Function Repository Resource:

TensorCoordinateTransform

Transform components of tensors with arbitrary rank with regard to their transformation behavior under any given mapping

Contributed by: Lars Ulke-Winter

ResourceFunction["TensorCoordinateTransform"][tensor,mat]

transform tensor components according to their transformation behavior matrix mat. All tensor slots are considered to be contravariant and not normalized.

Details and Options

Generally tensor components (with mixed n×m-rank) transform from one system to another (i→i') according to:

The mapping from the old system to the new one is described in the matrix

for covariant transformation behavior (tensor components with lower indices) and

for so-called contravariant tensor components (depicted with superscript indices). Summation must be done for the same but opposing indices about the dimensions of the tensor coordinates. The fundamental relationship of the contrary transformation behavior

ensures invariance of the tensor object T in all systems after contracting all opposing indices (e.g.

). Due to this fact, a distinction of covariant and contravariant transformation behavior is not required in transformation relations with J=(J)^-T, e.g. at transformations between orthonormal reference systems.

The input tensor can be of any rank and should be a List or an object of StructuredArray.

The resulting format is always a normal List.

Unlike TransformedField, the ResourceFunction["TensorCoordinateTransform"] result is not given with respect to a normalized basis by default. Using the option Normalize → True forces a normalized basis.

The matrix mat defines the mapping between two coordinate systems (often their transposed Jacobian).

The following options are supported:

"TransformationBehavior"

"AllContravariant"

transformation behavior of individual components "AllContravariant" or "AllCovariant" or {"Con","Cov", …}

Normalize

False

whether the components are represented with respect to a normalized basis

Examples

Basic Examples

Apply a rotation transformation to a tensor:

In[1]:=

$ResourceFunction[ "TensorCoordinateTransform"][{Subscript[v, 1], Subscript[v, 2], Subscript[v, 3]}, ( { {Cos[\[Phi]], -Sin[\[Phi]], 0}, {Sin[\[Phi]], Cos[\[Phi]], 0}, {0, 0, 1} } )]$

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Scope

Tensor Components with Respect to Rotated Base Vectors

Rotate vector components about the three-axes counterclockwise:

In[2]:=

$ResourceFunction[ "TensorCoordinateTransform"][{Subscript[v, 1], Subscript[v, 2], Subscript[v, 3]}, ( { {Cos[\[Phi]], -Sin[\[Phi]], 0}, {Sin[\[Phi]], Cos[\[Phi]], 0}, {0, 0, 1} } )] // Simplify // MatrixForm$

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Change the axis:

In[3]:=

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Rotate the rank-2 tensor about the three-axes counterclockwise:

In[4]:=

$ResourceFunction["TensorCoordinateTransform"][( { {Subscript[\[Sigma], 11], Subscript[\[Sigma], 12], Subscript[\[Sigma], 13]}, {Subscript[\[Sigma], 21], Subscript[\[Sigma], 22], Subscript[\[Sigma], 23]}, {Subscript[\[Sigma], 31], Subscript[\[Sigma], 32], Subscript[\[Sigma], 33]} } ), RotationMatrix[\[Theta], {0, 0, 1}]] // Simplify // MatrixForm$

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Rotate a rank-4 tensor with three symmetries about the three-axes counterclockwise :

In[5]:=

$SeedRandom[314]; t = SymmetrizedArray[ pos_ :> RandomInteger[10], {3, 3, 3, 3}, {{{2, 1, 3, 4}, 1}, {{1, 2, 4, 3}, 1}, {{3, 4, 1, 2}, 1}}]; tRot = ResourceFunction["TensorCoordinateTransform"][t, RotationMatrix[45 \[Degree], {0, 0, 1}]]; tRot // Simplify // MatrixForm$

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Transform Vector Components Regarding Another Affine Frame of Reference

Contravariant vector components with respect to covariant bases:

a=v^jb_j=v^i'b_i'

Old system is Cartesian (a^j=v^j):

In[7]:=

Transformation between new e_i' and old e_j covariant base vectors:

e_1'=2 e₁+ 3 e₂+1e₃

e_2'=1 e₁+ 2 e₂+2e₃

e_3'=1 e₁+2e₃

Contravariant components v^i' regarding the new system:

In[8]:=

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They should be the same vector:

In[9]:=

In[10]:=

$(\!$ \*UnderoverscriptBox[\(\[Sum]$, $i = 1$, $3$]$vNewCon[\([$$i$$]$] \ bNewCov[$[$$i$$]$]\)\))$

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In[11]:=

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Covariant vector components with respect to contravariant bases: a=v^jb_j=v_i'b^j'

In[12]:=

vNewCov = ResourceFunction["TensorCoordinateTransform"][{2, 8, 7}, ( {
{2, 3, 1},
{1, 2, 2},
{1, 0, 1}
} ), "TransformationBehavior" -> "AllCovariant"]

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Find related contravariant base vectors b^i', by transforming the covariant base in contravariant sense (index juggling b_i'→b^i'):

In[13]:=

bNewCon = ResourceFunction["TensorCoordinateTransform"][( {
{2, 3, 1},
{1, 2, 2},
{1, 0, 1}
} ), ( {
{2, 3, 1},
{1, 2, 2},
{1, 0, 1}
} ), "TransformationBehavior" -> "AllContravariant"]

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In[14]:=

$(\!$ \*UnderoverscriptBox[\(\[Sum]$, $i = 1$, $3$]$vNewCov[\([$$i$$]$] \ bNewCon[$[$$i$$]$]\)\))$

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Note: covariant and contravariant base vectors are inverse to each other

In[16]:=

$(bNewCon.bNewCov\[Transpose]) // MatrixForm$

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A Local Base on the Manifold of a Torus Surface (Curvilinear Coordinate System)

Example equation: aⁱb_i≡c^ijkls_klt_jb_i=c^i'j'k'l's_k'l't_j'b_i'≡a^i'b_i'→invariant

Tensors in Cartesian frame:

In[17]:=

cFourRankOld = RandomInteger[{-5, 5}, {3, 3, 3, 3}];
sTwoRankOld = RandomInteger[{-5, 5}, {3, 3}];
tOneRankOld = RandomInteger[{-5, 5}, {3}];

Cartesian → local torus coordinates (i →i^' ):

In[18]:=

$x[r_, \[Alpha]_, \[Beta]_] := (R + r Cos[\[Beta]]) Cos[\[Alpha]] y[r_, \[Alpha]_, \[Beta]_] := (R + r Cos[\[Beta]]) Sin[\[Alpha]] z[r_, \[Alpha]_, \[Beta]_] := r Sin[\[Beta]]$

The torus surface at α=β=0⋯2π at R and r constant:

In[19]:=

$ParametricPlot3D[({x[r, \[Alpha], \[Beta]], y[r, \[Alpha], \[Beta]], z[r, \[Alpha], \[Beta]]} /. {r -> 1, R -> 3}), {\[Alpha], 0, 2 \[Pi]}, {\[Beta], 0, 2 \[Pi]}]$

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Transform tensors’ different ranks regarding the local torus base with respect to transformation behavior (Hint: this is faster than transformation of an entire assembled 7-rank tensor):

In[20]:=

$cFourRankNew = ResourceFunction["TensorCoordinateTransform"][cFourRankOld, Transpose@(jacobianTorus = D[{x[r, \[Alpha], \[Beta]], y[r, \[Alpha], \[Beta]], z[r, \[Alpha], \[Beta]]}, {{r, \[Alpha], \[Beta]}}]), "TransformationBehavior" -> "AllContravariant"]; sTwoRankNew = ResourceFunction["TensorCoordinateTransform"][sTwoRankOld, jacobianTorus\[Transpose], "TransformationBehavior" -> "AllCovariant"]; tOneRankNew = ResourceFunction["TensorCoordinateTransform"][tOneRankOld, jacobianTorus\[Transpose], "TransformationBehavior" -> {"Cov"}];$

Local covariant base vector b_i':

In[21]:=

Assemble the tensor term by contracting the right indices c^i'j'k'l's_k'l't_j'b_i≡a^i'b_i':

In[22]:=

$aNew = TensorContract[ cFourRankNew\[TensorProduct]sTwoRankNew\[TensorProduct]tOneRankNew,\ {{2, 7}, {3, 5}, {4, 6}}] // Simplify$

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In[23]:=

$((\!$ \*UnderoverscriptBox[\(\[Sum]$, $i = 1$, $3$]$aNew[\([$$i$$]$] bNew[$[$$i$$]$]\)\))) \ // Simplify$

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The result should be the same in Cartesian system:

In[24]:=

$TensorContract[ cFourRankOld\[TensorProduct]sTwoRankOld\[TensorProduct]tOneRankOld, \ {{2, 7}, {3, 5}, {4, 6}}]$

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Options

TransformationBehavior

If the base vectors are not normalized ("Normalize"→False), i.e. have local-dependent lengths (and are not orthogonal in general reference systems either), the corresponding tensor components are transformed according to their co- and contravariant transformation behaviors. This leads to different representations of the same tensor object:

Here is an example of all four transforming possibilities of a rank-2 tensor to a local, non-normalized cylindrical system:

In[25]:=

$jacobianCylTransposed = Transpose@(CoordinateTransformData["Cylindrical" -> "Cartesian", "MappingJacobian", {r, \[Phi], z}]);$

All coordinates transform the contravariant t^ij (default):

In[26]:=

$tConCon = ResourceFunction["TensorCoordinateTransform"][( { {Subscript[\[Sigma], 11], Subscript[\[Sigma], 12], Subscript[\[Sigma], 13]}, {Subscript[\[Sigma], 21], Subscript[\[Sigma], 22], Subscript[\[Sigma], 23]}, {Subscript[\[Sigma], 31], Subscript[\[Sigma], 32], Subscript[\[Sigma], 33]} } ), jacobianCylTransposed] // Simplify[#, r > 0] & // TraditionalForm$

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All coordinates transform the covariant t_ij:

In[27]:=

$tCovCov = ResourceFunction["TensorCoordinateTransform"][( { {Subscript[\[Sigma], 11], Subscript[\[Sigma], 12], Subscript[\[Sigma], 13]}, {Subscript[\[Sigma], 21], Subscript[\[Sigma], 22], Subscript[\[Sigma], 23]}, {Subscript[\[Sigma], 31], Subscript[\[Sigma], 32], Subscript[\[Sigma], 33]} } ), jacobianCylTransposed, "TransformationBehavior" -> "AllCovariant"] // Simplify[#, r > 0] & // TraditionalForm$

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A tensor object with coordinates of mixed-transformation behavior t_i^j:

In[28]:=

$tConCov = ResourceFunction["TensorCoordinateTransform"][( { {Subscript[\[Sigma], 11], Subscript[\[Sigma], 12], Subscript[\[Sigma], 13]}, {Subscript[\[Sigma], 21], Subscript[\[Sigma], 22], Subscript[\[Sigma], 23]}, {Subscript[\[Sigma], 31], Subscript[\[Sigma], 32], Subscript[\[Sigma], 33]} } ), jacobianCylTransposed, "TransformationBehavior" -> {"Cov", "Con"}] // Simplify[#, r > 0] & // TraditionalForm$

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A tensor object with coordinates of opposite mixed-transformation behavior t_jⁱ:

In[29]:=

$tCovCon = ResourceFunction["TensorCoordinateTransform"][( { {Subscript[\[Sigma], 11], Subscript[\[Sigma], 12], Subscript[\[Sigma], 13]}, {Subscript[\[Sigma], 21], Subscript[\[Sigma], 22], Subscript[\[Sigma], 23]}, {Subscript[\[Sigma], 31], Subscript[\[Sigma], 32], Subscript[\[Sigma], 33]} } ), jacobianCylTransposed, "TransformationBehavior" -> {"Con", "Cov"}] // Simplify[#, r > 0] & // TraditionalForm$

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Nomenclature referencing all indices together or each index separately is interchangeable:

In[30]:=

$(ResourceFunction["TensorCoordinateTransform"][( { {Subscript[\[Sigma], 11], Subscript[\[Sigma], 12], Subscript[\[Sigma], 13]}, {Subscript[\[Sigma], 21], Subscript[\[Sigma], 22], Subscript[\[Sigma], 23]}, {Subscript[\[Sigma], 31], Subscript[\[Sigma], 32], Subscript[\[Sigma], 33]} } ), jacobianCylTransposed] == ResourceFunction["TensorCoordinateTransform"][( { {Subscript[\[Sigma], 11], Subscript[\[Sigma], 12], Subscript[\[Sigma], 13]}, {Subscript[\[Sigma], 21], Subscript[\[Sigma], 22], Subscript[\[Sigma], 23]}, {Subscript[\[Sigma], 31], Subscript[\[Sigma], 32], Subscript[\[Sigma], 33]} } ), jacobianCylTransposed, "TransformationBehavior" -> {"Con", "Con"}] == ResourceFunction["TensorCoordinateTransform"][( { {Subscript[\[Sigma], 11], Subscript[\[Sigma], 12], Subscript[\[Sigma], 13]}, {Subscript[\[Sigma], 21], Subscript[\[Sigma], 22], Subscript[\[Sigma], 23]}, {Subscript[\[Sigma], 31], Subscript[\[Sigma], 32], Subscript[\[Sigma], 33]} } ), jacobianCylTransposed, "TransformationBehavior" -> "AllContravariant"]) // Simplify[#, r > 0] &$

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This is analogous for the covariant transformation:

In[31]:=

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Normalize

Transformed contravariant vector components with respect to normalized base vectors:

In[32]:=

vNewConN = ResourceFunction["TensorCoordinateTransform"][vOld = {2, 8, 7}, mappingToNew = ( {
{2, 3, 1},
{1, 2, 2},
{1, 0, 1}
} ), "Normalize" -> True]

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The same transformed contravariant vector components with respect to non-normalized base vectors (default):

In[33]:=

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Transformed covariant normalized base vectors:

In[34]:=

bNewCovN = ResourceFunction["TensorCoordinateTransform"][{e1, e2, e3}, mappingToNew, "TransformationBehavior" -> "AllCovariant", "Normalize" -> True] /. {e1 -> {1, 0, 0}, e2 -> {0, 1, 0}, e3 -> {0, 0, 1}}

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The contraction of the assigned tensors should be produce invariance:

In[35]:=

$(\!$ \*UnderoverscriptBox[\(\[Sum]$, $i = 1$, $3$]$vNewConN[\([$$i$$]$] bNewCovN[$[$$i$$]$]\ \)\)) == vOld$

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Tensor components with respect to an orthonormal cylindrical system:

In[36]:=

$tCylConCon = ResourceFunction["TensorCoordinateTransform"][( { {Subscript[\[Sigma], 11], Subscript[\[Sigma], 12], Subscript[\[Sigma], 13]}, {Subscript[\[Sigma], 21], Subscript[\[Sigma], 22], Subscript[\[Sigma], 23]}, {Subscript[\[Sigma], 31], Subscript[\[Sigma], 32], Subscript[\[Sigma], 33]} } ), Transpose@ CoordinateTransformData["Cylindrical" -> "Cartesian", "MappingJacobian", {r, \[Phi], z}], "Normalize" -> True] // Simplify[#, r > 0] &; tCylConCon // TraditionalForm$

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TransformedField also assumes a normalized base:

In[37]:=

$TransformedField["Cartesian" -> "Cylindrical", ( { {Subscript[\[Sigma], 11], Subscript[\[Sigma], 12], Subscript[\[Sigma], 13]}, {Subscript[\[Sigma], 21], Subscript[\[Sigma], 22], Subscript[\[Sigma], 23]}, {Subscript[\[Sigma], 31], Subscript[\[Sigma], 32], Subscript[\[Sigma], 33]} } ), {x1, x2, x3} -> {r, \[Phi], z}] // Simplify // TraditionalForm$

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In[38]:=

$(tCylConCon == TransformedField["Cartesian" -> "Cylindrical", ( { {Subscript[\[Sigma], 11], Subscript[\[Sigma], 12], Subscript[\[Sigma], 13]}, {Subscript[\[Sigma], 21], Subscript[\[Sigma], 22], Subscript[\[Sigma], 23]}, {Subscript[\[Sigma], 31], Subscript[\[Sigma], 32], Subscript[\[Sigma], 33]} } ), {x1, x2, x3} -> {r, \[Phi], z}]) // Simplify$

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With orthonormal reference systems (e.g. normalized cylindrical systems), no distinction between co- and contravariant transformation behavior is required:

In[39]:=

$(tCylConCon == ResourceFunction["TensorCoordinateTransform"][( { {Subscript[\[Sigma], 11], Subscript[\[Sigma], 12], Subscript[\[Sigma], 13]}, {Subscript[\[Sigma], 21], Subscript[\[Sigma], 22], Subscript[\[Sigma], 23]}, {Subscript[\[Sigma], 31], Subscript[\[Sigma], 32], Subscript[\[Sigma], 33]} } ), Transpose@ CoordinateTransformData["Cylindrical" -> "Cartesian", "MappingJacobian", {r, \[Phi], z}], "TransformationBehavior" -> "AllCovariant", "Normalize" -> True]) // Simplify[#, r > 0] &$

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Applications

Stiffness Components of the rank-4 Elasticity Tensor with Different Material Symmetries

In[40]:=

Hook's general law describes a linear relationship between the components of the rank-2 stress tensor σ and the two-stage strain tensor ϵ using the rank-4 tensor C→σ_ij=C_ijklϵ^kl with the symmetries σ_ij=σ_ji ; ϵ^kl=ϵ^lk; E_ijkl=E_jikl=E_ijlk=E_klij :

Components of the general, fully anisotropic stiffness tensor with initial symmetries:

In[41]:=

$cFull = Normal@ SymmetrizedArray[ pos_ :> Subscript[c, pos], {3, 3, 3, 3}, {{{2, 1, 3, 4}, 1}, {{1, 2, 4, 3}, 1}, {{3, 4, 1, 2}, 1}}]; cFull // MatrixForm$

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The number of independent components:

In[42]:=

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One plane of symmetry, rotated by 180 degrees, results in the same stiffness:

In[43]:=

$rot3 = RotationMatrix[180 \[Degree], {0, 0, 1}]; rot3 // MatrixForm$

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This results in a stiffness tensor with fewer independent components:

In[45]:=

monoclinic = cFull /. (Flatten@(Solve[Thread@(
Flatten@cFull == Flatten@ResourceFunction["TensorCoordinateTransform"][cFull,
rot3, "TransformationBehavior" -> "AllCovariant"])]));
monoclinic // MatrixForm

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Determination and comparison of different material symmetries:

In[46]:=

$orthotrop = Fold[cFull /. (Solve[ Thread@(Flatten@cFull == Flatten@ ResourceFunction["TensorCoordinateTransform"][#1, #2, "TransformationBehavior" -> "AllCovariant"])] // Flatten) &, cFull, {RotationMatrix[180 \[Degree], {0, 0, 1}], RotationMatrix[180 \[Degree], {0, 1, 0}]}];$

In[47]:=

$transversalIso = Fold[cFull /. Quiet@ (* \[Alpha] only between 0\[Degree] and 360\[Degree] \ *)(Solve[Thread@(Flatten@cFull == Flatten@ ResourceFunction["TensorCoordinateTransform"][#1, #2, "TransformationBehavior" -> "AllCovariant"])] // Flatten) &, cFull, {RotationMatrix[180 \[Degree], {0, 0, 1}], RotationMatrix[180 \[Degree], {0, 1, 0}], RotationMatrix[\[Alpha], {1, 0, 0}]}];$

In[48]:=

$isotropic = Fold[cFull /. Quiet@(Solve[ Thread@(Flatten@cFull == Flatten@ ResourceFunction["TensorCoordinateTransform"][#1, #2, "TransformationBehavior" -> "AllCovariant"])] // Flatten) &, cFull, {RotationMatrix[\[Alpha]1, {1, 0, 0}], RotationMatrix[\[Alpha]2, {0, 1, 0}], RotationMatrix[\[Alpha]3, {0, 0, 1}]}];$

Summary:

In[49]:=

$(<|"Material behaviour" -> #2, "Stiffness Tensor" -> (#1 // MatrixForm), "Independent Components" -> (Cases[#1, _Subscript, \[Infinity]] // Union), "Number of independent components" -> Length[Cases[#1, _Subscript, \[Infinity]] // Union], "Description" -> #3|>) & @@@ {{monoclinic, "Monoclinic", "One symmetric plane"}, {orthotrop, "Orthotropic", "Two (three) symmetric planes"}, {transversalIso, "Transversal isotropic", "Two symmetric planes transversal isometric"}, {isotropic, "Isotropic", "All planes isometric"}} // Dataset$

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Resource History

Date Created: 11 February 2020

Source Metadata

Citation:
- Grinfeld, Pavel: “Introduction to Tensor Analysis and the Calculus of Moving Surfaces”, Springer, (2013), ISBN 978-1-4614-7867-6

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License

Wolfram Function Repository

TensorCoordinateTransform

Details and Options

Examples

Basic Examples

Scope

Tensor Components with Respect to Rotated Base Vectors

Transform Vector Components Regarding Another Affine Frame of Reference

A Local Base on the Manifold of a Torus Surface (Curvilinear Coordinate System)

Options

TransformationBehavior

Normalize

Applications

Stiffness Components of the rank-4 Elasticity Tensor with Different Material Symmetries

Related Links

Resource History

Source Metadata

License Information

TensorCoordinateTransform

Details and Options

Examples

Basic Examples

Scope

Tensor Components with Respect to Rotated Base Vectors

Transform Vector Components Regarding Another Affine Frame of Reference

A Local Base on the Manifold of a Torus Surface (Curvilinear Coordinate System)

Options

TransformationBehavior

Normalize

Applications

Stiffness Components of the rank-4 Elasticity Tensor with Different Material Symmetries

Related Links

Resource History

Source Metadata

Related Symbols

License Information