Lode Coordinates

Lode coordinates $(z,r,\theta)$ or Haigh–Westergaard coordinates $(\xi,\rho,\theta)$.Menetrey, Philippe; Willam, K. J.; 1995, "Triaxial Failure Criterion for Concrete and its Generalization", ''Structural Journal'', American Concrete Institute, Volume 92, Issue 3, pages 311-318, DOI: 10.14359/1132 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 $(\sigma_1, \sigma_2, \sigma_3)$ or the set of kinematic invariants $(I_1, J_2, J_3)$. 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 $(\sigma_1,\sigma_2,\sigma_3)=(1,1,1)$.

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, $\boldsymbol$, and the stress deviator, $\boldsymbol$, and are given byAsaro, Robert J.; Lubarda, Vlado A.; 2006, ''Mechanics of Solids and Materials'', Cambridge University Press

:$I_1 = \mathrm(\boldsymbol)$

: J_2 = \frac\left text(\boldsymbol^2) - \frac\text(\boldsymbol)^2\right= \frac\mathrm\left(\boldsymbol\cdot\boldsymbol\right) = \frac\lVert \boldsymbol \rVert^2

:$J_3 = \mathrm(\boldsymbol) = \frac\mathrm\left(\boldsymbol\cdot\boldsymbol\cdot\boldsymbol\right)$

which can be written equivalently in Einstein notation

:$I_1 = \sigma_$

: J_2 = \frac\left text(\boldsymbol^2) - \frac\text(\boldsymbol)^2\right= \fracs_s_ = \fracs_s_

:$J_3 = \frac\epsilon_\epsilon_\sigma_\sigma_\sigma_ = \fracs_s_s_$

where $\epsilon$ is the Levi-Civita symbol (or permutation symbol) and the last two forms for $J_2$ are equivalent because $\boldsymbol$ is symmetric ($s_=s_$).

The gradients of these invariantsBrannon, Rebecca M.; 2009, ''KAYENTA: Theory and User's Guide'', Sandia National Laboratories, Albuquerque, New Mexico. can be calculated by

:$\frac = \boldsymbol$

:$\frac = \boldsymbol = \boldsymbol - \frac\boldsymbol$

:$\frac = \boldsymbol = \boldsymbol\cdot\boldsymbol - \frac\boldsymbol$

where $\boldsymbol$ is the second-order identity tensor and $\boldsymbol$ is called the Hill tensor.

Axial coordinate $(z)$

The $z$-coordinate is found by calculating the magnitude of the orthogonal projection of the stress state onto the hydrostatic axis.

:$z = \boldsymbol \colon \boldsymbol = \frac = \frac$

where

:$\boldsymbol = \frac = \frac$

is the unit normal in the direction of the hydrostatic axis.

Radial coordinate $(r)$

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

:$r = \boldsymbol\colon \boldsymbol = \lVert \boldsymbol \rVert = \sqrt$

where

:$\boldsymbol = \frac$

:

is a unit tensor in the direction of the radial component.

Lode angle – angular coordinate $(\theta)$

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,Chakrabarty, Jagabanduhu; 2006, ''Theory of Plasticity: Third edition'', Elsevier, Amsterdam. negative sine,de Souza Neto, Eduardo A.; Peric, Djordje; Owen, David R. J.; 2008, ''Computational Methods for Plasticity: Theory and Applications'', Wiley and positive cosineHan, D. J.; Chen, Wai-Fah; 1985, "A Nonuniform Hardening Plasticity Model for Concrete Materials", ''Mechanics of Materials'', Volume 4, Issues 3–4, December 1985, Pages 283-302, doi: 10.1016/0167-6636(85)90025-0 (here denoted $\theta_s$, $\bar_s$, and $\theta_c$, respectively)

:$\sin(3\theta_s) = -\sin(3\bar_) = \cos(3\theta_c) = \frac\left(\frac\right)^$

and are related by

:$\theta_s = \frac - \theta_c \qquad \qquad \theta_s = -\bar_s$

:

These definitions are all defined for a range of $\pi/3$.

The unit normal in the angular direction which completes the orthonormal basis can be calculated for $\theta_s$Brannon, Rebecca M.; 2007, ''Elements of Phenomenological Plasticity: Geometrical Insight, Computational Algorithms, and Topics in Shock Physics'', Shock Wave Science and Technology Reference Library: Solids I, Springer-New York and $\theta_c$Bigoni, Davide; Piccolroaz, Andrea; 2004, "Yield criteria for quasibrittle and frictional materials", ''International Journal of Solids and Structures'', Volume 41, Issues 11–12, June 2004, Pages 2855-2878, doi: 10.1016/j.ijsolstr.2003.12.024 using

:$\boldsymbol = \frac \qquad \boldsymbol = \frac$.

Meridional profile

The meridional profile is a 2D plot of $(z,r)$ holding $\theta$ constant and is sometimes plotted using scalar multiples of $(z,r)$. It is commonly used to demonstrate the pressure dependence of a yield surface or the pressure-shear trajectory of a stress path. Because $r$ is non-negative the plot usually omits the negative portion of the $r$-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 $(z,r)$ 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 $(z,r)$ by constant multiples of equal absolute value are also isomorphic with respect to principal stress space. As an example, pressure $p=-I1/3$ and the Von Mises stress $\sigma_v = \sqrt$ are not an isomorphic coordinate pair and, therefore, distort the yield surface because

:$p = -\fracz$

:$\sigma_v = \sqrt r$

and, finally, $, -1/\sqrt, \neq , \sqrt,$.

Octahedral profile

The octahedral profile is a 2D plot of $(r,\theta)$ holding $z$ 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 spaceKeryvin, Vincent; 2008, "Indentation as a probe for pressure sensitivity of metallic glasses", ''Journal of Physics: Condensed Matter'', Volume 20, Number 11, DOI: 10.1088/0953-8984/20/11/114119 and the cylindrical Lode coordinate spaceČervenka, Jan; Papanikolaou, Vassilis K.; 2008, "Three dimensional combined fracture–plastic material model for concrete", ''International Journal of Plasticity'', Volume 24, Issue 12, December 2008, Pages 2192-2220,
doi: 10.1016/j.ijplas.2008.01.004Piccolroaz, Andrea; Bigoni, Davide; 2009, "Yield criteria for quasibrittle and frictional materials: A generalization to surfaces with corners", ''International Journal of Solids and Structures'', Volume 46, Issue 20, 1 October 2009, Pages 3587-3596, doi: 10.1016/j.ijsolstr.2009.06.006

See also

* Yield (engineering)
* Plasticity (physics)
* Stress
* Henri Tresca
* von Mises stress
* Mohr–Coulomb theory
* Strain
* Strain tensor
* Stress–energy tensor
* Stress concentration
* 3-D elasticity

References

Solid mechanics
Materials science