The diagrams and matrices refer to y and z, but in ANSYS, 2D is in X and Y directions.

Are you trying to solve 2D problems assuming Plane Stress or Plane Strain?

For applying tension or compression forces along the X axis, assume Quarter Symmetry. That means consider the lower left corner of the square to be the center of a square twice as large. Apply a Displacement of X = 0, Y = Free to the left edge.  Apply an X component of force to the right edge. Apply a Displacement of X = Free, Y = 0 to the bottom edge.  This will work for a single element or a mesh of many elements, however it is not useful for shear strain since that is not a symmetric load.

Note that applying forces to an element will not result in the strain matrix you show.  You are showing a strain matrix where the strain in the Y direction is zero.  That is an unusual constraint.  If there are no forces acting in the Y direction, but only in the X direction, there will be a strain in the X direction of Stress/E but the strain in the Y direction will occur due to the Poisson's Ratio.

If you want zero strain in the Y direction as shown in your matrix, then you need to apply Y=0 Displacement constraint to the top edge. If you want to apply strain in the X direction instead of force, you do that by using a Displacement on the right edge that is nonzero in X and leave Y free. For a strain of 0.01 use a Displacement that is 1% of the X dimension of the surface.

We can discuss Shear Strain in a later post.