Return diagonal tensor defined by magnitudes of principal stretches
Details
The eigenvalues \(\lambda\) of the deformation matrix are the quadratic forms of the principal stretches \(s\) (\(s = 1 + \epsilon = l/l_0\)).
See also
Other ellipsoid:
ellipsoid-class,
ellipsoid-params
Examples
el <- ellipsoid_from_stretch(4, 3, 1)
principal_stretch(el)
#> S1 S2 S3
#> 4 3 1
lapply(seq.int(nrow(hossack1968)), function(i) {
ellipsoid_from_stretch(hossack1968[i, 3], hossack1968[i, 2], hossack1968[i, 1])
})
#> [[1]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 334.89 0.00 0
#> [2,] 0.00 123.21 0
#> [3,] 0.00 0.00 1
#>
#> [[2]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 1024 0.00 0
#> [2,] 0 17.64 0
#> [3,] 0 0.00 1
#>
#> [[3]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 900 0.00 0
#> [2,] 0 5.29 0
#> [3,] 0 0.00 1
#>
#> [[4]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 275.56 0 0
#> [2,] 0.00 16 0
#> [3,] 0.00 0 1
#>
#> [[5]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 2948.49 0.00 0
#> [2,] 0.00 56.25 0
#> [3,] 0.00 0.00 1
#>
#> [[6]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 912.04 0.00 0
#> [2,] 0.00 54.76 0
#> [3,] 0.00 0.00 1
#>
#> [[7]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 1024 0.00 0
#> [2,] 0 4.41 0
#> [3,] 0 0.00 1
#>
#> [[8]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 1592.01 0.00 0
#> [2,] 0.00 249.64 0
#> [3,] 0.00 0.00 1
#>
#> [[9]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 1288.81 0.00 0
#> [2,] 0.00 11.56 0
#> [3,] 0.00 0.00 1
#>
#> [[10]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 196 0.00 0
#> [2,] 0 23.04 0
#> [3,] 0 0.00 1
#>
#> [[11]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 1317.69 0.00 0
#> [2,] 0.00 17.64 0
#> [3,] 0.00 0.00 1
#>
#> [[12]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 1190.25 0.00 0
#> [2,] 0.00 67.24 0
#> [3,] 0.00 0.00 1
#>
#> [[13]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 556.96 0.00 0
#> [2,] 0.00 20.25 0
#> [3,] 0.00 0.00 1
#>
#> [[14]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 1505.44 0.00 0
#> [2,] 0.00 10.24 0
#> [3,] 0.00 0.00 1
#>
#> [[15]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 1361.61 0.00 0
#> [2,] 0.00 6.25 0
#> [3,] 0.00 0.00 1
#>
#> [[16]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 625 0.00 0
#> [2,] 0 4.41 0
#> [3,] 0 0.00 1
#>
#> [[17]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 590.49 0.00 0
#> [2,] 0.00 4.41 0
#> [3,] 0.00 0.00 1
#>
#> [[18]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 148.84 0.00 0
#> [2,] 0.00 6.76 0
#> [3,] 0.00 0.00 1
#>
#> [[19]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 453.69 0.00 0
#> [2,] 0.00 32.49 0
#> [3,] 0.00 0.00 1
#>
#> [[20]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 676 0.00 0
#> [2,] 0 8.41 0
#> [3,] 0 0.00 1
#>
#> [[21]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 625 0.00 0
#> [2,] 0 5.29 0
#> [3,] 0 0.00 1
#>
#> [[22]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 484 0.00 0
#> [2,] 0 34.81 0
#> [3,] 0 0.00 1
#>
#> [[23]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 357.21 0.00 0
#> [2,] 0.00 327.61 0
#> [3,] 0.00 0.00 1
#>
#> [[24]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 392.04 0.00 0
#> [2,] 0.00 8.41 0
#> [3,] 0.00 0.00 1
#>
#> [[25]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 262.44 0.00 0
#> [2,] 0.00 19.36 0
#> [3,] 0.00 0.00 1
#>
#> [[26]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 492.84 0.00 0
#> [2,] 0.00 24.01 0
#> [3,] 0.00 0.00 1
#>
#> [[27]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 237.16 0 0
#> [2,] 0.00 9 0
#> [3,] 0.00 0 1
#>
#> [[28]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 729 0.00 0
#> [2,] 0 19.36 0
#> [3,] 0 0.00 1
#>
#> [[29]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 268.96 0.00 0
#> [2,] 0.00 14.44 0
#> [3,] 0.00 0.00 1
#>
#> [[30]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 404.01 0.00 0
#> [2,] 0.00 7.29 0
#> [3,] 0.00 0.00 1
#>
#> [[31]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 2237.29 0.00 0
#> [2,] 0.00 10.89 0
#> [3,] 0.00 0.00 1
#>
#> [[32]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 470.89 0.00 0
#> [2,] 0.00 12.25 0
#> [3,] 0.00 0.00 1
#>
#> [[33]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 187.69 0.00 0
#> [2,] 0.00 37.21 0
#> [3,] 0.00 0.00 1
#>
#> [[34]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 313.29 0.00 0
#> [2,] 0.00 47.61 0
#> [3,] 0.00 0.00 1
#>
#> [[35]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 712.89 0.00 0
#> [2,] 0.00 60.84 0
#> [3,] 0.00 0.00 1
#>
#> [[36]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 313.29 0 0
#> [2,] 0.00 16 0
#> [3,] 0.00 0 1
#>
#> [[37]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 1681 0 0
#> [2,] 0 16 0
#> [3,] 0 0 1
#>
#> [[38]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 3708.81 0.00 0
#> [2,] 0.00 19.36 0
#> [3,] 0.00 0.00 1
#>
#> [[39]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 424.36 0.00 0
#> [2,] 0.00 12.96 0
#> [3,] 0.00 0.00 1
#>
#> [[40]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 610.09 0.00 0
#> [2,] 0.00 10.24 0
#> [3,] 0.00 0.00 1
#>
#> [[41]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 1183.36 0.00 0
#> [2,] 0.00 268.96 0
#> [3,] 0.00 0.00 1
#>
#> [[42]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 466.56 0.00 0
#> [2,] 0.00 184.96 0
#> [3,] 0.00 0.00 1
#>
#> [[43]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 1049.76 0.00 0
#> [2,] 0.00 309.76 0
#> [3,] 0.00 0.00 1
#>
#> [[44]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 457.96 0.00 0
#> [2,] 0.00 32.49 0
#> [3,] 0.00 0.00 1
#>
#> [[45]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 630.01 0.00 0
#> [2,] 0.00 20.25 0
#> [3,] 0.00 0.00 1
#>
#> [[46]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 231.04 0.00 0
#> [2,] 0.00 43.56 0
#> [3,] 0.00 0.00 1
#>
#> [[47]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 1036.84 0.00 0
#> [2,] 0.00 24.01 0
#> [3,] 0.00 0.00 1
#>
#> [[48]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 571.21 0.00 0
#> [2,] 0.00 33.64 0
#> [3,] 0.00 0.00 1
#>
#> [[49]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 256 0.00 0
#> [2,] 0 59.29 0
#> [3,] 0 0.00 1
#>
#> [[50]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 789.61 0.00 0
#> [2,] 0.00 29.16 0
#> [3,] 0.00 0.00 1
#>
#> [[51]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 432.64 0.00 0
#> [2,] 0.00 31.36 0
#> [3,] 0.00 0.00 1
#>
#> [[52]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 77.44 0.00 0
#> [2,] 0.00 8.41 0
#> [3,] 0.00 0.00 1
#>
#> [[53]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 16.81 0 0
#> [2,] 0.00 16 0
#> [3,] 0.00 0 1
#>
#> [[54]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 26.01 0.00 0
#> [2,] 0.00 17.64 0
#> [3,] 0.00 0.00 1
#>
#> [[55]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 50.41 0.00 0
#> [2,] 0.00 12.25 0
#> [3,] 0.00 0.00 1
#>
#> [[56]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 39.69 0.00 0
#> [2,] 0.00 9.61 0
#> [3,] 0.00 0.00 1
#>
#> [[57]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 33.64 0.00 0
#> [2,] 0.00 7.84 0
#> [3,] 0.00 0.00 1
#>
#> [[58]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 59.29 0.00 0
#> [2,] 0.00 14.44 0
#> [3,] 0.00 0.00 1
#>
#> [[59]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 88.36 0 0
#> [2,] 0.00 16 0
#> [3,] 0.00 0 1
#>
#> [[60]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 342.25 0.00 0
#> [2,] 0.00 34.81 0
#> [3,] 0.00 0.00 1
#>
#> [[61]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 46.24 0.00 0
#> [2,] 0.00 12.96 0
#> [3,] 0.00 0.00 1
#>
#> [[62]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 50.41 0.00 0
#> [2,] 0.00 15.21 0
#> [3,] 0.00 0.00 1
#>
#> [[63]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 112.36 0.00 0
#> [2,] 0.00 20.25 0
#> [3,] 0.00 0.00 1
#>
#> [[64]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 64 0 0
#> [2,] 0 16 0
#> [3,] 0 0 1
#>
#> [[65]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 51.84 0.00 0
#> [2,] 0.00 17.64 0
#> [3,] 0.00 0.00 1
#>
#> [[66]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 98.01 0 0
#> [2,] 0.00 16 0
#> [3,] 0.00 0 1
#>
#> [[67]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 16 0.00 0
#> [2,] 0 12.25 0
#> [3,] 0 0.00 1
#>
#> [[68]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 40.96 0.00 0
#> [2,] 0.00 7.29 0
#> [3,] 0.00 0.00 1
#>
#> [[69]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 67.24 0.00 0
#> [2,] 0.00 12.96 0
#> [3,] 0.00 0.00 1
#>
#> [[70]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 118.81 0.00 0
#> [2,] 0.00 18.49 0
#> [3,] 0.00 0.00 1
#>
#> [[71]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 59.29 0.00 0
#> [2,] 0.00 14.44 0
#> [3,] 0.00 0.00 1
#>
#> [[72]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 88.36 0.00 0
#> [2,] 0.00 16.81 0
#> [3,] 0.00 0.00 1
#>
#> [[73]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 72.25 0.00 0
#> [2,] 0.00 12.96 0
#> [3,] 0.00 0.00 1
#>
#> [[74]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 72.25 0.00 0
#> [2,] 0.00 17.64 0
#> [3,] 0.00 0.00 1
#>
#> [[75]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 163.84 0.00 0
#> [2,] 0.00 21.16 0
#> [3,] 0.00 0.00 1
#>
#> [[76]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 158.76 0.00 0
#> [2,] 0.00 15.21 0
#> [3,] 0.00 0.00 1
#>
#> [[77]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 182.25 0.00 0
#> [2,] 0.00 20.25 0
#> [3,] 0.00 0.00 1
#>
#> [[78]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 204.49 0.00 0
#> [2,] 0.00 27.04 0
#> [3,] 0.00 0.00 1
#>
#> [[79]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 262.44 0.00 0
#> [2,] 0.00 17.64 0
#> [3,] 0.00 0.00 1
#>
#> [[80]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 201.64 0.00 0
#> [2,] 0.00 70.56 0
#> [3,] 0.00 0.00 1
#>
#> [[81]]
#> Ellipsoid tensor
#> [,1] [,2] [,3]
#> [1,] 102.01 0.00 0
#> [2,] 0.00 26.01 0
#> [3,] 0.00 0.00 1
#>