yield surface

File:YieldSurfacerhoxitheta.svg

The first principal invariant ($I\_1$) of the Cauchy stress ($\backslash boldsymbol\{\backslash sigma\}$), and the second and third principal invariants ($J\_2,\; J\_3$) of the deviatoric part ($\backslash boldsymbol\{s\}$) of the Cauchy stress are defined as:

:

:$$

\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}

where ($\backslash sigma\_1,\; \backslash sigma\_2\; ,\; \backslash sigma\_3$) are the principal values of $\backslash boldsymbol\{\backslash sigma\}$, ($s\_1,\; s\_2,\; s\_3$) are the principal values of $\backslash boldsymbol\{s\}$, and

:$$

\boldsymbol{s} = \boldsymbol{\sigma}-\tfrac{I_1}{3}\,\boldsymbol{I}

where $\backslash boldsymbol\{I\}$ is the identity matrix.

A related set of quantities, ($p,\; q,\; r\backslash ,$), are usually used to describe yield surfaces for cohesive frictional materials such as rocks, soils, and ceramics. These are defined as

:$$

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}

where $\backslash sigma\_\backslash mathrm\{eq\}$ is the equivalent stress. However, the possibility of negative values of $J\_3$ and the resulting imaginary $r$ makes the use of these quantities problematic in practice.

Another related set of widely used invariants is ($\backslash xi,\; \backslash rho,\; \backslash theta\backslash ,$) which describe a cylindrical coordinate system (the Haigh–Westergaard coordinates). These are defined as:

:$$

\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}}

The $\backslash xi-\backslash rho\backslash ,$ plane is also called the Rendulic plane. The angle $\backslash theta$ is called stress angle, the value $\backslash cos(3\backslash theta)$ is sometimes called the Lode parameterLode, W. (1925). Versuche über den Einfluß der mittleren Hauptspannug auf die Fließgrenze. ZAMM 5(2), pp. 142–144Lode, 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.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 and the relation between $\backslash theta$ and $J\_2,J\_3$ was first given by Novozhilov V.V. in 1951,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. see also 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.

The principal stresses and the Haigh–Westergaard coordinates are related by

:$$

\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} \,.

A different definition of the Lode angle can also be found in the literature:Chakrabarty, J., 2006, Theory of Plasticity: Third edition, Elsevier, Amsterdam.

:$$

\sin(3\theta) = ~\tfrac{3\sqrt{3}}{2}~\cfrac{J_3}{J_2^{3/2}}

in which case the ordered principal stresses (where $\backslash sigma\_1\; \backslash geq\; \backslash sigma\_2\; \backslash geq\; \backslash sigma\_3$) are related byBrannon, R.M., 2009, KAYENTA: Theory and User's Guide, Sandia National Laboratories, Albuquerque, New Mexico.

:$$

\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}

\,.

There are several different yield surfaces known in engineering, and those most popular are listed below.

The Tresca yield criterion is taken to be the work of Henri Tresca.Tresca, H. (1864). Mémoire sur l'écoulement des corps solides soumis à de fortes pressions. C. R. Acad. Sci. Paris, vol. 59, p. 754. It is also known as the maximum shear stress theory (MSST) and the Tresca–GuestGuest (TG) criterion. In terms of the principal stresses the Tresca criterion is expressed as

:$\backslash tfrac\{1\}\{2\}\{\backslash max(|\backslash sigma\_1\; -\; \backslash sigma\_2|\; ,\; |\backslash sigma\_2\; -\; \backslash sigma\_3|\; ,\; |\backslash sigma\_3\; -\; \backslash sigma\_1|\; )\; =\; S\_\{sy\}\; =\; \backslash tfrac\{1\}\{2\}S\_y\}\backslash !$

Where $S\_\{sy\}$ is the yield strength in shear, and $S\_y$ 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 $\backslash sigma\_1,\; \backslash sigma\_2$ plane.

Image:Tresca Guest Yield Surface 3D.png

Image:Tresca Guest Yield Surface 2D.png

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{{Main|von Mises yield criterion}}

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

:$\{(\backslash sigma\_1\; -\; \backslash sigma\_2)^2\; +\; (\backslash sigma\_2\; -\; \backslash sigma\_3)^2\; +\; (\backslash sigma\_3\; -\; \backslash sigma\_1)^2\; =\; 2\; \{S\_y\}^2\; \}\backslash !$

where $S\_y$ 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 $\backslash sigma\_1,\; \backslash sigma\_2$ produces the elliptical shape of the yield surface.

Image:Mises Yield Surface 3D.png

Image:Tresca stress 2D.png

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This criterionBurzyński, W. (1929). Über die Anstrengungshypothesen. Schweizerische Bauzeitung, 94 (21), pp. 259–262.Yagn, Yu. I. (1931). New methods of strength prediction (in Russ.: Novye metody pascheta na prochnost'). Vestnik inzhenerov i tekhnikov, 6, pp. 237–244. reformulated as the function of the hydrostatic nodes with the coordinates $1/\backslash gamma\_1$ and $1/\backslash gamma\_2$

:$3I\_2\text{'}\; =$

\frac{\sigma_\mathrm{eq}-\gamma_1I_1}{1-\gamma_1}

\frac{\sigma_\mathrm{eq}-\gamma_2I_1}{1-\gamma_2}

represents the general equation of a second order surface of revolution about the hydrostatic axis. Some special case are: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

• cylinder $\backslash gamma\_1\; =\; \backslash gamma\_2\; =\; 0$ (Maxwell (1865), Huber (1904), von Mises (1913), Hencky (1924)),
• cone $\backslash gamma\_1\; =\; \backslash gamma\_2\; \backslash in\; ]0,1[$ (Botkin (1940), Drucker-Prager (1952), Mirolyubov (1953)),
• paraboloid $\backslash gamma\_1\; \backslash in\; ]0,1[,\; \backslash gamma\_2\; =\; 0$ (Burzyński (1928), Balandin (1937), Torre (1947)),
• ellipsoid centered of symmetry plane $I\_1\; =\; 0$, $\backslash gamma\_1\; =\; -\; \backslash gamma\_2\; \backslash in\; ]0,1[$ (Beltrami (1885)),
• ellipsoid centered of symmetry plane $I\_1\; =\; \backslash frac\{1\}\{2\}\backslash ,\backslash bigg(\backslash frac\{1\}\{\backslash gamma\_1\}+\backslash frac\{1\}\{\backslash gamma\_2\}\; \backslash bigg)$ with (Schleicher (1926)),
• hyperboloid of two sheets $\backslash gamma\_1\; \backslash in\; ]0,1[,\; \backslash gamma\_2\; \backslash in\; ]0,\backslash gamma\_1[$ (Burzynski (1928), Yagn (1931)),
• hyperboloid of one sheet centered of symmetry plane $I\_1\; =\; 0$, $\backslash gamma\_1=-\backslash gamma\_2\; =a\backslash ,i$, $i\; =\backslash sqrt\{-1\}$ (Kuhn (1980))
• hyperboloid of one sheet $\backslash gamma\_\{1,2\}=\; b\; \backslash pm\; a\backslash ,i$, $i\; =\backslash sqrt\{-1\}$ (Filonenko-Boroditsch (1960), Gol’denblat-Kopnov (1968), Filin (1975)).

The relations compression-tension and torsion-tension can be computed to

:$\backslash frac\{\backslash sigma\_-\}\{\backslash sigma\_+\}\; =\backslash frac\{1\}\{1-\backslash gamma\_1-\backslash gamma\_2\},\; \backslash qquad\; \backslash bigg(\backslash sqrt\{3\}\backslash ,\backslash frac\{\backslash tau\_*\}\{\backslash sigma\_+\}\backslash bigg)^2\; =\; \backslash frac\{1\}\{(1-\backslash gamma\_1)(1-\backslash gamma\_2)\}$

The Poisson's ratios at tension and compression are obtained using

:$\backslash nu\_+^\backslash mathrm\{in\}\; =$

\frac{-1+2(\gamma_1+\gamma_2)-3\gamma_1\gamma_2}{-2+\gamma_1+\gamma_2}

:$\backslash nu\_-^\backslash mathrm\{in\}\; =\; -\; \backslash frac\{-1+$

\gamma_1^2+\gamma_2^2-\gamma_1\,\gamma_2}

{(-2+\gamma_1+\gamma_2)\,(-1+\gamma_1+\gamma_2)}

For ductile materials the restriction

:$\backslash nu\_+^\backslash mathrm\{in\}\backslash in\; \backslash bigg[\backslash ,0.48,\backslash ,\backslash frac\{1\}\{2\}\backslash ,\backslash bigg]$

is important. The application of rotationally symmetric criteria for brittle failure with

:$\backslash nu\_+^\backslash mathrm\{in\}\backslash in\; ]-1,~\backslash nu\_+^\backslash mathrm\{el\}\backslash ,]$

has not been studied sufficiently.Beljaev, N. M. (1979). Strength of Materials. Mir Publ., Moscow

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.: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

:$3I\_2\text{'}\; \backslash frac\{1+c\_3\; \backslash cos\; 3\backslash theta+c\_6\; \backslash cos^2\; 3\backslash theta\}\{1+c\_3+$

c_6} =

\frac{\sigma_\mathrm{eq}-\gamma_1I_1}{1-\gamma_1}

\frac{\sigma_\mathrm{eq}-\gamma_2I_1}{1-\gamma_2}

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The Huber criterion consists of the Beltrami ellipsoid and a scaled von Mises cylinder in the principal stress space,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, 80-81Föppl, A., Föppl, L. (1920). Drang und Zwang: eine höhere Festigkeitslehre für Ingenieure. R. Oldenbourg, MünchenBurzyński, W. (1929). Über die Anstrengungshypothesen. Schweizerische Bauzeitung

94(21):259–262Kuhn, P. (1980). Grundzüge einer allgemeinen Festigkeitshypothese, Auszug aus Antrittsvorlesung

des Verfassers vom 11. Juli, 1980 Vom Konstrukteur und den Festigkeitshypothesen.

Inst. für Maschinenkonstruktionslehre, Karlsruhe see alsoKolupaev, 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):79–82Kolupaev, V.A. (2018). Equivalent Stress Concept for Limit State Analysis, Springer, Cham.

:$$

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.

with $\backslash gamma\_1\backslash in[0,\; 1[$. The transition between the surfaces in the cross section $I\_1=0$ is continuously differentiable.

The criterion represents the "classical view" with respect to inelastic material behavior:

• pressure-sensitive material behavior for $I\_1>0$ with $\backslash nu\_+^\backslash mathrm\{in\}\backslash in\backslash left]-1,\backslash ,1/2\backslash right]$ and
• pressure-insensitive material behavior for with $\backslash nu\_-^\backslash mathrm\{in\}=1/2$

The Huber criterion can be used as a yield surface with an empirical restriction for Poisson's ratio at tension $\backslash nu\_+^\backslash mathrm\{in\}\backslash in[0.48,\; 1/2]$, which leads to $\backslash gamma\_1\backslash in[0,\; 0.1155]$.

[[File:Burzynski_Huber_F2_nuP0.jpg|800px|thumb|left|Huber criterion with $\backslash gamma\_1=1/\backslash sqrt\{3\}$ and modified Huber criterion with $\backslash gamma\_1=(1+\backslash sqrt\{5\})/6$ and $\backslash gamma\_2=(1-\backslash sqrt\{5\})/6$ in the Burzyński-plane: setting according the normal stress hypothesis ($\backslash nu\_+^\backslash mathrm\{in\}=0$). The von Mises criterion ($\backslash nu\_-^\backslash mathrm\{in\}=\backslash nu\_+^\backslash mathrm\{in\}=1/2$) 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.]]

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The modified Huber criterion,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 see also,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. cf. DiMaggio, F.L., Sandler, I.S. (1971) Material model for granular soils, Journal of the Engineering Mechanics Division, 97(3), 935-950

:$$

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.

consists of the Schleicher ellipsoid with the restriction of Poisson's ratio at compression

:$\backslash nu\_-^\backslash mathrm\{in\}\; =\; -\; \backslash frac\{-1+$

\gamma_1^2+\gamma_2^2-\gamma_1\,\gamma_2}

{(-2+\gamma_1+\gamma_2)\,(-1+\gamma_1+\gamma_2)}=\frac{1}{2}

and a cylinder with the $C^1$-transition in the cross section $I\_1=-d\backslash ,\backslash sigma\_\backslash mathrm\{+\}$.

The second setting for the parameters $\backslash gamma\_1\backslash in[0,\; 1[$ and follows with the compression / tension relation

:$d=\backslash frac\{\backslash sigma\_-\}\{\backslash sigma\_+\}\; =\backslash frac\{1\}\{1-\backslash gamma\_1-\backslash gamma\_2\}\; \backslash geq1$

The modified Huber criterion can be better fitted to the measured data as the Huber criterion. For setting $\backslash nu\_+^\backslash mathrm\{in\}=0.48$ it follows $\backslash gamma\_1=0.0880$ and $\backslash gamma\_2=-0.0747$.

The Huber criterion and the modified Huber criterion should be preferred to the von Mises criterion since one obtains safer results in the region $I\_1>\backslash sigma\_\backslash mathrm\{+\}$.

For practical applications the third invariant of the deviator $I\_3\text{'}$ should be considered in these criteria.

{{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:

:$$

\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}

where

:$m\; =\; \backslash frac\; \{S\_\{yc\}\}\{S\_\{yt\}\};\; K\; =\; \backslash frac\; \{m-1\}\{m+1\}$

and the parameters $S\_\{yc\}$ and $S\_\{yt\}$ are the yield (failure) stresses of the material in uniaxial compression and tension, respectively. The formula reduces to the Tresca criterion if $S\_\{yc\}=S\_\{yt\}$.

Figure 5 shows Mohr–Coulomb yield surface in the three-dimensional space of principal stresses. It is a conical prism and $K$ determines the inclination angle of conical surface. Figure 6 shows Mohr–Coulomb yield surface in two-dimensional stress space. In Figure 6 $R\_\{r\}$ and $R\_\{c\}$ is used for $S\_\{yt\}$ and $S\_\{yc\}$, respectively, in the formula. It is a cross section of this conical prism on the plane of $\backslash sigma\_1,\; \backslash sigma\_2$. In Figure 6 Rr and Rc are used for Syc and Syt, respectively, in the formula.

Image:MH Yield Surface 3D.png

Image:MH Surface 2D.png

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{{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

:$\backslash bigg(\backslash frac\; \{m-1\}\{2\}\backslash bigg)\; (\; \backslash sigma\_1\; +\; \backslash sigma\_2\; +\; \backslash sigma\_3\; )\; +\; \backslash bigg(\backslash frac\{m+1\}\{2\}\backslash bigg)\backslash sqrt\{\backslash frac\{(\backslash sigma\_1\; -\; \backslash sigma\_2)^2\; +\; (\backslash sigma\_2\; -\; \backslash sigma\_3)^2\; +\; (\backslash sigma\_3\; -\; \backslash sigma\_1)^2\}\{2\}\}\; =\; S\_\{yc\}$

where

:$m\; =\; \backslash frac\{S\_\{yc\}\}\{S\_\{yt\}\}$

and $S\_\{yc\}$, $S\_\{yt\}$ are the uniaxial yield stresses in compression and tension respectively. The formula reduces to the von Mises equation if $S\_\{yc\}=S\_\{yt\}$.

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 $\backslash sigma\_1,\; \backslash sigma\_2$; 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 $\backslash sigma\_1\; =\; -\backslash sigma\_2$ line, but usually selected by convention to be those in the compression regime.Khan and Huang. (1995), Continuum Theory of Plasticity. J.Wiley. 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 $\backslash sigma\_1\; =\; \backslash sigma\_2$ (biaxial fit).Neto, Periç, Owen. (2008), The mathematical Theory of Plasticity. J.Wiley. The Drucker-Prager yield criterion is also commonly expressed in terms of the material cohesion and friction angle.

Image:Drucker Prager Yield Surface 3D.png

Image:Drucker Prager WIKI.png

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{{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

:$$

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

where $c\_0,\; c\_1,\; c\_2$ are material constants. The additional parameter $c\_2$ gives the yield surface an ellipsoidal cross section when viewed from a direction perpendicular to its axis. If $\backslash sigma\_c$ is the yield stress in uniaxial compression, $\backslash sigma\_t$ is the yield stress in uniaxial tension, and $\backslash sigma\_b$ is the yield stress in biaxial compression, the parameters can be expressed as

:$$

\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}

Image:Bresler Pister Yield Surface 3D.png

Image:Bresler Pister Surface 2D.png

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{{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

:$$

f(I_1, J_2, J_3) = 0 ~.

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

:$$

f(\xi, \rho, \theta) = 0 ~.

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.

Image:Willam Warnke Yield Surface 3Da.png Image:Willam Warnke Yield Surface 3Db.png

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Normalized with respect to the uniaxial tensile stress $\backslash sigma\_\backslash mathrm\{eq\}=\backslash sigma\_+$, the Podgórski criterion Podgórski, J. (1984). Limit state condition and the dissipation function for isotropic materials, Archives of Mechanics 36(3), pp. 323-342. as function of the stress angle $\backslash theta$ reads

:$$

\sigma_\mathrm{eq}=\sqrt{3\,I_2'}\,\frac{\Omega_3(\theta, \beta_3, \chi_3)}{\Omega_3(0, \beta_3, \chi_3)},

with the shape function of trigonal symmetry in the $\backslash pi$-plane

:$$

\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].

It contains the criteria of von Mises (circle in the $\backslash pi$-plane, $\backslash beta\_3=[0,\backslash ,1]$, $\backslash chi\_3=0$), Tresca (regular hexagon, $\backslash beta\_3=1/2$, $\backslash chi\_3=\backslash \{1,\; -1\backslash \}$), Mariotte (regular triangle, $\backslash beta\_3=\backslash \{0,\; 1\backslash \}$, $\backslash chi\_3=\backslash \{1,\; -1\backslash \}$), Ivlev 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. (regular triangle, $\backslash beta\_3=\backslash \{1,\; 0\backslash \}$, $\backslash chi\_3=\backslash \{1,\; -1\backslash \}$) and also the cubic criterion of Sayir Sayir, M. (1970). Zur Fließbedingung der Plastizitätstheorie, Ingenieur-Archiv 39(6), pp. 414-432. (the Ottosen criterion Ottosen, N. S. (1975). Failure and Elasticity of Concrete, Danish Atomic Energy Commission, Research Establishment Risö, Engineering Department, Report Risö-M-1801, Roskilde.) with $$

\beta_3=\{0, 1\} and the isotoxal (equilateral) hexagons of the Capurso criterionCapurso, M. (1967). Yield conditions for incompressible isotropic and orthotropic materials with different yield stress in tension and compression, Meccanica 2(2), pp. 118--125. with $\backslash chi\_3=\backslash \{1,\; -1\backslash \}$. The von Mises - Tresca transition Lemaitre J., Chaboche J.L. (1990). Mechanics of Solid Materials, Cambridge University Press, Cambridge. follows with $\backslash beta\_3=1/2$, $\backslash chi\_3=[0,\; 1]$. The isogonal (equiangular) hexagons of the Haythornthwaite criterion Candland C.T. (1975). Implications of macroscopic failure criteria which are independent of hydrostatic stress, Int. J. Fracture 11(3), pp. 540–543.Haythornthwaite R.M. (1961). Range of yield condition in ideal plasticity, Proc ASCE J Eng Mech Div, EM6, 87, pp. 117–133. containing the Schmidt-Ishlinsky criterion (regular hexagon) cannot be described with the Podgórski ctiterion.

The Rosendahl criterion 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.Rosendahl, P. L. (2020). From bulk to structural failure: Fracture of hyperelastic materials, Diss., Technische Universität Darmstadt.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. reads

:$$

\sigma_\mathrm{eq}=\sqrt{3\,I_2'}\,\frac{\Omega_6(\theta, \beta_6, \chi_6)}{\Omega_6(0, \beta_6, \chi_6)},

with the shape function of hexagonal symmetry in the $\backslash pi$-plane

:$$

\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].

It contains the criteria of von Mises (circle, $\backslash beta\_6=[0,\backslash ,1]$, $\backslash chi\_6=0$), Tresca (regular hexagon, $\backslash beta\_6=\backslash \{1,\; 0\backslash \}$, $\backslash chi\_6=\backslash \{1,\; -1\backslash \}$), Schmidt—Ishlinsky (regular hexagon, $\backslash beta\_6=\backslash \{0,\; 1\backslash \}$, $\backslash chi\_6=\backslash \{1,\; -1\backslash \}$), Sokolovsky (regular dodecagon, $\backslash beta\_6=1/2$, $\backslash chi\_6=\backslash \{1,\; -1\backslash \}$), and also the bicubic criterion 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.Lagzdin, A. (1997). Smooth convex limit surfaces in the space of symmetric second-rank tensors, Mechanics of Composite Materials, 3(2), 119-127. with $\backslash beta\_6=0$ or equally with $\backslash beta\_6=1$ and the isotoxal dodecagons of the unified yield criterion of Yu 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. with $\backslash chi\_6=\backslash \{1,\; -1\backslash \}$. 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 $Re$ can be introduced to the shape function: $Re(\backslash Omega\_\{3\})$ and $Re(\backslash Omega\_\{6\})$. The generalization in the form $\backslash Omega\_\{3n\}$ is relevant for theoretical investigations.

A pressure-sensitive extension of the criteria can be obtained with the linear $I\_1$-substitution

:$$

\sigma_\mathrm{eq}\rightarrow

\frac{\sigma_\mathrm{eq}-\gamma_1\,I_1}{1-\gamma_1} \qquad\mbox{with}\qquad \gamma_1\in[0,\,1[,

which is sufficient for many applications, e.g. metals, cast iron, alloys, concrete, unreinforced polymers, etc.

[[File:pi-Plane_3-6-9-12-2.jpg|600px|thumb|left|Basic cross sections described

by a circle and regular polygons of trigonal or hexagonal symmetries in the $\backslash pi$-plane.]]

{{Clear}}

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

:$$

f(p,q,\theta) = F(p) + \frac{q}{g(\theta)} = 0,

where $F(p)$ is the "meridian" function

:$$

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.

:$$

\phi = \frac{p + c}{p_c + c},

describing the pressure-sensitivity and $g(\backslash theta)$ is the "deviatoric" function

Podgórski, J. (1984). Limit state condition and the dissipation function for isotropic materials. Archives of Mechanics, 36 (3), pp. 323–342.

:$$

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

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

:$$

\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)}},

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.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.

{{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 = $\backslash pi$-plane

| caption2 = In the $\backslash pi$-plane

}}

For the formulation of the strength criteria the stress angle

:$\backslash cos\; 3\backslash theta\; =\; \backslash frac\{3\backslash sqrt\{3\}\}\{2\}\backslash frac\{I\_3\text{'}\}\{I\_2\text{'}^\{\backslash frac\{3\}\{2\}\}\}$

can be used.

The following criterion of isotropic material behavior

:$(3I\_2\text{'})^3\; \backslash frac\{1+c\_3\; \backslash cos\; 3\backslash theta+c\_6\; \backslash cos^2\; 3\backslash theta\}\{1+c\_3+$

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

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

Parameters $c\_3$ and $c\_6$ describe the geometry of the surface in the $\backslash pi$-plane. They are subject to the constraints

:$c\_6=\backslash frac\{1\}\{4\}(2+c\_3),\; \backslash qquad\; c\_6=\backslash frac\{1\}\{4\}(2-c\_3),\; \backslash qquad\; c\_6\backslash ge\; \backslash frac\{5\}\{12\}\backslash ,c\_3^2-\backslash frac\{1\}\{3\},$

which follow from the convexity condition. A more precise formulation of the third constraints is proposed in.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.Kolupaev, V.A. (2018). Equivalent Stress Concept for Limit State Analysis, Springer, Cham.

Parameters $\backslash gamma\_1\backslash in[0,\backslash ,1[$ and $\backslash gamma\_2$ 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 $\backslash gamma\_2\backslash in[0,\backslash ,\backslash gamma\_1[$. Otherwise for materials which fail at hydrostatic pressure (hard foams, ceramics, sintered materials, etc.) it follows .

The integer powers $l\backslash geq0$ and $m\backslash geq0$, describe the curvature of the meridian. The meridian with $l=m=0$ is a straight line and with $l=0$ – a parabola.

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:

:$\backslash Phi\; =\; \backslash Phi\; \text{'}(X\text{'})\; +\; \backslash Phi$(X) = 2{\bar \sigma ^a} :File:Yld2000-2D.png

: where $\backslash bar\; \backslash sigma$ is the effective stress. and $X\text{'}$ and $X\text{'}\text{'}$ are the transformed matrices (by linear transformation C or L):

:$\backslash begin\{array\}\{l\}$

X' = C'.s = L'.\sigma \\

X = C.s = L''.\sigma

\end{array}

: where s is the deviatoric stress tensor.

for principal values of X’ and X”, the model could be expressed as:

:$\backslash begin\{array\}\{l\}$

\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}\

and:

:$\backslash left[\; \{\backslash begin\{array\}\{*\{20\}\{c\}\}$

{{{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]

where $\backslash alpha\; \_1\; ...\; \backslash alpha\; \_8$ are eight parameters of the Barlat's Yld2000-2D model to be identified with a set of experiments.

{{div col}}

• Yield (engineering)
• Plasticity (physics)
• Stress
• Henri Tresca
• von Mises stress
• Mohr–Coulomb theory
• Hill yield criterion
• Hosford yield criterion
• Strain
• Strain tensor
• Stress–energy tensor
• Stress concentration
• 3-D elasticity
• Frederic Barlat

{{Div col end}}

{{reflist}}

{{Topics in continuum mechanics}}

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Category:Plasticity (physics)

Category:Solid mechanics

Category:Continuum mechanics

Category:Materials science

Category:Structural analysis