Yield surface
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Yield surface is described in three dimensional space of principal stresses (<math> \sigma_1, \sigma_2 , \sigma_3</math>), and encompasses the elastic region of material behavior. The states of stress of material inside the yield surface are elastic, when the stress crosses this surface it reaches the yield point after which the material behaviour is plastic.
There are several different yield surfaces known in engineering, and those most popular are listed below.
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[edit] List of symbols used in this article
Following symbols are used below:
- <math>\sigma_1</math> - principal stress in direction along first axis
- <math>\sigma_2</math> - principal stress in direction along second axis
- <math>\sigma_3</math> - principal stress in direction along third axis
- <math>\sigma_0</math> - yield stress for single parametric yield surface
- <math>R_c</math> - yield strength during compression
- <math>R_r</math> - yield strength during tension
- <math>m</math> - ratio of yield strengths
- <math>c</math> - material cohesion
- <math>\alpha</math> - stress coefficient
- <math>K</math> - material stiffness for two parametric yield surface (<math>R_c, R_r</math>)
[edit] Tresca - Guest yield surface
This is the most simple yield surface, Henri Tresca is assumed as it's author. But also it is referred as TG criterion. In principal stresses it is expressed as follows:
- <math>{\max(|\sigma_1 - \sigma_2| , |\sigma_2 - \sigma_3| , |\sigma_3 - \sigma_1| ) = \sigma_0 }\!</math>
Figure 1 shows TG criterion in three dimensional space of principal stresses. It is a prism of infinite length and six sides. This means that material remains elastic when all three principal stresses are roughly equiwalent (a hydrostatic pressure), no matter how much compressed or stretched. But when the material is subject to shearing, one of principal stresses becomes smaller (or bigger), then the yield surface is crossed and material enters plastic domain.
Figure 2 shows Tresca-Guest criterion in two dimensional space, it is a cross section of prism along the <math> \sigma_1, \sigma_2</math> plane.
[edit] Huber - Mises - Hencky, also known as Prandtl - Rauss yield surface
This is another simple yield surface, this explains why it has so many authors. Who is the real author depends on the university, although often it is credited to Maximilian Huber and Richard von Mises (see von Mises stress). It is also referred as HMH criterion. It is expressed as follows:
- <math>{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 = 2 {\sigma_0}^2 }\!</math>
Also it can be expressed in non-principal stresses as below:
- <math>(\sigma_{11} - \sigma_{22})^2 + (\sigma_{22} - \sigma_{33})^2 + (\sigma_{33} - \sigma_{11})^2 + 6 ({\sigma_{12}}^2 + {\sigma_{13}}^2 + {\sigma_{23}}^2 + ) = 2 {\sigma_0}^2 </math>
Figure 3 shows HMH criterion in three dimensional space of principal stresses. It is a circular cylinder of infinite length, with the same angle to all three axes.
Figure 4 shows Huber-Mises-Hencky criterion in two dimensional space compared with Tresca-Guest criterion. HMH is a cross section of this cylinder on the plane of <math> \sigma_1, \sigma_2</math>, which produces an ellipse.
[edit] Mohr - Coulomb yield surface
It is a first two-parametric yield surface, the parameters are <math>R_c</math> and <math>R_r</math> which are the maximum values for compression and tension for given material. This model is ofen used to model concrete, soil or granular materials. This model is the first one that takes shearing into account. It is expressaed as follows:
- <math>m = \frac {R_c}{R_r} </math>
- <math>K = \frac {m-1}{m+1} </math>
- <math>c = {R_c} \frac {1}{m+1} = {R_r} \frac {m}{m+1} </math>
- <math>
\left\{{\begin{matrix}
\pm\cfrac{\sigma_1 - \sigma_2}{2} = c - K \cfrac{\sigma_1 + \sigma_2}{2} \\
\pm\cfrac{\sigma_2 - \sigma_3}{2} = c - K \cfrac{\sigma_2 + \sigma_3}{2} \\
\pm\cfrac{\sigma_3 - \sigma_1}{2} = c - K \cfrac{\sigma_3 + \sigma_1}{2}
\end{matrix}}\right. </math>
To plot this surface on Fig. 5 the following formula was used:
- <math>\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</math>
Figure 5 shows Mohr-Coulomb criterion in three dimensional space of principal stresses. It is a conical prism. If <math>K=0</math> then it becomes Tresca-Guest criterion, thus <math>K</math> determines the inclination angle of conical surface.
Figure 6 shows Mohr-Coulomb criterion in two dimensional space, it is a cross section of this conical prism on the plane of <math> \sigma_1, \sigma_2</math>, which produces a shape shown below.
[edit] Drucker - Prager yield surface
This criterion is most often used for concrete, both normal and shear stresses are taken into account.
- <math>m = \frac {R_c}{R_r} </math>
- <math>K = \frac {2 \cdot R_c}{\sqrt{3}\cdot(m+1)} </math>
- <math>\alpha = \frac {m-1}{\sqrt{3}\cdot(m+1)} </math>
- <math>\alpha \cdot \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 </math>
Figure 7 shows Drucker-Prager criterion in three dimensional space of principal stresses. It is a regular cone.
Figure 8 shows Drucker-Prager criterion in two dimensional space, it is a cross section of this cone on the plane of <math> \sigma_1, \sigma_2</math>, which produces an ellipsioidal shape. It is compared here with Mohr-Colulomb criterion.
[edit] Brestler - Pister criterion
This criterion is a first criterion that uses three parameters. It is similar to HMH criterion but additional parameter affects the cylinder radius using an <math>x^2</math> function. Thus cylinder's section along its axis is no longer a rectangle (or rather two parallel lines, since the cylinder has infinite length) but a parabola.
[edit] Willam - Warnke criterion
This is the most advanced yield surface, it takes the idea from Brestler - Pister a bit further and applies it to Mohr-Colulomb criterion. The resulting surface is smooth (unlike Mohr-Colulumb) and has first and second derivative fully defined on every point of its surface which is an important property. This smoothness allows optimisations during calculations when searching for a yield point on the surface (using gradient method for instance).
