Yield surface

{{Short description|Geometric representation of material yield}}  right|300px|thumb|Surfaces on which the invariants <math>I_1</math>, <math>J_2</math>, <math>J_3</math> are constant. Plotted in principal stress space. 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.<ref name=Simo>Simo, J. C. and Hughes, T,. J. R., (1998), Computational Inelasticity, Springer.</ref>

The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space (<math> \sigma_1, \sigma_2 , \sigma_3</math>), a two- or three-dimensional space spanned by stress invariants (<math> I_1, J_2, J_3</math>) 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:

*<math> f(\sigma_1,\sigma_2,\sigma_3) = 0 \,</math> where <math>\sigma_i</math> are the principal stresses. *<math> f(I_1, J_2, J_3) = 0 \,</math> where <math>I_1</math> is the first principal invariant of the Cauchy stress and <math>J_2, J_3</math> are the second and third principal invariants of the deviatoric part of the Cauchy stress. *<math> f(p, q, r) = 0 \,</math> where <math>p, q</math> are scaled versions of <math>I_1</math> and <math>J_2</math> and <math>r</math> is a function of <math>J_2, J_3</math>. *<math>f(\xi,\rho,\theta) = 0 \,</math> where <math>\xi,\rho</math> are scaled versions of <math>I_1</math> and <math>J_2</math>, and <math>\theta</math> is the stress angle<ref>Yu, M.-H. (2004), ''Unified strength theory and its applications''. Springer, Berlin</ref> or '''Lode angle'''<ref>Zienkiewicz O.C., Pande, G.N. (1977), Some useful forms of isotropic yield surfaces for soil and rock mechanics. In: Gudehus, G. (ed.) ''Finite Elements in Geomechanics''. Wiley, New York, pp. 179–198</ref>

== Invariants used to describe yield surfaces == right|300px|thumb|Surfaces on which the invariants <math>\xi</math>, <math>\rho</math>, <math>\theta</math> are constant. Plotted in principal stress space. The first principal invariant (<math>I_1</math>) of the Cauchy stress (<math>\boldsymbol{\sigma}</math>), and the second and third principal invariants (<math>J_2, J_3</math>) of the ''deviatoric'' part (<math>\boldsymbol{s}</math>) of the Cauchy stress are defined as: : :<math> \begin{align} I_1 & = \text{Tr}(\boldsymbol{\sigma}) = \sigma_1 + \sigma_2 + \sigma_3 \\ J_2 & = \tfrac{1}{2} \boldsymbol{s}:\boldsymbol{s} = \tfrac{1}{6}\left[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right] \\ J_3 & = \det(\boldsymbol{s}) = \tfrac{1}{3} (\boldsymbol{s}\cdot\boldsymbol{s}):\boldsymbol{s} = s_1 s_2 s_3 \end{align} </math> where (<math> \sigma_1, \sigma_2 , \sigma_3</math>) are the principal values of <math>\boldsymbol{\sigma}</math>, (<math>s_1, s_2, s_3</math>) are the principal values of <math>\boldsymbol{s}</math>, and :<math> \boldsymbol{s} = \boldsymbol{\sigma}-\tfrac{I_1}{3}\,\boldsymbol{I} </math> where <math>\boldsymbol{I}</math> is the identity matrix.

A related set of quantities, (<math>p, q, r\,</math>), are usually used to describe yield surfaces for cohesive frictional materials such as rocks, soils, and ceramics. These are defined as :<math> p = \tfrac{1}{3}~I_1 ~:~~ q = \sqrt{3~J_2} = \sigma_\mathrm{eq} ~;~~ r = 3\left(\tfrac{1}{2}\,J_3\right)^{1/3} </math> where <math>\sigma_\mathrm{eq}</math> is the '''equivalent stress'''. However, the possibility of negative values of <math>J_3</math> and the resulting imaginary <math>r</math> makes the use of these quantities problematic in practice.

Another related set of widely used invariants is (<math>\xi, \rho, \theta\,</math>) which describe a cylindrical coordinate system (the '''Haigh–Westergaard''' coordinates). These are defined as: :<math> \xi = \tfrac{1}{\sqrt{3}}~I_1 = \sqrt{3}~p ~;~~ \rho = \sqrt{2 J_2} = \sqrt{\tfrac{2}{3}}~q ~;~~ \cos(3\theta) = \left(\tfrac{r}{q}\right)^3 = \tfrac{3\sqrt{3}}{2}~\cfrac{J_3}{J_2^{3/2}} </math> The <math>\xi-\rho\,</math> plane is also called the '''Rendulic plane'''. The angle <math>\theta</math> is called stress angle, the value <math>\cos(3\theta)</math> is sometimes called the '''Lode parameter'''<ref>Lode, W. (1925). Versuche über den Einfluß der mittleren Hauptspannug auf die Fließgrenze. ''ZAMM'' 5(2), pp. 142–144</ref><ref>Lode, W. (1926). '' Versuche über den Einfuss der mittleren Hauptspannung auf das Fliessen der Metalle Eisen Kupfer und Nickel''. ''Zeitung Phys.'', vol. 36, pp. 913–939.</ref><ref>Lode, W. (1928). ''Der Einfluß der mittleren Hauptspannung auf das Fließen der Metalle''. Dissertation, Universität zu Göttingen. Forschungsarbeiten auf dem Gebiete des Ingenieurwesens, Heft 303, VDI, Berlin</ref> and the relation between <math>\theta</math> and <math>J_2,J_3</math> was first given by Novozhilov V.V. in 1951,<ref>Novozhilov, V.V. (1951). On the principles of the statical analysis of the experimental results for isotropic materials (in Russ.: O prinzipakh obrabotki rezultatov staticheskikh ispytanij izotropnykh materialov). ''Prikladnaja Matematika i Mekhanika'', XV(6):709–722.</ref> see also <ref>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.</ref>

The principal stresses and the Haigh–Westergaard coordinates are related by :<math> \begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \end{bmatrix} = \tfrac{1}{\sqrt{3}} \begin{bmatrix} \xi \\ \xi \\ \xi \end{bmatrix} + \sqrt{\tfrac{2}{3}}~\rho~\begin{bmatrix} \cos\theta \\ \cos\left(\theta-\tfrac{2\pi}{3}\right) \\ \cos\left(\theta+\tfrac{2\pi}{3}\right) \end{bmatrix} = \tfrac{1}{\sqrt{3}} \begin{bmatrix} \xi \\ \xi \\ \xi \end{bmatrix} + \sqrt{\tfrac{2}{3}}~\rho~\begin{bmatrix} \cos\theta \\ -\sin\left(\tfrac{\pi}{6}-\theta\right) \\ -\sin\left(\tfrac{\pi}{6}+\theta\right) \end{bmatrix} \,. </math> A different definition of the Lode angle can also be found in the literature:<ref name=chak>Chakrabarty, J., 2006, ''Theory of Plasticity: Third edition'', Elsevier, Amsterdam.</ref> :<math> \sin(3\theta) = ~\tfrac{3\sqrt{3}}{2}~\cfrac{J_3}{J_2^{3/2}} </math> in which case the ordered principal stresses (where <math>\sigma_1 \geq \sigma_2 \geq \sigma_3</math>) are related by<ref name=kayenta>Brannon, R.M., 2009, ''KAYENTA: Theory and User's Guide'', Sandia National Laboratories, Albuquerque, New Mexico.</ref> :<math> \begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \end{bmatrix} = \tfrac{1}{\sqrt{3}} \begin{bmatrix} \xi \\ \xi \\ \xi \end{bmatrix} + \tfrac{\rho}{\sqrt{2}}~\begin{bmatrix} \cos\theta - \tfrac{\sin\theta}{\sqrt{3}} \\ \tfrac{2\sin\theta}{\sqrt{3}} \\ -\tfrac{\sin\theta}{\sqrt{3}} - \cos\theta \end{bmatrix} \,. </math> '''Note'''. It is incorrect (false) information that Chakrabarty did define ''shear stress mode angle'' with sin(3Teta). Chakrabarty adopted Lode parameter with opposite sign than it has in the original Lode paper from 1926, see Page 59, formula (3) in the book «Chakrabarty, Jagabanduhu; 2006, Theory of Plasticity: Third edition, Elsevier, Amsterdam.». Actually, Chakrabarty defined shear stress mode angle with -sin(3Teta) just like it did for the first time V.V. Novozhilov in his work from 1951, Novozhilov V.V., «O связи между напряжениями и деформациями в нелинейно упругой среде» (On relations between stresses and strains in non-linear elastic media), Прикладная математика и механика, 1951, p. 183-194; <nowiki>https://pmm.ipmnet.ru/ru/get/1951/15-2/183-194</nowiki>. The first work in which rational arguments and discussion is delivered why it is ''wise and useful'' to define shear stress mode angle with sin(.) function, regardless of the sign of the angle theta, is the Ziolkowski’s work «Parametrization of Cauchy Stress Tensor Treated as Autonomous Object Using Isotropy Angle and Skewness Angle» from 2022, <nowiki>https://et.ippt.pan.pl/index.php/et/article/view/2210</nowiki>, where it is elucidated that adopting definition of ''shear stress mode angle'' with sin(.) function physically means accepting ''pure shears'' as ''comparison reference states'', and it is explained why it is very beneficial.<ref>{{Cite journal |last=Ziółkowski |first=Andrzej Grzegorz |date=2022-08-09 |title=Parametrization of Cauchy Stress Tensor Treated as Autonomous Object Using Isotropy Angle and Skewness Angle |url=https://et.ippt.pan.pl/index.php/et/article/view/2210 |journal=Engineering Transactions |language=en |volume=70 |issue=3 |pages=239–286 |doi=10.24423/EngTrans.2210.20220809 |issn=2450-8071}}</ref>

== 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.<ref>Tresca, H. (1864). ''Mémoire sur l'écoulement des corps solides soumis à de fortes pressions.'' C. R. Acad. Sci. Paris, vol. 59, p. 754.</ref> It is also known as the ''maximum shear stress theory'' (MSST) and the Tresca–Guest<ref>{{cite journal |last=Guest |first=James |date=1900 |title=V. On the strength of ductile materials under combined stress |url=https://www.tandfonline.com/doi/pdf/10.1080/14786440009463892 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=50 |issue=302 |pages=69-132 |archive-url=https://web.archive.org/web/20171028005317/http://www.tandfonline.com/doi/pdf/10.1080/14786440009463892 |archive-date=October 28, 2017 |access-date=November 16, 2025 |doi=10.1080/14786440009463892 |url-status=bot: unknown }}</ref> (TG) criterion. In terms of the principal stresses the Tresca criterion is expressed as :<math>\tfrac{1}{2}{\max(|\sigma_1 - \sigma_2| , |\sigma_2 - \sigma_3| , |\sigma_3 - \sigma_1| ) = S_{sy} = \tfrac{1}{2}S_y}\!</math> Where <math>S_{sy}</math> is the yield strength in shear, and <math>S_y</math> 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 <math> \sigma_1, \sigma_2</math> plane.

left|400px|thumb|Figure 1: View of Tresca–Guest yield surface in 3D space of principal stresses none|200px|thumb|Figure 2: Tresca–Guest yield surface in 2D space (<math> \sigma_1, \sigma_2</math>) {{Clear}}

=== von Mises yield surface === {{Main|von Mises yield criterion}} The von Mises yield criterion is expressed in the principal stresses as :<math> {(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 = 2 {S_y}^2 }\!</math> where <math>S_y</math> 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 <math> \sigma_1, \sigma_2</math> produces the elliptical shape of the yield surface.

left|400px|thumb|Figure 3: View of Huber–Mises–Hencky yield surface in 3D space of principal stresses none|200px|thumb|Figure 4: Comparison of Tresca–Guest and Huber–Mises–Hencky criteria in 2D space (<math> \sigma_1, \sigma_2</math>) {{Clear}}

=== Burzyński-Yagn criterion=== This criterion<ref>Burzyński, W. (1929). '' Über die Anstrengungshypothesen''. Schweizerische Bauzeitung, 94 (21), pp. 259–262.</ref><ref>Yagn, Yu. I. (1931). ''New methods of strength prediction (in Russ.: Novye metody pascheta na prochnost')''. Vestnik inzhenerov i tekhnikov, 6, pp. 237–244.</ref> reformulated as the function of the hydrostatic nodes with the coordinates <math> 1/\gamma_1 </math> and <math> 1/\gamma_2 </math>

:<math> 3I_2' = \frac{\sigma_\mathrm{eq}-\gamma_1I_1}{1-\gamma_1} \frac{\sigma_\mathrm{eq}-\gamma_2I_1}{1-\gamma_2} </math>

represents the general equation of a second order surface of revolution about the hydrostatic axis. Some special case are:<ref>Altenbach, H., Kolupaev, V.A. (2014) Classical and Non-Classical Failure Criteria, in Altenbach, H., Sadowski, Th., eds., ''Failure and Damage Analysis of Advanced Materials'', in press, Springer, Heidelberg (2014), pp. 1–66</ref>

* cylinder <math> \gamma_1 = \gamma_2 = 0 </math> (Maxwell (1865), Huber (1904), von Mises (1913), Hencky (1924)), * cone <math> \gamma_1 = \gamma_2 \in ]0,1[ </math> (Botkin (1940), Drucker-Prager (1952), Mirolyubov (1953)), * paraboloid <math>\gamma_1 \in ]0,1[, \gamma_2 = 0 </math> (Burzyński (1928), Balandin (1937), Torre (1947)), * ellipsoid centered of symmetry plane <math>I_1 = 0 </math>, <math>\gamma_1 = - \gamma_2 \in ]0,1[ </math> (Beltrami (1885)), * ellipsoid centered of symmetry plane <math>I_1 = \frac{1}{2}\,\bigg(\frac{1}{\gamma_1}+\frac{1}{\gamma_2} \bigg) </math> with <math>\gamma_1 \in ]0,1[, \gamma_2<0 </math> (Schleicher (1926)), * hyperboloid of two sheets <math>\gamma_1 \in ]0,1[, \gamma_2 \in ]0,\gamma_1[ </math> (Burzynski (1928), Yagn (1931)), * hyperboloid of one sheet centered of symmetry plane <math>I_1 = 0 </math>, <math>\gamma_1=-\gamma_2 =a\,i </math>, <math> i =\sqrt{-1} </math> (Kuhn (1980)) * hyperboloid of one sheet <math>\gamma_{1,2}= b \pm a\,i </math>, <math> i =\sqrt{-1} </math> (Filonenko-Boroditsch (1960), Gol’denblat-Kopnov (1968), Filin (1975)).

The relations compression-tension and torsion-tension can be computed to :<math> \frac{\sigma_-}{\sigma_+} =\frac{1}{1-\gamma_1-\gamma_2}, \qquad \bigg(\sqrt{3}\,\frac{\tau_*}{\sigma_+}\bigg)^2 = \frac{1}{(1-\gamma_1)(1-\gamma_2)} </math>

The Poisson's ratios at tension and compression are obtained using :<math> \nu_+^\mathrm{in} = \frac{-1+2(\gamma_1+\gamma_2)-3\gamma_1\gamma_2}{-2+\gamma_1+\gamma_2} </math> :<math> \nu_-^\mathrm{in} = - \frac{-1+ \gamma_1^2+\gamma_2^2-\gamma_1\,\gamma_2} {(-2+\gamma_1+\gamma_2)\,(-1+\gamma_1+\gamma_2)} </math>

For ductile materials the restriction :<math>\nu_+^\mathrm{in}\in \bigg[\,0.48,\,\frac{1}{2}\,\bigg] </math> is important. The application of rotationally symmetric criteria for brittle failure with :<math>\nu_+^\mathrm{in}\in ]-1,~\nu_+^\mathrm{el}\,] </math> has not been studied sufficiently.<ref>Beljaev, N. M. (1979). ''Strength of Materials''. Mir Publ., Moscow</ref>

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.:<ref>Bolchoun, A., Kolupaev, V. A., Altenbach, H. (2011) Convex and non-convex yield surfaces (in German: Konvexe und nichtkonvexe Fließflächen), ''Forschung im Ingenieurwesen'', 75 (2), pp. 73–92</ref>

:<math> 3I_2' \frac{1+c_3 \cos 3\theta+c_6 \cos^2 3\theta}{1+c_3+ c_6} = \frac{\sigma_\mathrm{eq}-\gamma_1I_1}{1-\gamma_1} \frac{\sigma_\mathrm{eq}-\gamma_2I_1}{1-\gamma_2} </math> {{Clear}}

=== Huber criterion=== The Huber criterion consists of the Beltrami ellipsoid and a scaled von Mises cylinder in the principal stress space,<ref>Huber, M. T. (1904). Specific strain work as a measure of material effort (in Polish: Właściwa praca odkształcenia jako miara wytężenia materyału), ''Czasopismo Techniczne'', Lwów, Organ Towarzystwa Politechnicznego we Lwowie, v. 22. pp. 34-40, 49-50, 61-62, 80-81</ref><ref>Föppl, A., Föppl, L. (1920). ''Drang und Zwang: eine höhere Festigkeitslehre für Ingenieure''. R. Oldenbourg, München</ref><ref>Burzyński, W. (1929). Über die Anstrengungshypothesen. ''Schweizerische Bauzeitung'' 94(21):259–262</ref><ref>Kuhn, P. (1980). ''Grundzüge einer allgemeinen Festigkeitshypothese'', Auszug aus Antrittsvorlesung des Verfassers vom 11. Juli, 1980 Vom Konstrukteur und den Festigkeitshypothesen. Inst. für Maschinenkonstruktionslehre, Karlsruhe</ref> see also<ref>Kolupaev, V.A., Moneke M., Becker F. (2004). Stress appearance during creep. Calculation of plastic parts (in German: Spannungsausprägung beim Kriechen: Berechnung von Kunststoffbauteilen). Kunststoffe 94(11):79–82</ref><ref name=":1 ">Kolupaev, V.A. (2018). ''Equivalent Stress Concept for Limit State Analysis'', Springer, Cham.</ref> :<math> 3\,I_2' = \left\{ \begin{array}{ll} \displaystyle\frac{\sigma_\mathrm{eq}-\gamma_1 \,I_1}{1-\gamma_1} \, \frac{\sigma_\mathrm{eq}+\gamma_1 \,I_1}{1+\gamma_1}, & I_1>0 \\[1em] \displaystyle\frac{\sigma_\mathrm{eq}}{1-\gamma_1}\, \frac{\sigma_\mathrm{eq}}{1+\gamma_1}, & I_1\leq 0 \end{array} \right. </math> with <math>\gamma_1\in[0, 1[</math>. The transition between the surfaces in the cross section <math>I_1=0</math> is continuously differentiable. The criterion represents the "classical view" with respect to inelastic material behavior: * pressure-sensitive material behavior for <math>I_1>0</math> with <math>\nu_+^\mathrm{in}\in\left]-1,\,1/2\right]</math> and * pressure-insensitive material behavior for <math>I_1<0</math> with <math>\nu_-^\mathrm{in}=1/2</math>

The Huber criterion can be used as a yield surface with an empirical restriction for Poisson's ratio at tension <math>\nu_+^\mathrm{in}\in[0.48, 1/2]</math>, which leads to <math>\gamma_1\in[0, 0.1155]</math>.

800px|thumb|left|Huber criterion with <math>\gamma_1=1/\sqrt{3}</math> and modified Huber criterion with <math>\gamma_1=(1+\sqrt{5})/6</math> and <math>\gamma_2=(1-\sqrt{5})/6</math> in the Burzyński-plane: setting according the normal stress hypothesis (<math>\nu_+^\mathrm{in}=0</math>). The von Mises criterion (<math>\nu_-^\mathrm{in}=\nu_+^\mathrm{in}=1/2</math>) is shown for comparison. C - uniaxial compression, Cc - biaxial compression in the stress relation 1:2, CC - equibiaxial compression, CCC - hydrostatic compression, S or TC - shear, T - uniaxial tension, Tt - biaxial tension in the stress relation 1:2, TT - equibiaxial tension, TTT - hydrostatic tension. {{Clear}}

The modified Huber criterion,<ref name=":12354">Kolupaev, V. A., (2006). ''3D-Creep Behaviour of Parts Made of Non-Reinforced Thermoplastics (in German: Dreidimensionales Kriechverhalten von Bauteilen aus unverstärkten Thermoplasten)'', Diss., Martin-Luther-Universität Halle-Wittenberg, Halle-Saale</ref><ref name=":1 "/> see also,<ref>Memhard, D,., Andrieux, F., Sun, D.-Z., Häcker, R. (2011) Development and verification of a material model for prediction of containment safety of exhaust turbochargers, ''8th European LS-DYNA Users Conference'', Strasbourg, May 2011, 11 p.</ref> cf. <ref>DiMaggio, F.L., Sandler, I.S. (1971) Material model for granular soils, ''Journal of the Engineering Mechanics Division'', 97(3), 935-950</ref> :<math> 3\,I_2' = \left\{ \begin{array}{ll} \displaystyle\frac{\sigma_\mathrm{eq}-\gamma_1 \,I_1}{1-\gamma_1} \, \frac{\sigma_\mathrm{eq}-\gamma_2 \,I_1}{1-\gamma_2}, & I_1>-d\,\sigma_\mathrm{+} \\[1em] \displaystyle\frac{\sigma_\mathrm{eq}^2}{(1-\gamma_1-\gamma_2)^2}, & I_1\leq -d\,\sigma_\mathrm{+} \end{array} \right. </math> consists of the Schleicher ellipsoid with the restriction of Poisson's ratio at compression :<math> \nu_-^\mathrm{in} = - \frac{-1+ \gamma_1^2+\gamma_2^2-\gamma_1\,\gamma_2} {(-2+\gamma_1+\gamma_2)\,(-1+\gamma_1+\gamma_2)}=\frac{1}{2} </math> and a cylinder with the <math>C^1</math>-transition in the cross section <math>I_1=-d\,\sigma_\mathrm{+}</math>. The second setting for the parameters <math>\gamma_1\in[0, 1[</math> and <math>\gamma_2<0</math> follows with the compression / tension relation :<math> d=\frac{\sigma_-}{\sigma_+} =\frac{1}{1-\gamma_1-\gamma_2} \geq1 </math> The modified Huber criterion can be better fitted to the measured data as the Huber criterion. For setting <math>\nu_+^\mathrm{in}=0.48</math> it follows <math>\gamma_1=0.0880</math> and <math>\gamma_2=-0.0747</math>.

The Huber criterion and the modified Huber criterion should be preferred to the von Mises criterion since one obtains safer results in the region <math>I_1>\sigma_\mathrm{+} </math>. For practical applications the third invariant of the deviator <math>I_3'</math> should be considered in these criteria.<ref name=":1 "/>

===Mohr–Coulomb yield surface=== {{Main|Mohr–Coulomb theory}}

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: :<math> \frac{m+1}{2}\max \Big(|\sigma_1 - \sigma_2|+K(\sigma_1 + \sigma_2) ~,~~ |\sigma_1 - \sigma_3|+K(\sigma_1 + \sigma_3) ~,~~ |\sigma_2 - \sigma_3|+K(\sigma_2 + \sigma_3) \Big) = S_{yc} </math> where :<math> m = \frac {S_{yc}}{S_{yt}}; K = \frac {m-1}{m+1}</math>

and the parameters <math>S_{yc}</math> and <math>S_{yt}</math> are the yield (failure) stresses of the material in uniaxial compression and tension, respectively. The formula reduces to the Tresca criterion if <math>S_{yc}=S_{yt}</math>.

Figure 5 shows Mohr–Coulomb yield surface in the three-dimensional space of principal stresses. It is a conical prism and <math>K</math> determines the inclination angle of conical surface. Figure 6 shows Mohr–Coulomb yield surface in two-dimensional stress space. In Figure 6 <math>R_{r}</math> and <math>R_{c}</math> is used for <math>S_{yt}</math> and <math>S_{yc}</math>, respectively, in the formula. It is a cross section of this conical prism on the plane of <math> \sigma_1, \sigma_2</math>. In Figure 6 Rr and Rc are used for Syc and Syt, respectively, in the formula.

400px|left|thumb|Figure 5: View of Mohr–Coulomb yield surface in 3D space of principal stresses 250px|none|thumb|Figure 6: Mohr–Coulomb yield surface in 2D space (<math> \sigma_1, \sigma_2</math>) {{Clear}}

=== Drucker–Prager yield surface=== {{Main|Drucker Prager yield criterion}}

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 :<math> \bigg(\frac {m-1}{2}\bigg) ( \sigma_1 + \sigma_2 + \sigma_3 ) + \bigg(\frac{m+1}{2}\bigg)\sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2}{2}} = S_{yc} </math> where :<math> m = \frac{S_{yc}}{S_{yt}} </math> and <math>S_{yc}</math>, <math>S_{yt}</math> are the uniaxial yield stresses in compression and tension respectively. The formula reduces to the von Mises equation if <math>S_{yc}=S_{yt}</math>.

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 <math> \sigma_1, \sigma_2</math>; 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 <math> \sigma_1 = -\sigma_2 </math> line, but usually selected by convention to be those in the compression regime.<ref>Khan and Huang. (1995), Continuum Theory of Plasticity. J.Wiley.</ref> 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 <math> \sigma_1 = \sigma_2 </math> (biaxial fit).<ref>Neto, Periç, Owen. (2008), The mathematical Theory of Plasticity. J.Wiley.</ref> The Drucker-Prager yield criterion is also commonly expressed in terms of the material cohesion and friction angle.

400px|left|thumb|Figure 7: View of Drucker–Prager yield surface in 3D space of principal stresses

740px|none|thumb|Figure 8: View of Drucker–Prager yield surface in 2D space of principal stresses {{Clear}}

===Bresler–Pister yield surface=== {{Main|Bresler Pister yield criterion}}

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 :<math> S_{yc} = \tfrac{1}{\sqrt{2}}\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 </math> where <math>c_0, c_1, c_2 </math> are material constants. The additional parameter <math>c_2</math> gives the yield surface an ellipsoidal cross section when viewed from a direction perpendicular to its axis. If <math>\sigma_c</math> is the yield stress in uniaxial compression, <math>\sigma_t</math> is the yield stress in uniaxial tension, and <math>\sigma_b</math> is the yield stress in biaxial compression, the parameters can be expressed as :<math> \begin{align} c_1 = & \left(\cfrac{\sigma_t-\sigma_c}{(\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}{(\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 = & c_1\sigma_c -c_2\sigma_c^2 \end{align} </math>

400px|left|thumb|Figure 9: View of Bresler–Pister yield surface in 3D space of principal stresses 200px|none|thumb|Figure 10: Bresler–Pister yield surface in 2D space (<math> \sigma_1, \sigma_2</math>) {{Clear}}

===Willam–Warnke yield surface=== {{Main|Willam Warnke yield criterion}}

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 :<math> f(I_1, J_2, J_3) = 0 ~. </math> However, it is more commonly expressed in Haigh–Westergaard coordinates as :<math> f(\xi, \rho, \theta) = 0 ~. </math>

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.

300px|left|thumb|Figure 11: View of Willam–Warnke yield surface in 3D space of principal stresses 300px|none|thumb|Figure 12: Willam–Warnke yield surface in the {{nowrap|<math>\pi</math>-plane}} {{Clear}}

===Podgórski and Rosendahl trigonometric yield surfaces === Normalized with respect to the uniaxial tensile stress <math>\sigma_\mathrm{eq}=\sigma_+</math>, the Podgórski criterion <ref>Podgórski, J. (1984). Limit state condition and the dissipation function for isotropic materials, ''Archives of Mechanics'' 36(3), pp. 323-342.</ref> as function of the stress angle <math>\theta</math> reads :<math> \sigma_\mathrm{eq}=\sqrt{3\,I_2'}\,\frac{\Omega_3(\theta, \beta_3, \chi_3)}{\Omega_3(0, \beta_3, \chi_3)}, </math> with the shape function of trigonal symmetry in the <math>\pi</math>-plane :<math> \Omega_3(\theta, \beta_3, \chi_3)=\cos\left[\displaystyle\frac{1}{3}\left(\pi \beta_3 -\arccos [\,\sin (\chi_3\,\frac{\pi}{2}) \,\!\cos 3\,\theta\,]\right)\right], \qquad \beta_3\in[0,\,1], \quad \chi_3\in[-1,\,1]. </math> It contains the criteria of von Mises (circle in the <math>\pi</math>-plane, <math>\beta_3=[0,\,1]</math>, <math>\chi_3=0</math>), Tresca (regular hexagon, <math>\beta_3=1/2</math>, <math>\chi_3=\{1, -1\}</math>), Mariotte (regular triangle, <math>\beta_3=\{0, 1\}</math>, <math>\chi_3=\{1, -1\}</math>), Ivlev <ref name=":2">Ivlev, D. D. (1959). The theory of fracture of solids (in Russ.: K teorii razrusheniia tverdykh tel), ''J. of Applied Mathematics and Mechanics'', 23(3), pp. 884-895.</ref> (regular triangle, <math>\beta_3=\{1, 0\}</math>, <math>\chi_3=\{1, -1\}</math>) and also the cubic criterion of Sayir <ref name=":4">Sayir, M. (1970). Zur Fließbedingung der Plastizitätstheorie, ''Ingenieur-Archiv'' 39(6), pp. 414-432.</ref> (the Ottosen criterion <ref>Ottosen, N. S. (1975). Failure and Elasticity of Concrete, ''Danish Atomic Energy Commission'', Research Establishment Risö, Engineering Department, Report Risö-M-1801, Roskilde.</ref>) with <math> \beta_3=\{0, 1\}</math> and the isotoxal (equilateral) hexagons of the Capurso criterion<ref name=":2" /><ref name=":4" /><ref>Capurso, M. (1967). Yield conditions for incompressible isotropic and orthotropic materials with different yield stress in tension and compression, ''Meccanica'' 2(2), pp. 118--125.</ref> with <math>\chi_3=\{1, -1\}</math>. The von Mises - Tresca transition <ref>Lemaitre J., Chaboche J.L. (1990). ''Mechanics of Solid Materials'', Cambridge University Press, Cambridge.</ref> follows with <math>\beta_3=1/2</math>, <math>\chi_3=[0, 1]</math>. The isogonal (equiangular) hexagons of the Haythornthwaite criterion <ref name=":1 "/><ref>Candland C.T. (1975). Implications of macroscopic failure criteria which are independent of hydrostatic stress, ''Int. J. Fracture'' 11(3), pp. 540–543.</ref><ref>Haythornthwaite R.M. (1961). Range of yield condition in ideal plasticity, ''Proc ASCE J Eng Mech Div'', EM6, 87, pp. 117–133.</ref> containing the Schmidt-Ishlinsky criterion (regular hexagon) cannot be described with the Podgórski criterion.

The Rosendahl criterion <ref name=":0">Rosendahl, P. L., Kolupaev, V A., Altenbach, H. (2019). Extreme Yield Figures for Universal Strength Criteria, in Altenbach, H., Öchsner, A., eds., ''State of the Art and Future Trends in Material Modeling'', Advanced Structured Materials STRUCTMAT, Springer, Cham, pp. 259-324.</ref><ref>Rosendahl, P. L. (2020). ''From bulk to structural failure: Fracture of hyperelastic materials'', Diss., Technische Universität Darmstadt.</ref><ref>Altenbach, H., Kolupaev, V. A. (2024). Reviewing yield criteria in plasticity theory, in Altenbach, H., Hohe, J., Mittelsted, Ch., eds., ''Progress in Structural Mechanics'', Springer, Cham, pp. 19-106.</ref> reads :<math> \sigma_\mathrm{eq}=\sqrt{3\,I_2'}\,\frac{\Omega_6(\theta, \beta_6, \chi_6)}{\Omega_6(0, \beta_6, \chi_6)}, </math> with the shape function of hexagonal symmetry in the <math>\pi</math>-plane :<math> \Omega_6(\theta, \beta_6, \chi_6)=\cos\left[\displaystyle\frac{1}{6}\left(\pi \beta_6 -\arccos [\,\sin (\chi_6\,\frac{\pi}{2})\,\!\cos 6\,\theta\,]\right)\right], \qquad \beta_6\in[0,\,1], \quad \chi_6\in[-1,\,1]. </math> It contains the criteria of von Mises (circle, <math>\beta_6=[0,\,1]</math>, <math>\chi_6=0</math>), Tresca (regular hexagon, <math>\beta_6=\{1, 0\}</math>, <math>\chi_6=\{1, -1\}</math>), Schmidt—Ishlinsky (regular hexagon, <math>\beta_6=\{0, 1\}</math>, <math>\chi_6=\{1, -1\}</math>), Sokolovsky (regular dodecagon, <math>\beta_6=1/2</math>, <math>\chi_6=\{1, -1\}</math>), and also the bicubic criterion <ref name=":1 "/><ref name=":0" /><ref>Szwed, A. (2000). ''Strength Hypotheses and Constitutive Relations of Materials Including Degradation Effects'', (in Polish: Hipotezy Wytężeniowe i Relacje Konstytutywne Materiałów z Uwzględnieniem Efektów Degradacji), Praca Doctorska, Wydział Inąynierii Lądowej Politechniki Warszawskiej, Warszawa.</ref><ref>Lagzdin, A. (1997). Smooth convex limit surfaces in the space of symmetric second-rank tensors, ''Mechanics of Composite Materials'', 3(2), 119-127.</ref> with <math>\beta_6=0</math> or equally with <math>\beta_6=1</math> and the isotoxal dodecagons of the unified yield criterion of Yu <ref>Yu M.-H. (2002). Advances in strength theories for materials under complex stress state in the 20th century, ''Applied Mechanics Reviews'', 55(5), pp. 169-218.</ref> with <math>\chi_6=\{1, -1\}</math>. The isogonal dodecagons of the multiplicative ansatz criterion of hexagonal symmetry <ref name=":1 "/> 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 <math>Re</math> can be introduced to the shape function: <math>Re(\Omega_{3})</math> and <math>Re(\Omega_{6})</math>. The generalization in the form <math>\Omega_{3n}</math><ref name=":0" /> is relevant for theoretical investigations.

A pressure-sensitive extension of the criteria can be obtained with the linear <math>I_1</math>-substitution <ref name=":1 "/> :<math> \sigma_\mathrm{eq}\rightarrow \frac{\sigma_\mathrm{eq}-\gamma_1\,I_1}{1-\gamma_1} \qquad\mbox{with}\qquad \gamma_1\in[0,\,1[, </math> which is sufficient for many applications, e.g. metals, cast iron, alloys, concrete, unreinforced polymers, etc.

600px|thumb|left|Basic cross sections described by a circle and regular polygons of trigonal or hexagonal symmetries in the <math>\pi</math>-plane. {{Clear}}

===Bigoni–Piccolroaz yield surface=== The Bigoni–Piccolroaz yield criterion<ref>Bigoni, D. Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Cambridge University Press, 2012 . {{ISBN|9781107025417}}.</ref><ref name=BP>Bigoni, D. and Piccolroaz, A., (2004), Yield criteria for quasibrittle and frictional materials, ''International Journal of Solids and Structures'' '''41''', 2855–2878.</ref> is a seven-parameter surface defined by

:<math> f(p,q,\theta) = F(p) + \frac{q}{g(\theta)} = 0, </math>

where <math>F(p)</math> is the "meridian" function

:<math> F(p) = \left\{ \begin{array}{ll} -M p_c \sqrt{(\phi - \phi^m)[2(1 - \alpha)\phi + \alpha]}, & \phi \in [0,1], \\ +\infty, & \phi \notin [0,1], \end{array} \right. </math>

:<math> \phi = \frac{p + c}{p_c + c}, </math>

describing the pressure-sensitivity and <math>g(\theta)</math> is the "deviatoric" function <ref>Podgórski, J. (1984). Limit state condition and the dissipation function for isotropic materials. ''Archives of Mechanics'', 36 (3), pp. 323–342.</ref>

:<math> g(\theta) = \frac{1}{\cos[\beta \frac{\pi}{6} - \frac{1}{3} \cos^{-1}(\gamma \cos 3\theta)]}, </math>

describing the Lode-dependence of yielding. The seven, non-negative material parameters:

:<math> \underbrace{M > 0,~ p_c > 0,~ c \geq 0,~ 0 < \alpha < 2,~ m > 1}_{\mbox{defining}~\displaystyle{F(p)}},~~~ \underbrace{0\leq \beta \leq 2,~ 0 \leq \gamma < 1}_{\mbox{defining}~\displaystyle{g(\theta)}}, </math>

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.<ref name=BP2>Piccolroaz, A. and Bigoni, D. (2009), Yield criteria for quasibrittle and frictional materials: a generalization to surfaces with corners, ''International Journal of Solids and Structures'' '''46''', 3587–3596.</ref>

{{multiple image | align = none | footer = Bigoni-Piccolroaz yield surface | image1 = Supbp1.png | width1 = 350 | alt1 = 3D | caption1 = In 3D space of principal stresses | image2 = Supbp2.png | width2 = 280 | alt2 = <math>\pi</math>-plane | caption2 = In the <math>\pi</math>-plane }}

=== Cosine Ansatz (Altenbach-Bolchoun-Kolupaev) === For the formulation of the strength criteria the stress angle :<math>\cos 3\theta = \frac{3\sqrt{3}}{2}\frac{I_3'}{I_2'^{\frac{3}{2}}}</math> can be used.

The following criterion of isotropic material behavior :<math> (3I_2')^3 \frac{1+c_3 \cos 3\theta+c_6 \cos^2 3\theta}{1+c_3+ c_6}= \displaystyle \left(\frac{\sigma_\mathrm{eq}-\gamma_1\,I_1}{1-\gamma_1}\right)^{6-l-m}\, \left(\frac{\sigma_\mathrm{eq}-\gamma_2\,I_1}{1-\gamma_2}\right)^l \, \sigma_\mathrm{eq}^m </math> contains a number of other well-known less general criteria, provided suitable parameter values are chosen.

Parameters <math>c_3</math> and <math>c_6</math> describe the geometry of the surface in the <math>\pi</math>-plane. They are subject to the constraints :<math> c_6=\frac{1}{4}(2+c_3), \qquad c_6=\frac{1}{4}(2-c_3), \qquad c_6\ge \frac{5}{12}\,c_3^2-\frac{1}{3}, </math> which follow from the convexity condition. A more precise formulation of the third constraints is proposed in.<ref>Altenbach, H., Bolchoun, A., Kolupaev, V.A. (2013). Phenomenological Yield and Failure Criteria, in Altenbach, H., Öchsner, A., eds., ''Plasticity of Pressure-Sensitive Materials'', Serie ASM, Springer, Heidelberg, pp. 49–152.</ref><ref>Kolupaev, V.A. (2018). Equivalent Stress Concept for Limit State Analysis, Springer, Cham.</ref>

Parameters <math>\gamma_1\in[0,\,1[</math> and <math>\gamma_2</math> 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 <math>\gamma_2\in[0,\,\gamma_1[</math>. Otherwise for materials which fail at hydrostatic pressure (hard foams, ceramics, sintered materials, etc.) it follows <math>\gamma_2<0</math>.

The integer powers <math>l\geq0</math> and <math>m\geq0</math>, <math>l+m< 6</math> describe the curvature of the meridian. The meridian with <math>l=m=0</math> is a straight line and with <math>l=0</math> – 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: :<math> \Phi = \Phi '(X') + \Phi ''(X'') = 2{\bar \sigma ^a} </math> :350px|thumb|right|The Yld2000-2D yield loci for a AA6022 T4 sheet. : where <math> \bar \sigma </math> is the effective stress. and <math> X' </math> and <math> X'' </math> are the transformed matrices (by linear transformation C or L): :<math> \begin{array}{l} X' = C'.s = L'.\sigma \\ X'' = C''.s = L''.\sigma \end{array} </math> : where s is the deviatoric stress tensor. for principal values of X’ and X”, the model could be expressed as: :<math> \begin{array}{l} \Phi ' = {\left| {{{X'}_1} + {{X'}_2}} \right|^a}\\ \Phi '' = {\left| {2{{X''}_2} + {{X''}_1}} \right|^a} + {\left| {2{{X''}_1} + {{X''}_2}} \right|^a} \end{array}\ </math> and: :<math> \left[ {\begin{array}{*{20}{c}} {{{L'}_{11}}}\\ {{{L'}_{12}}}\\ {{{L'}_{21}}}\\ {{{L'}_{22}}}\\ {{{L'}_{66}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {2/3}&0&0\\ { - 1/3}&0&0\\ 0&{ - 1/3}&0\\ 0&{ - 2/3}&0\\ 0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\alpha _1}}\\ {{\alpha _2}}\\ {{\alpha _7}} \end{array}} \right], \left[ {\begin{array}{*{20}{c}} {{{L''}_{11}}}\\ {{{L''}_{12}}}\\ {{{L''}_{21}}}\\ {{{L''}_{22}}}\\ {{{L''}_{66}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 2}&2&8&{ - 2}&0\\ 1&{ - 4}&{ - 4}&4&0\\ 4&{ - 4}&{ - 4}&4&0\\ { - 2}&8&2&{ - 2}&0\\ 0&0&0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\alpha _3}}\\ {{\alpha _4}}\\ {{\alpha _5}}\\ {{\alpha _6}}\\ {{\alpha _8}} \end{array}} \right] </math> where <math> \alpha _1 ... \alpha _8 </math> are eight parameters of the Barlat's Yld2000-2D model to be identified with a set of experiments.

== References == {{reflist}}

Category:Plasticity (physics) Category:Solid mechanics Category:Continuum mechanics Category:Materials science Category:Structural analysis