Lode coordinates

( ξ , ρ , θ )

[1] are a set of tensor invariants that span the space of real, symmetric, second-order, 3-dimensional tensors and are isomorphic with respect to principal stress space.

This right-handed orthogonal coordinate system is named in honor of the German scientist Dr. Walter Lode because of his seminal paper written in 1926 describing the effect of the middle principal stress on metal plasticity.

The Lode coordinate system can be described as a cylindrical coordinate system within principal stress space with a coincident origin and the z-axis parallel to the vector

The Lode coordinates are most easily computed using the mechanics invariants.

, and are given by[3] which can be written equivalently in Einstein notation where

-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.

-coordinate is found by calculating the magnitude of the stress deviator (the orthogonal projection of the stress state into the deviatoric plane).

in terms of the isotropic and deviatoric parts while expanding the magnitude of

Which leaves us with Applying the identity

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,[5] negative sine,[6] and positive cosine[7] (here denoted

The unit normal in the angular direction which completes the orthonormal basis can be calculated for

[9] using The meridional profile is a 2D plot of

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.

is non-negative the plot usually omits the negative portion of the

-axis, but can be included to illustrate effects at opposing Lode angles (usually triaxial extension and triaxial compression).

One of the benefits of plotting the meridional profile with

is that it is a geometrically accurate depiction of the yield surface.

[8] 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.

are not an isomorphic coordinate pair and, therefore, distort the yield surface because and, finally,

Plotting the yield surface in the octahedral plane demonstrates the level of Lode angle dependence.

[11] 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.

The term Haigh-Westergaard space is ambiguously used in the literature to mean both the Cartesian principal stress space[12][13] and the cylindrical Lode coordinate space[14][15]

Surfaces on which the invariants , , are constant. Plotted in principal stress space. The red plane represents a meridional plane and the yellow plane an octahedral plane.
This plot demonstrates that an intuitive approximation for the Lode angle is the relative position of the middle principal stress with respect to the low and high principal stresses.
This plot shows a typical meridional profile of several plasticity models: von Mises , linear Drucker–Prager , Mohr–Coulomb , Gurson , and Bigoni–Piccolroaz . The upper portion of the plot depicts yield surface behavior in triaxial extension and the lower portion depicts yield surface behavior in triaxial compression.
This plot shows a typical octahedral profile of several plasticity models: von Mises , linear Drucker–Prager , Mohr–Coulomb , Gurson , and Bigoni–Piccolroaz . This plot has omitted Lode angle values in favor of loading descriptions because of the preponderance of definitions of the Lode angle. The radial coordinate is .