GeoShear Demonstration #3 Strain and Matrix Algebra. 1. Linking Pure Shear to the Transformation Matrix- - - Diagonal Matrices.

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1 GeoShear Demonstration #3 Strain and Matrix Algebra Paul Karabinos Department of Geosciences, Williams College, Williamstown, MA This demonstration is suitable for an undergraduate structural geology or linear algebra course. Suggested Demonstrations 1. Linking Pure Shear to the Transformation Matrix- - - Diagonal Matrices. a. Open GeoShear, navigate to the GeoShear_2.1/Pebble Files folder, and load the Initial_Circle_Square.tab file. b. Click in the X box under pure shear in the Control Panel on the left, and use the up- - - arrow to extend the square and circle in the X- - - direction and shorten them in the Y- - - direction. Each click of the up- - - arrow increases the X value by Holding down the up- - - arrow on most keyboards increases the value quickly. Make sure the chain icon is in the closed position; this forces the Y value to be the reciprocal of the X value, thus area is preserved. c. As the square and circle are deformed, the corresponding linear transformation is shown in the red 2x2 matrix in the Control Panel. The axial ration of the strain ellipse, Rs, is shown along with the orientation of the long axis, Phi, in degrees. 1

2 d. For Pure Shear, the transformation takes the form of a diagonal matrix. The upper- - - left value corresponds to the extension in the X- - - direction. The lower- - - right value corresponds to the shortening in the Y- - - direction. Again, note that X = 1/Y; the area of the rectangle does not change during deformation. The off diagonal values are both zero. This corresponds to the irrotational characteristic of the transformation; the long and short axes of the strain ellipse remain in the same orientation, i.e. the eigen vectors have a constant orientation with increasing deformation. e. Click on the chain icon to unlock the constant area constrain. to make the X and Y values for pure shear independent. Now area can increase X * Y > 1 ore decrease X * Y < 1. An increase in area could be caused by injections of veins or dikes. A decrease in area is possible by pressure solution during cleavage formation. 2. Linking Simple Shear to the Transformation Matrix- - - Triangular Matrices. a. Click the Reset button in the Control Panel to return to the square and circle, or, if you used the Apply button to lock in the deformation, use the Remove button undo the strain. b. Click in the X box under simple shear in the Control Panel and use the up- - - arrow to shear the square into a parallelogram and the circle into an ellipse. c. Note that the very first 0.01 increment of shear strain, shown in the lower- - - left entry in the red transformation matrix, barely affects the shape of the circle the axial ratio is 1.01 but the orientation of the long axis is approximately degrees. The first infinitesimal strain increment applied results in Phi = 45 degrees. 2

3 d. Increase the X value to 0.3 and note that this corresponds to the value in the lower- - - left of the transformation matrix shown in red. The diagonal values remain 1, and the upper- - - right value is 0. This is a triangular matrix. This is right- - - handed shear or dextral shear. Note that lines parallel to the X- - - axis remain horizontal, whereas lines initially parallel to the Y- - - axis are rotated to the right. This is an example of rotational strain. e. Use the Reset button to return to the square and circle. Click in the X box under simple shear again, but use the down- - - arrow to apply increasingly negative values. This corresponds to left- - - handed, sinistral shear in the X- - - direction. Note that the lower- - - left value in the transformation matrix is a negative number. Again, lines remain parallel to the X- - - axis, but lines that started off parallel to the Y- - - axis are rotated to the left. 3

4 f. Use the Reset button to return to the square and circle. Click in the Y box under simple shear, and use the up- - - arrow to apply increasingly positive values. This corresponds to left- - - handed, sinistral shear in the Y- - - direction. Lines remain parallel to the Y- - - axis, whereas lines initially parallel to the X- - - axis are rotated g. Use the Reset button to return to the square and circle. Click in the Y box under simple shear, and use the down- - - arrow to apply increasingly negative values. This corresponds to right- - - handed, dextral shear in the Y- - - direction. 3. Linking Rotation to the Transformation Matrix- - - Anti- - - symmetric Matrices. a. Click the Reset button in the Control Panel to return to the square and circle, or, if you used the Apply button to lock in the deformation, use the Remove button undo the strain. b. Click in the deg. box under rotation in the Control Panel and use the up- - - arrow to rotate the square and circle. Each click of the up- - - arrow key produces a one degree rotation. If you prefer radians [360 4

5 degrees = 2apple radians] click in the rad. box; each click of the up- - - arrow key causes 0.01 radians of rotation. c. After 30 degrees of counter- - - clockwise rotation, the red transformation matrix has the following values: the diagonal values are both cos(30), the upper- - - right is sin(30), and the lower- - - left is sin(30). This is, of course, an anti- - - symmetric matrix. d. Note that the value of Rs = 1, in other words there is no shape change. Phi = 30, which is just the amount of rotation applied. 4. Combining Pure Shear and Rotation or Pure Shear and Simple Shear to produce any general transformation. a. Pure shear requires a diagonal transformation matrix, simple shear is produced by a triangular matrix, and rotation by an anti- - - symmetric matrix. Any general transformation matrix can be accomplished by combining pure shear and rotation. If area is not conserved, the diagonal values of the pure shear transformation cannot be reciprocals. b. Click the Reset button in the Control Panel to return to the square and circle, or, if you used the Apply button to lock in the deformation, use the Remove button undo the strain. c. Click in the X box under pure shear in the Control Panel on the left, and use the up- - - arrow to extend the square and circle in the X- - - direction and shorten them in the Y- - - direction. The exact amount of extension is not critical, and the chain icon can be locked or unlocked. In the example below, Rs =

6 d. Click the Apply button to lock in the strain. Then click in the deg. box under rotation and use the up- - - arrow key to rotate the ellipse 30 degrees counter- - - clockwise. Before you click the Apply button to lock in the rotation, note the three matrices in the Control Panel. e. The red matrix shows the rotation transformation that is pending. The black matrix shows the pure shear transformation that was applied in the previous step. The blue matrix shows the transformation that will result from combining the pure shear and rotation. Click the Apply button to lock in the rotation. f. Once you click the Apply button, the red matrix becomes gray, the blue matrix disappears, and the resultant transformation is shown in black. 6

7 g. From the GeoShear Main menu, click on Windows and select Deformation Matrix. In the lower- - - right, bottom row, is the identity matrix, which represents the undeformed state of the square and circle. In the upper row to the left is the pure shear diagonal matrix. The identity matrix is multiplied by the pure shear matrix to produce the transformation recorded in the middle of the bottom row. This transformation is, in turn, multiplied by the rotation matrix, which is shown in the left side of the upper row, to produce the final transformation shown in the left side of the bottom row. Any general transformation can be achieved by combining the correct combination of pure shear and rotation. 7