Yield surface

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A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The state of stress of inside the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and the material is said to have become plastic. Further deformation of the material causes the stress state to remain on the yield surface, even though the surface itself may change shape and size as the plastic deformation evolves.

The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space ( $σ 1,σ 2,σ 3$ ), a two- or three-dimensional space spanned by stress invariants ( $I 1, J 2, J 3$ ) or a version of the three-dimensional Haigh-Westergaard space. Thus we may write the equation of the yield surface (that is, the yield function) in the forms:

$f(\sigma_1,\sigma_2,\sigma_3) = 0 \,$ where $σ i$ are the principal stresses.
$f(I_1, J_2, J_3) = 0 \,$ where $I 1$ is the first invariant of the Cauchy stress and $J 2, J 3$ are the second and third invariants of the deviatoric part of the Cauchy stress.
$f(p, q, r) = 0 \,$ where $p, q$ are scaled versions of $I 1$ and $J 2$ and $r$ is a function of $J 2, J 3$ .
$f(\xi,\rho,\theta) = 0 \,$ where $ξ,ρ$ are scaled versions of $I 1$ and $J 2$ , and $θ$ is the Lode angle.

1 Invariants used to describe yield surfaces
2 Examples of yield surfaces
3 References
4 See also

[edit] Invariants used to describe yield surfaces

The first invariant of the Cauchy stress ( $I 1$ ), and the second and third invariants of the deviatoric part of the Cauchy stress ( $J 2, J 3$ ) are defined as

$\begin{align} I_1 & = \text{Tr}(\boldsymbol{\sigma}) = \sigma_1 + \sigma_2 + \sigma_3 \\ J_2 & = \cfrac{1}{2} \boldsymbol{s}:\boldsymbol{s} = \cfrac{1}{6}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right] \\ J_3 & = \det(\boldsymbol{s}) = \cfrac{1}{3} (\boldsymbol{s}\cdot\boldsymbol{s}):\boldsymbol{s} = s_1 s_2 s_3 \end{align}$

where $\boldsymbol{\sigma}$ is the Cauchy stress and $σ 1,σ 2,σ 3$ are its principal values, $\boldsymbol{s}$ is the deviatoric part of the Cauchy stress and $s 1, s 2, s 3$ are its principal values.

The quantities $p, q, r\,$ are usually used to describe yield surfaces for cohesive frictional materials such as rocks, soils, and ceramics. These quantities are defined as

$p = \cfrac{1}{3}~I_1 ~:~~ q = \sqrt{3~J_2} = \sigma_{eq} ~;~~ r = 3~\left(\cfrac{J_3}{2}\right)^{1/3}$

where $σ e q$ is the equivalent stress.

The quantities $\xi, \rho, \theta\,$ describe a cylindrical coordinate system (the Haigh-Westergaard coordinates) and are defined as

$\xi = \cfrac{1}{\sqrt{3}}~I_1 = \sqrt{3}~p ~;~~ \rho = \sqrt{2 J_2} = \sqrt{\cfrac{2}{3}}~q ~;~~ \cos(3\theta) = \left(\cfrac{r}{q}\right)^3 = \cfrac{3\sqrt{3}}{2}~\cfrac{J_3}{J_2^{3/2}}$

The $\xi-\rho\,$ plane is also called the Rendulic plane. The angle $θ$ is called the Lode angle^[1] and the relation between $θ$ and $J 2, J 3$ was first given by Nayak and Zienkiewicz in 1972 ^[2]

The principal stresses and the Haigh-Westergaard coordinates are related by

$\begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \end{bmatrix} = \cfrac{1}{\sqrt{3}} \begin{bmatrix} \xi \\ \xi \\ \xi \end{bmatrix} + \sqrt{\cfrac{2}{3}}~\rho~\begin{bmatrix} \cos\theta \\ \cos\left(\theta-\cfrac{2\pi}{3}\right) \\ \cos\left(\theta+\cfrac{2\pi}{3}\right) \end{bmatrix}$

[edit] Examples of yield surfaces

There are several different yield surfaces known in engineering, and those most popular are listed below.

[edit] Tresca yield surface

The Tresca ^[3] or maximum shear stress yield criterion is taken to be the work of Henri Tresca. It is also referred as the Tresca-Guest (TG) criterion. The functional form of this yield criterion is

$f(\sigma_1,\sigma_2,\sigma_3) = 0 ~.$

In terms of the principal stresses the Tresca criterion is expressed as

${\max(|\sigma_1 - \sigma_2| , |\sigma_2 - \sigma_3| , |\sigma_3 - \sigma_1| ) = \sigma_0 }\!$

Figure 1 shows the Tresca-Guest yield surface in the three-dimensional space of principal stresses. It is a prism of six sides and having infinite length. This means that the material remains elastic when all three principal stresses are roughly equivalent (a hydrostatic pressure), no matter how much it is compressed or stretched. However, when one of principal stresses becomes smaller (or larger) than the others the material is subject to shearing. In such situations, if the shear stress reaches the yield limit then the material enters the plastic domain. Figure 2 shows the Tresca-Guest yield surface in two-dimensional stress space, it is a cross section of the prism along the $σ 1,σ 2$ plane.

Figure 1: View of Tresca-Guest yield surface in 3D space of principal stresses

Figure 2: Tresca-Guest yield surface in 2D space (

σ 1,σ 2

)

[edit] von Mises yield surface

The von Mises yield criterion (also known as Prandtl-Reuss yield criterion) has the functional form

$f(J_2) = 0 ~.$

This yield criterion is often credited to Maximilian Huber and Richard von Mises (see von Mises stress). It is also referred to as the Huber-von Mises-Hencky (HMH) criterion.

The von Mises yield criterion is expressed in the principal stresses as

$\sqrt{3J_2} = \sigma_y \quad \text{or}, \quad {(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 = 2 {\sigma_y}^2 }\!$

where $σ y$ is the yield stress in uniaxial tension.

Figure 3 shows the von Mises yield surface in the three-dimensional space of principal stresses. It is a circular cylinder of infinite length with its axis inclined at equal angles to the three principal stresses. Figure 4 shows the von Mises yield surface in two-dimensional space compared with Tresca-Guest criterion. A cross section of the von Mises cylinder on the plane of $σ 1,σ 2$ produces the elliptical shape of the yield surface.

Figure 3: View of Huber-Mises-Hencky yield surface in 3D space of principal stresses

Figure 4: Comparison of Tresca-Guest and Huber-Mises-Hencky criteria in 2D space (

σ 1,σ 2

)

[edit] Mohr-Coulomb yield surface

The Mohr-Coulomb yield (failure) criterion is a two-parameter yield criterion which has the functional form

f (σ 1,σ 2,σ 3) = 0

This model is often used to model concrete, soil or granular materials.

The Mohr-Coulomb yield criterion may be expressed as:

$\pm\cfrac{\sigma_1 - \sigma_2}{2} = c - K \left(\cfrac{\sigma_1 + \sigma_2}{2}\right) ~;~~ \pm\cfrac{\sigma_2 - \sigma_3}{2} = c - K \left(\cfrac{\sigma_2 + \sigma_3}{2}\right) ~;~~ \pm\cfrac{\sigma_3 - \sigma_1}{2} = c - K \left(\cfrac{\sigma_3 + \sigma_1}{2}\right)$

where

$m = \frac {\sigma_c}{\sigma_t} ~,~~ K = \frac{m-1}{m+1} ~;~~c = \left(\frac {1}{m+1}\right)\sigma_c = \left(\frac {m}{m+1}\right)\sigma_t$

and the parameters $σ c$ and $σ t$ are the yield (failure) stresses of the material in uniaxial compression and tension, respectively. If $K = 0$ then the Mohr-Coulomb criterion reduces to the Tresca-Guest criterion.

Figure 5 shows Mohr-Coulomb yield surface in the three-dimensional space of principal stresses. It is a conical prism and $K$ determines the inclination angle of conical surface. Figure 6 shows Mohr-Coulomb yield surface in two-dimensional stress space. It is a cross section of this conical prism on the plane of $σ 1,σ 2$ .

Figure 5: View of Mohr-Coulomb yield surface in 3D space of principal stresses

Figure 6: Mohr-Coulomb yield surface in 2D space (

σ 1,σ 2

)

The following formula was used to plot the surface in Fig. 5 :

$\max\left(\cfrac{|\sigma_1 - \sigma_2|}{2}\ - c + K \cfrac{\sigma_1 + \sigma_2}{2}\ ,\ \cfrac{|\sigma_2 - \sigma_3|}{2}\ - c + K \cfrac{\sigma_2 + \sigma_3}{2}\ ,\ \cfrac{|\sigma_3 - \sigma_1|}{2}\ - c + K \cfrac{\sigma_3 + \sigma_1}{2} \right) = 0$

[edit] Drucker-Prager yield surface

The Drucker-Prager yield criterion has the function form

$f(I_1, J_2) = 0 ~.$

This criterion is most often used for concrete where both normal and shear stresses can determine failure. The Drucker-Prager yield criterion may be expressed as

$\alpha \left( \sigma_1 + \sigma_2 + \sigma_3 \right) + \sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2}{6}} = K$

where

$m = \frac {\sigma_c}{\sigma_t} ~;~~ K = \frac {2 \sigma_c}{\sqrt{3} (m+1)} ~;~~ \alpha = \frac {m-1}{\sqrt{3}(m+1)}$

and $σ c,σ t$ are the uniaxial yield stresses in compression and tension respectively.

Figure 7 shows Drucker-Prager yield surface in the three-dimensional space of principal stresses. It is a regular cone. Figure 8 shows Drucker-Prager yield surface in two-dimensional space. The ellipsoidal-shaped elastic domain is a cross section of the cone on the plane of $σ 1,σ 2$ and encloses the elastic domain for the Mohr-Coulomb yield criterion.

Figure 7: View of Drucker-Prager yield surface in 3D space of principal stresses

Figure 8: Drucker-Prager and Mohr-Coulomb yield surface in 2D space (

σ 1,σ 2

)

[edit] Bresler-Pister yield surface

The Bresler-Pister yield criterion is an extension of the Drucker-Prager yield criterion that uses three parameters.

The Bresler-Pister yield surface has the functional form

$f(I_1,J_2) = 0 ~.$

In terms of the principal stresses, this yield criterion may be expressed as

$f := \cfrac{1}{\sqrt{6}}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right]^{1/2} - c_0 - c_1~(\sigma_1+\sigma_2+\sigma_3) - c_2~(\sigma_1+\sigma_2+\sigma_3)^2$

where $c 0, c 1, c 2$ are material constants. The additional parameter $c 2$ gives the yield surface a ellipsoidal cross section when viewed from a direction perpendicular to its axis. If $σ c$ is the yield stress in uniaxial compression, $σ t$ is the yield stress in uniaxial tension, and $σ b$ is the yield stress in biaxial compression, the parameters can be expressed as

$\begin{align} c_1 = & \left(\cfrac{\sigma_t-\sigma_c}{\sqrt{3}(\sigma_t+\sigma_c)}\right) \left(\cfrac{4\sigma_b^2 - \sigma_b(\sigma_c+\sigma_t) + \sigma_c\sigma_t}{4\sigma_b^2 + 2\sigma_b(\sigma_t-\sigma_c) - \sigma_c\sigma_t} \right) \\ c_2 = & \left(\cfrac{1}{\sqrt{3}(\sigma_t+\sigma_c)}\right) \left(\cfrac{\sigma_b(3\sigma_t-\sigma_c) -2\sigma_c\sigma_t}{4\sigma_b^2 + 2\sigma_b(\sigma_t-\sigma_c) - \sigma_c\sigma_t} \right) \\ c_0 = & \cfrac{\sigma_c}{\sqrt{3}} + c_1\sigma_c -c_2\sigma_c^2 \end{align}$

Figure 9: View of Bresler-Pister yield surface in 3D space of principal stresses

Figure 10: Bresler-Pister yield surface in 2D space (

σ 1,σ 2

)

[edit] Willam-Warnke yield surface

The Willam-Warnke yield criterion is a three-parameter smoothed version of the Mohr-Coulomb yield criterion that has similarities in form to the Drucker-Prager and Bresler-Pister yield criteria.

The yield criterion has the functional form

$f(I_1, J_2, J_3) = 0 ~.$

However, it is more commonly expressed in Haigh-Westergaard coordinates as

$f(\xi, \rho, \theta) = 0 ~.$

The cross-section of the surface when viewed along its axis is a smoothed triangle (unlike Mohr-Coulumb). The Willam-Warnke yield surface is convex and has unique and well defined first and second derivatives on every point of its surface. Therefore the Willam-Warnke model is computationally robust and has been used for a variety of cohesive-frictional materials.

Figure 11: View of Willam-Warnke yield surface in 3D space of principal stresses

Figure 12: Willam-Warnke yield surface in the

π

-plane

[edit] References

^ Lode, W. (1926). Versuche ueber den Einfuss der mitt leren Hauptspannung auf das Fliessen der Metalle Eisen Kupfer und Nickel. Zeitung Phys., vol. 36, pp. 913-939.
^ Nayak, G. C. and Zienkiewicz, O.C. (1972). Convenient forms of stress invariants for plasticity. Proceedings of the ASCE Journal of the Structural Division, vol. 98, no. ST4, pp. 949–954.
^ Tresca, H. (1864). Memoire sur l'ecoulement des corps solides soumis a defortes pressions. C.R. Acad. Sci. Paris, vol. 59, p. 754.

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Yield surface

From Wikipedia, the free encyclopedia

Contents

[edit] Invariants used to describe yield surfaces

[edit] Examples of yield surfaces

[edit] Tresca yield surface

[edit] von Mises yield surface

[edit] Mohr-Coulomb yield surface

[edit] Drucker-Prager yield surface

[edit] Bresler-Pister yield surface

[edit] Willam-Warnke yield surface

[edit] References

[edit] See also

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