|
|
| Line 1: |
Line 1: |
| {{Unreferenced|date=December 2009}}
| | Hello. Allow me introduce the author. Her title is Refugia Shryock. For many years he's been residing in North Dakota and his family members enjoys it. Doing ceramics is what her family and her appreciate. My day occupation is a meter reader.<br><br>my web blog ... at home std test ([http://musica.togyu.com/Canto/97974 read the article]) |
| In [[mathematics]], '''Voigt notation''' or '''Voigt form''' in [[multilinear algebra]] is a way to represent a [[symmetric tensor]] by reducing its order. There are a few variants and associated names for this idea: '''Mandel notation''', '''Mandel–Voigt notation''' and '''Nye notation''' are others found. '''Kelvin notation''' is a revival by Helbig (1994) of old ideas of [[Lord Kelvin]]. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application.
| |
| | |
| For example, a 2×2 symmetric tensor '''''X''''' has only three distinct elements, the two on the diagonal and the other being off-diagonal. Thus it can be expressed as the vector | |
| | |
| :<math>\langle x_{1 1}, x_{2 2}, x_{1 2}\rangle</math>.
| |
| | |
| As another example:
| |
| | |
| The stress tensor (in matrix notation) is given as
| |
| :<math>\boldsymbol{\sigma}=
| |
| \left[{\begin{matrix}
| |
| \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
| |
| \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
| |
| \sigma_{zx} & \sigma_{zy} & \sigma_{zz}
| |
| \end{matrix}}\right].
| |
| </math>
| |
| | |
| In Voigt notation it is simplified to a 6-dimensional vector:
| |
| :<math>\tilde\sigma= (\sigma_{xx}, \sigma_{yy}, \sigma_{zz},
| |
| \sigma_{yz},\sigma_{xz},\sigma_{xy}) \equiv (\sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5, \sigma_6).
| |
| </math>
| |
| | |
| The strain tensor, similar in nature to the stress tensor -- both are symmetric second-order tensors --, is given in matrix form as
| |
| :<math>\boldsymbol{\epsilon}=
| |
| \left[{\begin{matrix}
| |
| \epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz} \\
| |
| \epsilon_{yx} & \epsilon_{yy} & \epsilon_{yz} \\
| |
| \epsilon_{zx} & \epsilon_{zy} & \epsilon_{zz}
| |
| \end{matrix}}\right].
| |
| </math>
| |
| | |
| Its representation in Voigt notation is
| |
| :<math>\tilde\epsilon= (\epsilon_{xx}, \epsilon_{yy}, \epsilon_{zz},
| |
| \gamma_{yz},\gamma_{xz},\gamma_{xy}) \equiv (\epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4, \epsilon_5, \epsilon_6),
| |
| </math>
| |
| where <math>\gamma_{xy}=2\epsilon_{xy}</math>, <math>\gamma_{yz}=2\epsilon_{yz}</math>, and <math>\gamma_{zx}=2\epsilon_{zx}</math> are engineering shear strains.
| |
| | |
| The benefit of using different representations for stress and strain is that the scalar invariance
| |
| :<math> \boldsymbol{\sigma}\cdot\boldsymbol{\epsilon} = \sigma_{ij}\epsilon_{ij} = \tilde\sigma \cdot \tilde\epsilon
| |
| </math>
| |
| is preserved.
| |
| | |
| Likewise, a three-dimensional symmetric fourth-order tensor can be reduced to a 6×6 matrix.
| |
| | |
| == Mnemonic rule ==
| |
| An easy [[Mnemonic|Mnemonic rule]] for memorizing Voigt notation is as follows:
| |
| | |
| * Write down the second order tensor in matrix form (in the Example the Stress Tensor)
| |
| * Strike out the diagonal
| |
| * Continue on the third column
| |
| * Go back to the first element along the first row.
| |
| | |
| Voigt indexes are numbered consecutively from the starting point to the end (in Example the numbers in blue).
| |
| | |
| [[File:Voigt notation Mnemonic rule.png]] | |
| | |
| ==Mandel notation==
| |
| For a symmetric tensor of second rank
| |
| :<math> \boldsymbol{\sigma}=
| |
| \left[{\begin{matrix}
| |
| \sigma_{11} & \sigma_{12} & \sigma_{13} \\
| |
| \sigma_{21} & \sigma_{22} & \sigma_{23} \\
| |
| \sigma_{31} & \sigma_{32} & \sigma_{33}
| |
| \end{matrix}}\right]
| |
| </math>
| |
| | |
| only six components are distinct, the three on the diagonal and the other being off-diagonal.
| |
| Thus it can be expressed, in Mandel notation, as the vector
| |
| :<math>
| |
| \tilde \sigma ^M=
| |
| \langle \sigma_{11},
| |
| \sigma_{22},
| |
| \sigma_{33},
| |
| \sqrt 2 \sigma_{12},
| |
| \sqrt 2 \sigma_{23},
| |
| \sqrt 2 \sigma_{13}
| |
| \rangle. </math>
| |
| | |
| The main advantage of Mandel notation is to allow the use of the same conventional operations used with vectors,
| |
| for example:
| |
| :<math> \tilde \sigma : \tilde \sigma = \tilde \sigma^M \cdot \tilde \sigma^M =
| |
| \sigma_{11}^2 +
| |
| \sigma_{22}^2 +
| |
| \sigma_{33}^2 +
| |
| 2 \sigma_{12}^2 +
| |
| 2 \sigma_{23}^2 +
| |
| 2 \sigma_{13}^2.
| |
| </math>
| |
| | |
| A symmetric tensor of rank four satisfying <math> D_{ijkl} = D_{jikl} </math> and <math> D_{ijkl} = D_{ijlk} </math> has 81 components in four-dimensional space, but only 36
| |
| components are distinct. Thus, in Mandel notation, it can be expressed as
| |
| :<math> \tilde D^M=
| |
| \begin{pmatrix}
| |
| D_{1111} & D_{1122} & D_{1133} & \sqrt 2 D_{1112} & \sqrt 2 D_{1123} & \sqrt 2 D_{1113} \\
| |
| D_{2211} & D_{2222} & D_{2233} & \sqrt 2 D_{2212} & \sqrt 2 D_{2223} & \sqrt 2 D_{2213} \\
| |
| D_{3311} & D_{3322} & D_{3333} & \sqrt 2 D_{3312} & \sqrt 2 D_{3323} & \sqrt 2 D_{3313} \\
| |
| \sqrt 2 D_{1211} & \sqrt 2 D_{1222} & \sqrt 2 D_{1233} & 2 D_{1212} & 2 D_{1223} & 2 D_{1213} \\
| |
| \sqrt 2 D_{2311} & \sqrt 2 D_{2322} & \sqrt 2 D_{2333} & 2 D_{2312} & 2 D_{2323} & 2 D_{2313} \\
| |
| \sqrt 2 D_{1311} & \sqrt 2 D_{1322} & \sqrt 2 D_{1333} & 2 D_{1312} & 2 D_{1323} & 2 D_{1313} \\
| |
| \end{pmatrix}.
| |
| </math>
| |
| | |
| ==Applications==
| |
| The notation is named after physicist [[Woldemar Voigt]]. 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]].
| |
| | |
| Hooke's law has a symmetric fourth-order stiffness tensor with 81 components (3×3×3×3). Voigt notation enables this to be simplified to a 6×6 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 be found in Helnwein (2001).
| |
| | |
| ==See also==
| |
| * [[Vectorization (mathematics)]]
| |
| * [[Hooke's law]]
| |
| * P. Helnwein (2001). Some Remarks on the Compressed Matrix Representation of Symmetric Second-Order and Fourth-Order Tensors. Computer Methods in Applied Mechanics and Engineering, 190(22–23):2753–2770
| |
| | |
| {{tensors}}
| |
| {{DEFAULTSORT:Voigt Notation}}
| |
| | |
| [[Category:Tensors]]
| |
| [[Category:Mathematical notation]]
| |
| [[Category:Solid mechanics]]
| |
Hello. Allow me introduce the author. Her title is Refugia Shryock. For many years he's been residing in North Dakota and his family members enjoys it. Doing ceramics is what her family and her appreciate. My day occupation is a meter reader.
my web blog ... at home std test (read the article)