Draw a 2inch \(\times\) 2inch square \(\mathrm{OABC}\) on the engineering paper. The coordinates of \(O\) are \((0,0)\) and those of \(B\) are \((2,2)\). Using the displacement field in problem 16 , determine the \(u\) and \(v\) displacements of the corners of the square. Let the deformed square be denoted as \(\mathrm{O}^{\prime} \mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}\) '.

a. Determine the change in lengths of \(O A\) and \(O C\). Relate the changes to the strain components.

b. Determine the change in \(\angle A O C\). Relate the change to the shear strain.

c. Determine the change in length in the diagonal \(O B\). How is it related to the strain(s)?

d. Show that the relative change in the area of the square (change in area/original area) is given by \(\Delta A / A=\varepsilon_{x x}+\varepsilon_{y y}=\varepsilon_{1}+\varepsilon_{2}\).