As defined above J2 ≥ 0. In many material models, the most relevant invariants are
I1,
J2, and
J3.
I1 represents the effect of mean stress,
J2 represents the magnitude of shear stress, and
J3 contains information about the direction of the shear stress.
The principal stresses are the roots of the characteristic equation (Cayley–Hamilton theorem)
The expressions described in Equation 3-9 and
Equation 3-10 enter into expressions that define various kind of yield and failure surfaces. A yield surface is a surface in the 3D space of principal stresses that circumscribe an elastic state of stress.
The Lode angle is bounded to 0 ≤ θ ≤ π/3 when the principal stresses are sorted as
σp1 ≥ σp2 ≥ σp3 (
Ref. 51). Following this convention,
θ =
0 corresponds to the
tensile meridian, and
θ =
π/3
corresponds to the
compressive meridian.
The principal stresses (σp1,
σp2, and
σp3) are then written in terms of the invariants
I1,
J2, and the Lode angle
θ as:
The octahedral plane (also called
π-plane or deviatoric plane) lies perpendicular to the hydrostatic axis in the
Haigh–Westergaard Coordinates. The stress normal to this plane is
σoct =
I1/
3, also called
mean stress or
hydrostatic stress, and the shear stress on that plane is defined by

.
The meridional plane, also called Rendulic plane, is the profile of the yield criterion at a given Lode angle, typically at the compressive meridian
θ =
π/3.
In the Haigh–Westergaard Coordinates, the meridional plane is given by the coordinates

and

, but it also common to visualize criteria in the
pq-plane, given by the coordinates

and

, or the plane given by the axes

and

.