The velocity gradient tensor L can be decomposed into a symmetric matrix S (the rate or stretching tensor) and the skew-symmetric matrix W (the spin or vorticity tensor).
Details
The velocity gradient tensor\(\mathbf{L}\) can be decomposed into the sum of a symmetric matrix \(\mathbf{\dot{S}}\) and a skew-symmetric matrix \(\mathbf{W}\) $$\mathbf{L} = \mathbf{\dot{S}} + \mathbf{W}$$
where \(\mathbf{\dot{S}}\) is the stretching tensor (or strain-rate tensor) that describes the portion of the deformation that over time produces strain. \(\mathbf{W}\) is the vorticity or spin tensor and describes the internal rotation (rate) during the deformation.
Examples
R <- defgrad_from_comp(xx = 2, xy = 1, zz = 0.5)
L <- velgrad(R, time = 10)
velgrad_rate(L)
#> Velocity gradient tensor
#> [,1] [,2] [,3]
#> [1,] 0.06931472 0.03465736 0.00000000
#> [2,] 0.03465736 0.00000000 0.00000000
#> [3,] 0.00000000 0.00000000 -0.06931472
velgrad_spin(L)
#> Velocity gradient tensor
#> [,1] [,2] [,3]
#> [1,] 0.00000000 0.03465736 0
#> [2,] -0.03465736 0.00000000 0
#> [3,] 0.00000000 0.00000000 0