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.Simo, J. C. and Hughes, T,. J. R., (1998), Computational Inelasticity, Springer.
The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space (), a two- or three-dimensional space spanned by stress invariants () 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:
where are the principal stresses. where is the first principal invariant of the Cauchy stress and are the second and third principal invariants of the deviatoric part of the Cauchy stress. where are scaled versions of and and is a function of . where are scaled versions of and , and is the stress angleYu, M.-H. (2004), Unified strength theory and its applications. Springer, Berlin or Lode angleZienkiewicz 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
Invariants used to describe yield surfaces
The first principal invariant () of the Cauchy stress (), and the second and third principal invariants () of the deviatoric part () of the Cauchy stress are defined as:
where () are the principal values of , () are the principal values of , and
where is the identity matrix.
A related set of quantities, (), are usually used to describe yield surfaces for cohesive frictional materials such as rocks, soils, and ceramics. These are defined as
where is the equivalent stress. However, the possibility of negative values of and the resulting imaginary makes the use of these quantities problematic in practice.
Another related set of widely used invariants is () which describe a cylindrical coordinate system (the Haigh–Westergaard coordinates). These are defined as:
The plane is also called the Rendulic plane. The angle is called stress angle, the value 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 and 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
A different definition of the Lode angle can also be found in the literature:Chakrabarty, J., 2006, Theory of Plasticity: Third edition, Elsevier, Amsterdam.
in which case the ordered principal stresses (where ) are related byBrannon, R.M., 2009, KAYENTA: Theory and User's Guide, Sandia National Laboratories, Albuquerque, New Mexico.
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; https://pmm.ipmnet.ru/ru/get/1951/15-2/183-194. 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, https://et.ippt.pan.pl/index.php/et/article/view/2210, 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.
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.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–Guest (TG) criterion. In terms of the principal stresses the Tresca criterion is expressed as
Where is the yield strength in shear, and 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 plane.
von Mises yield surface
The von Mises yield criterion is expressed in the principal stresses as
where 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 produces the elliptical shape of the yield surface.
Burzyński-Yagn criterion 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 and
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 (Maxwell (1865), Huber (1904), von Mises (1913), Hencky (1924)), cone (Botkin (1940), Drucker-Prager (1952), Mirolyubov (1953)), paraboloid (Burzyński (1928), Balandin (1937), Torre (1947)), ellipsoid centered of symmetry plane , (Beltrami (1885)), ellipsoid centered of symmetry plane with (Schleicher (1926)), hyperboloid of two sheets (Burzynski (1928), Yagn (1931)), hyperboloid of one sheet centered of symmetry plane , , (Kuhn (1980)) hyperboloid of one sheet , (Filonenko-Boroditsch (1960), Gol’denblat-Kopnov (1968), Filin (1975)).
The relations compression-tension and torsion-tension can be computed to
The Poisson's ratios at tension and compression are obtained using
For ductile materials the restriction
is important. The application of rotationally symmetric criteria for brittle failure with
has not been studied sufficiently.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
Huber criterion 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.
with . The transition between the surfaces in the cross section is continuously differentiable. The criterion represents the "classical view" with respect to inelastic material behavior: pressure-sensitive material behavior for with and pressure-insensitive material behavior for with
The Huber criterion can be used as a yield surface with an empirical restriction for Poisson's ratio at tension , which leads to .
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
consists of the Schleicher ellipsoid with the restriction of Poisson's ratio at compression
and a cylinder with the -transition in the cross section . The second setting for the parameters and follows with the compression / tension relation
The modified Huber criterion can be better fitted to the measured data as the Huber criterion. For setting it follows and .