Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 24

Section Stress and strain tensors – Voigt matrices

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: Stress and strain tensors – Voigt matrices

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The stress and strain tensors are symmetric because body torques and rotations are not taken into account, respectively (see Sections 1.3.1[link] and 1.3.2[link] ). Their components are usually represented using Voigt's one-index notation.

  • (i) Strain tensor [\left. \matrix{S_{1} = S_{11}\semi\hfill &S_{2} = S_{22}\semi\hfill &S_{3} = S_{33}\semi\hfill \cr S_{4} = S_{23} + S_{32}\semi\hfill &S_{5} = S_{31} + S_{13}\semi\hfill &S_{6} = S_{12} + S_{21}\semi\hfill \cr S_{4} = 2S_{23} = 2S_{32}\semi &S_{5} = 2S_{31} = 2S_{13}\semi &S_{6} = 2S_{12} = 2S_{21}.\cr}\right\} \eqno(]The Voigt components [S_{\alpha}] form a Voigt matrix: [\pmatrix{S_{1} &S_{6} &S_{5}\cr &S_{2} &S_{4}\cr & &S_{3}\cr}.]The terms of the leading diagonal represent the elongations (see Section 1.3.1[link] ). It is important to note that the non-diagonal terms, which represent the shears, are here equal to twice the corresponding components of the strain tensor. The components [S_{\alpha}] of the Voigt strain matrix are therefore not the components of a tensor.

  • (ii) Stress tensor [\left. \matrix{T_{1}= T_{11}\semi\hfill &T_{2} = T_{22}\semi\hfill &T_{3} = T_{33}\semi\hfill\cr T_{4} = T_{23} = T_{32}\semi &T_{5} = T_{31} = T_{13}\semi &T_{6} = T_{12} =T_{21}.\cr}\right\}]The Voigt components [T_{\alpha}] form a Voigt matrix: [\pmatrix{T_{1} &T_{6} &T_{5}\cr &T_{2} &T_{4}\cr & & T_{3}\cr}.]The terms of the leading diagonal correspond to principal normal constraints and the non-diagonal terms to shears (see Section 1.3.2[link] ).

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