von Mises yield criterion

The maximum distortion criterion (also von Mises yield criterion[1]) states that yielding of a ductile
material begins when the second invariant of deviatoric stress reaches a critical value.[2] It is a part of
plasticity theory that mostly applies to ductile materials, such as some metals. Prior to yield, material
response can be assumed to be of a nonlinear elastic, viscoelastic, or linear elastic behavior.

In materials science and engineering von Mises yield criterion is also formulated in terms of the von Mises
stress or equivalent tensile stress, . This is a scalar value of stress that can be computed from the Cauchy
stress tensor. In this case, a material is said to start yielding when the von Mises stress reaches a value
known as yield strength, . The von Mises stress is used to predict yielding of materials under complex
loading from the results of uniaxial tensile tests. The von Mises stress satisfies the property where two stress
states with equal distortion energy have an equal von Mises stress.

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.[8] For the above reasons this criterion is also referred to as the "Maxwell–Huber–Hencky–von Mises
theory".

Contents
Mathematical formulation
Reduced von Mises equation for different stress conditions
Uniaxial (1D) stress
Multi-axial (2D or 3D) stress
Summary
Physical interpretation of the von Mises yield criterion
Practical engineering usage of the von Mises yield criterion
See also
References

Mathematical formulation
Mathematically the von Mises yield criterion is expressed as:
Here is yield stress of the material in
pure shear. As shown later in this
article, at the onset of yielding, the
magnitude of the shear yield stress in
pure shear is √3 times lower than the
tensile yield stress in the case of simple
tension. Thus, we have:

where is tensile yield strength of the

material. 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:

The von Mises yield surfaces in principal stress coordinates

or circumscribes a cylinder with radius around the hydrostatic

axis. Also shown is Tresca's hexagonal yield surface.

Substituting with the Cauchy stress tensor components, we get

where is called deviatoric stress. 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 , or . This
implies that the yield condition is independent of hydrostatic stresses.

Reduced von Mises equation for different stress conditions

Uniaxial (1D) stress

In the case of uniaxial stress or simple tension, , the von Mises criterion simply
reduces to

which means the material starts to yield when reaches the yield strength of the material , in agreement
with the definition of tensile (or compressive) yield strength.

Multi-axial (2D or 3D) stress

An equivalent tensile stress or equivalent von-Mises stress, is used to predict yielding of materials under
multiaxial loading conditions using results from simple uniaxial tensile tests. Thus, we define
 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.

where are components of stress deviator tensor :

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. In view of the stress
tensor, which fully describes the stress state, this difference manifests in six degrees of freedom, because the
stress tensor has six independent components. 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.

In the case of pure shear stress, , while all other , von Mises criterion becomes:

.
This means that, at the onset of yielding, the magnitude of the shear stress in pure shear is times lower
than the yield stress in the case of simple tension. The von Mises yield criterion for pure shear stress,
expressed in principal stresses, is

In the case of principal plane stress, and , the von Mises criterion becomes:

This equation represents an ellipse in the plane .

Summary

State of Boundary
von Mises equations
stress conditions

General No restrictions

Principal
stresses

General
plane
stress
Principal
plane
stress

Pure
shear

Uniaxial

Physical interpretation of the von Mises yield criterion

Hencky (1924) offered a physical interpretation of von Mises criterion suggesting that yielding begins when
the elastic energy of distortion reaches a critical value.[6] For this reason, the von Mises criterion is also
known as the maximum distortion strain energy criterion. This comes from the relation between and the
elastic strain energy of distortion :

with the elastic shear modulus .

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 and the octahedral shear stress, , which by definition is
thus we have

Strain energy density consists of two components - volumetric or dialational and

distortional. Volumetric component is responsible for change in volume without any change
in shape. Distortional component is responsible for shear deformation or change in shape.

Practical engineering usage of the von Mises yield criterion

As shown in the equations above, the use of the von Mises criterion as a yield criterion is only exactly
applicable when homogeneous material properties are equal to

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 yield margin of safety is written as

Although the given criterion is based on a yield phenomenon, extensive testing has shown that use of a "von
Mises" stress is applicable at ultimate loading [10]

See also
Yield surface Hoek–Brown failure criterion
Huber's equation Yield (engineering)
Henri Tresca Stress
Stephen Timoshenko Strain
Mohr–Coulomb theory 3-D elasticity

References
1. "Von Mises Criterion (Maximum Distortion Energy Criterion)" (https://www.engineersedge.co
m/material_science/von_mises.htm). Engineer's edge. Retrieved 8 February 2018.
2. von Mises, R. (1913). "Mechanik der festen Körper im plastisch-deformablen Zustand" (http://
www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002503697). Nachrichten von der
Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse. 1913
(1): 582–592.
3. "Deformation Theory of Plasticity, p. 151, Section 4.5.6" (https://books.google.com/books?id=
kiCVc3AJhVwC&pg=PA152). Retrieved 2017-06-11.
4. Ford (1963). Advanced Mechanics of Materials. London: Longmans.
 5. Huber, M. T. (1904). "Właściwa praca odkształcenia jako miara wytezenia materiału".
Czasopismo Techniczne. Lwów. 22. Translated as "Specific Work of Strain as a Measure of
Material Effort" (http://am.ippt.pan.pl/am/article/viewFile/v56p173/pdf). Archives of
Mechanics. 56: 173–190. 2004.
6. Hill, R. (1950). The Mathematical Theory of Plasticity. Oxford: Clarendon Press.
7. Timoshenko, S. (1953). History of strength of materials. New York: McGraw-Hill.
8. Hencky, H. (1924). "Zur Theorie plastischer Deformationen und der hierdurch im Material
hervorgerufenen Nachspannngen". Z. Angew. Math. Mech. 4: 323–334.
doi:10.1002/zamm.19240040405 (https://doi.org/10.1002%2Fzamm.19240040405).
9. S. M. A. Kazimi. (1982). Solid Mechanics. Tata McGraw-Hill. ISBN 0-07-451715-5
10. Stephen P. Timoshenko, Strength of Materials, Part I, 2nd ed., 1940

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