# Resolution of the elastic problem of a sphere in an infinite matrix


```julia
using TensND, LinearAlgebra, SymPy, Tensors, OMEinsum, Rotations
```

## Definition of the coordinate system, base vectors...


```julia
Spherical = coorsys_spherical()
θ, ϕ, r = getcoords(Spherical) # Note the order of coordinates not `r, θ, ϕ` but `θ, ϕ, r` so that the frame `(𝐞ᶿ, 𝐞ᵠ, 𝐞ʳ)` coincides with the canonical frame for null angles
𝐞ᶿ, 𝐞ᵠ, 𝐞ʳ = unitvec(Spherical) # "\bfe<TAB>" to write bold `𝐞` and "\^\theta<TAB><TAB>" to write superscript `ᶿ`
ℬ = getbasis(Spherical) ; # "\scrB<TAB>" to write `ℬ` 
```


```julia
𝟏, 𝟙, 𝕀, 𝕁, 𝕂 = init_isotropic() # Defines usual isotropic Tensors
k, μ = symbols("k μ", positive = true)
ℂ = 3k * 𝕁 + 2μ * 𝕂 # Isotropic stiffness tensor "\bbC<TAB>" to write `ℂ`
display(ℂ)
```


    TensND.TensndCanonical{4, 3, Sym, SymmetricTensor{4, 3, Sym, 36}}



$\left[ \begin{array}{rrrrrr}k + \frac{4 μ}{3}&k - \frac{2 μ}{3}&k - \frac{2 μ}{3}&0&0&0\\k - \frac{2 μ}{3}&k + \frac{4 μ}{3}&k - \frac{2 μ}{3}&0&0&0\\k - \frac{2 μ}{3}&k - \frac{2 μ}{3}&k + \frac{4 μ}{3}&0&0&0\\0&0&0&2 μ&0&0\\0&0&0&0&2 μ&0\\0&0&0&0&0&2 μ\end{array}\right]$



    # data: 


$\left[ \begin{array}{rrr}1&0&0\\0&1&0\\0&0&1\end{array}\right]$




    (:cont, :cont, :cont, :cont)


    # basis: # var: 

## General resolution

### Hydrostatic loading
\begin{equation}
\mathbf{u}=𝐄
\end{equation}


```julia
u = SymFunction("u", real = true)(r)
𝐮 = u * 𝐞ʳ
```

a8c54b742234f63254a4fb4dfd944a9e36a67334
ipynb
Jupyter Notebook
notebooks/.ipynb_checkpoints/sphere-checkpoint.ipynb
jfbarthelemy/TensND.jl
668c22129e9b074ab67ca8752f518976266ef0c3
MIT
2021-08-19T19:59:09.000Z
2021-09-12T23:16:22.000Z
notebooks/.ipynb_checkpoints/sphere-checkpoint.ipynb
jfbarthelemy/TensND.jl
668c22129e9b074ab67ca8752f518976266ef0c3
MIT
notebooks/.ipynb_checkpoints/sphere-checkpoint.ipynb
jfbarthelemy/TensND.jl
668c22129e9b074ab67ca8752f518976266ef0c3
MIT
Qwen/Qwen-72B
1. YES
2. YES