Voigt Form

Tensorial code usually does not need Voigt or Mandel arrays: tensor operations can be written directly with tensor types. The conversion functions are useful at boundaries, for example when exchanging data with a solver or file format that expects vectors and matrices.

Second-order tensors

tovoigt converts a tensor to its Voigt vector. For symmetric 3D second-order tensors, the default order is (11, 22, 33, 23, 13, 12):

julia> σ = SymmetricSecondOrderTensor{3}((2.0, 0.4, 0.2, 1.2, 0.1, 0.9))3×3 SymmetricSecondOrderTensor{3, Float64, 6}:
 2.0  0.4  0.2
 0.4  1.2  0.1
 0.2  0.1  0.9
julia> v = tovoigt(σ)6-element StaticArraysCore.SVector{6, Float64} with indices SOneTo(6):
 2.0
 1.2
 0.9
 0.1
 0.2
 0.4
julia> fromvoigt(SymmetricSecondOrderTensor{3}, v) ≈ σtrue

The inverse conversion must use the same order and scaling that were used to create the Voigt vector.

Mandel scaling

For symmetric tensors, Mandel form scales off-diagonal components by √2. This makes the Euclidean dot product of the vector representation match the tensor inner product:

julia> m = tomandel(σ)6-element StaticArraysCore.SVector{6, Float64} with indices SOneTo(6):
 2.0
 1.2
 0.9
 0.14142135623730953
 0.28284271247461906
 0.5656854249492381
julia> m ≈ tovoigt(σ; offdiagscale = sqrt(2.0))true
julia> frommandel(SymmetricSecondOrderTensor{3}, m) ≈ σtrue

Use tovoigt(...; offdiagscale = 2) for engineering-strain style scaling, and use the same offdiagscale in fromvoigt to reconstruct the tensor.

Fourth-order tensors

Fourth-order tensors are converted to matrices in Voigt or Mandel form:

julia> I = one(SymmetricFourthOrderTensor{3});
julia> M = tomandel(I);
julia> size(M)(6, 6)
julia> frommandel(SymmetricFourthOrderTensor{3}, M) ≈ Itrue

For Voigt and Mandel conversion docstrings, see Voigt and Mandel API.