von Mises yield criterion

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The von Mises yield criterion[1] 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 material science and engineering the von Mises yield criterion can be also formulated in terms of the von Mises stress, \ \sigma_v \,, 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, \ \sigma_y. 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, \ I_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. Although formulated by Maxwell in 1865, it is generally attributed to von Mises (1913).[2] Huber (1904), in a paper in Polish, anticipated to some extent this criterion.[3] This criterion is referred also as the Maxwell–Huber–Hencky–von Mises theory.

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[edit] Mathematical formulation

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.
The von Mises yield surfaces in principal stress coordinates circumscribes a cylinder with radius \ \sqrt \tfrac{2}{3} \sigma_y around the hydrostatic axis. Also shown is Tresca's hexagonal yield surface.

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

\ f(J_2) = \sqrt{J_2} - k = 0

An alternative form is:

\ f(J_2) = J_2 - k^2 = 0

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 stress in pure shear is \sqrt{3} times lower than the tensile stress in the case of simple tension . Thus, we have

\ k = \frac{\sigma_y}{\sqrt{3}}

Furthermore, if we define the von Mises stress as \ \sigma_v=\sqrt{3J_2}, the von Mises yield criterion can be expressed as:

\ \begin{align} f(J_2) &= \sqrt{3J_2} - \sigma_y \\ &= \sigma_v - \sigma_y =0 \\ \end{align}

Substituting \ J_2 in terms of the principal stresses into the von Mises criterion equation we have

\ (\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_1 - \sigma_3)^2 = 6k^2=2\sigma_y^2

or

\ (\sigma_1^2 + \sigma_2^2 + \sigma_3^2) - \sigma_1\sigma_2 - \sigma_2\sigma_3 - \sigma_1\sigma_3 = 3k^2=\sigma_y^2

or as a function of the stress tensor components

\ (\sigma_{11} - \sigma_{22})^2 + (\sigma_{22} - \sigma_{33})^2 + (\sigma_{11} - \sigma_{33})^2 + 6(\sigma_{23}^2 + \sigma_{31}^2 + \sigma_{12}^2) = 6k^2

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 \ \sqrt{2}k, or \ \sqrt \tfrac{2}{3} \sigma_y . This implies that the yield condition is independent of hydrostatic stresses.

[edit] Von Mises criterion for different stress conditions

Projection of the von Mises yield criterion into the σ1,σ2 plane
Projection of the von Mises yield criterion into the σ1,σ2 plane

In the case of uniaxial stress or simple tension, \ \sigma_1 \neq 0, \ \sigma_3 = \sigma_2=0, the von Mises criterion reduces to

\ \sigma_1 = \sigma_y.

Therefore, the material starts to yield, when \ \sigma_1 reaches the yield strength of the material \ \sigma_y, which is a characteristic material property. In practice, this parameter is, indeed, determined in a tensile test satisfying the uniaxial stress condition.

It is also convenient to define an Equivalent tensile stress or von Mises stress, \ \sigma_v \,, 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_{33} - \sigma_{11})^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_3 - \sigma_1)^2 } {2}} \\ &= \sqrt{\textstyle{\frac{3}{2}}\;s_{ij}s_{ij}} \end{align}

where \ s_{ij} are the components of the stress deviator tensor \boldsymbol{\sigma}^{dev} :

\boldsymbol{\sigma}^{dev} = \boldsymbol{\sigma} - \frac{1}{3}\left( \boldsymbol{\sigma}\cdot\mathbf{I} \right) \mathbf{I} .

In this case, yielding occurs when the equivalent stress, \sigma_v \,, reaches the yield strength of the material in simple tension, \ \sigma_y. 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 specimen 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, \ \sigma_{12} = \sigma_{21}\neq0, while all other \ \sigma_{ij} = 0, von Mises criterion becomes:

\ \sigma_{12} = k=\frac{\sigma_y}{\sqrt{3}}.

This means that, at the onset of yielding, the magnitude of the shear stress in pure shear is \sqrt{3} 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

\ (\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_1 - \sigma_3)^2 = 6\sigma_{12}^2.

In the case of plane stress, \ \sigma_3 = 0, the von Mises criterion becomes:

\ \sigma_1^2- \sigma_1\sigma_2+ \sigma_2^2 = 3k^2=\sigma_y^2

This equation represents an ellipse in the plane \ \sigma_1-\sigma_2, as shown in the Figure above.

[edit] 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. [4] 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:

\ W_D = \frac{J_2}{2G} with the elastic shear modulus \ G = \frac{E}{2(1+\nu)}.

In 1937 [5] 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, \ \tau_{oct}, which by definition is

\tau_{oct} = \sqrt{\tfrac{2}{3}J_2}

thus we have

\ \tau_{oct}=\tfrac{\sqrt 2}{3} \sigma_y

[edit] Comparison with Tresca yield criterion

Also shown in the figure is Tresca's maximum shear stress criterion (dashed line). Observe that Tresca's yield surface circumscribed by von Mises'. Therefore, it predicts plastic yielding already for stress states that are still elastic according to the von Mises criterion. As a model for plastic material behavior Tresca's criterion is therefore more conservative, which basically means on the safer side.

[edit] See also

[edit] References

  1. ^ von Mises, R. (1913). Mechanik der Festen Korper im plastisch deformablen Zustand. Gottin. Nacher. Math. Phys., vol. 1, pp. 582–592.
  2. ^ Ford, Advanced Mechanics of Materials, Longmans, London, 1963
  3. ^ Hill, R. (1950). The Mathematical Theory of Plasticity. Oxford, Clarendon Press
  4. ^ Hill, R. (1950). The Mathematical Theory of Plasticity. Oxford, Clarendon Press
  5. ^ S. M. A. Kazimi. (1982). Solid Mechanics. Tata McGraw-Hill. ISBN 0074517155