Editing Voigt notation (section)

==Applications==
It is useful, for example, in calculations involving constitutive models to simulate materials, such as the generalized [[Hooke's law]], as well as [[finite element analysis]],<ref>{{Cite book
  | author1 = O.C. Zienkiewicz
  | author2 = R.L. Taylor
  | author3 = J.Z. Zhu
  | title = The Finite Element Method: Its Basis and Fundamentals
  | year = 2005
  | edition = 6
  | publisher =  Elsevier Butterworth—Heinemann
  | isbn = 978-0-7506-6431-8
}}</ref> and [[Diffusion MRI]].<ref>{{cite book
  | title = Visualization and Processing of Tensor Fields
  | chapter = The Algebra of Fourth-Order Tensors with Application to Diffusion MRI
  | author = Maher Moakher
  | series = Mathematics and Visualization
 | year = 2009
  | pages = 57–80
  | publisher = Springer Berlin Heidelberg
  | doi = 10.1007/978-3-540-88378-4_4
| isbn = 978-3-540-88377-7
 }}</ref>

Hooke's law has a symmetric fourth-order [[stiffness tensor]] with 81 components (3&times;3&times;3&times;3), but because the application of such a rank-4 tensor to a symmetric rank-2 tensor must yield another symmetric rank-2 tensor, not all of the 81 elements are independent. Voigt notation enables such a rank-4 tensor to be ''represented'' by a 6&times;6 matrix. However, Voigt's form does not preserve the sum of the squares, which in the case of Hooke's law has geometric significance. This explains why weights are introduced (to make the mapping an [[isometry]]).

A discussion of invariance of Voigt's notation and Mandel's notation can be found in Helnwein (2001).<ref name="Helnwein">{{cite journal
  | title = Some Remarks on the Compressed Matrix Representation of Symmetric Second-Order and Fourth-Order Tensors
  | author = Peter Helnwein
  | journal = Computer Methods in Applied Mechanics and Engineering
  | volume = 190
  | issue = 22–23
  | pages = 2753–2770
  | date = February 16, 2001
  | doi=10.1016/s0045-7825(00)00263-2| bibcode = 2001CMAME.190.2753H
 }}</ref>