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Yield surfaceYield surface is described in three dimensional space of stresses, and encompasses the elastic region of material behavior. The states of stress of material inside the yield surface are elastic, when the stress reaches this surface it reaches the yield point. Then the material behaviour becomes plastic, because the stress cannot cross this surface. Useful means of describing yield surface include expressing it in the terms of principal stresses (σ_{1},σ_{2},σ_{3}), or using stress invariants (I_{1},I_{2},I_{3}). There are several different yield surfaces known in engineering, and those most popular are listed below. Additional recommended knowledge
List of symbols used in this articleFollowing symbols are used below:
Tresca  Guest yield surfaceThis is the most simple yield surface, and it is taken to be the work of Henri Tresca. It is also referred as the TG criterion. In terms of the principal stresses it is expressed as Figure 1 shows the TG criterion 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 the material is subject to shearing and one of principal stresses becomes smaller (or larger) than the others, then the yield surface is reached and material enters the plastic domain. Figure 2 shows the TrescaGuest criterion in two dimensional space, it is a cross section of the prism along the σ_{1},σ_{2} plane. Huber  Mises  Hencky, also known as Prandtl  Reuss yield surfaceThis is another simple yield surface, which perhaps explains why it is credited to 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 to as the HMH criterion. It is expressed in the principal stresses as In the nonprincipal stresses, it takes the form of Figure 3 shows the HMH criterion in the 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 the HuberMisesHencky criterion in two dimensional space compared with TrescaGuest criterion. HMH is a cross section of this cylinder on the plane of σ_{1},σ_{2}, which produces an ellipse.
Mohr  Coulomb yield surfaceIt is a first twoparametric yield surface, the parameters are R_{c} and R_{r} which are the maximum values for compression and tension for given material. This model is often used to model concrete, soil or granular materials. This model is the first one that takes shearing into account. It is expressaed as follows: To plot this surface on Fig. 5 the following formula was used: Figure 5 shows MohrCoulomb criterion in three dimensional space of principal stresses. It is a conical prism. If K = 0 then it becomes TrescaGuest criterion, thus K determines the inclination angle of conical surface. Figure 6 shows MohrCoulomb criterion in two dimensional space, it is a cross section of this conical prism on the plane of σ_{1},σ_{2}, which produces a shape shown below. Drucker  Prager yield surfaceThis criterion is most often used for concrete, both normal and shear stresses are taken into account.
Figure 8 shows DruckerPrager criterion in two dimensional space, it is a cross section of this cone on the plane of σ_{1},σ_{2}, which produces an ellipsioidal shape. It is compared here with MohrColulomb criterion. Brestler  Pister criterionThis criterion is a first criterion that uses three parameters. It is similar to HMH criterion but additional parameter affects the cylinder radius using an x^{2} 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. Willam  Warnke criterionThis is the most advanced yield surface, it takes the idea from Brestler  Pister a bit further and applies it to MohrColulomb criterion. The resulting surface is smooth (unlike MohrColulumb) 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). See also
Categories: Continuum mechanics  Materials science 

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Yield_surface". A list of authors is available in Wikipedia. 