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Von Mises Definition

Von Mises failure criteria describes a scalar function called the Von Mises stress or Mises stress that can be used to predict the onset and amount of plastic deformation in ductile materials under triaxial loading based on the results of a simple uniaxial tensile test. Plastic yield occurs when the Von Mises stress reaches the initial yield stress in uniaxial tension. The Von Mises stress can also be used to predict failure by ductile tearing but not failure from crack propagation or fatigue. It is expressed mathematically in terms of principal stresses or the deviatoric stress tensor. The Von Mises yield criterion was first formulated by Maxwell in 1865 but is generally attributed to Von Mises in 1913 and states

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329 views3 pages

Von Mises Definition

Von Mises failure criteria describes a scalar function called the Von Mises stress or Mises stress that can be used to predict the onset and amount of plastic deformation in ductile materials under triaxial loading based on the results of a simple uniaxial tensile test. Plastic yield occurs when the Von Mises stress reaches the initial yield stress in uniaxial tension. The Von Mises stress can also be used to predict failure by ductile tearing but not failure from crack propagation or fatigue. It is expressed mathematically in terms of principal stresses or the deviatoric stress tensor. The Von Mises yield criterion was first formulated by Maxwell in 1865 but is generally attributed to Von Mises in 1913 and states

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Von Mises failure criteria

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Von Mises stress, , or simply Mises stress, is a scalar function of the deviatoric
components of the stress tensor that gives an appreciation of the overall magnitude of
the shear components of the tensor. This allows the onset and amount of plastic
deformation under triaxial loading to be predicted from the results of a simple
uniaxial tensile test. It is most applicable to ductile materials.

; where τ0 is the allowable shear. [1]

Plastic yield initiates when the Mises stress reaches the initial yield stress in uniaxial
tension and, for hardening materials, will continue provided the Mises stress is equal
to the current yield stress and tending to increase. Mises stress can then be used to
predict failure by ductile tearing. It is not appropriate for failure by crack propagation
or fatigue, which depend on the maximum principal stress.

In 3-D, the Mises stress can be expressed as:

where σ1,σ2,σ3 are the principal stresses. In 1-D, this reduces to the uniaxial stress.

Or, in terms of an arbitrary cartesian coordinate system (x,y,z):

Or, in terms of the deviatoric stress tensor :

Von Mises yield criterion


Von Mises stress in two dimensions.

This criterion for the onset of yield in ductile materials was first formulated by
Maxwell in 1865 but is generally attributed to von Mises in 1913.[2] Originally
suggested by Maxwell purely on the grounds of mathematical simplicity, the
corresponding yield function Φ is the simplest function that meets certain physical
requirements for yielding, taking the form:

Φ = J2 - k2

where J2 is the second deviatoric invariant of the stress tensor and k is the yield stress
in shear. The use of von Mises yield criterion is therefore sometimes called J2 flow
theory.

Von Mises yield criterion can be interpreted physically in terms of the maximum
distortion strain energy, octahedral shear stress theory, or Maxwell-Huber-Hencky-
von Mises theory. This states that yielding in 3-D occurs when the distortion strain
energy reaches that required for yielding in uniaxial loading. Mathematically, this is
expressed as:

In the case of plane stress, σ3 = 0, von Mises criterion reduces to:

In the 2-D stress space shown in the figure above, this equation represents the interior
of an ellipse. Stress states σ1,σ2 not touching the boundary of the ellipse produce only
elastic deformation. Yielding initiates when the stress state pushes against the
boundary. This ellipse is the projection, onto a plane, of the yield surface in 3-D stress
space, which takes the form of a cylinder equiaxial to the three stress axes.
Also shown on the figure is Tresca's maximum shear stress criterion (dashed line).
This is more conservative than von Mises' criterion since it lies inside the von Mises
ellipse.

In addition to bounding the principal stresses to prevent ductile failure, von Mises'
criterion sometimes gives a reasonable estimation of fatigue life, especially with
complex loading (mode II & III loading).

See also
 Yield surface
 Henri Tresca
 Mohr-Coulomb theory
 Yield (engineering)
 Stress
 Strain
 3-D elasticity

References
1. ^ A. Nadai, Theory of Flow and Fracture of Solids, Volume 1, 2nd edition,
McGraw-Hill, 1950
2. ^ Ford, Advanced Mechanics of Materials, Longmans, London, 1963

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