Lode coordinates explained

Lode coordinates

(z,r,\theta) or Haigh–Westergaard coordinates (\xi,\rho,\theta) .^[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.^[2] 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{\sigma} , and the stress deviator, \boldsymbol{s} , and are given by^[3] I _{1} = tr (\boldsymbol{\sigma}) J _{2} = (1)/(2) \left[ tr (\boldsymbol{\sigma} ^{2)} - (1)/(3) tr (\boldsymbol{\sigma}) ^{2\right]} = (1)/(2) tr \left(\boldsymbol{s} ⋅ \boldsymbol{s}\right) = (1)/(2) \lVert \boldsymbol{s} \rVert ^{2} J _{3} = det (\boldsymbol{s}) = (1)/(3) tr \left(\boldsymbol{s} ⋅ \boldsymbol{s} ⋅ \boldsymbol{s}\right)

which can be written equivalently in Einstein notation

I _{1} = \sigma _{kk} J _{2} = (1)/(2) \left[ tr (\boldsymbol{\sigma} ^{2)} - (1)/(3) tr (\boldsymbol{\sigma}) ^{2\right]} = (1)/(2) s _{ij} s _{ji} = (1)/(2) s _{ij} s _{ij} J _{3} = (1)/(6) \epsilon _{ijk} \epsilon _{pqr} \sigma _{ip} \sigma _{jq} \sigma _{kr} = (1)/(3) s _{ij} s _{jk} s _{ki}

where

\epsilon is the Levi-Civita symbol (or permutation symbol) and the last two forms for J _{2} are equivalent because \boldsymbol{s} is symmetric (s _{ij} = s _{ji} ).

The gradients of these invariants^[4] can be calculated by

(\partial I _{1})/(\partial \boldsymbol{\sigma) } = \boldsymbol (\partial J _{2})/(\partial \boldsymbol{\sigma) } = \boldsymbol = \boldsymbol - \frac\boldsymbol(\partial J _{3})/(\partial \boldsymbol{\sigma) } = \boldsymbol = \boldsymbol\cdot\boldsymbol - \frac\boldsymbol

where

\boldsymbol{I} is the second-order identity tensor and \boldsymbol{T} 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{E _{z}} \colon \boldsymbol{\sigma} = (tr (\boldsymbol{\sigma)/()}{\sqrt{ 3 }}) = (I _{1})/(\sqrt{ 3) }

where

\boldsymbol{E _{z}} = (\boldsymbol{I)/() } = \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{E _{r}\colon} \boldsymbol{\sigma} = \lVert \boldsymbol{s} \rVert = \sqrt{ 2 J _{2}}

where

\boldsymbol{E _{r}} = (\boldsymbol{s)/() }

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

\theta _{s} , \bar{\theta} _{s} , and \theta _{c} , respectively)\sin( 3 \theta _{s)} = - \sin( 3 \bar{\theta} _{s} ) = \cos( 3 \theta _{c)} = (J _{3})/(\left( 2) (3)/(J _{2}) \right) ^{3/2}

and are related by

\theta _{s} = (\pi)/(6) - \theta _{c} \theta _{s} = - \bar{\theta} _{s}

These definitions are all defined for a range of

\pi/ 3 .

	Stress State \sigma _{1} \geq \sigma _{2} \geq \sigma _{3}	\theta _{s}	\bar{\theta} _{s}	\theta _{c}
range		- (\pi)/(6) \leq \theta _{s} \leq (\pi)/(6)	- (\pi)/(6) \leq \theta _{s} \leq (\pi)/(6)	0 \leq \theta _{c} \leq (\pi)/(3)
Triaxial Compression (TXC)	\sigma _{1} \geq \sigma _{2} = \sigma _{3}	- (\pi)/(6)	(\pi)/(6)	(\pi)/(3)
Shear (SHR)	\sigma _{2=(\sigma} _{1+\sigma} _{3)/2}	0	0	(\pi)/(6)
Triaxial Extension (TXE)	\sigma _{1} = \sigma _{2} \geq \sigma _{3}	(\pi)/(6)	- (\pi)/(6)	0

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

\theta _{s} ^[8] and \theta _{c} ^[9] using\boldsymbol{E _{\theta} } = \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.^[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 (z,r) by constant multiples of equal absolute value are also isomorphic with respect to principal stress space. As an example, pressure p =- I 1 / 3 and the Von Mises stress \sigma _{v} = \sqrt{ 3 J _{2}} are not an isomorphic coordinate pair and, therefore, distort the yield surface becausep =- (1)/(\sqrt{ 3) }z \sigma _{v} = \sqrt{ (3)/(2) } r

and, finally,

| -1 /\sqrt{ 3 }| ≠ |\sqrt{ 3 / 2 }| .

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'^[10] or 'deviatoric plane'.^[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.

A note on terminology

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]}

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

1. Menetrey, P.H., Willam, K.J., 1995, Triaxial Failure Criterion for Concrete and Its Generalization, ACI Structural Journal
2. Lode, W. (1926). Versuche über den Einfuss der mittleren Hauptspannung auf das Fliessen der Metalle Eisen Kupfer und Nickel. Zeitung Phys., vol. 36, pp. 913–939.
3. Asaro, R.J., Lubarda, V.A., 2006, Mechanics of Solids and Materials, Cambridge University Press
4. Brannon, R.M., 2009, KAYENTA: Theory and User's Guide, Sandia National Laboratories, Albuquerque, New Mexico.
5. Chakrabarty, J., 2006, Theory of Plasticity: Third edition, Elsevier, Amsterdam.
6. de Souza Neto, E.A., Peric, D., Owen, D.R.J., 2008, Computational Methods for Plasticity, Wiley
7. Han, D.J., Chen, W.F., 1985, A Nonuniform Hardening Plasticity Model for Concrete Materials, Mechanics of Materials
8. Brannon, R.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
9. Bigoni, D., Piccolroaz, A., 2004, Yield criteria for quasibrittle and frictional materials, Int. J. Solids Struct.
10. Lubliner, J., 1990, Plasticity Theory, Pearson Education
11. Chaboche, J.L., 2008, A review of some plasticity and viscoplasticity theories, Int. J. Plasticity
12. Mouazen, A.M., Nemenyi, M., 1998, A review of the finite element modelling techniques of soil tillage, Mathematics and Computers in Simulation
13. Keryvin, V., 2008, Indentation as a probe for pressure sensitivity of metallic glasses, J. Phys.: Condens. Matter
14. Cervenka, J., Papanikolaou, V.K., 2008, Three dimensional combined fracture-plastic material model for concrete, Int. J. of Plasticity
15. Piccolroaz, A., Bigoni, D., 2009, Yield criteria for quasibrittle and frictional materials: A generalization to surfaces with corners, Int. J. of Solids and Struc.