The Lode coordinates are most easily computed using the mechanics invariants. These invariants are a mixture of the invariants of the Cauchy stress tensor,, and the stress deviator,, and are given by which can be written equivalently in Einstein notation where is the Levi-Civita symbol and the last two forms for are equivalent because is symmetric. The gradients of these invariants can be calculated by where is the 3x3 identity matrix and is called the Hill tensor.
Axial coordinate
The -coordinate is found by calculating the magnitude of the orthogonal projection of the stress state onto the hydrostatic axis. where is the unit normal in the direction of the hydrostatic axis.
Radial coordinate
The -coordinate is found by calculating the magnitude of the stress deviator. where and writing in terms of the isotropic and deviatoric parts while expanding the magnitude of Because is isotropic and is deviatoric, their product is zero. Which leaves us with Applying the identity and using the definition of is a unit tensor in the direction of the radial component.
The Lode angle can be considered, rather loosely, a measure of loading type. The Lode angle varies with respect to the middle eigenvalue of the stress. There are many definitions of Lode angle that each utilize different trigonometric functions: the positive sine, negative sine, and positive cosine and are related by Because cosine is an even function and the range of the inverse cosine is usually we take the negative possible value for the term, thus ensuring that is positive. These definitions are all defined for a range of.
The unit normal in the angular direction which completes the orthonormal basis can be calculated for and using
Meridional profile
The meridional profile is a 2D plot of holding constant and is sometimes plotted using scalar multiples of. It is commonly used to demonstrate the pressure dependence of a yield surface or the pressure-shear trajectory of a stress path. Because is non-negative the plot usually omits the negative portion of the -axis, but can be included to illustrate effects at opposing Lode angles. One of the benefits of plotting the meridional profile with is that it is a geometrically accurate depiction of the yield surface. If a non-isomorphic pair is used for the meridional profile then the normal to the yield surface will not appear normal in the meridional profile. Any pair of coordinates that differ from by constant multiples of equal absolute value are also isomorphic with respect to principal stress space. As an example, pressure and the Von Mises stress are not an isomorphic coordinate pair and, therefore, distort the yield surface because and, finally,.
Octahedral profile
The octahedral profile is a 2D plot of holding constant. Plotting the yield surface in the octahedral plane demonstrates the level of Lode angle dependence. The octahedral plane is sometimes referred to as the 'pi plane' or 'deviatoric plane'. The octahedral profile is not necessarily constant for different values of pressure with the notable exceptions of the von Mises yield criterion and the Tresca yield criterion which are constant for all values of pressure.
A note on terminology
The term Haigh-Westergaard space is ambiguously used in the literature to mean both the Cartesian principal stress space and the cylindrical Lode coordinate space
