Deformation Gradient Tensor

Usage

is.defgrad(x)

as.defgrad(object)

defgrad(x, time, steps, ...)

defgrad_from_ratio(Rxy = 1, Ryz = 1)

defgrad_from_shearstrain(Rxy = 1, Ryz = 1)

# S3 method for class 'Pair'
defgrad(x, ...)

defgrad_from_vectors(v1, v2)

defgrad_from_axisangle(axis, angle)

defgrad_from_comp(
  xx = 1,
  xy = 0,
  xz = 0,
  yx = 0,
  yy = 1,
  yz = 0,
  zx = 0,
  zy = 0,
  zz = 1
)

defgrad_from_simpleshear(gamma)

defgrad_from_pureshear(k)

defgrad_from_generalshear(k, gamma)

defgrad_from_dilation(dilation = 0)

# Default S3 method
defgrad(x, ...)

# S3 method for class 'velgrad'
defgrad(x, time, steps, ...)

Source

apsg by O. Lexa: https://apsg.readthedocs.io/

Arguments

x: object of class "Pair", "velgrad" or 3x3 "matrix"
object: 3x3 "matrix"
time: numeric. Total time (default is 1)
steps: numeric. Time increments (default is 1)
...: parameters passed to function call
Rxy, Ryz: numeric. the XY and YZ strain ratio to create a strain tensor with axial stretches.Values must be greater than or equal to 1.
v1, v2: spherical objects. Deformation gradient results from the rotation around axis perpendicular to both vectors to rotate v1 to v2.
axis, angle: rotation axis and angle, axis can be an object of class "Vec3", "Line", "Ray", or "Plane", or a three-element vector. Angle in degrees when axis is a object of class "Line", "Ray", or "Plane", and radians otherwise. Counterclockwise rotation for positive angles.
xx, xy, xz, yx, yy, yz, zx, zy, zz: numeric. Directly specify components of the tensor. Identity matrix by default.
gamma: numeric. shear strain in x-direction
k: numeric. Horizontal pure shear component in the y-direction
dilation: numeric. Volume increase

Value

object of class "defgrad", i.e. a 3x3 matrix.

If x is a Pair object, then defgrad() creates "defgrad" tensor representing rotation defined by "Pair". Rotation brings x-axis to lineation and z-axis to normal to plane
defgrad_by_comp creates an defined by individual components (default is identity tensor)
defgrad_by_ratio() creates an isochoric "defgrad" tensor with axial stretches defined by strain ratios (default is identity tensor).
defgrad_from_vectors() creates "defgrad" tensor representing rotation around the axis perpendicular to both vectors and rotate v1 to v2.
defgrad_from_axisangle creates "defgrad" tensor representing a rigid-body rotation about an axis and an angle.
defgrad_from_pureshear creates an isochoric coaxial "defgrad" tensor.
defgrad_from_simpleshear creates an isochoric non-coaxial "defgrad" tensor.
defgrad_from_generalshear creates an isochoric "defgrad" tensor, where transtension is \(k>1\) and \(\gamma \neq 0\), and transpression is \(k<1\) and \(\gamma \neq 0\).
defgrad_from_dilation creates "defgrad" tensor representing the volume change in z-direction.

References

Fossen, H., & Tikoff, B. (1993). The deformation matrix for simultaneous simple shearing, pure shearing and volume change, and its application to transpression-transtension tectonics. Journal of Structural Geology, 15(3–5), 413–422. doi:10.1016/0191-8141(93)90137-Y

Sanderson, D. J., & Marchini, W. R. D. (1984). Transpression. Journal of Structural Geology, 6(5), 449–458. doi:10.1016/0191-8141(84)90058-0

Examples

defgrad_from_ratio(2, 3)
#> Deformation gradient tensor
#>          [,1]     [,2]      [,3]
#> [1,] 2.289428 0.000000 0.0000000
#> [2,] 0.000000 1.144714 0.0000000
#> [3,] 0.000000 0.000000 0.3815714
defgrad_from_axisangle(Line(120, 50), 60)
#> Deformation gradient tensor
#>            [,1]        [,2]      [,3]
#> [1,]  0.5516470 -0.75286916 0.3589897
#> [2,]  0.5739587  0.65494097 0.4915523
#> [3,] -0.6051917 -0.06511807 0.7934120
defgrad_from_vectors(Line(120, 50), Line(270, 80))
#> Deformation gradient tensor
#>            [,1]       [,2]       [,3]
#> [1,]  0.8165776 0.03175915  0.2514224
#> [2,]  0.1158354 0.69202890 -0.5070905
#> [3,] -0.2253975 0.51918204  0.6588077
defgrad(Pair(40, 20, 75, 16))
#> Deformation gradient tensor
#>           [,1]       [,2]       [,3]
#> [1,] 0.2487928 -0.9331095 -0.2620026
#> [2,] 0.9285075  0.3060065 -0.2198463
#> [3,] 0.2756374 -0.1885752  0.9396926
defgrad_from_generalshear(k = 2, gamma = tan(30 * pi / 180))
#> Deformation gradient tensor
#>      [,1]      [,2] [,3]
#> [1,]    1 0.8329404  0.0
#> [2,]    0 2.0000000  0.0
#> [3,]    0 0.0000000  0.5

# combine deformation by matrix multiplication
D1 <- defgrad_from_ratio(5, 1)
D2 <- defgrad_from_axisangle(Line(0, 90), 30)

# Matrix multiplication is not commutative!!!!
D12 <- D2 %*% D1 # here: D1 is applied first

# Apply deformation of orientation data
set.seed(20250411)
l <- rvmf(100, mu = Line(0, 90), k = 100)
l_trans <- transform_linear(l, D12)

axes <- Vec3(c(1, 0, 0), c(0, 1, 0), c(0, 0, 1))
plot(l, col = "darkgrey")
points(l_trans, col = "red")
points(axes, pch = 15)
text(axes, labels = c("x", "y", "z"), pos = 1)