Lode coordinates

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Lode coordinates or Haigh–Westergaard coordinates .[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 or the set of kinematic invariants . 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 .

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.

Mechanics invariants

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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[3]

 
 
 

which can be written equivalently in Einstein notation

 
 
 

where   is the Levi-Civita symbol (or permutation symbol) and the last two forms for   are equivalent because   is symmetric ( ).

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

 
 
 

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

Axial coordinate

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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

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The  -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

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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.

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  ,  , and  , respectively)

 

and are related by

 

These definitions are all defined for a range of  .

Stress State        
range      
Triaxial Compression (TXC)        
Shear (SHR)        
Triaxial Extension (TXE)        

The unit normal in the angular direction which completes the orthonormal basis can be calculated for  [8] and  [9] using

 .

Meridional profile

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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.

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 (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. 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

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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  .

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'[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

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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

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References

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  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. ^ a b 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.