von Mises yield criterion

In continuum mechanics, the maximum distortion energy criterion (also von Mises yield criterion[1]) states that yielding of a ductile material begins when the second invariant of deviatoric stress

[2] It is a part of plasticity theory that mostly applies to ductile materials, such as some metals.

In materials science and engineering, the von Mises yield criterion is also formulated in terms of the von Mises stress or equivalent tensile stress,

The von Mises stress is used to predict yielding of materials under complex loading from the results of uniaxial tensile tests.

Because the von Mises yield criterion is independent of the first stress invariant,

, it is applicable for the analysis of plastic deformation for ductile materials such as metals, as onset of yield for these materials does not depend on the hydrostatic component of the stress tensor.

Although it has been believed it was formulated by James Clerk Maxwell in 1865, Maxwell only described the general conditions in a letter to William Thomson (Lord Kelvin).

[3] Richard Edler von Mises rigorously formulated it in 1913.

[2][4] Tytus Maksymilian Huber (1904), in a paper written in Polish, anticipated to some extent this criterion by properly relying on the distortion strain energy, not on the total strain energy as his predecessors.

[5][6][7] Heinrich Hencky formulated the same criterion as von Mises independently in 1924.

Mathematically the von Mises yield criterion is expressed as: Here

is yield stress of the material in pure shear.

If we set the von Mises stress equal to the yield strength and combine the above equations, the von Mises yield criterion is written as: or Substituting

This equation defines the yield surface as a circular cylinder (See Figure) whose yield curve, or intersection with the deviatoric plane, is a circle with radius

This implies that the yield condition is independent of hydrostatic stresses.

, the von Mises criterion simply reduces to which means the material starts to yield when

, in agreement with the definition of tensile (or compressive) yield strength.

is used to predict yielding of materials under multiaxial loading conditions using results from simple uniaxial tensile tests.

: In this case, yielding occurs when the equivalent stress,

, reaches the yield strength of the material in simple tension,

As an example, the stress state of a steel beam in compression differs from the stress state of a steel axle under torsion, even if both specimens are of the same material.

Therefore, it is difficult to tell which of the two specimens is closer to the yield point or has even reached it.

However, by means of the von Mises yield criterion, which depends solely on the value of the scalar von Mises stress, i.e., one degree of freedom, this comparison is straightforward: A larger von Mises value implies that the material is closer to the yield point.

times lower than the yield stress in the case of simple tension.

, the von Mises criterion becomes: This equation represents an ellipse in the plane

Hencky (1924) offered a physical interpretation of von Mises criterion suggesting that yielding begins when the elastic energy of distortion reaches a critical value.

: In 1937 [9] Arpad L. Nadai suggested that yielding begins when the octahedral shear stress reaches a critical value, i.e. the octahedral shear stress of the material at yield in simple tension.

In this case, the von Mises yield criterion is also known as the maximum octahedral shear stress criterion in view of the direct proportionality that exists between

, which by definition is thus we have As shown in the equations above, the use of the von Mises criterion as a yield criterion is only exactly applicable when the following material properties are isotropic, and the ratio of the shear yield strength to the tensile yield strength has the following value:[10] Since no material will have this ratio precisely, in practice it is necessary to use engineering judgement to decide what failure theory is appropriate for a given material.

Alternately, for use of the Tresca theory, the same ratio is defined as 1/2.

The von Mises yield surfaces in principal stress coordinates circumscribes a cylinder with radius around the hydrostatic axis. Also shown is Tresca 's hexagonal yield surface.
Von Mises yield criterion in 2D (planar) loading conditions: if stress in the third dimension is zero ( ), no yielding is predicted to occur for stress coordinates within the red area. Because Tresca's criterion for yielding is within the red area, Von Mises' criterion is more lax.