Lode coordinates

Lode coordinates formula_1 or Haigh–Westergaard coordinates formula_2. 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. Other examples of sets of tensor invariants are the set of principal stresses formula_3 or the set of kinematic invariants formula_4. 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 formula_5.
Mechanics invariants.
The Lode coordinates are most easily computed using the mechanics invariants. These invariants are a mixture of the invariants of the Cauchy stress tensor, formula_6, and the stress deviator, formula_7, and are given by
which can be written equivalently in Einstein notation
where formula_14 is the Levi-Civita symbol (or permutation symbol) and the last two forms for formula_15 are equivalent because formula_7 is symmetric (formula_17).
The gradients of these invariants can be calculated by
where formula_21 is the second-order identity tensor and formula_22 is called the Hill tensor.
Axial coordinate formula_23.
The formula_24-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 formula_27.
The formula_28-coordinate is found by calculating the magnitude of the stress deviator (the orthogonal projection of the stress state into the deviatoric plane).
where
is a unit tensor in the direction of the radial component.
Lode angle – angular coordinate formula_31.
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 (here denoted formula_32, formula_33, and formula_34, respectively)
and are related by
These definitions are all defined for a range of formula_37.
The unit normal in the angular direction which completes the orthonormal basis can be calculated for formula_32 and formula_34 using
Meridional profile.
The meridional profile is a 2D plot of formula_41 holding formula_42 constant and is sometimes plotted using scalar multiples of formula_41. It is commonly used to demonstrate the pressure dependence of a yield surface or the pressure-shear trajectory of a stress path. Because formula_28 is non-negative the plot usually omits the negative portion of the formula_28-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 formula_41 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 formula_41 by constant multiples of equal absolute value are also isomorphic with respect to principal stress space. As an example, pressure formula_48 and the Von Mises stress formula_49 are not an isomorphic coordinate pair and, therefore, distort the yield surface because
and, finally, formula_52.
Octahedral profile.
The octahedral profile is a 2D plot of formula_53 holding formula_24 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