Yield surface

A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and 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 shape and size of the surface may change as the plastic deformation evolves. This is because stress states that lie outside the yield surface are non-permissible in rate-independent plasticity, though not in some models of viscoplasticity.
The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space (formula_1), a two- or three-dimensional space spanned by stress invariants (formula_2) or a version of the three-dimensional Haigh–Westergaard stress space. Thus we may write the equation of the yield surface (that is, the yield function) in the forms:
Invariants used to describe yield surfaces.
The first principal invariant (formula_6) of the Cauchy stress (formula_20), and the second and third principal invariants (formula_7) of the "deviatoric" part (formula_22) of the Cauchy stress are defined as:
where (formula_1) are the principal values of formula_20, (formula_26) are the principal values of formula_22, and 
where formula_29 is the identity matrix.
A related set of quantities, (formula_30), are usually used to describe yield surfaces for cohesive frictional materials such as rocks, soils, and ceramics. These are defined as
where formula_32 is the equivalent stress. However, the possibility of negative values of formula_33 and the resulting imaginary formula_12 makes the use of these quantities problematic in practice.
Another related set of widely used invariants is (formula_35) which describe a cylindrical coordinate system (the Haigh–Westergaard coordinates). These are defined as:
The formula_37 plane is also called the Rendulic plane. The angle formula_18 is called stress angle, the value formula_39 is sometimes called the Lode parameter and the relation between formula_18 and formula_41 was first given by Novozhilov V.V. in 1951, see also 
The principal stresses and the Haigh–Westergaard coordinates are related by
A different definition of the Lode angle can also be found in the literature:
in which case the ordered principal stresses (where formula_44) are related by
Examples of yield surfaces.
There are several different yield surfaces known in engineering, and those most popular are listed below.
Tresca yield surface.
The Tresca yield criterion is taken to be the work of Henri Tresca. It is also known as the "maximum shear stress theory" (MSST) and the Tresca–Guest (TG) criterion. In terms of the principal stresses the Tresca criterion is expressed as
Where formula_47 is the yield strength in shear, and formula_48 is the tensile yield strength.
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 the 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 formula_49 plane.
von Mises yield surface.
The von Mises yield criterion is expressed in the principal stresses as
where formula_48 is the yield strength 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 formula_49 produces the elliptical shape of the yield surface.
Burzyński-Yagn criterion.
This criterion
represents the general equation of a second order surface of revolution about the hydrostatic axis. Some special case are:
The relations compression-tension and torsion-tension can be computed to
The Poisson's ratios at tension and compression are obtained using
For ductile materials the restriction
is important. The application of rotationally symmetric criteria for brittle failure with
has not been studied sufficiently.
The Burzyński-Yagn criterion is well suited for academic purposes. For practical applications, the third invariant of the deviator in the odd and even power should be introduced in the equation, e.g.:
Huber criterion.
The Huber criterion consists of the Beltrami ellipsoid and a scaled von Mises cylinder in the principal stress space, see also
with formula_74. The transition between the surfaces in the cross section formula_75 is continuously differentiable.
The criterion represents the "classical view" with respect to inelastic material behavior:
The Huber criterion can be used as a yield surface with an empirical restriction for Poisson's ratio at tension formula_80, which leads to formula_81.
The modified Huber criterion, see also 
consists of the Schleicher ellipsoid with the restriction of Poisson's ratio at compression
and a cylinder with the formula_84-transition in the cross section formula_85. 
The second setting for the parameters formula_74 and formula_87 follows with the compression / tension relation
The modified Huber criterion can be better fitted to the measured data as the Huber criterion. For setting formula_89 it follows formula_90 and formula_91.
The Huber criterion and the modified Huber criterion should be preferred to the von Mises criterion since one obtains safer results in the region formula_92.
For practical applications the third invariant of the deviator formula_93 should be considered in these criteria.
Mohr–Coulomb yield surface.
The Mohr–Coulomb yield (failure) criterion is similar to the Tresca criterion, with additional provisions for materials with different tensile and compressive yield strengths. This model is often used to model concrete, soil or granular materials. The Mohr–Coulomb yield criterion may be expressed as:
where
and the parameters formula_96 and formula_97 are the yield (failure) stresses of the material in uniaxial compression and tension, respectively. The formula reduces to the Tresca criterion if formula_98.
Figure 5 shows Mohr–Coulomb yield surface in the three-dimensional space of principal stresses. It is a conical prism and formula_99 determines the inclination angle of conical surface. Figure 6 shows Mohr–Coulomb yield surface in two-dimensional stress space. In Figure 6 formula_100 and formula_101 is used for formula_97 and formula_96, respectively, in the formula. It is a cross section of this conical prism on the plane of formula_49. In Figure 6 Rr and Rc are used for Syc and Syt, respectively, in the formula.
Drucker–Prager yield surface.
The Drucker–Prager yield criterion is similar to the von Mises yield criterion, with provisions for handling materials with differing tensile and compressive yield strengths. 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
where
and formula_96, formula_97 are the uniaxial yield stresses in compression and tension respectively. The formula reduces to the von Mises equation if formula_98.
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 elliptical elastic domain is a cross section of the cone on the plane of formula_49; it can be chosen to intersect the Mohr–Coulomb yield surface in different number of vertices. One choice is to intersect the Mohr–Coulomb yield surface at three vertices on either side of the formula_111 line, but usually selected by convention to be those in the compression regime. Another choice is to intersect the Mohr–Coulomb yield surface at four vertices on both axes (uniaxial fit) or at two vertices on the diagonal formula_112 (biaxial fit). The Drucker-Prager yield criterion is also commonly expressed in terms of the material cohesion and friction angle.
Bresler–Pister yield surface.
The Bresler–Pister yield criterion is an extension of the Drucker Prager yield criterion that uses three parameters, and has additional terms for materials that yield under hydrostatic compression.
In terms of the principal stresses, this yield criterion may be expressed as
where formula_114 are material constants. The additional parameter formula_115 gives the yield surface an ellipsoidal cross section when viewed from a direction perpendicular to its axis. If formula_116 is the yield stress in uniaxial compression, formula_117 is the yield stress in uniaxial tension, and formula_118 is the yield stress in biaxial compression, the parameters can be expressed as
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
However, it is more commonly expressed in Haigh–Westergaard coordinates as
The cross-section of the surface when viewed along its axis is a smoothed triangle (unlike Mohr–Coulomb). 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.
Podgórski and Rosendahl trigonometric yield surfaces.
Normalized with respect to the uniaxial tensile stress formula_122, the Podgórski criterion as function of the stress angle formula_18 reads
with the shape function of trigonal symmetry in the formula_125-plane
It contains the criteria of von Mises (circle in the formula_125-plane, formula_128, formula_129), Tresca (regular hexagon, formula_130, formula_131), Mariotte (regular triangle, formula_132, formula_131), Ivlev (regular triangle, formula_134, formula_131) and also the cubic criterion of Sayir (the Ottosen criterion ) with formula_136 and the isotoxal (equilateral) hexagons of the Capurso criterion with formula_131. The von Mises - Tresca transition follows with formula_130, formula_139. The isogonal (equiangular) hexagons of the Haythornthwaite criterion containing the Schmidt-Ishlinsky criterion (regular hexagon) cannot be described with the Podgórski ctiterion.
The Rosendahl criterion reads
with the shape function of hexagonal symmetry in the formula_125-plane
It contains the criteria of von Mises (circle, formula_143, formula_144), Tresca (regular hexagon, formula_145, formula_146), Schmidt—Ishlinsky (regular hexagon, formula_147, formula_146), Sokolovsky (regular dodecagon, formula_149, formula_146), and also the bicubic criterion with formula_151 or equally with formula_152 and the isotoxal dodecagons of the unified yield criterion of Yu with formula_146. The isogonal dodecagons of the multiplicative ansatz criterion of hexagonal symmetry containing the Ishlinsky-Ivlev criterion (regular dodecagon) cannot be described by the Rosendahl criterion.
The criteria of Podgórski and Rosendahl describe single surfaces in principal stress space without any additional outer contours and plane intersections. Note that in order to avoid numerical issues the real part function formula_154 can be introduced to the shape function: formula_155 and formula_156. The generalization in the form formula_157 is relevant for theoretical investigations.
A pressure-sensitive extension of the criteria can be obtained with the linear formula_6-substitution 
which is sufficient for many applications, e.g. metals, cast iron, alloys, concrete, unreinforced polymers, etc.
Bigoni–Piccolroaz yield surface.
The Bigoni–Piccolroaz yield criterion is a seven-parameter surface defined by
where formula_161 is the "meridian" function
describing the pressure-sensitivity and formula_164 is the "deviatoric" function
describing the Lode-dependence of yielding. The seven, non-negative material parameters:
define the shape of the meridian and deviatoric sections.
This criterion represents a smooth and convex surface, which is closed both in hydrostatic tension and compression and has a 
drop-like shape, particularly suited to describe frictional and granular materials. This criterion has also been generalized to the case of surfaces with corners.
Cosine Ansatz (Altenbach-Bolchoun-Kolupaev).
For the formulation of the strength criteria the stress angle
can be used.
The following criterion of isotropic material behavior
contains a number of other well-known less general criteria, provided suitable parameter values are chosen.
Parameters formula_169 and formula_170 describe the geometry of the surface in the formula_125-plane. They are subject to the constraints
which follow from the convexity condition. A more precise formulation of the third constraints is proposed in.
Parameters formula_173 and formula_174 describe the position of the intersection points of the yield surface with hydrostatic axis (space diagonal in the principal stress space). These intersections points are called hydrostatic nodes.
In the case of materials which do not fail at hydrostatic pressure (steel, brass, etc.) one gets formula_175. Otherwise for materials which fail at hydrostatic pressure (hard foams, ceramics, sintered materials, etc.) it follows formula_87.
The integer powers formula_177 and formula_178, formula_179 describe the curvature of the meridian. The meridian with formula_180 is a straight line and with formula_181 – a parabola.
Barlat's Yield Surface.
For the anisotropic materials, depending on the direction of the applied process (e.g., rolling) the mechanical properties vary and, therefore, using an anisotropic yield function is crucial. Since 1989 Frederic Barlat has developed a family of yield functions for constitutive modelling of plastic anisotropy. Among them, Yld2000-2D yield criteria has been applied for a wide range of sheet metals (e.g., aluminum alloys and advanced high-strength steels). The Yld2000-2D model is a non-quadratic type yield function based on two linear transformation of the stress tensor:
for principal values of X’ and X”, the model could be expressed as: 
where formula_188 are eight parameters of the Barlat's Yld2000-2D model to be identified with a set of experiments.