### Von Mises Criterion ( Maximum Distortion Energy Criterion ) Strength ( Mechanics ) of Materials

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This criterion is based on the determination of the distortion energy in a given material, i.e., of the energy associated with changes in the shape in that material( as opposed to the energy associated with the changes in volume in the same material. According to this criterion, named after German-American applied mathematician Richard von Mises (1883-1953), a given structural material is safe as long as the maximum value of the distortion energy per unit volume in that material remains smaller than the distortion energy per unit volume required to cause yield in a tensile-test specified of the same material.

Mathematically the yield function for the von Mises condition is expressed as:

An alternative form is:

where *k* can be shown to be the yield stress of the material in pure shear. As it will become evident later in the 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

Furthermore, if we define the von Mises stress as , the von Mises yield criterion can be expressed as:

Substituting *J*_{2} in terms of the principal stresses into the von Mises criterion equation we have

or

or as a function of the stress tensor components

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.

Von Mises criterion for different stress conditions

The Von Mises yield criterion suggests that the yielding of materials begins when the second deviatoric stress invariant J_{2} reaches a critical value k. For this reason, it is sometimes called the J_{2}-plasticity or J_{2} flow theory. It is part of a plasticity theory that applies best to ductile materials, such as metals. Prior to yield, material response is assumed to be elastic.

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

Because the von Mises yield criterion is independent of the first stress invariant, J_{1}, it is applicable for the analysis of plastic deformation for ductile materials such as metals, as the onset of yield for these materials does not depend on the hydrostatic component of the stress tensor.

**Interpretation of 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.^{} For this, the von Mises criterion is also known as the **maximum distortion strain energy criterion**. This comes from the relation between *J*_{2} and the elastic strain energy of distortion *W*_{D}:

- with the elastic shear modulus .

In 1937 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 exist between *J*_{2} and the octahedral shear stress, τ_{oct}, which by definition is

thus we have

- References:

- von Mises, R. (1913). Mechanik der festen Körper im plastisch deformablen Zustand. Göttin. Nachr. Math. Phys., vol. 1, pp. 582–592.
- Ford, Advanced Mechanics of Materials, Longmans, London, 1963
- a b Hill, R. (1950). The Mathematical Theory of Plasticity. Oxford, Clarendon Press
- S. M. A. Kazimi. (1982). Solid Mechanics. Tata McGraw-Hill. ISBN 0074517155
- Wikipedia