can separate stress tensor into two components – hydrostatic stress (also called dilatational or volumetric) and deviatoric stress. Hydrostatic stress purely corresponds to a change in volume of the object without any changes in the overall shape and resembles scaling an object. It is the average of the three normal stress components of any stress tensor:

*σ _{Hyd} = σ_{11} + σ_{22} + σ_{33}*

Instead, deviatoric stress changes the shape only and corresponds to the shearing and distortion effects observed.

Deviatoric stress is left after subtracting out the hydrostatic stress. It corresponds to the shearing and distortion effects observed. The deviatoric stress will be represented by σ′. For example:

*σ′ = σ − σ _{Hyd}*

Von Mises yield criterion is another concept important for understanding the theory of von Mises stress. It develops the method of ductile materials behavior prediction for any complex, 3D loading condition. **Mathematically, the von Mises yield criterion is expressed as**:

*J _{2} = K²*

Here, K is the yield stress of the material in pure shear.

Yield stress is the point at which the material behavior transforms from elastic to plastic. It is often said that the material yields if the stress is greater than the yield strength. The stress tensors are more generic in real-life applications and not essentially uniaxial. Each component of the stress tensor is likely non-zero. The stress tensor has six independent components, and similarly, the strain tensor can also be decomposed into analog strains. The yield criterion must relate the full stress tensor to the deformation strain energy density. For this total strain energy can be first noted, which has units of Energy/Volume, W, is:

\(W = \int \sigma:d\epsilon \)

For linear elastic materials, the equation will look like this:

\(W = \frac{1}{2} \sigma:\epsilon = \frac{1}{2}(\sigma_{xx}\epsilon_{xx}+\sigma_{yy}\epsilon_{yy}+\sigma_{zz}\epsilon_{zz}+2(\sigma_{xy}\epsilon_{xy}+\sigma_{yz}\epsilon_{yz}+\sigma_{xz}\epsilon_{xz})) \)

Hooke’s law states that the force needed to extend or compress a spring by some distance is proportional to that distance. Hooke’s law is linear and isotropic (having equal stiffness in every direction.) It is the 1st order linearization of any hyperelastic material law, including nonlinear ones, as long as the law is also isotropic. So it can be applied to rubber as long as the strains are small. It is the standard for metals in the elastic range.

Strain is a relative change in the position of points within a deformed body. The deformation, expressed by strain, arises throughout the material as the particles (molecules, atoms, ions) of which the material is composed are slightly displaced from their normal position. The square deformed to a parallelepiped can be the classic two-dimensional example. Normal strains and shear strains depend on the forces that cause the deformation. Forces perpendicular to planes or cross-sectional areas of the material, such as in a volume that is under pressure on all sides or in a rod that is pulled or compressed lengthwise, cause a normal strain.

A shear strain is caused by forces that are parallel to, and lie in, planes or cross-sectional areas, for example, in a short metal tube that is twisted about its longitudinal axis.

The strain energy density is a non-negative scalar-valued function of a tensorial strain measure. Strain energy density is defined as:

\(W = \frac{1}{2} \sigma\epsilon\)

In other words, this is the total strain energy stored in each differential volume of the body. If this strain energy is summed over all the differential volumes, we can obtain the total strain energy stored in the body. The strain energy in the solid may not distribute uniformly throughout it. Below, the concept of strain energy density, which is strain energy per unit volume, is introduced and denoted by ** U_{0}**. The strain energy in the body can be obtained by integration as follows:

\(U = \iiint\limits_{V}U_{0}(x,y,z)dV\)

We can think of the strain energy density as consisting of two components: one due to dilation or change in volume (dilatational strain energy) and the other due to distortion or change in shape (distortional energy). Many experiments have shown that ductile materials can outstand hydrostatic stresses levels beyond their ultimate strength in compression without failure because the hydrostatic state of stress reduces the volume of the specimen without changing its shape.

In structural engineering and the strength of materials, a member or component may be subject to different types of forces or moments or a complex combination of them. Theories of failure try to ascertain the cause of failure of a particular material due to the exceedance of different parameters. There is only one dominant mode of failure for any material and others are not valid. This is usually checked by applying different theories of failure in different loading conditions. The theory that accurately predicts failure under different conditions is accepted for that material. Hence, it is crucial to differentiate between von Mises stress and principal stress. The maximum principal stress theory states that failure in any material occurs when the principal stress in that material due to any loading exceeds the principal stress at which failure occurs in the one-dimensional loading test (universal tensile test in the case of mild steel). And von Mises failure criteria theory states that failure in any material occurs when the shear strain energy per unit volume stored in that material due to any loading exceeds the shear strain energy per unit volume stored in that material in the one-dimensional loading test (universal tensile test in the case of mild steel). Von Mises stress equations calculate what is known as the distortion energy density at a particular point in the system. Principal stress, on the other hand, is obtained by suitably transforming (rotating) the stress element such that the rotated element is subjected to no shear stress. Most of the ductile material failures can be predicted using von Mises criteria. Principal stress theory, on the other hand, has very limited applications and can be used for brittle materials like cast iron. For failure prediction tasks von Mises stress function is used a lot more than principal stress function. Von Mises stress function can be easily calculated from principal stresses, but principal stresses cannot be calculated from von Mises stress values.

**Application of von Mises stress in SDC Verifier**

SDC Verifier uses nominal stress output results from Ansys, Femap, Simcenter 3D as variables for formulas creation and calculation of von Mises stress by the general state of stress:

\(\sqrt{\frac{(\sigma_{11}-\sigma_{22})^2+(\sigma_{22}-\sigma_{33})^2+(\sigma_{33}-\sigma_{11})^2+6(\sigma_{12}^2+\sigma_{23}^2+\sigma_{31}^2)}{2}}\)

SDC Verifier uses equivalent stresses, principal stresses, nominal stresses for checks:

FEA programs calculate nominal stress values and SDC Verifier software uses them in the formulas for failure modes checks to assure the construction’s stability.

SDC Verifier uses von Mises stress in checks according to different industry standards such as: DIN15018, F.E.M. 1.001, DNV 1995 & 2010, Eurocode 3, EN13001, ABS 2004, 2014 2022 and 2024, FKM.