A square plate, $1$m long on a side, is loaded in a state of plane strain, and has a linear displacement field, as shown in the following figure.

$1)$ Write expressions for the displacements u and v for any point on the plate.

$2)$ Determine the components of the Green strain:

$3)$ Determine the magnification factor M at point B for a line segment in the direction of O$->$B$.$

$4)$ Determine the magnification factor M at point B for a line segment in the direction of B$->$A$.$

$5)$ Determine the shear strain between the line segments O$->$B and B$->$A$.$

$6)$ For point B$,$ compare the components of strain from $2)$ to the components of strain for small$-$displacements $.$

$7)$ Compare the strain determined in $3)$ to the corresponding strain using small$-$displacement theory.

$[$hint: assume the displacements vary $$by$-$linearly$,$ meaning are functions of x$,$ y and xy$]$