von Mises yield surface

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

where is the elastic limit in uniaxial tension.

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

Burzyński-Yagn criterion

This criterion takes the form[5][6]​.

and represents the general equation of a second-order surface of revolution around the hydrostatic axis. Some special cases are:[7]​.

Compression-tension and torsion-tension relationships can be calculated for.

Poisson's ratios in tension and compression are obtained using.

For ductile materials the restriction.

It's important. The application of rotationally symmetric criteria for brittle failure with.

It has not been sufficiently studied.[8]​.

The Burzyński-Yagn criterion is very suitable for academic purposes. For practical applications, the third invariant of the deviator in even and odd power must be introduced into the equation, for example: [9]​.

Huber criterion

The Huber criterion consists of the Beltrami ellipsoid and a von Mises cylinder scaled in the principal stress space,[10]​[11]​[12]​[13]​ see also[14]​[15]​.

with . The transition between the surfaces in the cross section is continuously differentiable.

The criterion represents the "classical view" regarding inelastic material behavior:

The Huber criterion can be used as a yield surface with an empirical constraint for the Poisson ratio in stress, leading to .

The modified Huber criterion,[16]​[15]​ see also,[17]​ cf.[18]​.

It consists of the Schleicher ellipsoid with the constraint of the Poisson ratio in compression.

and a cylinder with the transition in the cross section. The second configuration of the parameters and continues with the compression/tension relationship.

The modified Huber criterion may be better suited to measured data as the Huber criterion. To configure follow and .

The Huber criterion and the modified Huber criterion should be preferred to the von Mises criterion, since safer results are obtained in the region.

For practical applications, the third invariant of the deviator should be considered in these criteria.[15]​.

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, soils or granular materials. The Mohr-Coulomb creep criterion can be expressed as:.

where.

and the parameters and are the yield stresses (failure) of the material in compression and uniaxial tension, respectively. The formula reduces to the Tresca criterion if .

Figure 5 shows the Mohr–Coulomb yield surface in three-dimensional principal stress space. It is a conical prism and determines the angle of inclination of the conical surface. Figure 6 shows the Mohr–Coulomb yield surface in a two-dimensional stress space. In Figure 6, and are used for and , respectively, in the formula. It is a cross section of this conical prism in the plane of . 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 different 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 creep criterion can be expressed as.

where.

and , are the uniaxial yield stresses in compression and tension respectively. The formula reduces to the von Mises equation if .

Figure 7 shows the Drucker-Prager yield surface in three-dimensional principal stress space. It is a normal "Cone (geometry)" cone. Figure 8 shows the Drucker-Prager yield surface in two-dimensional space. The elliptical elastic domain is a cross section of the cone in the plane of ; can be chosen to cross the Mohr-Coulomb yield surface at different numbers of vertices. One option is to intersect the Mohr-Coulomb yield surface at three vertices on either side of the line, but these are usually selected by convention to be those in the compression regime.[19] Another option is to intersect the Mohr-Coulomb yield surface at four vertices on both axes (uniaxial fit) or at two vertices on the diagonal (biaxial fit).[20] Drucker-Prager is also commonly expressed in terms of the cohesion of the material and the angle of friction.

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 can be expressed as:.

where are material constants. The additional parameter gives the yield surface an ellipsoidal cross section when viewed from a direction perpendicular to its axis. If it is the elastic limit in uniaxial compression, it is the elastic limit in uniaxial tension and it is the elastic limit 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 creep criterion has the functional form.

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

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

Trigonometric yield surfaces of Podgórski and Rosendahl

Normalized with respect to the uniaxial tensile stress, the Podgórski criterion[21]​ depending on the tension angle states that.

with the form function of trigonal symmetry in the plane.

It contains the von Mises criteria (circle in the plane, , ), Tresca (regular hexagon, , ), Mariotte (regular triangle, , ), Ivlev[22]​ (regular triangle, , ) and also the Sayir cubic criterion[23]​ (the Ottosen criterion[24]​) with and the isotoxal (equilateral) hexagons of the criterion Capurso[22][23][25]​ with . The von Mises - Tresca transition[26]​ is followed with , . The isogonal (equiangular) hexagons of the Haythornthwaite criterion[15]​[27]​[28]​ that contain the Schmidt-Ishlinsky criterion (regular hexagon) cannot be described by the Podgórski criterion.

The Rosendahl criterion[29][30]​ says that.

with the shape function of hexagonal symmetry in the plane.

It contains the von Mises criteria (circle, , ), Tresca (regular hexagon, , ), Schmidt-Ishlinsky (regular hexagon, , ), Sokolovsky (regular dodecagon, , ), and also the bicubic criterion[15]​[29]​[31]​[32]​ with or equally with and the isotoxal dodecagons of the unified creep criterion of Yu[33]​ with . The isogonal dodecagons of the ansatz multiplicative criterion of hexagonal symmetry[15]​ that contain the Ishlinsky-Ivlev criterion (regular dodecagon) cannot be described by the Rosendahl criterion.

The Podgórski and Rosendahl criteria describe individual surfaces in the principal stress space without additional outer contours or planar intersections. Note that to avoid numerical problems, the real part function can be introduced in the form function: and . Generalization in the form [29]​ is relevant for theoretical investigations.

A pressure-sensitive extension of the criteria can be obtained with linear substitution [15]​.

which is sufficient for many applications, such as various metals, cast iron, alloys, concrete, unreinforced polymers and others.

Bigoni-Piccolroaz yield surface

The Bigoni-Piccolroaz yield criterion")[34]​[35]​ is a seven-parameter surface defined by.

where is the "meridian" function.

which describes the pressure sensitivity and is the "deviator" function[36].

describing the Lode dependence of creep. The seven non-negative material parameters are:.

defining the shape of the meridian and deviator sections.

This criterion represents a smooth and convex surface, which is closed in both tension and hydrostatic compression and has a drop shape, especially suitable for describing granular and friction materials. This criterion has also been generalized to the case of surfaces with corners.[37]​.

Cosine Ansatz (Altenbach-Bolchoun-Kolupaev)

For the formulation of strength criteria, the stress angle can be used:.

The following criterion for the behavior of isotropic material.

contains a number of other well-known less general criteria, provided that appropriate parameter values ​​are chosen.

The parameters and describe the geometry of the surface in the plane. They are subject to restrictions.

which follow from the convexity condition. A more precise formulation of the third restriction is proposed in [38][39].

The parameters and describe the position of the intersection points of the yield surface with the hydrostatic axis (diagonal space in the principal stress space). These intersection points are called hydrostatic nodes. In the case of materials that do not fail under hydrostatic pressure (steel, brass, etc.), . Otherwise, for materials that fail under hydrostatic pressure (hard foams, ceramics, sintered materials, etc.), it follows that .

The integer powers and , describe the curvature of the meridian. The meridian with is a straight line and with is a parabola.

Barlat yield surface

For anisotropic materials, depending on the direction of the applied process (e.g. lamination), the mechanical properties vary and therefore it is crucial to use an anisotropic yield function. Since 1989 Frédéric Barlat") developed a family of yield functions for constitutive modeling of plastic anisotropy. Among them, the Yld2000-2D yield criterion has been applied for a wide range of metal sheets (e.g., aluminum alloys and advanced high-strength steels). The Yld2000-2D model is a non-quadratic type yield function based on two linear transformations of the tension:.

and:.

where are eight parameters of the Barlat Yld2000-2D model that will be identified with a set of experiments.

Tresca yield surface

The Tresca yield criterion is considered the work of Henri Tresca.[3] It is also known as "maximum shear stress theory" (MSST) and Tresca-Guest criterion[4]​ (TG). In terms of the main tensions, the Tresca criterion is expressed as.

Where is the elastic limit in shear and is the elastic limit in tension.

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

von Mises yield surface

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

where is the elastic limit in uniaxial tension.

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

Burzyński-Yagn criterion

This criterion takes the form[5][6]​.

and represents the general equation of a second-order surface of revolution around the hydrostatic axis. Some special cases are:[7]​.

Compression-tension and torsion-tension relationships can be calculated for.

Poisson's ratios in tension and compression are obtained using.

For ductile materials the restriction.

It's important. The application of rotationally symmetric criteria for brittle failure with.

It has not been sufficiently studied.[8]​.

The Burzyński-Yagn criterion is very suitable for academic purposes. For practical applications, the third invariant of the deviator in even and odd power must be introduced into the equation, for example: [9]​.

Huber criterion

The Huber criterion consists of the Beltrami ellipsoid and a von Mises cylinder scaled in the principal stress space,[10]​[11]​[12]​[13]​ see also[14]​[15]​.

with . The transition between the surfaces in the cross section is continuously differentiable.

The criterion represents the "classical view" regarding inelastic material behavior:

The Huber criterion can be used as a yield surface with an empirical constraint for the Poisson ratio in stress, leading to .

The modified Huber criterion,[16]​[15]​ see also,[17]​ cf.[18]​.

It consists of the Schleicher ellipsoid with the constraint of the Poisson ratio in compression.

and a cylinder with the transition in the cross section. The second configuration of the parameters and continues with the compression/tension relationship.

The modified Huber criterion may be better suited to measured data as the Huber criterion. To configure follow and .

The Huber criterion and the modified Huber criterion should be preferred to the von Mises criterion, since safer results are obtained in the region.

For practical applications, the third invariant of the deviator should be considered in these criteria.[15]​.

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, soils or granular materials. The Mohr-Coulomb creep criterion can be expressed as:.

where.

and the parameters and are the yield stresses (failure) of the material in compression and uniaxial tension, respectively. The formula reduces to the Tresca criterion if .

Figure 5 shows the Mohr–Coulomb yield surface in three-dimensional principal stress space. It is a conical prism and determines the angle of inclination of the conical surface. Figure 6 shows the Mohr–Coulomb yield surface in a two-dimensional stress space. In Figure 6, and are used for and , respectively, in the formula. It is a cross section of this conical prism in the plane of . 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 different 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 creep criterion can be expressed as.

where.

and , are the uniaxial yield stresses in compression and tension respectively. The formula reduces to the von Mises equation if .

Figure 7 shows the Drucker-Prager yield surface in three-dimensional principal stress space. It is a normal "Cone (geometry)" cone. Figure 8 shows the Drucker-Prager yield surface in two-dimensional space. The elliptical elastic domain is a cross section of the cone in the plane of ; can be chosen to cross the Mohr-Coulomb yield surface at different numbers of vertices. One option is to intersect the Mohr-Coulomb yield surface at three vertices on either side of the line, but these are usually selected by convention to be those in the compression regime.[19] Another option is to intersect the Mohr-Coulomb yield surface at four vertices on both axes (uniaxial fit) or at two vertices on the diagonal (biaxial fit).[20] Drucker-Prager is also commonly expressed in terms of the cohesion of the material and the angle of friction.

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 can be expressed as:.

where are material constants. The additional parameter gives the yield surface an ellipsoidal cross section when viewed from a direction perpendicular to its axis. If it is the elastic limit in uniaxial compression, it is the elastic limit in uniaxial tension and it is the elastic limit 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 creep criterion has the functional form.

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

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

Trigonometric yield surfaces of Podgórski and Rosendahl

Normalized with respect to the uniaxial tensile stress, the Podgórski criterion[21]​ depending on the tension angle states that.

with the form function of trigonal symmetry in the plane.

It contains the von Mises criteria (circle in the plane, , ), Tresca (regular hexagon, , ), Mariotte (regular triangle, , ), Ivlev[22]​ (regular triangle, , ) and also the Sayir cubic criterion[23]​ (the Ottosen criterion[24]​) with and the isotoxal (equilateral) hexagons of the criterion Capurso[22][23][25]​ with . The von Mises - Tresca transition[26]​ is followed with , . The isogonal (equiangular) hexagons of the Haythornthwaite criterion[15]​[27]​[28]​ that contain the Schmidt-Ishlinsky criterion (regular hexagon) cannot be described by the Podgórski criterion.

The Rosendahl criterion[29][30]​ says that.

with the shape function of hexagonal symmetry in the plane.

It contains the von Mises criteria (circle, , ), Tresca (regular hexagon, , ), Schmidt-Ishlinsky (regular hexagon, , ), Sokolovsky (regular dodecagon, , ), and also the bicubic criterion[15]​[29]​[31]​[32]​ with or equally with and the isotoxal dodecagons of the unified creep criterion of Yu[33]​ with . The isogonal dodecagons of the ansatz multiplicative criterion of hexagonal symmetry[15]​ that contain the Ishlinsky-Ivlev criterion (regular dodecagon) cannot be described by the Rosendahl criterion.

The Podgórski and Rosendahl criteria describe individual surfaces in the principal stress space without additional outer contours or planar intersections. Note that to avoid numerical problems, the real part function can be introduced in the form function: and . Generalization in the form [29]​ is relevant for theoretical investigations.

A pressure-sensitive extension of the criteria can be obtained with linear substitution [15]​.

which is sufficient for many applications, such as various metals, cast iron, alloys, concrete, unreinforced polymers and others.

Bigoni-Piccolroaz yield surface

The Bigoni-Piccolroaz yield criterion")[34]​[35]​ is a seven-parameter surface defined by.

where is the "meridian" function.

which describes the pressure sensitivity and is the "deviator" function[36].

describing the Lode dependence of creep. The seven non-negative material parameters are:.

defining the shape of the meridian and deviator sections.

This criterion represents a smooth and convex surface, which is closed in both tension and hydrostatic compression and has a drop shape, especially suitable for describing granular and friction materials. This criterion has also been generalized to the case of surfaces with corners.[37]​.

Cosine Ansatz (Altenbach-Bolchoun-Kolupaev)

For the formulation of strength criteria, the stress angle can be used:.

The following criterion for the behavior of isotropic material.

contains a number of other well-known less general criteria, provided that appropriate parameter values ​​are chosen.

The parameters and describe the geometry of the surface in the plane. They are subject to restrictions.

which follow from the convexity condition. A more precise formulation of the third restriction is proposed in [38][39].

The parameters and describe the position of the intersection points of the yield surface with the hydrostatic axis (diagonal space in the principal stress space). These intersection points are called hydrostatic nodes. In the case of materials that do not fail under hydrostatic pressure (steel, brass, etc.), . Otherwise, for materials that fail under hydrostatic pressure (hard foams, ceramics, sintered materials, etc.), it follows that .

The integer powers and , describe the curvature of the meridian. The meridian with is a straight line and with is a parabola.

Barlat yield surface

For anisotropic materials, depending on the direction of the applied process (e.g. lamination), the mechanical properties vary and therefore it is crucial to use an anisotropic yield function. Since 1989 Frédéric Barlat") developed a family of yield functions for constitutive modeling of plastic anisotropy. Among them, the Yld2000-2D yield criterion has been applied for a wide range of metal sheets (e.g., aluminum alloys and advanced high-strength steels). The Yld2000-2D model is a non-quadratic type yield function based on two linear transformations of the tension:.

and:.

where are eight parameters of the Barlat Yld2000-2D model that will be identified with a set of experiments.