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 Lode angle is bounded to 0 ≤ θ ≤ π/3 when the principal stresses are sorted as
σp1 ≥ σp2 ≥ σp3 (
Ref. 1).
Following this convention, θ =
0 corresponds to the
tensile meridian, and
θ =
π/3
corresponds to the
compressive meridian. The Lode angle is part of a cylindrical coordinate system (the Haigh–Westergaard coordinates) with height (hydrostatic axis)
and radius
.
The octahedral plane (also called π-plane) is defined perpendicular to the hydrostatic axis in the Haigh–Westergaard coordinate system. The stress normal to this plane is
σoct =
I1/
3, and the shear stress on that plane is defined by
The functions 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 principal stresses (σp1,
σp2, and
σp3) can, when sorted as
σp1 ≥ σp2 ≥ σp3, be written by using the invariants
I1 and
J2 and the Lode angle (
Ref. 1):