Von Mises Yield Criterion - Reduced Von Mises Equation For Different Stress Conditions

Reduced Von Mises Equation For Different Stress Conditions

The above equation can be reduced and reorganized for practical use in different loading scenarios.

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

It is also convenient to define an Equivalent tensile stress or von Mises stress, which is used to predict yielding of materials under multiaxial loading conditions using results from simple uniaxial tensile tests. Thus, we define

\begin{align} \sigma_v &= \sqrt{3J_2} \\ &= \sqrt{\frac{(\sigma_{11} - \sigma_{22})^2 + (\sigma_{22} - \sigma_{33})^2 + (\sigma_{11} - \sigma_{33})^2 + 6(\sigma_{12}^2 + \sigma_{23}^2 + \sigma_{31}^2)}{2}} \\ &= \sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_1 - \sigma_3)^2 } {2}} \\ &= \sqrt{\textstyle{\frac{3}{2}}\;s_{ij}s_{ij}} \end{align} \,\!

where are the components of the stress deviator tensor :

\boldsymbol{\sigma}^{dev} = \boldsymbol{\sigma} - \frac{1}{3} \left(\mbox{tr} \ \boldsymbol{\sigma} \right) \mathbf{I} \,\!.

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 tensile 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 plane stress, the von Mises criterion becomes:

This equation represents an ellipse in the plane, as shown in the Figure above.

The following table summarizes von Mises yield criterion for the different stress conditions.

Load scenario Restrictions Simplified von Mises equation
General No restrictions
Principal stresses
Plane stress

Pure shear

Uniaxial

Notes:

  • subscripts 1,2,3 can be replaced with x,y,z, or other orthogonal coordinate system

Read more about this topic:  Von Mises Yield Criterion

Famous quotes containing the words reduced, von, equation, stress and/or conditions:

    “Drill and uniforms impose an architecture on the crowd. An army’s beautiful. But that’s not all; it panders to lower instincts than the aesthetic. The spectacle of human beings reduced to automatism satisfies the lust for power. Looking at mechanized slaves, one fancies oneself a master.”
    —Aldous Huxley (1894–1963)

    “I curse all negative purism that tells me not to use a word from another language that either expresses something that my own language cannot or does that in a more delicate manner.”
    —Johann Wolfgang Von Goethe (1749–1832)

    “A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
    —Norman Mailer (b. 1923)

    “Anyone who wishes to combine domestic responsibilities and paid employment with the least stress and most enjoyment might start by pondering this paradox: the first step to better functioning is to stop blaming herself for not functioning well enough.”
    —Faye J. Crosby (20th century)

    “No great idea in its beginning can ever be within the law. How can it be within the law? The law is stationary. The law is fixed. The law is a chariot wheel which binds us all regardless of conditions or place or time.”
    —Emma Goldman (1869–1940)