Problem $5(10$ Extra Credits $10$ Points$)$:

Use Hooke's law to develop the strain transformation equations from the stress transformation equations.

Strain transformation equations:

$lo{n}_{x^{\text{'}}}=\frac{lo{n}_x+lo{n}_y}{2}+\frac{lo{n}_x-lo{n}_y}{2}cos2+\frac{{}_{xy}}{2}sin2$

$\frac{{}_{x^{\text{'}}{y}^{\text{'}}}}{2}=-(\frac{lo{n}_x-lo{n}_y}{2})sin2+\frac{{}_{xy}}{2}cos2$

${}_{x^{\text{'}}}=\frac{{}_x+{}_y}{2}+\frac{{}_x-{}_y}{2}cos2+{}_{xy}sin2$

${}_{x^{\text{'}}{y}^{\text{'}}}=-\frac{{}_x-{}_y}{2}sin2+{}_{xy}cos2$