A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of inside the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and the material is said to have become plastic. Further deformation of the material causes the stress state to remain on the yield surface, even though the shape and size of the surface may change as the plastic deformation evolves. This is because stress states that lie outside the yield surface are non-permissible in rate-independent plasticity, though not in some models of viscoplasticity. The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space ( σ 1 , σ 2 , σ 3 {\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}} ), a two- or three-dimensional space spanned by stress invariants ( I 1 , J 2 , J 3 {\displaystyle I_{1},J_{2},J_{3}} ) 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:
f ( σ 1 , σ 2 , σ 3 ) = 0 {\displaystyle f(\sigma _{1},\sigma _{2},\sigma _{3})=0\,} where σ i {\displaystyle \sigma _{i}} are the principal stresses.
f ( I 1 , J 2 , J 3 ) = 0 {\displaystyle f(I_{1},J_{2},J_{3})=0\,} where I 1 {\displaystyle I_{1}} is the first principal invariant of the Cauchy stress and J 2 , J 3 {\displaystyle J_{2},J_{3}} are the second and third principal invariants of the deviatoric part of the Cauchy stress.
f ( p , q , r ) = 0 {\displaystyle f(p,q,r)=0\,} where p , q {\displaystyle p,q} are scaled versions of I 1 {\displaystyle I_{1}} and J 2 {\displaystyle J_{2}} and r {\displaystyle r} is a function of J 2 , J 3 {\displaystyle J_{2},J_{3}} .
f ( ξ , ρ , θ ) = 0 {\displaystyle f(\xi ,\rho ,\theta )=0\,} where ξ , ρ {\displaystyle \xi ,\rho } are scaled versions of I 1 {\displaystyle I_{1}} and J 2 {\displaystyle J_{2}} , and θ {\displaystyle \theta } is the stress angle or Lode angle
Invariants used to describe yield surfaces
The first principal invariant ( I 1 {\displaystyle I_{1}} ) of the Cauchy stress ( σ {\displaystyle {\boldsymbol {\sigma }}} ), and the second and third principal invariants ( J 2 , J 3 {\displaystyle J_{2},J_{3}} ) of the deviatoric part ( s {\displaystyle {\boldsymbol {s}}} ) of the Cauchy stress are defined as:
I 1 = Tr ( σ ) = σ 1 + σ 2 + σ 3 J 2 = 1 2 s : s = 1 6 [ ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 ] J 3 = det ( s ) = 1 3 ( s ⋅ s ) : s = s 1 s 2 s 3 {\displaystyle {\begin{aligned}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{aligned}}}
where ( σ 1 , σ 2 , σ 3 {\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}} ) are the principal values of σ {\displaystyle {\boldsymbol {\sigma }}} , ( s 1 , s 2 , s 3 {\displaystyle s_{1},s_{2},s_{3}} ) are the principal values of s {\displaystyle {\boldsymbol {s}}} , and
s = σ − I 1 3 I {\displaystyle {\boldsymbol {s}}={\boldsymbol {\sigma }}-{\tfrac {I_{1}}{3}}\,{\boldsymbol {I}}}
where I {\displaystyle {\boldsymbol {I}}} is the identity matrix. A related set of quantities, ( p , q , r {\displaystyle p,q,r\,} ), are usually used to describe yield surfaces for cohesive frictional materials such as rocks, soils, and ceramics. These are defined as
p = 1 3 I 1 : q = 3 J 2 = σ e q ; r = 3 ( 1 2 J 3 ) 1 / 3 {\displaystyle 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 σ e q {\displaystyle \sigma _{\mathrm {eq} }} is the equivalent stress. However, the possibility of negative values of J 3 {\displaystyle J_{3}} and the resulting imaginary r {\displaystyle r} makes the use of these quantities problematic in practice. Another related set of widely used invariants is ( ξ , ρ , θ {\displaystyle \xi ,\rho ,\theta \,} ) which describe a cylindrical coordinate system (the Haigh–Westergaard coordinates). These are defined as:
ξ = 1 3 I 1 = 3 p ; ρ = 2 J 2 = 2 3 q ; cos ( 3 θ ) = ( r q ) 3 = 3 3 2 J 3 J 2 3 / 2 {\displaystyle \xi ={\tfrac {1}{\sqrt {3}}}~I_{1}={\sqrt {3}}~p~;~~\rho ={\sqrt {2J_{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 ξ − ρ {\displaystyle \xi -\rho \,} plane is also called the Rendulic plane. The angle θ {\displaystyle \theta } is called stress angle, the value cos ( 3 θ ) {\displaystyle \cos(3\theta )} is sometimes called the Lode parameter and the relation between θ {\displaystyle \theta } and J 2 , J 3 {\displaystyle J_{2},J_{3}} was first given by Novozhilov V.V. in 1951, see also The principal stresses and the Haigh–Westergaard coordinates are related by
[ σ 1 σ 2 σ 3 ] = 1 3 [ ξ ξ ξ ] + 2 3 ρ [ cos θ cos ( θ − 2 π 3 ) cos ( θ + 2 π 3 ) ] = 1 3 [ ξ ξ ξ ] + 2 3 ρ [ cos θ − sin ( π 6 − θ ) − sin ( π 6 + θ ) ] . {\displaystyle {\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:
sin ( 3 θ ) = 3 3 2 J 3 J 2 3 / 2 {\displaystyle \sin(3\theta )=~{\tfrac {3{\sqrt {3}}}{2}}~{\cfrac {J_{3}}{J_{2}^{3/2}}}}
in which case the ordered principal stresses (where σ 1 ≥ σ 2 ≥ σ 3 {\displaystyle \sigma _{1}\geq \sigma _{2}\geq \sigma _{3}} ) are related by
[ σ 1 σ 2 σ 3 ] = 1 3 [ ξ ξ ξ ] + ρ 2 [ cos θ − sin θ 3 2 sin θ 3 − sin θ 3 − cos θ ] . {\displaystyle {\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}}\,.}
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. 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
1 2 max ( | σ 1 − σ 2 | , | σ 2 − σ 3 | , | σ 3 − σ 1 | ) = S s y = 1 2 S y {\displaystyle {\tfrac {1}{2}}{\max(|\sigma _{1}-\sigma _{2}|,|\sigma _{2}-\sigma _{3}|,|\sigma _{3}-\sigma _{1}|)=S_{sy}={\tfrac {1}{2}}S_{y}}\!}
Where S s y {\displaystyle S_{sy}} is the yield strength in shear, and S y {\displaystyle 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 σ 1 , σ 2 {\displaystyle \sigma _{1},\sigma _{2}} plane.
von Mises yield surface
The von Mises yield criterion is expressed in the principal stresses as
( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 = 2 S y 2 {\displaystyle {(\sigma _{1}-\sigma _{2})^{2}+(\sigma _{2}-\sigma _{3})^{2}+(\sigma _{3}-\sigma _{1})^{2}=2{S_{y}}^{2}}\!}
where S y {\displaystyle 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 σ 1 , σ 2 {\displaystyle \sigma _{1},\sigma _{2}} produces the elliptical shape of the yield surface.
Burzyński-Yagn criterion This criterion reformulated as the function of the hydrostatic nodes with the coordinates 1 / γ 1 {\displaystyle 1/\gamma _{1}} and 1 / γ 2 {\displaystyle 1/\gamma _{2}}
3 I 2 ′ = σ e q − γ 1 I 1 1 − γ 1 σ e q − γ 2 I 1 1 − γ 2 {\displaystyle 3I_{2}'={\frac {\sigma _{\mathrm {eq} }-\gamma _{1}I_{1}}{1-\gamma _{1}}}{\frac {\sigma _{\mathrm {eq} }-\gamma _{2}I_{1}}{1-\gamma _{2}}}}
represents the general equation of a second order surface of revolution about the hydrostatic axis. Some special case are:
cylinder γ 1 = γ 2 = 0 {\displaystyle \gamma _{1}=\gamma _{2}=0} (Maxwell (1865), Huber (1904), von Mises (1913), Hencky (1924)), cone γ 1 = γ 2 ∈ ] 0 , 1 [ {\displaystyle \gamma _{1}=\gamma _{2}\in ]0,1[} (Botkin (1940), Drucker-Prager (1952), Mirolyubov (1953)), paraboloid γ 1 ∈ ] 0 , 1 [ , γ 2 = 0 {\displaystyle \gamma _{1}\in ]0,1[,\gamma _{2}=0} (Burzyński (1928), Balandin (1937), Torre (1947)), ellipsoid centered of symmetry plane I 1 = 0 {\displaystyle I_{1}=0} , γ 1 = − γ 2 ∈ ] 0 , 1 [ {\displaystyle \gamma _{1}=-\gamma _{2}\in ]0,1[} (Beltrami (1885)), ellipsoid centered of symmetry plane I 1 = 1 2 ( 1 γ 1 + 1 γ 2 ) {\displaystyle I_{1}={\frac {1}{2}}\,{\bigg (}{\frac {1}{\gamma _{1}}}+{\frac {1}{\gamma _{2}}}{\bigg )}} with γ 1 ∈ ] 0 , 1 [ , γ 2 < 0 {\displaystyle \gamma _{1}\in ]0,1[,\gamma _{2}<0} (Schleicher (1926)), hyperboloid of two sheets γ 1 ∈ ] 0 , 1 [ , γ 2 ∈ ] 0 , γ 1 [ {\displaystyle \gamma _{1}\in ]0,1[,\gamma _{2}\in ]0,\gamma _{1}[} (Burzynski (1928), Yagn (1931)), hyperboloid of one sheet centered of symmetry plane I 1 = 0 {\displaystyle I_{1}=0} , γ 1 = − γ 2 = a i {\displaystyle \gamma _{1}=-\gamma _{2}=a\,i} , i = − 1 {\displaystyle i={\sqrt {-1}}} (Kuhn (1980)) hyperboloid of one sheet γ 1 , 2 = b ± a i {\displaystyle \gamma _{1,2}=b\pm a\,i} , i = − 1 {\displaystyle i={\sqrt {-1}}} (Filonenko-Boroditsch (1960), Gol’denblat-Kopnov (1968), Filin (1975)). The relations compression-tension and torsion-tension can be computed to
σ − σ +