Continuum Mechanics

Tensor operations for continuum mechanics.

Statistics.mean — Method

mean(::AbstractSecondOrderTensor)
mean(::AbstractSymmetricSecondOrderTensor)

Compute the mean value of diagonal entries of a square tensor.

Examples

julia> x = rand(Mat{3,3})
3×3 Tensor{Tuple{3,3},Float64,2,9}:
 0.590845  0.460085  0.200586
 0.766797  0.794026  0.298614
 0.566237  0.854147  0.246837

julia> mean(x)
0.5439025118691712

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Deviatoric–volumetric additive split

Tensorial.vol — Function

vol(::AbstractSecondOrderTensor{3})
vol(::AbstractSymmetricSecondOrderTensor{3})

Compute the volumetric part of a square tensor. Support only for tensors in 3D.

Examples

julia> x = rand(Mat{3,3})
3×3 Tensor{Tuple{3,3},Float64,2,9}:
 0.590845  0.460085  0.200586
 0.766797  0.794026  0.298614
 0.566237  0.854147  0.246837

julia> vol(x)
3×3 Tensor{Tuple{3,3},Float64,2,9}:
 0.543903  0.0       0.0
 0.0       0.543903  0.0
 0.0       0.0       0.543903

julia> vol(x) + dev(x) ≈ x
true

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Tensorial.dev — Function

dev(::AbstractSecondOrderTensor{3})
dev(::AbstractSymmetricSecondOrderTensor{3})

Compute the deviatoric part of a square tensor. Support only for tensors in 3D.

Examples

julia> x = rand(Mat{3,3})
3×3 Tensor{Tuple{3,3},Float64,2,9}:
 0.590845  0.460085  0.200586
 0.766797  0.794026  0.298614
 0.566237  0.854147  0.246837

julia> dev(x)
3×3 Tensor{Tuple{3,3},Float64,2,9}:
 0.0469421  0.460085   0.200586
 0.766797   0.250123   0.298614
 0.566237   0.854147  -0.297065

julia> tr(dev(x))
0.0

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Stress invariants

Tensorial.stress_invariants — Function

stress_invariants(::AbstractSecondOrderTensor{3})
stress_invariants(::AbstractSymmetricSecondOrderTensor{3})

Return NamedTuple storing stress invariants.

\[\begin{aligned} I_1(\bm{\sigma}) &= \mathrm{tr}(\bm{\sigma}) \\ I_2(\bm{\sigma}) &= \frac{1}{2} (\mathrm{tr}(\bm{\sigma})^2 - \mathrm{tr}(\bm{\sigma}^2)) \\ I_3(\bm{\sigma}) &= \det(\bm{\sigma}) \end{aligned}\]

julia> σ = rand(SymmetricSecondOrderTensor{3})
3×3 Tensor{Tuple{Symmetry{Tuple{3,3}}},Float64,2,6}:
 0.590845  0.766797  0.566237
 0.766797  0.460085  0.794026
 0.566237  0.794026  0.854147

julia> I₁, I₂, I₃ = stress_invariants(σ)
(I1 = 1.9050765715072775, I2 = -0.3695921176777066, I3 = -0.10054272199258936)

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Tensorial.deviatoric_stress_invariants — Function

deviatoric_stress_invariants(::AbstractSecondOrderTensor{3})
deviatoric_stress_invariants(::AbstractSymmetricSecondOrderTensor{3})

Return NamedTuple storing deviatoric stress invariants.

\[\begin{aligned} J_1(\bm{\sigma}) &= \mathrm{tr}(\mathrm{dev}(\bm{\sigma})) = 0 \\ J_2(\bm{\sigma}) &= -\frac{1}{2} \mathrm{tr}(\mathrm{dev}(\bm{\sigma})^2) \\ J_3(\bm{\sigma}) &= \frac{1}{3} \mathrm{tr}(\mathrm{dev}(\bm{\sigma})^3) \end{aligned}\]

julia> σ = rand(SymmetricSecondOrderTensor{3})
3×3 Tensor{Tuple{Symmetry{Tuple{3,3}}},Float64,2,6}:
 0.590845  0.766797  0.566237
 0.766797  0.460085  0.794026
 0.566237  0.794026  0.854147

julia> J₁, J₂, J₃ = deviatoric_stress_invariants(σ)
(J1 = 0.0, J2 = 1.5793643654463476, J3 = 0.6463152097154271)

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